Rhind Mathematical Papyrus

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href="#Book_III_–_Miscellany"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Book III – Miscellany</span> </div> </a> <ul id="toc-Book_III_–_Miscellany-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unit_concordance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unit_concordance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Unit concordance</span> </div> </a> <ul id="toc-Unit_concordance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Content" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Content"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Content</span> </div> </a> <ul id="toc-Content-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rhind Mathematical Papyrus</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 40 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%B1%D8%AF%D9%8A%D8%A9_%D8%B1%D9%8A%D9%86%D8%AF_%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9" title="بردية ريند الرياضية – Arabic" lang="ar" hreflang="ar" data-title="بردية ريند الرياضية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Papiru_d%27Ahmes" title="Papiru d'Ahmes – Asturian" lang="ast" hreflang="ast" data-title="Papiru d'Ahmes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Papir_de_Rhind" title="Papir de Rhind – Catalan" lang="ca" hreflang="ca" data-title="Papir de Rhind" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Papyrus_Rhind" title="Papyrus Rhind – German" lang="de" hreflang="de" data-title="Papyrus Rhind" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Papiro_de_Ahmes" title="Papiro de Ahmes – Spanish" lang="es" hreflang="es" data-title="Papiro de Ahmes" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Papiruso_de_Rhind" title="Papiruso de Rhind – Esperanto" lang="eo" hreflang="eo" data-title="Papiruso de Rhind" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ahmesen_papiroa" title="Ahmesen papiroa – Basque" lang="eu" hreflang="eu" data-title="Ahmesen papiroa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%A7%D9%BE%DB%8C%D8%B1%D9%88%D8%B3_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C_%D8%B1%DB%8C%D9%86%D8%AF" title="پاپیروس ریاضی ریند – Persian" lang="fa" hreflang="fa" data-title="پاپیروس ریاضی ریند" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Papyrus_Rhind" title="Papyrus Rhind – French" lang="fr" hreflang="fr" data-title="Papyrus Rhind" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Papiro_de_Rhind" title="Papiro de Rhind – Galician" lang="gl" hreflang="gl" data-title="Papiro de Rhind" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%B0%EB%93%9C_%EC%88%98%ED%95%99_%ED%8C%8C%ED%94%BC%EB%A3%A8%EC%8A%A4" title="린드 수학 파피루스 – Korean" lang="ko" hreflang="ko" data-title="린드 수학 파피루스" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BA%D5%A1%D5%BA%D5%AB%D6%80%D5%B8%D6%82%D5%BD%D5%B6%D5%A5%D6%80" title="Մաթեմատիկական պապիրուսներ – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական պապիրուսներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Papirus_Matematika_Rhind" title="Papirus Matematika Rhind – Indonesian" lang="id" hreflang="id" data-title="Papirus Matematika Rhind" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Papiro_di_Rhind" title="Papiro di Rhind – Italian" lang="it" hreflang="it" data-title="Papiro di Rhind" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A4%D7%99%D7%A8%D7%95%D7%A1_%D7%A8%D7%99%D7%A0%D7%93" title="פפירוס רינד – Hebrew" lang="he" hreflang="he" data-title="פפירוס רינד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BF%D0%B0%D0%BF%D0%B8%D1%80%D1%83%D1%81%D1%82%D0%B0%D1%80" title="Математикалық папирустар – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық папирустар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rhind-papirusz" title="Rhind-papirusz – Hungarian" lang="hu" hreflang="hu" data-title="Rhind-papirusz" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rhind-papyrus" title="Rhind-papyrus – Dutch" lang="nl" hreflang="nl" data-title="Rhind-papyrus" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%B9%E0%A4%AE_%E0%A4%AA%E0%A5%8D%E0%A4%AF%E0%A4%AA%E0%A4%BF%E0%A4%B0%E0%A4%B8" title="अहम प्यपिरस – Nepali" lang="ne" hreflang="ne" data-title="अहम प्यपिरस" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%B3%E3%83%89%E6%95%B0%E5%AD%A6%E3%83%91%E3%83%94%E3%83%AB%E3%82%B9" title="リンド数学パピルス – Japanese" lang="ja" hreflang="ja" data-title="リンド数学パピルス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rhind-papyrusen" title="Rhind-papyrusen – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Rhind-papyrusen" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Rhind-papyursen" title="Rhind-papyursen – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Rhind-papyursen" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Papir%C3%BAs_Rhind" title="Papirús Rhind – Occitan" lang="oc" hreflang="oc" data-title="Papirús Rhind" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Papirus_Rhinda" title="Papirus Rhinda – Polish" lang="pl" hreflang="pl" data-title="Papirus Rhinda" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Papiro_de_Rhind" title="Papiro de Rhind – Portuguese" lang="pt" hreflang="pt" data-title="Papiro de Rhind" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Papirusul_Rhind" title="Papirusul Rhind – Romanian" lang="ro" hreflang="ro" data-title="Papirusul Rhind" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BF%D0%B8%D1%80%D1%83%D1%81_%D0%90%D1%85%D0%BC%D0%B5%D1%81%D0%B0" title="Папирус Ахмеса – Russian" lang="ru" hreflang="ru" data-title="Папирус Ахмеса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BB%E0%B6%BA%E0%B7%92%E0%B6%B1%E0%B7%8A%E0%B6%A9%E0%B7%8A_%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B6%B4%E0%B7%90%E0%B6%B4%E0%B7%92%E0%B6%BB%E0%B7%83%E0%B6%BA" title="රයින්ඩ් ගණිතමය පැපිරසය – Sinhala" lang="si" hreflang="si" data-title="රයින්ඩ් ගණිතමය පැපිරසය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus – Simple English" lang="en-simple" hreflang="en-simple" data-title="Rhind Mathematical Papyrus" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rhindov_papyrus" title="Rhindov papyrus – Slovak" lang="sk" hreflang="sk" data-title="Rhindov papyrus" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://sl.wikipedia.org/wiki/Rhindov_matemati%C4%8Dni_papirus" title="Rhindov matematični papirus – Slovenian" lang="sl" hreflang="sl" data-title="Rhindov matematični papirus" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D1%85%D0%BC%D0%B5%D1%81%D0%BE%D0%B2_%D0%BF%D0%B0%D0%BF%D0%B8%D1%80%D1%83%D1%81" title="Ахмесов папирус – Serbian" lang="sr" hreflang="sr" data-title="Ахмесов папирус" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rhindov_matemati%C4%8Dki_papirus" title="Rhindov matematički papirus – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Rhindov matematički papirus" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rhindin_papyrus" title="Rhindin papyrus – Finnish" lang="fi" hreflang="fi" data-title="Rhindin papyrus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rhindpapyrusen" title="Rhindpapyrusen – Swedish" lang="sv" hreflang="sv" data-title="Rhindpapyrusen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B0%E0%AF%88%E0%AE%A9%E0%AF%8D%E0%AE%9F%E0%AF%8D_%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%B0%E0%AE%9A%E0%AF%81" title="ரைன்ட் கணிதப் பப்பிரசு – Tamil" lang="ta" hreflang="ta" data-title="ரைன்ட் கணிதப் பப்பிரசு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Rhind_Papir%C3%BCs%C3%BC" title="Rhind Papirüsü – Turkish" lang="tr" hreflang="tr" data-title="Rhind Papirüsü" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BF%D1%96%D1%80%D1%83%D1%81_%D0%A0%D0%B0%D0%B9%D0%BD%D0%B4%D0%B0" title="Папірус Райнда – Ukrainian" lang="uk" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Ancient Egyptian mathematical document</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard vevent" style="text-align:left; font-size: 88%; width: 300px;"><tbody><tr><th colspan="2" class="infobox-above fn summary" style="text-align:center; font-size:125%;">Rhind Mathematical Papyrus</th></tr><tr><td colspan="2" class="infobox-subheader"><a href="/wiki/British_Museum" title="British Museum">British Museum</a>, London</td></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Rhind_Mathematical_Papyrus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/300px-Rhind_Mathematical_Papyrus.jpg" decoding="async" width="300" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/450px-Rhind_Mathematical_Papyrus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/600px-Rhind_Mathematical_Papyrus.jpg 2x" data-file-width="750" data-file-height="449" /></a></span><div class="infobox-caption">A portion of the Rhind Papyrus</div></td></tr><tr><th scope="row" class="infobox-label">Date</th><td class="infobox-data" style="text-align:left"><a href="/wiki/Second_Intermediate_Period_of_Egypt" title="Second Intermediate Period of Egypt">Second Intermediate Period of Egypt</a></td></tr><tr><th scope="row" class="infobox-label">Place of origin</th><td class="infobox-data" style="text-align:left"><a href="/wiki/Thebes,_Egypt" title="Thebes, Egypt">Thebes, Egypt</a></td></tr><tr><th scope="row" class="infobox-label">Language(s)</th><td class="infobox-data" style="text-align:left"><a href="/wiki/Egyptian_language" title="Egyptian language">Egyptian</a> (<a href="/wiki/Hieratic" title="Hieratic">Hieratic</a>)</td></tr><tr><th scope="row" class="infobox-label">Scribe(s)</th><td class="infobox-data" style="text-align:left"><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></td></tr><tr><th scope="row" class="infobox-label">Material</th><td class="infobox-data" style="text-align:left"><a href="/wiki/Papyrus" title="Papyrus">Papyrus</a></td></tr><tr><th scope="row" class="infobox-label">Size</th><td class="infobox-data" style="text-align:left">First section (<a rel="nofollow" class="external text" href="https://www.britishmuseum.org/collection/object/Y_EA10057">BM 10057 </a>): <div><ul><li>Length: 295.5 cm (116.3 in)</li><li>Width: 32 cm (13 in)</li></ul></div> Second section (<a rel="nofollow" class="external text" href="https://www.britishmuseum.org/collection/object/Y_EA10058">BM 10058 </a>): <div><ul><li>Length: 199.5 cm (78.5 in)</li><li>Width: 32 cm (13 in)</li></ul></div></td></tr></tbody></table> <p>The <b>Rhind Mathematical Papyrus</b> (<b>RMP</b>; also designated as papyrus <a href="/wiki/British_Museum" title="British Museum">British Museum</a> 10057, pBM 10058, and <a href="/wiki/Brooklyn_Museum" title="Brooklyn Museum">Brooklyn Museum</a> 37.1784Ea-b) is one of the best known examples of <a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">ancient Egyptian mathematics</a>. </p><p>It is one of two well-known mathematical papyri, along with the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>. The Rhind Papyrus is the larger, but younger, of the two.<sup id="cite_ref-Spalinger_1-0" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In the papyrus' opening paragraphs Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues: </p> <blockquote><p>This book was copied in regnal year 33, month 4 of <a href="/wiki/Season_of_the_Inundation" title="Season of the Inundation">Akhet</a>, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy.<sup id="cite_ref-Clagett_2-0" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p></blockquote> <p>Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out.<sup id="cite_ref-Spalinger_1-1" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i>The Rhind Papyrus</i> was published in 1923 by the English <a href="/wiki/Egyptologist" class="mw-redirect" title="Egyptologist">Egyptologist</a> <a href="/wiki/T._Eric_Peet" title="T. Eric Peet">T. Eric Peet</a> and contains a discussion of the text that followed <a href="/wiki/Francis_Llewellyn_Griffith" title="Francis Llewellyn Griffith">Francis Llewellyn Griffith</a>'s Book I, II and III outline.<sup id="cite_ref-peet_3-0" class="reference"><a href="#cite_note-peet-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Chace published a compendium in 1927–29 which included photographs of the text.<sup id="cite_ref-Chace,_Arnold_Buffum_1929_4-0" class="reference"><a href="#cite_note-Chace,_Arnold_Buffum_1929-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Rhind Mathematical Papyrus dates to the <a href="/wiki/Second_Intermediate_Period" class="mw-redirect" title="Second Intermediate Period">Second Intermediate Period</a> of <a href="/wiki/History_of_ancient_Egypt" title="History of ancient Egypt">Egypt</a>. It was copied by the scribe <a href="/wiki/Ahmes" title="Ahmes">Ahmes</a> (i.e., Ahmose; <i>Ahmes</i> is an older <a href="/wiki/Transcription_(linguistics)" title="Transcription (linguistics)">transcription</a> favoured by historians of mathematics) from a now-lost text from the reign of the <a href="/wiki/Twelfth_dynasty_of_Egypt" class="mw-redirect" title="Twelfth dynasty of Egypt">12th dynasty</a> <a href="/wiki/Pharaoh" title="Pharaoh">king</a> <a href="/wiki/Amenemhat_III" title="Amenemhat III">Amenemhat III</a>. </p><p>It dates to around 1550 BC.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The document is dated to Year 33 of the <a href="/wiki/Hyksos" title="Hyksos">Hyksos</a> king <a href="/wiki/Apepi_I" class="mw-redirect" title="Apepi I">Apophis</a> and also contains a separate later historical note on its <a href="/wiki/Verso" class="mw-redirect" title="Verso">verso</a> likely dating from "Year 11" of his successor, <a href="/wiki/Khamudi" title="Khamudi">Khamudi</a>.<sup id="cite_ref-Schneider_6-0" class="reference"><a href="#cite_note-Schneider-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Alexander_Henry_Rhind" title="Alexander Henry Rhind">Alexander Henry Rhind</a>, a <a href="/wiki/Scotland" title="Scotland">Scottish</a> antiquarian, purchased two parts of the <a href="/wiki/Papyrus" title="Papyrus">papyrus</a> in 1858 in <a href="/wiki/Luxor,_Egypt" class="mw-redirect" title="Luxor, Egypt">Luxor, Egypt</a>;<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> it was stated to have been found in "one of the small buildings near the <a href="/wiki/Ramesseum" title="Ramesseum">Ramesseum</a>", near Luxor.<sup id="cite_ref-peet_3-1" class="reference"><a href="#cite_note-peet-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The British Museum, where the majority of the papyrus is now kept, acquired it in 1865 along with the <a href="/wiki/Egyptian_Mathematical_Leather_Roll" title="Egyptian Mathematical Leather Roll">Egyptian Mathematical Leather Roll</a>, also owned by Henry Rhind.<sup id="cite_ref-Clagett_2-1" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Fragments of the text were independently purchased in Luxor by American Egyptologist <a href="/wiki/Edwin_Smith_(Egyptologist)" title="Edwin Smith (Egyptologist)">Edwin Smith</a> in the mid 1860s, were donated by his daughter in 1906 to the New York Historical Society,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and are now held by the <a href="/wiki/Brooklyn_Museum" title="Brooklyn Museum">Brooklyn Museum</a>.<sup id="cite_ref-Spalinger_1-2" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> An 18 cm (7.1 in) central section is missing. </p><p>The papyrus began to be transliterated and mathematically translated in the late 19th century. The mathematical-translation aspect remains incomplete in several respects.<sup id="cite_ref-Schneider_6-1" class="reference"><a href="#cite_note-Schneider-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Books">Books</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=2" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Book_I_–_Arithmetic_and_Algebra"><span id="Book_I_.E2.80.93_Arithmetic_and_Algebra"></span>Book I – Arithmetic and Algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=3" title="Edit section: Book I – Arithmetic and Algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Rhind_Mathematical_Papyrus_2/n_table" title="Rhind Mathematical Papyrus 2/n table">Rhind Mathematical Papyrus 2/n table</a> and <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</a></div> <p>The first part of the Rhind papyrus consists of reference tables and a collection of 21 arithmetic and 20 <a href="/wiki/Algebra" title="Algebra">algebraic</a> problems. The problems start out with simple fractional expressions, followed by completion (<i>sekem</i>) problems and more involved linear equations (<a href="/wiki/Moscow_Mathematical_Papyrus#Aha_problems" title="Moscow Mathematical Papyrus"><i>aha</i> problems</a>).<sup id="cite_ref-Spalinger_1-3" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The first part of the papyrus is taken up by the <a href="/wiki/Rhind_Mathematical_Papyrus_2/n_table" title="Rhind Mathematical Papyrus 2/n table">2/<i>n</i> table</a>. The fractions 2/<i>n</i> for odd <i>n</i> ranging from 3 to 101 are expressed as sums of <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">unit fractions</a>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8338b5d556b65da6e16c359fdfd224a8e9663088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.422ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{15}}={\frac {1}{10}}+{\frac {1}{30}}}"></span>. The decomposition of 2/<i>n</i> into unit fractions is never more than 4 terms long as in for example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{101}}={\frac {1}{101}}+{\frac {1}{202}}+{\frac {1}{303}}+{\frac {1}{606}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>101</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>101</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>202</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>303</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>606</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{101}}={\frac {1}{101}}+{\frac {1}{202}}+{\frac {1}{303}}+{\frac {1}{606}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9acefc5cef6ff2c3c27dc4ae1074d9a41c339222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.237ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{101}}={\frac {1}{101}}+{\frac {1}{202}}+{\frac {1}{303}}+{\frac {1}{606}}}"></span></dd></dl> <p>This table is followed by a much smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. For instance the division of 7 by 10 is recorded as: </p> <dl><dd>7 divided by 10 yields 2/3 + 1/30</dd></dl> <p>After these two tables, the papyrus records 91 problems altogether, which have been designated by moderns as problems (or numbers) 1–87, including four other items which have been designated as problems 7B, 59B, 61B and 82B. Problems 1–7, 7B and 8–40 are concerned with arithmetic and elementary algebra. </p><p>Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4, and 1 + 2/3 + 1/3 = 2 by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘‘aha’’ problems; these are <a href="/wiki/Linear_equations" class="mw-redirect" title="Linear equations">linear equations</a>. Problem 32 for instance corresponds (in modern notation) to solving x + 1/3 x + 1/4 x = 2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian <a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">unit</a> of volume. Beginning at this point, assorted units of measurement become much more important throughout the remainder of the papyrus, and indeed a major consideration throughout the rest of the papyrus is <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. Problems 39 and 40 compute the division of loaves and use <a href="/wiki/Arithmetic_progressions" class="mw-redirect" title="Arithmetic progressions">arithmetic progressions</a>.<sup id="cite_ref-Clagett_2-2" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Book_II_–_Geometry"><span id="Book_II_.E2.80.93_Geometry"></span>Book II – Geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=4" title="Edit section: Book II – Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_(1065x1330).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png/220px-Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png/330px-Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png/440px-Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png 2x" data-file-width="1065" data-file-height="1330" /></a><figcaption>A portion of the Rhind Papyrus</figcaption></figure> <p>The second part of the Rhind papyrus, being problems 41–59, 59B and 60, consists of <a href="/wiki/Geometry" title="Geometry">geometry</a> problems. Peet referred to these problems as "mensuration problems".<sup id="cite_ref-Spalinger_1-4" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Volumes">Volumes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=5" title="Edit section: Volumes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Problems 41–46 show how to find the volume of both cylindrical and rectangular granaries. In problem 41 Ahmes computes the volume of a cylindrical granary. Given the diameter d and the height h, the volume V is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\left[\right(1-1/9\left)d\right]^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow /> <mo>(</mo> </mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> <msup> <mrow> <mo>)</mo> <mi>d</mi> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\left[\right(1-1/9\left)d\right]^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5a3a40effc46bd46f4a32c16e30c2b66959210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.475ex; height:3.343ex;" alt="{\displaystyle V=\left[\right(1-1/9\left)d\right]^{2}h}"></span></dd></dl> <p>In modern mathematical notation (and using d = 2r) this gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=(8/9)^{2}d^{2}h=(256/81)r^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>256</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>81</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=(8/9)^{2}d^{2}h=(256/81)r^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046c920dc22a1f2cb70f3c3cb34993c2e456370f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.172ex; height:3.176ex;" alt="{\displaystyle V=(8/9)^{2}d^{2}h=(256/81)r^{2}h}"></span>. The fractional term 256/81 approximates the value of π as being 3.1605..., an error of less than one percent. </p><p>Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of "100 quadruple heqats" is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of <a href="/wiki/Eye_of_Horus" title="Eye of Horus">Horus eye</a> fractions, sometimes also using a much smaller unit of volume known as a "quadruple ro". The quadruple heqat and the quadruple ro are units of volume derived from the simpler heqat and ro, such that these four units of volume satisfy the following relationships: 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. Thus, </p> <dl><dd>100/10 quadruple heqat = 10 quadruple heqat</dd> <dd>100/20 quadruple heqat = 5 quadruple heqat</dd> <dd>100/30 quadruple heqat = (3 + 1/4 + 1/16 + 1/64) quadruple heqat + (1 + 2/3) quadruple ro</dd> <dd>100/40 quadruple heqat = (2 + 1/2) quadruple heqat</dd> <dd>100/50 quadruple heqat = 2 quadruple heqat</dd> <dd>100/60 quadruple heqat = (1 + 1/2 + 1/8 + 1/32) quadruple heqat + (3 + 1/3) quadruple ro</dd> <dd>100/70 quadruple heqat = (1 + 1/4 + 1/8 + 1/32 + 1/64) quadruple heqat + (2 + 1/14 + 1/21 + 1/42) quadruple ro</dd> <dd>100/80 quadruple heqat = (1 + 1/4) quadruple heqat</dd> <dd>100/90 quadruple heqat = (1 + 1/16 + 1/32 + 1/64) quadruple heqat + (1/2 + 1/18) quadruple ro</dd> <dd>100/100 quadruple heqat = 1 quadruple heqat <sup id="cite_ref-Clagett_2-3" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Areas">Areas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=6" title="Edit section: Areas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Problems 48–55 show how to compute an assortment of <a href="/wiki/Area" title="Area">areas</a>. Problem 48 is notable in that it succinctly computes the <a href="/wiki/Area_of_a_disk" class="mw-redirect" title="Area of a disk">area of a circle</a> by approximating <a href="/wiki/Pi" title="Pi">π</a>. Specifically, problem 48 explicitly reinforces the convention (used throughout the geometry section) that "a circle's area stands to that of its circumscribing square in the ratio 64/81." Equivalently, the papyrus approximates π as 256/81, as was already noted above in the explanation of problem 41. </p><p>Other problems show how to find the area of rectangles, triangles and trapezoids. </p> <div class="mw-heading mw-heading4"><h4 id="Pyramids">Pyramids</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=7" title="Edit section: Pyramids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The final six problems are related to the slopes of <a href="/wiki/Pyramid" title="Pyramid">pyramids</a>. A <a href="/wiki/Seked" title="Seked">seked</a> problem is reported as follows:<sup id="cite_ref-Maor-7_10-0" class="reference"><a href="#cite_note-Maor-7-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its <i>seked</i>?"</dd></dl> <p>The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity found for the seked is the cotangent of the angle to the base of the pyramid and its face.<sup id="cite_ref-Maor-7_10-1" class="reference"><a href="#cite_note-Maor-7-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Book_III_–_Miscellany"><span id="Book_III_.E2.80.93_Miscellany"></span>Book III – Miscellany</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=8" title="Edit section: Book III – Miscellany"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The third part of the Rhind papyrus consists of the remainder of the 91 problems, being 61, 61B, 62–82, 82B, 83–84, and "numbers" 85–87, which are items that are not mathematical in nature. This final section contains more complicated tables of data (which frequently involve Horus eye fractions), several <i>pefsu</i> problems which are elementary algebraic problems concerning food preparation, and even an amusing problem (79) which is suggestive of geometric progressions, geometric series, and certain later problems and riddles in history. Problem 79 explicitly cites, "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." In particular problem 79 concerns a situation in which 7 houses each contain seven cats, which all eat seven mice, each of which would have eaten seven ears of grain, each of which would have produced seven measures of grain. The third part of the Rhind papyrus is therefore a kind of miscellany, building on what has already been presented. Problem 61 is concerned with multiplications of fractions. Problem 61B, meanwhile, gives a general expression for computing 2/3 of 1/n, where n is odd. In modern notation the formula given is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b8266b0a4d129f3775e7658e9442d51c17fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.119ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}}"></span></dd></dl> <p>The technique given in 61B is closely related to the derivation of the 2/n table. </p><p>Problems 62–68 are general problems of an algebraic nature. Problems 69–78 are all <i>pefsu</i> problems in some form or another. They involve computations regarding the strength of bread and beer, with respect to certain raw materials used in their production.<sup id="cite_ref-Clagett_2-4" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Problem 79 sums five terms in a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a>. Its language is strongly suggestive of the more modern riddle and nursery rhyme "<a href="/wiki/As_I_was_going_to_St_Ives" title="As I was going to St Ives">As I was going to St Ives</a>".<sup id="cite_ref-Spalinger_1-5" class="reference"><a href="#cite_note-Spalinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Problems 80 and 81 compute <a href="/wiki/Horus_eye" class="mw-redirect" title="Horus eye">Horus eye</a> fractions of hinu (or heqats). The last four mathematical items, problems 82, 82B and 83–84, compute the amount of feed necessary for various animals, such as fowl and oxen.<sup id="cite_ref-Clagett_2-5" class="reference"><a href="#cite_note-Clagett-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> However, these problems, especially 84, are plagued by pervasive ambiguity, confusion, and simple inaccuracy. </p><p>The final three items on the Rhind papyrus are designated as "numbers" 85–87, as opposed to "problems", and they are scattered widely across the papyrus's back side, or verso. They are, respectively, a small phrase which ends the document (and has a few possibilities for translation, given below), a piece of scrap paper unrelated to the body of the document, used to hold it together (yet containing words and Egyptian fractions which are by now familiar to a reader of the document), and a small historical note which is thought to have been written some time after the completion of the body of the papyrus's writing. This note is thought to describe events during the "<a href="/wiki/Hyksos" title="Hyksos">Hyksos</a> domination", a period of external interruption in ancient Egyptian society which is closely related with its second intermediary period. With these non-mathematical yet historically and philologically intriguing errata, the papyrus's writing comes to an end. </p> <div class="mw-heading mw-heading2"><h2 id="Unit_concordance">Unit concordance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=9" title="Edit section: Unit concordance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Much of the Rhind Papyrus's material is concerned with <a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">Ancient Egyptian units of measurement</a> and especially the dimensional analysis used to convert between them. A concordance of units of measurement used in the papyrus is given in the image. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Rhind_Papyrus_Unit_Concordance.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rhind_Papyrus_Unit_Concordance.png/220px-Rhind_Papyrus_Unit_Concordance.png" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rhind_Papyrus_Unit_Concordance.png/330px-Rhind_Papyrus_Unit_Concordance.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rhind_Papyrus_Unit_Concordance.png/440px-Rhind_Papyrus_Unit_Concordance.png 2x" data-file-width="2568" data-file-height="1996" /></a><figcaption>Units of measure used in the Rhind Papyrus.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Content">Content</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=10" title="Edit section: Content"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This table summarizes the content of the Rhind Papyrus by means of a concise modern paraphrase. It is based upon the two-volume exposition of the papyrus which was published by <a href="/wiki/Arnold_Buffum_Chace" title="Arnold Buffum Chace">Arnold Buffum Chace</a> in 1927, and in 1929.<sup id="cite_ref-Chace,_Arnold_Buffum_1929_4-1" class="reference"><a href="#cite_note-Chace,_Arnold_Buffum_1929-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In general, the papyrus consists of four sections: a title page, the 2/n table, a tiny "1–9/10 table", and 91 problems, or "numbers". The latter are numbered from 1 through 87 and include four mathematical items which have been designated by moderns as problems 7B, 59B, 61B, and 82B. Numbers 85–87, meanwhile, are not mathematical items forming part of the body of the document, but instead are respectively: a small phrase ending the document, a piece of "scrap-paper" used to hold the document together (having already contained unrelated writing), and a historical note which is thought to describe a time period shortly after the completion of the body of the papyrus. These three latter items are written on disparate areas of the papyrus's <a href="/wiki/Recto_and_verso" title="Recto and verso">verso</a> (back side), far away from the mathematical content. Chace therefore differentiates them by styling them as <i>numbers</i> as opposed to <i>problems</i>, like the other 88 numbered items. </p> <table class="wikitable"> <tbody><tr> <th>Section or Problem Numbers</th> <th>Statement of Problem, or Description</th> <th>Solution, or Description</th> <th>Notes </th></tr> <tr> <td>Title Page</td> <td>Ahmes identifies himself and his historical circumstances.</td> <td>"Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the fourth month of the inundation season, under the majesty of the king of Upper and Lower Egypt, 'A-user-Re', endowed with life, in likeness to writings of old made in the time of the king of Upper and Lower Egypt, Ne-ma'et-Re'. It is the scribe Ahmes who copies this writing."</td> <td>It is clear from the title page that Ahmes identifies both his own period, as well as the period of an older text or texts from which he is supposed to have copied, thereby creating the Rhind Papyrus. The papyrus has material written on both sides—that is, its <a href="/wiki/Recto_and_verso" title="Recto and verso">recto</a> and <a href="/wiki/Recto_and_verso" title="Recto and verso">verso</a>. See the picture for details. <figure class="mw-default-size mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:Rhind_Papyrus_Recto_and_Verso.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Rhind_Papyrus_Recto_and_Verso.png/220px-Rhind_Papyrus_Recto_and_Verso.png" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Rhind_Papyrus_Recto_and_Verso.png/330px-Rhind_Papyrus_Recto_and_Verso.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Rhind_Papyrus_Recto_and_Verso.png/440px-Rhind_Papyrus_Recto_and_Verso.png 2x" data-file-width="1733" data-file-height="889" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>2/n Table</td> <td>Express each of the quotients from 2/3 through 2/101 (where the denominator is always odd) as <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fractions</a>.</td> <td>See the <a href="/wiki/Rhind_Mathematical_Papyrus_2/n_table" title="Rhind Mathematical Papyrus 2/n table">Rhind Mathematical Papyrus 2/n table</a> article for summary and solutions of this section.</td> <td>Throughout the papyrus, most solutions are given as particular Egyptian fractional representations of a given real number. However, since every positive rational number has infinitely many representations as an Egyptian fraction, these solutions are not unique. Also bear in mind that the fraction 2/3 is the single exception, used in addition to integers, that Ahmes uses alongside all (positive) rational unit fractions to express Egyptian fractions. The 2/n table can be said to partially follow an algorithm (see problem 61B) for expressing 2/n as an Egyptian fraction of 2 terms, when n is composite. However, this fledgling algorithm is cast aside in many situations when n is prime. The method of solutions for the 2/n table, therefore, also suggests beginnings of <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and not merely <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>. </td></tr> <tr> <td>1–9/10 Table</td> <td>Write the quotients from 1/10 through 9/10 as Egyptian fractions.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}\;\;\;;\;\;\;{\frac {3}{10}}={\frac {1}{5}}+{\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}\;\;\;;\;\;\;{\frac {3}{10}}={\frac {1}{5}}+{\frac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cbf2479346310283d9ac26be94c8a9544c674d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.748ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}\;\;\;;\;\;\;{\frac {3}{10}}={\frac {1}{5}}+{\frac {1}{10}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{10}}={\frac {1}{3}}+{\frac {1}{15}}\;\;\;;\;\;\;{\frac {5}{10}}={\frac {1}{2}}\;\;\;;\;\;\;{\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{10}}={\frac {1}{3}}+{\frac {1}{15}}\;\;\;;\;\;\;{\frac {5}{10}}={\frac {1}{2}}\;\;\;;\;\;\;{\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd71d37ee7a37b222ab0b2215b25adda162f507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.587ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{10}}={\frac {1}{3}}+{\frac {1}{15}}\;\;\;;\;\;\;{\frac {5}{10}}={\frac {1}{2}}\;\;\;;\;\;\;{\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5086f50d5251ae63a13b526a67f95729744d9e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.429ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}"></span> </p> </td> <td> </td></tr> <tr> <td>Problems 1–6</td> <td>1, 2, 6, 7, 8 and 9 loaves of bread (respectively, in each problem) are divided among 10 men. In each case, represent each man's share of bread as an Egyptian fraction.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f242e95a30fb8c5debe1c8e7c946569f03ad78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.584ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10}}={\frac {1}{10}}\;\;\;;\;\;\;{\frac {2}{10}}={\frac {1}{5}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}\;\;\;;\;\;\;{\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}\;\;\;;\;\;\;{\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0537fc0a653a3d82aff4b1d15d926cadf3e25f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.424ex; height:5.176ex;" alt="{\displaystyle {\frac {6}{10}}={\frac {1}{2}}+{\frac {1}{10}}\;\;\;;\;\;\;{\frac {7}{10}}={\frac {2}{3}}+{\frac {1}{30}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9432a69f2fc4bc3219f41457c7559646306b059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.264ex; height:5.176ex;" alt="{\displaystyle {\frac {8}{10}}={\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{30}}\;\;\;;\;\;\;{\frac {9}{10}}={\frac {2}{3}}+{\frac {1}{5}}+{\frac {1}{30}}}"></span> </p> </td> <td>The first six problems of the papyrus are simple repetitions of the information already written in the 1–9/10 table, now in the context of story problems. </td></tr> <tr> <td>7, 7B, 8–20</td> <td>Let <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=1+1/2+1/4={\frac {7}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=1+1/2+1/4={\frac {7}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a750bf2f035240cb0323ccfe74e16034caf4a58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.513ex; height:5.176ex;" alt="{\displaystyle S=1+1/2+1/4={\frac {7}{4}}}"></span> and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=1+2/3+1/3=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=1+2/3+1/3=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df8d3fec235239561b7835a03f60b8b695e8761e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.814ex; height:2.843ex;" alt="{\displaystyle T=1+2/3+1/3=2}"></span>. </p><p>Then for the following multiplications, write the product as an Egyptian fraction. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;7B:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;8:{\frac {1}{4}}T={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>28</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>7</mn> <mi>B</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>28</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>8</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;7B:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;8:{\frac {1}{4}}T={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8044ce33e8d0281a7878870652aeefaa4ab1d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.64ex; height:6.176ex;" alt="{\displaystyle 7:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;7B:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;8:{\frac {1}{4}}T={\frac {1}{2}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9:{\bigg (}{\frac {1}{2}}+{\frac {1}{14}}{\bigg )}S=1\;\;\;;\;\;\;10:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;11:{\frac {1}{7}}S={\frac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>10</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>28</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>11</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9:{\bigg (}{\frac {1}{2}}+{\frac {1}{14}}{\bigg )}S=1\;\;\;;\;\;\;10:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;11:{\frac {1}{7}}S={\frac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1137b6c6710c0a9a35c1387a06254c4f4546939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.227ex; height:6.176ex;" alt="{\displaystyle 9:{\bigg (}{\frac {1}{2}}+{\frac {1}{14}}{\bigg )}S=1\;\;\;;\;\;\;10:{\bigg (}{\frac {1}{4}}+{\frac {1}{28}}{\bigg )}S={\frac {1}{2}}\;\;\;;\;\;\;11:{\frac {1}{7}}S={\frac {1}{4}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12:{\frac {1}{14}}S={\frac {1}{8}}\;\;\;;\;\;\;13:{\bigg (}{\frac {1}{16}}+{\frac {1}{112}}{\bigg )}S={\frac {1}{8}}\;\;\;;\;\;\;14:{\frac {1}{28}}S={\frac {1}{16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>13</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>112</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>14</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>28</mn> </mfrac> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12:{\frac {1}{14}}S={\frac {1}{8}}\;\;\;;\;\;\;13:{\bigg (}{\frac {1}{16}}+{\frac {1}{112}}{\bigg )}S={\frac {1}{8}}\;\;\;;\;\;\;14:{\frac {1}{28}}S={\frac {1}{16}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d56acd21b97a2db9a3447c9fd750dcf2d3a71b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.616ex; height:6.176ex;" alt="{\displaystyle 12:{\frac {1}{14}}S={\frac {1}{8}}\;\;\;;\;\;\;13:{\bigg (}{\frac {1}{16}}+{\frac {1}{112}}{\bigg )}S={\frac {1}{8}}\;\;\;;\;\;\;14:{\frac {1}{28}}S={\frac {1}{16}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15:{\bigg (}{\frac {1}{32}}+{\frac {1}{224}}{\bigg )}S={\frac {1}{16}}\;\;\;;\;\;\;16:{\frac {1}{2}}T=1\;\;\;;\;\;\;17:{\frac {1}{3}}T={\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>224</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>16</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>17</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15:{\bigg (}{\frac {1}{32}}+{\frac {1}{224}}{\bigg )}S={\frac {1}{16}}\;\;\;;\;\;\;16:{\frac {1}{2}}T=1\;\;\;;\;\;\;17:{\frac {1}{3}}T={\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7323f9dafa44acf4a1a489f0ca097844de1fe5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.729ex; height:6.176ex;" alt="{\displaystyle 15:{\bigg (}{\frac {1}{32}}+{\frac {1}{224}}{\bigg )}S={\frac {1}{16}}\;\;\;;\;\;\;16:{\frac {1}{2}}T=1\;\;\;;\;\;\;17:{\frac {1}{3}}T={\frac {2}{3}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18:{\frac {1}{6}}T={\frac {1}{3}}\;\;\;;\;\;\;19:{\frac {1}{12}}T={\frac {1}{6}}\;\;\;;\;\;\;20:{\frac {1}{24}}T={\frac {1}{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>18</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>19</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>20</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18:{\frac {1}{6}}T={\frac {1}{3}}\;\;\;;\;\;\;19:{\frac {1}{12}}T={\frac {1}{6}}\;\;\;;\;\;\;20:{\frac {1}{24}}T={\frac {1}{12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce5edaef9f60a011763c0d9268735f7767dc4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.279ex; height:5.176ex;" alt="{\displaystyle 18:{\frac {1}{6}}T={\frac {1}{3}}\;\;\;;\;\;\;19:{\frac {1}{12}}T={\frac {1}{6}}\;\;\;;\;\;\;20:{\frac {1}{24}}T={\frac {1}{12}}}"></span> </p> </td> <td>The same two multiplicands (here denoted as S and T) are used incessantly throughout these problems. Ahmes effectively writes the same problem thrice over (7, 7B, 10), sometimes approaching the same problem with different arithmetic work. </td></tr> <tr> <td>21–38</td> <td>For each of the following <a href="/wiki/Linear_Equation" class="mw-redirect" title="Linear Equation">linear equations</a> with variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, solve for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and express <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> as an Egyptian fraction.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21:{\bigg (}{\frac {2}{3}}+{\frac {1}{15}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 21:{\bigg (}{\frac {2}{3}}+{\frac {1}{15}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5702cca4f117b29d625f76987e8d2166c69d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.028ex; height:6.176ex;" alt="{\displaystyle 21:{\bigg (}{\frac {2}{3}}+{\frac {1}{15}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{15}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 22:{\bigg (}{\frac {2}{3}}+{\frac {1}{30}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>22</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 22:{\bigg (}{\frac {2}{3}}+{\frac {1}{30}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff283da30d30115587e47d669c08aa1c411eb10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.028ex; height:6.176ex;" alt="{\displaystyle 22:{\bigg (}{\frac {2}{3}}+{\frac {1}{30}}{\bigg )}+x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 23:{\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{10}}+{\frac {1}{30}}+{\frac {1}{45}}{\bigg )}+x={\frac {2}{3}}\;\;\;\rightarrow \;\;\;x={\frac {1}{9}}+{\frac {1}{40}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>23</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>45</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>40</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23:{\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{10}}+{\frac {1}{30}}+{\frac {1}{45}}{\bigg )}+x={\frac {2}{3}}\;\;\;\rightarrow \;\;\;x={\frac {1}{9}}+{\frac {1}{40}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1936c7bf772853a1730e78f5a8f1468fc01b2388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.706ex; height:6.176ex;" alt="{\displaystyle 23:{\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{10}}+{\frac {1}{30}}+{\frac {1}{45}}{\bigg )}+x={\frac {2}{3}}\;\;\;\rightarrow \;\;\;x={\frac {1}{9}}+{\frac {1}{40}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 24:x+{\frac {1}{7}}x=19\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{2}}+{\frac {1}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>19</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24:x+{\frac {1}{7}}x=19\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{2}}+{\frac {1}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34511811baef49cde752bf9dbf715372690ec370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.1ex; height:5.343ex;" alt="{\displaystyle 24:x+{\frac {1}{7}}x=19\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{2}}+{\frac {1}{8}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25:x+{\frac {1}{2}}x=16\;\;\;\rightarrow \;\;\;x=10+{\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>16</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25:x+{\frac {1}{2}}x=16\;\;\;\rightarrow \;\;\;x=10+{\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4425468d4e9abe77f65dc76cbdb74a84d807d7af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.261ex; height:5.176ex;" alt="{\displaystyle 25:x+{\frac {1}{2}}x=16\;\;\;\rightarrow \;\;\;x=10+{\frac {2}{3}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 26:x+{\frac {1}{4}}x=15\;\;\;\rightarrow \;\;\;x=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>26</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>15</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 26:x+{\frac {1}{4}}x=15\;\;\;\rightarrow \;\;\;x=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c46460f167b0b2081905f570eed2c0f6426377d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.422ex; height:5.176ex;" alt="{\displaystyle 26:x+{\frac {1}{4}}x=15\;\;\;\rightarrow \;\;\;x=12}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 27:x+{\frac {1}{5}}x=21\;\;\;\rightarrow \;\;\;x=17+{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>27</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>21</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>17</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 27:x+{\frac {1}{5}}x=21\;\;\;\rightarrow \;\;\;x=17+{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742cc090f0cc768ea47cf98e9854efc899b892f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.261ex; height:5.176ex;" alt="{\displaystyle 27:x+{\frac {1}{5}}x=21\;\;\;\rightarrow \;\;\;x=17+{\frac {1}{2}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 28:{\bigg (}x+{\frac {2}{3}}x{\bigg )}-{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}=10\;\;\;\rightarrow \;\;\;x=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>28</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 28:{\bigg (}x+{\frac {2}{3}}x{\bigg )}-{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}=10\;\;\;\rightarrow \;\;\;x=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43845dd033969e66778ebd2f94e6f645d9b36261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.439ex; height:6.176ex;" alt="{\displaystyle 28:{\bigg (}x+{\frac {2}{3}}x{\bigg )}-{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}=10\;\;\;\rightarrow \;\;\;x=9}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 29:{\frac {1}{3}}{\Bigg (}{\bigg (}x+{\frac {2}{3}}x{\bigg )}+{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}{\Bigg )}=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>29</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>13</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 29:{\frac {1}{3}}{\Bigg (}{\bigg (}x+{\frac {2}{3}}x{\bigg )}+{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}{\Bigg )}=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3d94d314b30289b90081bb687a775e453665b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.12ex; height:7.509ex;" alt="{\displaystyle 29:{\frac {1}{3}}{\Bigg (}{\bigg (}x+{\frac {2}{3}}x{\bigg )}+{\frac {1}{3}}{\bigg (}x+{\frac {2}{3}}x{\bigg )}{\Bigg )}=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{2}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30:{\bigg (}{\frac {2}{3}}+{\frac {1}{10}}{\bigg )}x=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{23}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>30</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>13</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>23</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 30:{\bigg (}{\frac {2}{3}}+{\frac {1}{10}}{\bigg )}x=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{23}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e064ed5b4ccdf168dd0e417d31f34fb1c6778c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.676ex; height:6.176ex;" alt="{\displaystyle 30:{\bigg (}{\frac {2}{3}}+{\frac {1}{10}}{\bigg )}x=10\;\;\;\rightarrow \;\;\;x=13+{\frac {1}{23}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 31:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=33\;\;\;\rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>31</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>33</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 31:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=33\;\;\;\rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca34ec0be153976211e216c3f6dff99b858f8bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.426ex; height:5.343ex;" alt="{\displaystyle 31:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=33\;\;\;\rightarrow }"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=14+{\frac {1}{4}}+{\frac {1}{56}}+{\frac {1}{97}}+{\frac {1}{194}}+{\frac {1}{388}}+{\frac {1}{679}}+{\frac {1}{776}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>14</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>56</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>97</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>194</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>388</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>679</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>776</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=14+{\frac {1}{4}}+{\frac {1}{56}}+{\frac {1}{97}}+{\frac {1}{194}}+{\frac {1}{388}}+{\frac {1}{679}}+{\frac {1}{776}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310fb3d4190ce9d30580b41897523014542176fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.251ex; height:5.343ex;" alt="{\displaystyle x=14+{\frac {1}{4}}+{\frac {1}{56}}+{\frac {1}{97}}+{\frac {1}{194}}+{\frac {1}{388}}+{\frac {1}{679}}+{\frac {1}{776}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 32:x+{\frac {1}{3}}x+{\frac {1}{4}}x=2\;\;\;\rightarrow \;\;\;x=1+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{114}}+{\frac {1}{228}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>32</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>114</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>228</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32:x+{\frac {1}{3}}x+{\frac {1}{4}}x=2\;\;\;\rightarrow \;\;\;x=1+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{114}}+{\frac {1}{228}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4490a77326f3e721dd434daaa6de584a6f5b0080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.434ex; height:5.176ex;" alt="{\displaystyle 32:x+{\frac {1}{3}}x+{\frac {1}{4}}x=2\;\;\;\rightarrow \;\;\;x=1+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{114}}+{\frac {1}{228}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 33:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=37\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{56}}+{\frac {1}{679}}+{\frac {1}{776}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>33</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>37</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>56</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>679</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>776</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 33:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=37\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{56}}+{\frac {1}{679}}+{\frac {1}{776}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7adcdf8754b1c195cc237d60ee6b1a83bc2dc47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:64.088ex; height:5.343ex;" alt="{\displaystyle 33:x+{\frac {2}{3}}x+{\frac {1}{2}}x+{\frac {1}{7}}x=37\;\;\;\rightarrow \;\;\;x=16+{\frac {1}{56}}+{\frac {1}{679}}+{\frac {1}{776}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 34:x+{\frac {1}{2}}x+{\frac {1}{4}}x=10\;\;\;\rightarrow \;\;\;x=5+{\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{14}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>34</mn> <mo>:</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>x</mi> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 34:x+{\frac {1}{2}}x+{\frac {1}{4}}x=10\;\;\;\rightarrow \;\;\;x=5+{\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{14}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5046af5fa9c2942951abd81199ad46c2d9cfd6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.108ex; height:5.343ex;" alt="{\displaystyle 34:x+{\frac {1}{2}}x+{\frac {1}{4}}x=10\;\;\;\rightarrow \;\;\;x=5+{\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{14}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 35:{\bigg (}3+{\frac {1}{3}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>35</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 35:{\bigg (}3+{\frac {1}{3}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5efcc584a18f1e58f17eefd9322a59a70c2a1d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.189ex; height:6.176ex;" alt="{\displaystyle 35:{\bigg (}3+{\frac {1}{3}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{5}}+{\frac {1}{10}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{5}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{53}}+{\frac {1}{106}}+{\frac {1}{212}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>53</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>106</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>212</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{5}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{53}}+{\frac {1}{106}}+{\frac {1}{212}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a603f7725a364f1f45243fa429ab52b4708ff4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.355ex; height:6.176ex;" alt="{\displaystyle 36:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{5}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{53}}+{\frac {1}{106}}+{\frac {1}{212}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 37:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{3}}\cdot {\frac {1}{3}}+{\frac {1}{9}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{32}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>37</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 37:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{3}}\cdot {\frac {1}{3}}+{\frac {1}{9}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{32}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aece92dc722653118fd3a920aa2d25bab3767ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.544ex; height:6.176ex;" alt="{\displaystyle 37:{\bigg (}3+{\frac {1}{3}}+{\frac {1}{3}}\cdot {\frac {1}{3}}+{\frac {1}{9}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{4}}+{\frac {1}{32}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 38:{\bigg (}3+{\frac {1}{7}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{6}}+{\frac {1}{11}}+{\frac {1}{22}}+{\frac {1}{66}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>38</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">→</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>22</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>66</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 38:{\bigg (}3+{\frac {1}{7}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{6}}+{\frac {1}{11}}+{\frac {1}{22}}+{\frac {1}{66}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2542cef4c0c0e61ce7f35476817a1f9da4aa4934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.191ex; height:6.176ex;" alt="{\displaystyle 38:{\bigg (}3+{\frac {1}{7}}{\bigg )}x=1\;\;\;\rightarrow \;\;\;x={\frac {1}{6}}+{\frac {1}{11}}+{\frac {1}{22}}+{\frac {1}{66}}}"></span> </p> </td> <td>Problem 31 has an especially onerous solution. Although the statement of problems 21–38 can at times appear complicated (especially in Ahmes' prose), each problem ultimately reduces to a simple linear equation. In some cases, a <a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">unit</a> of some kind has been omitted, being superfluous for these problems. These cases are problems 35–38, whose statements and "work" make the first mentions of units of volume known as a heqat and a ro (where 1 heqat = 320 ro), which will feature prominently throughout the rest of the papyrus. For the moment, however, their literal mention and usage in 35–38 is cosmetic. </td></tr> <tr> <td>39</td> <td>100 bread loaves will be distributed unequally among 10 men. 50 loaves will be divided equally among 4 men so that each of those 4 receives an equal share <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, while the other 50 loaves will be divided equally among the other 6 men so that each of those 6 receives an equal share <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. Find the difference of these two shares <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23667f02add9d6ce4dac94880b06f2b22d1b4aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle y-x}"></span> and express same as an Egyptian fraction.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-x=4+{\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-x=4+{\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300888f0c7822a15383ba0fbc78ce642ddd8b0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.425ex; height:5.176ex;" alt="{\displaystyle y-x=4+{\frac {1}{6}}}"></span></td> <td>In problem 39, the papyrus begins to consider situations with more than one variable. </td></tr> <tr> <td>40</td> <td>100 loaves of bread are to be divided among five men. The men's five shares of bread are to be in <a href="/wiki/Arithmetic_progressions" class="mw-redirect" title="Arithmetic progressions">arithmetic progression</a>, so that consecutive shares always differ by a fixed difference, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span>. Furthermore, the sum of the three largest shares is to be equal to seven times the sum of the two smallest shares. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> and write it as an Egyptian fraction.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =9+{\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>9</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =9+{\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac82521947dd2b32c25f24414a26a4a66a49744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.036ex; height:5.176ex;" alt="{\displaystyle \Delta =9+{\frac {1}{6}}}"></span></td> <td>Problem 40 concludes the arithmetic/algebraic section of the papyrus, to be followed by the geometry section. After problem 40, there is even a large section of blank space on the papyrus, which visually indicates the end of the section. As for problem 40 itself, Ahmes works out his solution by first considering the analogous case where the number of loaves is 60 as opposed to 100. He then states that in this case the difference is 5 1/2 and that the smallest share is equal to one, lists the others, and then scales his work back up to 100 to produce his result. Although Ahmes does not state the solution itself as it has been given here, the quantity is implicitly clear once he has re-scaled his first step by the multiplication 5/3 x 11/2, to list the five shares (which he does). It bears mentioning that this problem can be thought of as having four conditions: a) five shares sum to 100, b) the shares range from smallest to largest, c) consecutive shares have a constant difference and d) the sum of the three larger shares is equal to seven times the sum of the smaller two shares. Beginning with the first three conditions only, one can use elementary algebra and then consider whether adding the fourth condition yields a consistent result. It happens that once all four conditions are in place, the solution is unique. The problem is therefore a more elaborate case of linear equation solving than what has gone before, verging on <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>. </td></tr> <tr> <td>41</td> <td>Use the volume formula <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\bigg (}d-{\frac {1}{9}}d{\bigg )}^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>d</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\bigg (}d-{\frac {1}{9}}d{\bigg )}^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef2b7eee6b8eb5e8cd612ad380574cc0c08c2ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.971ex; height:6.509ex;" alt="{\displaystyle V={\bigg (}d-{\frac {1}{9}}d{\bigg )}^{2}h}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {64}{81}}d^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>81</mn> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {64}{81}}d^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4940b267fc8dda73ea77923cec623be167b429d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.226ex; height:5.176ex;" alt="{\displaystyle ={\frac {64}{81}}d^{2}h}"></span> </p><p>to calculate the volume of a cylindrical grain silo with a diameter of 9 <a href="/wiki/Cubit" title="Cubit">cubits</a> and a height of 10 cubits. Give the answer in terms of cubic cubits. Furthermore, given the following equalities among other units of volume, 1 cubic cubit = 3/2 khar = 30 heqats = 15/2 quadruple heqats, also express the answer in terms of khar and quadruple heqats. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=640\;\;\;cubit^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mn>640</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=640\;\;\;cubit^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff1ad1984066904c2d7faaede5b64b07a1bd264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.339ex; height:2.676ex;" alt="{\displaystyle V=640\;\;\;cubit^{3}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =960\;\;\;khar}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>960</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>k</mi> <mi>h</mi> <mi>a</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =960\;\;\;khar}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b88e655cc8d11bfa9a131ad26ec33b2adbdbf2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.705ex; height:2.176ex;" alt="{\displaystyle =960\;\;\;khar}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =4800\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>4800</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =4800\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee29a1567f4c444e9555ef962a6bde53442d3b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.705ex; height:2.509ex;" alt="{\displaystyle =4800\;\;\;quadruple\;\;\;heqat}"></span> </p> </td> <td>This problem opens up the papyrus's <a href="/wiki/Geometry" title="Geometry">geometry</a> section, and also gives its first factually incorrect result (albeit with a very good approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, differing by less than one percent). Other ancient Egyptian volume <a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">units</a> such as the quadruple heqat and the khar are later reported in this problem via unit conversion. Problem 41 is therefore also the first problem to treat significantly of <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. </td></tr> <tr> <td>42</td> <td>Reuse the volume formula and unit information given in 41 to calculate the volume of a cylindrical grain silo with a diameter of 10 cubits and a height of 10 cubits. Give the answer in terms of cubic cubits, khar, and <i>hundreds of</i> quadruple heqats, where 400 heqats = 100 quadruple heqats = 1 hundred-quadruple heqat, all as Egyptian fractions.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\bigg (}790+{\frac {1}{18}}+{\frac {1}{27}}+{\frac {1}{54}}+{\frac {1}{81}}{\bigg )}\;\;\;cubit^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>790</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>27</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>54</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>81</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\bigg (}790+{\frac {1}{18}}+{\frac {1}{27}}+{\frac {1}{54}}+{\frac {1}{81}}{\bigg )}\;\;\;cubit^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba86fd8c3430793c721fb805c10a3a066409936b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.766ex; height:6.176ex;" alt="{\displaystyle V={\bigg (}790+{\frac {1}{18}}+{\frac {1}{27}}+{\frac {1}{54}}+{\frac {1}{81}}{\bigg )}\;\;\;cubit^{3}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bigg (}1185+{\frac {1}{6}}+{\frac {1}{54}}{\bigg )}\;\;\;khar}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1185</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>54</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>k</mi> <mi>h</mi> <mi>a</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bigg (}1185+{\frac {1}{6}}+{\frac {1}{54}}{\bigg )}\;\;\;khar}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33753d9f07c24dd59db97374a37d7b7c0fbed70d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.129ex; height:6.176ex;" alt="{\displaystyle ={\bigg (}1185+{\frac {1}{6}}+{\frac {1}{54}}{\bigg )}\;\;\;khar}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bigg (}59+{\frac {1}{4}}+{\frac {1}{108}}{\bigg )}\;\;\;hundred\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>59</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>108</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bigg (}59+{\frac {1}{4}}+{\frac {1}{108}}{\bigg )}\;\;\;hundred\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6acd49e7911bcb4de94c74330603363ce58505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.367ex; height:6.176ex;" alt="{\displaystyle ={\bigg (}59+{\frac {1}{4}}+{\frac {1}{108}}{\bigg )}\;\;\;hundred\;\;\;quadruple\;\;\;heqat}"></span> </p> </td> <td>Problem 42 is effectively a repetition of 41, performing similar unit conversions at the end. However, although the problem does begin as stated, the arithmetic is considerably more involved, and certain of the given latter fractional terms are not actually present in the original document. However, the context is sufficient to fill in the gaps, and Chace has therefore taken license to add certain fractional terms in his mathematical translation (repeated here) which give rise to an internally consistent solution. </td></tr> <tr> <td>43</td> <td>Use the volume formula <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {2}{3}}{\Bigg (}{\bigg (}d-{\frac {1}{9}}d{\bigg )}+{\frac {1}{3}}{\bigg (}d-{\frac {1}{9}}d{\bigg )}{\Bigg )}^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>d</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>d</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {2}{3}}{\Bigg (}{\bigg (}d-{\frac {1}{9}}d{\bigg )}+{\frac {1}{3}}{\bigg (}d-{\frac {1}{9}}d{\bigg )}{\Bigg )}^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68bbdb33f8e54006080397b54548001a843870c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.182ex; height:8.009ex;" alt="{\displaystyle V={\frac {2}{3}}{\Bigg (}{\bigg (}d-{\frac {1}{9}}d{\bigg )}+{\frac {1}{3}}{\bigg (}d-{\frac {1}{9}}d{\bigg )}{\Bigg )}^{2}h}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {2048}{2187}}d^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2048</mn> <mn>2187</mn> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {2048}{2187}}d^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d75d8c31d56f420b14a5d4bebbf98add7c729fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.551ex; height:5.343ex;" alt="{\displaystyle ={\frac {2048}{2187}}d^{2}h}"></span> </p><p>to calculate the volume of a cylindrical grain silo with a diameter of 9 cubits and a height of 6 cubits, directly finding the answer in Egyptian fractional terms of khar, and later in Egyptian fractional terms of quadruple heqats and quadruple ro, where 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\bigg (}455+{\frac {1}{9}}{\bigg )}\;\;\;khar}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>455</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>k</mi> <mi>h</mi> <mi>a</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\bigg (}455+{\frac {1}{9}}{\bigg )}\;\;\;khar}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1265c97fd76b6c12f09bacc375223eb57c07d0af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.397ex; height:6.176ex;" alt="{\displaystyle V={\bigg (}455+{\frac {1}{9}}{\bigg )}\;\;\;khar}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bigg (}2275+{\frac {1}{2}}+{\frac {1}{32}}+{\frac {1}{64}}{\bigg )}\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>2275</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bigg (}2275+{\frac {1}{2}}+{\frac {1}{32}}+{\frac {1}{64}}{\bigg )}\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ca62f925e4588c1145ac0277a0d84d5bcb78f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.968ex; height:6.176ex;" alt="{\displaystyle ={\bigg (}2275+{\frac {1}{2}}+{\frac {1}{32}}+{\frac {1}{64}}{\bigg )}\;\;\;quadruple\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}2+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{36}}{\bigg )}\;\;\;quadruple\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>36</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}2+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{36}}{\bigg )}\;\;\;quadruple\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e45646c976f89edcbc001f208b9dc68498ac6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.288ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}2+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{36}}{\bigg )}\;\;\;quadruple\;\;\;ro}"></span> </p> </td> <td>Problem 43 represents the first serious mathematical mistake in the papyrus. Ahmes (or the source from which he may have been copying) attempted a shortcut in order to perform both the volume calculation and a unit conversion from cubic cubits to khar all in a single step, to avoid the need to use cubic cubits in an initial result. However, this attempt (which failed due to confusing part of the process used in 41 and 42 with that which was probably intended to be used in 43, giving consistent results by a different method) instead resulted in a new volume formula which is inconsistent with (and worse than) the approximation used in 41 and 42. </td></tr> <tr> <td>44, 45</td> <td>One cubic cubit is equal to 15/2 quadruple heqats. Consider (44) a cubic grain silo with a length of 10 cubits on every edge. Express its volume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> in terms of quadruple heqats. On the other hand, (45) consider a cubic grain silo which has a volume of 7500 quadruple heqats, and express its edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> in terms of cubits.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=7500\;\;\;quadruple\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mn>7500</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=7500\;\;\;quadruple\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa871ba5c3d7f6f354daffbfe571f837caabb840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.492ex; height:2.509ex;" alt="{\displaystyle V=7500\;\;\;quadruple\;\;heqat}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=10\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=10\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bc8155e0634d569ffe0392a922d686fb452518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.028ex; height:2.176ex;" alt="{\displaystyle l=10\;\;\;cubit}"></span> </p> </td> <td>Problem 45 is an exact reversal of problem 44, and they are therefore presented together here. </td></tr> <tr> <td>46</td> <td>A rectangular prism-grain silo has a volume of 2500 quadruple heqats. Describe its three dimensions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{1},l_{2},l_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{1},l_{2},l_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85523c9d374cfb6a83d646a0682b6046ebaffd70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.509ex;" alt="{\displaystyle l_{1},l_{2},l_{3}}"></span> in terms of cubits.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{1}=l_{2}=10\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{1}=l_{2}=10\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426e8cc438cf5c02ba9925fa23abbd3a0124fa2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.929ex; height:2.509ex;" alt="{\displaystyle l_{1}=l_{2}=10\;\;\;cubit}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{3}=3+{\frac {1}{3}}\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{3}=3+{\frac {1}{3}}\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75ad7850deea698a422a87cfca9569726b23e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.759ex; height:5.176ex;" alt="{\displaystyle l_{3}=3+{\frac {1}{3}}\;\;\;cubit}"></span> </p> </td> <td>This problem as stated has infinitely many solutions, but a simple choice of solution closely related to the terms of 44 and 45 is made. </td></tr> <tr> <td>47</td> <td>Divide the physical volume quantity of 100 quadruple heqats by each of the multiples of 10, from 10 through 100. Express the results in Egyptian fractional terms of quadruple heqat and quadruple ro, and present the results in a table.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {100}{10}}&q.\;heqat&=&10&q.\;heqat\\{\frac {100}{20}}&q.\;heqat&=&5&q.\;heqat\\{\frac {100}{30}}&q.\;heqat&=&(3+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}})&q.\;heqat\\&&+&(1+{\frac {2}{3}})&q.\;ro\\{\frac {100}{40}}&q.\;heqat&=&(2+{\frac {1}{2}})&q.\;heqat\\{\frac {100}{50}}&q.\;heqat&=&2&q.\;heqat\\{\frac {100}{60}}&q.\;heqat&=&(1+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}})&q.\;heqat\\&&+&(3+{\frac {1}{3}})&q.\;ro\\{\frac {100}{70}}&q.\;heqat&=&(1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&(2+{\frac {1}{14}}+{\frac {1}{21}}+{\frac {1}{42}})&q.\;ro\\{\frac {100}{80}}&q.\;heqat&=&(1+{\frac {1}{4}})&q.\;heqat\\{\frac {100}{90}}&q.\;heqat&=&(1+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&({\frac {1}{2}}+{\frac {1}{18}})&q.\;ro\\{\frac {100}{100}}&q.\;heqat&=&1&q.\;heqat\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>10</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>20</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>30</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>40</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>50</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>60</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>70</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>21</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>80</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>90</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>100</mn> </mfrac> </mrow> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>q</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {100}{10}}&q.\;heqat&=&10&q.\;heqat\\{\frac {100}{20}}&q.\;heqat&=&5&q.\;heqat\\{\frac {100}{30}}&q.\;heqat&=&(3+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}})&q.\;heqat\\&&+&(1+{\frac {2}{3}})&q.\;ro\\{\frac {100}{40}}&q.\;heqat&=&(2+{\frac {1}{2}})&q.\;heqat\\{\frac {100}{50}}&q.\;heqat&=&2&q.\;heqat\\{\frac {100}{60}}&q.\;heqat&=&(1+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}})&q.\;heqat\\&&+&(3+{\frac {1}{3}})&q.\;ro\\{\frac {100}{70}}&q.\;heqat&=&(1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&(2+{\frac {1}{14}}+{\frac {1}{21}}+{\frac {1}{42}})&q.\;ro\\{\frac {100}{80}}&q.\;heqat&=&(1+{\frac {1}{4}})&q.\;heqat\\{\frac {100}{90}}&q.\;heqat&=&(1+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&({\frac {1}{2}}+{\frac {1}{18}})&q.\;ro\\{\frac {100}{100}}&q.\;heqat&=&1&q.\;heqat\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30432d52339860ebe8f29f57e0b7b41356fb9003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -27.602ex; margin-bottom: -0.236ex; width:57.483ex; height:56.843ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {100}{10}}&q.\;heqat&=&10&q.\;heqat\\{\frac {100}{20}}&q.\;heqat&=&5&q.\;heqat\\{\frac {100}{30}}&q.\;heqat&=&(3+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}})&q.\;heqat\\&&+&(1+{\frac {2}{3}})&q.\;ro\\{\frac {100}{40}}&q.\;heqat&=&(2+{\frac {1}{2}})&q.\;heqat\\{\frac {100}{50}}&q.\;heqat&=&2&q.\;heqat\\{\frac {100}{60}}&q.\;heqat&=&(1+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}})&q.\;heqat\\&&+&(3+{\frac {1}{3}})&q.\;ro\\{\frac {100}{70}}&q.\;heqat&=&(1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&(2+{\frac {1}{14}}+{\frac {1}{21}}+{\frac {1}{42}})&q.\;ro\\{\frac {100}{80}}&q.\;heqat&=&(1+{\frac {1}{4}})&q.\;heqat\\{\frac {100}{90}}&q.\;heqat&=&(1+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}})&q.\;heqat\\&&+&({\frac {1}{2}}+{\frac {1}{18}})&q.\;ro\\{\frac {100}{100}}&q.\;heqat&=&1&q.\;heqat\\\end{bmatrix}}}"></span> </p> </td> <td>In problem 47, Ahmes is particularly insistent on representing more elaborate strings of fractions as <a href="/wiki/Eye_of_Horus" title="Eye of Horus">Horus eye</a> fractions, as far as he can. Compare problems 64 and 80 for similar preference of representation. To conserve space, "quadruple" has been shortened to "q." in all cases. </td></tr> <tr> <td>48</td> <td>Compare the area of a circle with diameter 9 to that of its circumscribing square, which also has a side length of 9. What is the ratio of the area of the circle to that of the square?</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {64}{81}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>81</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {64}{81}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3fbf96df491657f96846781727ffc6cb964277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {64}{81}}}"></span></td> <td>The statement and solution of problem 48 make explicitly clear this preferred method of approximating the area of a circle, which had been used earlier in problems 41–43. However, it is <a href="/wiki/Pi" title="Pi">erroneous</a>. The original statement of problem 48 involves the usage of a unit of area known as the setat, which will shortly be given further context in future problems. For the moment, it is cosmetic. </td></tr> <tr> <td>49</td> <td>One khet is a unit of length, being equal to 100 cubits. Also, a "cubit strip" is a rectangular strip-measurement of area, being 1 cubit by 100 cubits, or 100 square cubits (or a physical quantity of equal area). Consider a rectangular plot of land measuring 10 khet by 1 khet. Express its area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in terms of cubit strips.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=1000\;\;\;cubit\;\;\;strip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>1000</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=1000\;\;\;cubit\;\;\;strip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229ebfb79acbdc8d268b07d7767afa391e8d5ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.289ex; height:2.509ex;" alt="{\displaystyle A=1000\;\;\;cubit\;\;\;strip}"></span></td> <td>- </td></tr> <tr> <td>50</td> <td>One square khet is a unit of area equal to one setat. Consider a circle with a diameter of 9 khet. Express its area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in terms of setat.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=64\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>64</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=64\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718585335c5a101c598ba571c3ccff441184bb19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.185ex; height:2.176ex;" alt="{\displaystyle A=64\;\;\;setat}"></span></td> <td>Problem 50 is effectively a reinforcement of 48's 64/81 rule for a circle's area, which pervades the papyrus. </td></tr> <tr> <td>51</td> <td>A triangular tract of land has a base of 4 khet and an altitude of 10 khet. Find its area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in terms of setat.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=20\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>20</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=20\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71bd710b2567c0623707797e7227b038c05aeb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.185ex; height:2.176ex;" alt="{\displaystyle A=20\;\;\;setat}"></span></td> <td>The setup and solution of 51 recall the familiar formula for calculating a triangle's area, and per Chace it is paraphrased as such. However, the papyrus's triangular diagram, previous mistakes, and translation issues present ambiguity over whether the triangle in question is a right triangle, or indeed if Ahmes actually understood the conditions under which the stated answer is correct. Specifically, it is unclear whether the dimension of 10 khet was meant as an <i>altitude</i> (in which case the problem is correctly worked as stated) or whether "10 khet" simply refers to a <i>side</i> of the triangle, in which case the figure would have to be a right triangle in order for the answer to be factually correct and properly worked, as done. These problems and confusions perpetuate themselves throughout 51–53, to the point where Ahmes seems to lose understanding of what he is doing, especially in 53. </td></tr> <tr> <td>52</td> <td>A trapezoidal tract of land has two bases, being 6 khet and 4 khet. Its altitude is 20 khet. Find its area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in terms of setat.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=100\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>100</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=100\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989ee2590ea9513c125ce048dcb7fb6fa263b98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.347ex; height:2.176ex;" alt="{\displaystyle A=100\;\;\;setat}"></span></td> <td>Problem 52's issues are much the same as those of 51. The method of solution is familiar to moderns, and yet circumstances like those in 51 cast doubt over how well Ahmes or his source understood what they were doing. </td></tr> <tr> <td>53</td> <td>An isosceles triangle (a tract of land, say) has a base equal to 4 1/2 khet, and an altitude equal to 14 khet. Two line segments parallel to the base further partition the triangle into three sectors, being a bottom trapezoid, a middle trapezoid, and a top (similar) smaller triangle. The line segments cut the triangle's altitude at its midpoint (7) and further at a quarter-point (3 1/2) closer to the base, so that each trapezoid has an altitude of 3 1/2 khet, while the smaller similar triangle has an altitude of 7 khet. Find the lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{1},l_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{1},l_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9f9d870f9981b480b2ae35a0fed6f62e3786f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.529ex; height:2.509ex;" alt="{\displaystyle l_{1},l_{2}}"></span> of the two line segments, where they are the shorter and the longer line segments respectively, and express them in Egyptian fractional terms of khet. Furthermore, find the areas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},A_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},A_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18eb93c88961a573677555bffd4d27cc0557e70d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.46ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},A_{3}}"></span> of the three sectors, where they are the large trapezoid, the middle trapezoid, and the small triangle respectively, and express them in Egyptian fractional terms of setat and cubit strips. Use the fact that 1 setat = 100 cubit strips for unit conversions.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{1}={\bigg (}2+{\frac {1}{4}}{\bigg )}\;\;\;khet}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>k</mi> <mi>h</mi> <mi>e</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{1}={\bigg (}2+{\frac {1}{4}}{\bigg )}\;\;\;khet}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0aa810dd437f7d6e641183c685eabc2edc23f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.677ex; height:6.176ex;" alt="{\displaystyle l_{1}={\bigg (}2+{\frac {1}{4}}{\bigg )}\;\;\;khet}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{2}={\bigg (}3+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;khet}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>k</mi> <mi>h</mi> <mi>e</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{2}={\bigg (}3+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;khet}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49bb60f1715f0a3cf2958dfab16efef031a7e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.516ex; height:6.176ex;" alt="{\displaystyle l_{2}={\bigg (}3+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;khet}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}={\bigg (}13+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}3+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>13</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}={\bigg (}13+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}3+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60f945be7cf66506308fc4f5906c32c55eecbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.399ex; height:6.176ex;" alt="{\displaystyle A_{1}={\bigg (}13+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}3+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}={\bigg (}9+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}9+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>9</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>9</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}={\bigg (}9+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}9+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed34e09c996ffa5e43ae8e932b28da251d9a4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.076ex; height:6.176ex;" alt="{\displaystyle A_{2}={\bigg (}9+{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}\;\;\;setat+{\bigg (}9+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;cubit\;\;\;strip}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}={\bigg (}7+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}={\bigg (}7+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d900503c897d490552faa61b8753a3d54611963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.015ex; height:6.176ex;" alt="{\displaystyle A_{3}={\bigg (}7+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}{\bigg )}\;\;\;setat}"></span> </p> </td> <td>Problem 53, being more complex, is fraught with many of the same issues as 51 and 52—translation ambiguities and several numerical mistakes. In particular concerning the large bottom trapezoid, Ahmes seems to get stuck on finding the upper base, and proposes in the original work to subtract "one tenth, equal to 1 + 1/4 + 1/8 setat plus 10 cubit strips" from a rectangle being (presumably) 4 1/2 x 3 1/2 (khet). However, even Ahmes' answer here is inconsistent with the problem's other information. Happily the context of 51 and 52, together with the base, mid-line, and smaller triangle area (which <i>are</i> given as 4 + 1/2, 2 + 1/4 and 7 + 1/2 + 1/4 + 1/8, respectively) make it possible to interpret the problem and its solution as has been done here. The given paraphrase therefore represents a consistent best guess as to the problem's intent, which follows Chace. Ahmes also refers to the "cubit strips" again in the course of calculating for this problem, and we therefore repeat their usage here. It bears mentioning that neither Ahmes nor Chace explicitly give the area for the middle trapezoid in their treatments (Chace suggests that this is a triviality from Ahmes' point of view); liberty has therefore been taken to report it in a manner which is consistent with what Chace had thus far advanced. </td></tr> <tr> <td>54</td> <td>There are 10 plots of land. In each plot, a sector is partitioned off such that the sum of the area of these 10 new partitions is 7 setat. Each new partition has equal area. Find the area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of any one of these 10 new partitions, and express it in Egyptian fractional terms of setat and cubit strips.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}{\bigg )}\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}{\bigg )}\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6097d21a794c4ce740586a0ce987c4f668186f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.119ex; height:6.176ex;" alt="{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}{\bigg )}\;\;\;setat}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bigg (}{\frac {1}{2}}+{\frac {1}{8}}{\bigg )}\;\;\;setat+{\bigg (}7+{\frac {1}{2}}{\bigg )}\;\;\;cubit\;\;\;strip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bigg (}{\frac {1}{2}}+{\frac {1}{8}}{\bigg )}\;\;\;setat+{\bigg (}7+{\frac {1}{2}}{\bigg )}\;\;\;cubit\;\;\;strip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772ed9d5f461401846d555d95a68d7b8dfc91ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.791ex; height:6.176ex;" alt="{\displaystyle ={\bigg (}{\frac {1}{2}}+{\frac {1}{8}}{\bigg )}\;\;\;setat+{\bigg (}7+{\frac {1}{2}}{\bigg )}\;\;\;cubit\;\;\;strip}"></span> </p> </td> <td>- </td></tr> <tr> <td>55</td> <td>There are 5 plots of land. In each plot, a sector is partitioned off such that the sum of the area of these 5 new partitions is 3 setat. Each new partition has equal area. Find the area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of any one of these 5 new partitions, and express it in Egyptian fractional terms of setat and cubit strips.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{10}}{\bigg )}\;\;\;setat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{10}}{\bigg )}\;\;\;setat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802bf7ec4fc0ee53ad4123a036ce8ccb749255da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.281ex; height:6.176ex;" alt="{\displaystyle A={\bigg (}{\frac {1}{2}}+{\frac {1}{10}}{\bigg )}\;\;\;setat}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{2}}\;\;\;setat+10\;\;\;cubit\;\;\;strip}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}\;\;\;setat+10\;\;\;cubit\;\;\;strip}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea704ea8335d37b51a46308d9ab0a28dff1194a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.433ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}\;\;\;setat+10\;\;\;cubit\;\;\;strip}"></span> </p> </td> <td>- </td></tr> <tr> <td>56</td> <td>1) The unit of length known as a <a href="/wiki/Cubit" title="Cubit"><i>royal</i> cubit</a> is (and has been, throughout the papyrus) what is meant when we simply refer to a <i>cubit</i>. One <i>royal</i> cubit, or one cubit, is equal to seven palms, and one palm is equal to four fingers. In other words, the following equalities hold: 1 (royal) cubit = 1 cubit = 7 palms = 28 fingers. <p>2) Consider a <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">right regular square</a> <a href="/wiki/Pyramid" title="Pyramid">pyramid</a> whose base, the square face is coplanar with a plane (or the ground, say), so that any of the planes containing its triangular faces has the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> with respect to the ground-plane (that is, on the interior of the pyramid). In other words, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the angle of the triangular faces of the pyramid with respect to the ground. The <a href="/wiki/Seked" title="Seked">seked</a> of such a pyramid, then, having altitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and base edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, is defined as <i>that physical length</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S}{1\;\;\;royal\;\;\;cubit}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> <mi>y</mi> <mi>a</mi> <mi>l</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{1\;\;\;royal\;\;\;cubit}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48cb5fea850284a2d292fcb5860723b1050c9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.554ex; height:5.843ex;" alt="{\displaystyle {\frac {S}{1\;\;\;royal\;\;\;cubit}}=}"></span> <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot {\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot {\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07bdc6b7d518cf5b00cef06b8d677c9acde06e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.577ex; height:2.176ex;" alt="{\displaystyle \cot {\theta }}"></span></a>. Put another way, the seked of a pyramid can be interpreted as the ratio of its triangular faces' <i>run per one unit (cubit) rise</i>. Or, for the appropriate right triangle on a pyramid's interior having legs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,{\frac {b}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,{\frac {b}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cec4af66cd9d39cc6d705325c04c2e70ebd943f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.262ex; height:5.343ex;" alt="{\displaystyle a,{\frac {b}{2}}}"></span> and the perpendicular bisector of a triangular face as the hypotenuse, then the pyramid's seked <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot {\theta }={\frac {b}{2a}}={\frac {S}{1\;\;\;royal\;\;\;cubit}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> <mi>y</mi> <mi>a</mi> <mi>l</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot {\theta }={\frac {b}{2a}}={\frac {S}{1\;\;\;royal\;\;\;cubit}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4519c79079a1335e664e5e6f4005d805d6d9af06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.103ex; height:5.843ex;" alt="{\displaystyle \cot {\theta }={\frac {b}{2a}}={\frac {S}{1\;\;\;royal\;\;\;cubit}}}"></span>. Similar triangles are therefore described, and one can be scaled to the other. </p><p>3) A pyramid has an altitude of 250 (royal) cubits, and the side of its base has a length of 360 (royal) cubits. Find its <a href="/wiki/Seked" title="Seked">seked</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> in Egyptian fractional terms of (royal) cubits, and also in terms of palms. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{50}}{\bigg )}\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>50</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{50}}{\bigg )}\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705b2e0025212a6db432ce21030dee4e871e91ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.77ex; height:6.176ex;" alt="{\displaystyle S={\bigg (}{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{50}}{\bigg )}\;\;\;cubit}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\bigg (}5+{\frac {1}{25}}{\bigg )}\;\;\;palm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>25</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>p</mi> <mi>a</mi> <mi>l</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\bigg (}5+{\frac {1}{25}}{\bigg )}\;\;\;palm}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4745ceb93b0a5f43aa9203457aed313dbd695afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.107ex; height:6.176ex;" alt="{\displaystyle ={\bigg (}5+{\frac {1}{25}}{\bigg )}\;\;\;palm}"></span> </p> </td> <td>Problem 56 is the first of the "pyramid problems" or seked problems in the Rhind papyrus, 56–59, 59B and 60, which concern the notion of a pyramid's facial inclination with respect to a flat ground. In this connection, the concept of a <a href="/wiki/Seked" title="Seked">seked</a> suggests early beginnings of <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>. Unlike modern trigonometry however, note especially that a seked is found with respect to some pyramid, and is itself a <i>physical length measurement</i>, which may be given in terms of any physical length units. For obvious reasons however, we (and the papyrus) confine our attention to situations involving ancient Egygtian units. We have also clarified that royal cubits are used throughout the papyrus, to differentiate them from "short" cubits which were used elsewhere in ancient Egypt. One "short" cubit is equal to six palms. </td></tr> <tr> <td>57, 58</td> <td>The seked of a pyramid is 5 palms and 1 finger, and the side of its base is 140 cubits. Find (57) its altitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> in terms of cubits. On the other hand, (58), a pyramid's altitude is 93 + 1/3 cubits, and the side of its base is 140 cubits. Find its seked <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and express it in terms of palms and fingers.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\bigg (}93+{\frac {1}{3}}{\bigg )}\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>93</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\bigg (}93+{\frac {1}{3}}{\bigg )}\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa0079d0262aa443de897eeb9fc00145f8f994b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.825ex; height:6.176ex;" alt="{\displaystyle a={\bigg (}93+{\frac {1}{3}}{\bigg )}\;\;\;cubit}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=5\;\;\;palm+1\;\;\;finger}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>p</mi> <mi>a</mi> <mi>l</mi> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> <mi>e</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=5\;\;\;palm+1\;\;\;finger}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32124c93bffb1a7e10f68f7bb5a0563e674fe19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.491ex; height:2.509ex;" alt="{\displaystyle S=5\;\;\;palm+1\;\;\;finger}"></span> </p> </td> <td>Problem 58 is an exact reversal of problem 57, and they are therefore presented together here. </td></tr> <tr> <td>59, 59B</td> <td>A pyramid's (59) altitude is 8 cubits, and its base length is 12 cubits. Express its seked <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> in terms of palms and fingers. On the other hand, (59B), a pyramid's seked is five palms and one finger, and the side of its base is 12 cubits. Express its altitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> in terms of cubits.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=5\;\;\;palm+1\;\;\;finger}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>p</mi> <mi>a</mi> <mi>l</mi> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> <mi>e</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=5\;\;\;palm+1\;\;\;finger}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32124c93bffb1a7e10f68f7bb5a0563e674fe19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.491ex; height:2.509ex;" alt="{\displaystyle S=5\;\;\;palm+1\;\;\;finger}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=8\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>8</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=8\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7950059d17a1aaf1a4164c9a373ced870ff8a7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.402ex; height:2.176ex;" alt="{\displaystyle a=8\;\;\;cubit}"></span> </p> </td> <td>Problems 59 and 59B consider a case similar to 57 and 58, ending with familiar results. As exact reversals of each other, they are presented together here. </td></tr> <tr> <td>60</td> <td>If a "pillar" (that is, a cone) has an altitude of 30 cubits, and the side of its base (or diameter) has a length of 15 cubits, find its seked <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and express it in terms of cubits.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{4}}\;\;\;cubit}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>c</mi> <mi>u</mi> <mi>b</mi> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{4}}\;\;\;cubit}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27e8d758b98dcb372a1ed0dca1cb3a96eceed65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.508ex; height:5.176ex;" alt="{\displaystyle S={\frac {1}{4}}\;\;\;cubit}"></span></td> <td>Ahmes uses slightly different words to present this problem, which lend themselves to translation issues. However, the overall context of the problem, together with its accompanying diagram (which differs from the previous diagrams), leads Chace to conclude that a cone is meant. The notion of seked is easily generalized to the lateral face of a cone; he therefore reports the problem in these terms. Problem 60 concludes the geometry section of the papyrus. Moreover, it is the last problem on the <a href="/wiki/Recto_and_verso" title="Recto and verso">recto</a> (front side) of the document; all later content in this summary is present on the <a href="/wiki/Recto_and_verso" title="Recto and verso">verso</a> (back side) of the papyrus. The transition from 60 to 61 is thus both a thematic and physical shift in the papyrus. </td></tr> <tr> <td>61</td> <td>Seventeen multiplications are to have their products expressed as Egyptian fractions. The whole is to be given as a table.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {2}{3}}\cdot {\frac {2}{3}}={\frac {1}{3}}+{\frac {1}{9}}&;&{\frac {1}{3}}\cdot {\frac {2}{3}}={\frac {1}{6}}+{\frac {1}{18}}\\{\frac {2}{3}}\cdot {\frac {1}{3}}={\frac {1}{6}}+{\frac {1}{18}}&;&{\frac {2}{3}}\cdot {\frac {1}{6}}={\frac {1}{12}}+{\frac {1}{36}}\\{\frac {2}{3}}\cdot {\frac {1}{2}}={\frac {1}{3}}&;&{\frac {1}{3}}\cdot {\frac {1}{2}}={\frac {1}{6}}\\{\frac {1}{6}}\cdot {\frac {1}{2}}={\frac {1}{12}}&;&{\frac {1}{12}}\cdot {\frac {1}{2}}={\frac {1}{24}}\\{\frac {1}{9}}\cdot {\frac {2}{3}}={\frac {1}{18}}+{\frac {1}{54}}&;&{\frac {2}{3}}\cdot {\frac {1}{9}}={\frac {1}{18}}+{\frac {1}{54}}\\{\frac {1}{4}}\cdot {\frac {1}{5}}={\frac {1}{20}}&;&{\frac {2}{3}}\cdot {\frac {1}{7}}={\frac {1}{14}}+{\frac {1}{42}}\\{\frac {1}{2}}\cdot {\frac {1}{7}}={\frac {1}{14}}&;&{\frac {2}{3}}\cdot {\frac {1}{11}}={\frac {1}{22}}+{\frac {1}{66}}\\{\frac {1}{3}}\cdot {\frac {1}{11}}={\frac {1}{33}}&;&{\frac {1}{2}}\cdot {\frac {1}{11}}={\frac {1}{22}}\\{\frac {1}{4}}\cdot {\frac {1}{11}}={\frac {1}{44}}&&\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>36</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>54</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>18</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>54</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>22</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>66</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>33</mn> </mfrac> </mrow> </mtd> <mtd> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>22</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>44</mn> </mfrac> </mrow> </mtd> <mtd /> <mtd /> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {2}{3}}\cdot {\frac {2}{3}}={\frac {1}{3}}+{\frac {1}{9}}&;&{\frac {1}{3}}\cdot {\frac {2}{3}}={\frac {1}{6}}+{\frac {1}{18}}\\{\frac {2}{3}}\cdot {\frac {1}{3}}={\frac {1}{6}}+{\frac {1}{18}}&;&{\frac {2}{3}}\cdot {\frac {1}{6}}={\frac {1}{12}}+{\frac {1}{36}}\\{\frac {2}{3}}\cdot {\frac {1}{2}}={\frac {1}{3}}&;&{\frac {1}{3}}\cdot {\frac {1}{2}}={\frac {1}{6}}\\{\frac {1}{6}}\cdot {\frac {1}{2}}={\frac {1}{12}}&;&{\frac {1}{12}}\cdot {\frac {1}{2}}={\frac {1}{24}}\\{\frac {1}{9}}\cdot {\frac {2}{3}}={\frac {1}{18}}+{\frac {1}{54}}&;&{\frac {2}{3}}\cdot {\frac {1}{9}}={\frac {1}{18}}+{\frac {1}{54}}\\{\frac {1}{4}}\cdot {\frac {1}{5}}={\frac {1}{20}}&;&{\frac {2}{3}}\cdot {\frac {1}{7}}={\frac {1}{14}}+{\frac {1}{42}}\\{\frac {1}{2}}\cdot {\frac {1}{7}}={\frac {1}{14}}&;&{\frac {2}{3}}\cdot {\frac {1}{11}}={\frac {1}{22}}+{\frac {1}{66}}\\{\frac {1}{3}}\cdot {\frac {1}{11}}={\frac {1}{33}}&;&{\frac {1}{2}}\cdot {\frac {1}{11}}={\frac {1}{22}}\\{\frac {1}{4}}\cdot {\frac {1}{11}}={\frac {1}{44}}&&\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5656d128300a5b850b3d5bb472078d6c8a7ff1c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.505ex; width:41.755ex; height:36.176ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {2}{3}}\cdot {\frac {2}{3}}={\frac {1}{3}}+{\frac {1}{9}}&;&{\frac {1}{3}}\cdot {\frac {2}{3}}={\frac {1}{6}}+{\frac {1}{18}}\\{\frac {2}{3}}\cdot {\frac {1}{3}}={\frac {1}{6}}+{\frac {1}{18}}&;&{\frac {2}{3}}\cdot {\frac {1}{6}}={\frac {1}{12}}+{\frac {1}{36}}\\{\frac {2}{3}}\cdot {\frac {1}{2}}={\frac {1}{3}}&;&{\frac {1}{3}}\cdot {\frac {1}{2}}={\frac {1}{6}}\\{\frac {1}{6}}\cdot {\frac {1}{2}}={\frac {1}{12}}&;&{\frac {1}{12}}\cdot {\frac {1}{2}}={\frac {1}{24}}\\{\frac {1}{9}}\cdot {\frac {2}{3}}={\frac {1}{18}}+{\frac {1}{54}}&;&{\frac {2}{3}}\cdot {\frac {1}{9}}={\frac {1}{18}}+{\frac {1}{54}}\\{\frac {1}{4}}\cdot {\frac {1}{5}}={\frac {1}{20}}&;&{\frac {2}{3}}\cdot {\frac {1}{7}}={\frac {1}{14}}+{\frac {1}{42}}\\{\frac {1}{2}}\cdot {\frac {1}{7}}={\frac {1}{14}}&;&{\frac {2}{3}}\cdot {\frac {1}{11}}={\frac {1}{22}}+{\frac {1}{66}}\\{\frac {1}{3}}\cdot {\frac {1}{11}}={\frac {1}{33}}&;&{\frac {1}{2}}\cdot {\frac {1}{11}}={\frac {1}{22}}\\{\frac {1}{4}}\cdot {\frac {1}{11}}={\frac {1}{44}}&&\\\end{bmatrix}}}"></span> </p> </td> <td>The syntax of the original document and its repeated multiplications indicate a rudimentary understanding that multiplication is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>. </td></tr> <tr> <td>61B</td> <td>Give a general procedure for converting the product of 2/3 and the reciprocal of any (positive) odd number 2n+1 into an Egyptian fraction of two terms, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3}}\cdot {\frac {1}{2n+1}}={\frac {1}{p}}+{\frac {1}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\cdot {\frac {1}{2n+1}}={\frac {1}{p}}+{\frac {1}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4e675bb94b01c0a9935a6cba0c87b5dac3e361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.017ex; height:5.676ex;" alt="{\displaystyle {\frac {2}{3}}\cdot {\frac {1}{2n+1}}={\frac {1}{p}}+{\frac {1}{q}}}"></span> with natural p and q. In other words, find p and q in terms of n.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2(2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2(2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f02f0428c4de9cc6ae2b41411326908cbdf255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.889ex; height:2.843ex;" alt="{\displaystyle p=2(2n+1)}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=6(2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>6</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=6(2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec51f3ad94433e6e403fb30d1759efb5d6e2d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.7ex; height:2.843ex;" alt="{\displaystyle q=6(2n+1)}"></span> </p> </td> <td>Problem 61B, and the method of decomposition that it describes (and suggests) is closely related to the computation of the <a href="/wiki/Rhind_Mathematical_Papyrus_2/n_table" title="Rhind Mathematical Papyrus 2/n table">Rhind Mathematical Papyrus 2/n table</a>. In particular, every case in the 2/n table involving a denominator which is a multiple of 3 can be said to follow the example of 61B. 61B's statement and solution are also suggestive of a generality which most of the rest of the papyrus's more concrete problems do not have. It therefore represents an early suggestion of both <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>. </td></tr> <tr> <td>62</td> <td>A bag of three precious metals, gold, silver and lead, has been purchased for 84 sha'ty, which is a monetary unit. All three substances weigh the same, and a deben is a unit of weight. 1 deben of gold costs 12 sha'ty, 1 deben of silver costs 6 sha'ty, and 1 deben of lead costs 3 sha'ty. Find the common weight <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> of any of the three metals in the bag.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=4\;\;\;deben}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mn>4</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>e</mi> <mi>b</mi> <mi>e</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=4\;\;\;deben}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3f6fce309a562909a160b0e81dc2e67fa31c3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.407ex; height:2.176ex;" alt="{\displaystyle W=4\;\;\;deben}"></span></td> <td>Problem 62 becomes a division problem entailing a little dimensional analysis. Its setup involving standard weights renders the problem straightforward. </td></tr> <tr> <td>63</td> <td>700 loaves are to be divided unevenly among four men, in four unequal, weighted shares. The shares will be in the respective proportions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3}}:{\frac {1}{2}}:{\frac {1}{3}}:{\frac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}:{\frac {1}{2}}:{\frac {1}{3}}:{\frac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3745781cb9ab45becce793af71677d8659fe39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.806ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}:{\frac {1}{2}}:{\frac {1}{3}}:{\frac {1}{4}}}"></span>. Find each share.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 266+{\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>266</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 266+{\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a39acde59dd0134a64427256a8f7cb50b6dc59f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.326ex; height:5.176ex;" alt="{\displaystyle 266+{\frac {2}{3}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198350442fbcb39391818f80732eb8701439bad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 200}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 133+{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>133</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 133+{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ef714cd643056ecb5d393eed1b9bc981b06989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.326ex; height:5.176ex;" alt="{\displaystyle 133+{\frac {1}{3}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"></span> </p> </td> <td>- </td></tr> <tr> <td>64</td> <td>Recall that the heqat is a unit of volume. Ten heqat of barley are to be distributed among ten men in an arithmetic progression, so that consecutive men's shares have a difference of 1/8 heqats. Find the ten shares and list them in descending order, in Egyptian fractional terms of heqat.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}1+{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}1+{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75051410a4ff7fa0ee5d453c19ee781df0295af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.631ex; height:6.176ex;" alt="{\displaystyle {\bigg (}1+{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ce99f5f0ef25c818137534e70b680ef629afbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.47ex; height:6.176ex;" alt="{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/331760f8366d93a38b121e08f893729bbd0d2cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.631ex; height:6.176ex;" alt="{\displaystyle {\bigg (}1+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}1+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}1+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df3f5e5e3c50dfc5ee3e5f0c66b5d7fec2e8767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.631ex; height:6.176ex;" alt="{\displaystyle {\bigg (}1+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}1+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}1+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52540a9975de6cf9bdfe999cb74457a520fb056c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.792ex; height:6.176ex;" alt="{\displaystyle {\bigg (}1+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a797f609d9fe377fd4d5ec818bbc32d325e00e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.306ex; height:6.176ex;" alt="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b8d12878b6eace91f82d5c552e2ff86e18fc0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.467ex; height:6.176ex;" alt="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87325798a4ef607110a146969c9bc746284cebf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.467ex; height:6.176ex;" alt="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d116282f1e142e9b1d696274266bc4ea513908d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.628ex; height:6.176ex;" alt="{\displaystyle {\bigg (}{\frac {1}{2}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feedd9f0bf0d569823c1ed729252b5bd16de53a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.467ex; height:6.176ex;" alt="{\displaystyle {\bigg (}{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}{\bigg )}\;heqat}"></span> </p> </td> <td>Problem 64 is a variant of 40, this time involving an even number of unknowns. For quick modern reference apart from Egyptian fractions, the shares range from 25/16 down through 7/16, where the numerator decreases by consecutive odd numbers. The terms are given as <a href="/wiki/Eye_of_Horus" title="Eye of Horus">Horus eye</a> fractions; compare problems 47 and 80 for more of this. </td></tr> <tr> <td>65</td> <td>100 loaves of bread are to be unevenly divided among ten men. Seven of the men receive a single share, while the other three men, being a boatman, a foreman, and a door-keeper, each receive a double share. Express each of these two share amounts as Egyptian fractions.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7+{\frac {2}{3}}+{\frac {1}{39}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>39</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7+{\frac {2}{3}}+{\frac {1}{39}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2786b9fa11bbda19aa334c770ba6bd6e6b5e2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.003ex; height:5.176ex;" alt="{\displaystyle 7+{\frac {2}{3}}+{\frac {1}{39}}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15+{\frac {1}{3}}+{\frac {1}{26}}+{\frac {1}{78}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>26</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>78</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15+{\frac {1}{3}}+{\frac {1}{26}}+{\frac {1}{78}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60c72866092d084ac2be322c97e88f3278c84420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.167ex; height:5.343ex;" alt="{\displaystyle 15+{\frac {1}{3}}+{\frac {1}{26}}+{\frac {1}{78}}}"></span> </p> </td> <td>- </td></tr> <tr> <td>66</td> <td>Recall that the heqat is a unit of volume and that one heqat equals 320 ro. 10 heqat of fat are distributed to one person over the course of one year (365 days), in daily allowances of equal amount. Express the allowance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> as an Egyptian fraction in terms of heqat and ro.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {1}{64}}\;\;\;heqat+{\bigg (}3+{\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{2190}}{\bigg )}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2190</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {1}{64}}\;\;\;heqat+{\bigg (}3+{\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{2190}}{\bigg )}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0428a7b8ea47b2f4b42a0ceb639e1a0270ef2920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.689ex; height:6.176ex;" alt="{\displaystyle a={\frac {1}{64}}\;\;\;heqat+{\bigg (}3+{\frac {2}{3}}+{\frac {1}{10}}+{\frac {1}{2190}}{\bigg )}\;\;\;ro}"></span></td> <td>Problem 66 in its original form explicitly states that one year is equal to 365 days, and repeatedly uses the number 365 for its calculations. It is therefore <a href="/wiki/Primary_source" title="Primary source">primary</a> historical evidence of the ancient Egyptian understanding of the <a href="/wiki/Year" title="Year">year</a>. </td></tr> <tr> <td>67</td> <td>A shepherd had a flock of animals, and had to give a portion of his flock to a lord as tribute. The shepherd was told to give two-thirds OF one-third of his original flock as tribute. The shepherd gave 70 animals. Find the size of the shepherd's original flock.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 315}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>315</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 315}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f00f1c8b8d6466c47bec4ee684903fed31a3c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 315}"></span></td> <td>- </td></tr> <tr> <td>68</td> <td>Four overseers are in charge of four crews of men, being 12, 8, 6 and 4 men, respectively. Each crewman works at a fungible rate, to produce a single work-product: production (picking, say) of grain. Working on some interval of time, these four gangs collectively produced 100 units, or 100 quadruple heqats of grain, where each crew's work-product will be given to each crew's overseer. Express each crew's output <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{12},O_{8},O_{6},O_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{12},O_{8},O_{6},O_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167a18be9c5956b4fc9c2c0dad16e0aec782e037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.234ex; height:2.509ex;" alt="{\displaystyle O_{12},O_{8},O_{6},O_{4}}"></span> in terms of quadruple heqat.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{12}=40\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>40</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{12}=40\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2277030ed945f23e780677048f12507a868f0166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.675ex; height:2.509ex;" alt="{\displaystyle O_{12}=40\;\;\;quadruple\;\;\;heqat}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{8}=26+{\frac {2}{3}}\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mn>26</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{8}=26+{\frac {2}{3}}\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/946fa48b9851aef8310119af9ce9da07fecc81d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.692ex; height:5.176ex;" alt="{\displaystyle O_{8}=26+{\frac {2}{3}}\;\;\;quadruple\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{6}=20\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>20</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{6}=20\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c81f46be85578a6880aede55b7141d8ced0ab14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.853ex; height:2.509ex;" alt="{\displaystyle O_{6}=20\;\;\;quadruple\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{4}=13+{\frac {1}{3}}\;\;\;quadruple\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>13</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mi>r</mi> <mi>u</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{4}=13+{\frac {1}{3}}\;\;\;quadruple\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a3e867d6e018fdfc2ea6b1f9c87e6d00f17ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.692ex; height:5.176ex;" alt="{\displaystyle O_{4}=13+{\frac {1}{3}}\;\;\;quadruple\;\;\;heqat}"></span> </p> </td> <td>- </td></tr> <tr> <td>69</td> <td>1) Consider cooking and food preparation. Suppose that there is a standardized way of cooking, or a production process, which will take volume units, specifically <i>heqats</i> of raw food-material (in particular, some <i>one</i> raw food-material) and produce <i>units</i> of some <i>one</i> finished food product. The <i>pefsu</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> of the (one) finished food product with respect to the (one) raw food-material, then, is defined as <i>the quantity of finished food product units <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> yielded from exactly one heqat of raw food material.</i> In other words, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {p\;\;\;finished\;\;\;unit}{1\;\;\;heqat_{raw\;\;\;material}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>i</mi> <mi>s</mi> <mi>h</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>u</mi> <mi>n</mi> <mi>i</mi> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>a</mi> <mi>w</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {p\;\;\;finished\;\;\;unit}{1\;\;\;heqat_{raw\;\;\;material}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eba8c670107f80d8e3849abba0fd5f1788772ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.636ex; height:5.843ex;" alt="{\displaystyle P={\frac {p\;\;\;finished\;\;\;unit}{1\;\;\;heqat_{raw\;\;\;material}}}}"></span>. <p>2) 3 + 1/2 heqats of meal produce 80 loaves of bread. Find the meal per loaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> in heqats and ro, and find the pefsu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> of these loaves with respect to the meal. Express them as Egyptian fractions. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {1}{32}}\;\;\;heqat+4\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mn>4</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {1}{32}}\;\;\;heqat+4\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6178e6e67cf9f1f11a90966eed4de068d8e468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.911ex; height:5.176ex;" alt="{\displaystyle m={\frac {1}{32}}\;\;\;heqat+4\;\;\;ro}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\bigg (}22+{\frac {2}{3}}+{\frac {1}{7}}+{\frac {1}{21}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>22</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>21</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>f</mi> </mrow> <mrow> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\bigg (}22+{\frac {2}{3}}+{\frac {1}{7}}+{\frac {1}{21}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed959a68995a7a0d4c08d7d220fb16774f928901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.468ex; height:6.176ex;" alt="{\displaystyle P={\bigg (}22+{\frac {2}{3}}+{\frac {1}{7}}+{\frac {1}{21}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}"></span> </p> </td> <td>Problem 69 begins the "pefsu" problems, 69–78, in the context of food preparation. The notion of the pefsu assumes some standardized production process without accidents, waste, etc., and only concerns the relationship of one standardized finished food product to one particular raw material. That is, the pefsu is not immediately concerned with matters like production time, or (in any one given case) the relationship of other raw materials or equipment to the production process, etc. Still, the notion of the pefsu is another hint of abstraction in the papyrus, capable of being applied to <i>any</i> binary relationship between a food product (or finished good, for that matter) and a raw material. The concepts that the pefsu entails are thus typical of <a href="/wiki/Manufacturing" title="Manufacturing">manufacturing</a>. </td></tr> <tr> <td>70</td> <td>(7 + 1/2 + 1/4 + 1/8) heqats of meal produce 100 loaves of bread. Find the meal per loaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> in heqats and ro, and find the pefsu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> of these loaves with respect to the meal. Express them as Egyptian fractions.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\bigg (}{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\frac {1}{5}}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\bigg (}{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\frac {1}{5}}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452f6e9dc9309c49ace135a484905dfdb2050b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.17ex; height:6.176ex;" alt="{\displaystyle m={\bigg (}{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\frac {1}{5}}\;\;\;ro}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\bigg (}12+{\frac {2}{3}}+{\frac {1}{42}}+{\frac {1}{126}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>12</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>126</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>f</mi> </mrow> <mrow> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\bigg (}12+{\frac {2}{3}}+{\frac {1}{42}}+{\frac {1}{126}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4123787b1ec29f90df1f9d65acf5baf10481154b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.793ex; height:6.176ex;" alt="{\displaystyle P={\bigg (}12+{\frac {2}{3}}+{\frac {1}{42}}+{\frac {1}{126}}{\bigg )}{\frac {loaf}{heqat_{meal}}}}"></span> </p> </td> <td>- </td></tr> <tr> <td>71</td> <td>1/2 heqats of besha, a raw material, produces exactly one full des-measure (glass) of beer. Suppose that there is a production process for diluted glasses of beer. 1/4 of the glass just described is poured out, and what has just been poured out is captured and re-used later. This glass, which is now 3/4 full, is then diluted back to capacity with water, producing exactly one full diluted glass of beer. Find the pefsu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> of these diluted beer glasses with respect to the besha as an Egyptian fraction.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\bigg (}2+{\frac {2}{3}}{\bigg )}{\frac {des-measure}{heqat_{besha}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>e</mi> <mi>s</mi> <mo>−</mo> <mi>m</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>u</mi> <mi>r</mi> <mi>e</mi> </mrow> <mrow> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>e</mi> <mi>s</mi> <mi>h</mi> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\bigg (}2+{\frac {2}{3}}{\bigg )}{\frac {des-measure}{heqat_{besha}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6e0ac27aa2bd7cf432c8f814851b4b7ef50168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.239ex; height:6.176ex;" alt="{\displaystyle P={\bigg (}2+{\frac {2}{3}}{\bigg )}{\frac {des-measure}{heqat_{besha}}}}"></span></td> <td>Problem 71 describes intermediate steps in a production process, as well as a second raw material, water. These are irrelevant to the relationship between the <i>finished unit and the raw material</i> (besha in this case). </td></tr> <tr> <td>72</td> <td>100 bread loaves "of pefsu 10" are to be evenly exchanged for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> loaves "of pefsu 45". Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=450}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>450</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=450}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb29046c0c9c3ee2938f266854dc040bd5f0e7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.916ex; height:2.176ex;" alt="{\displaystyle x=450}"></span></td> <td>Now that the concept of the pefsu has been established, problems 72–78 explore even exchanges of different heaps of finished foods, having different pefsu. In general however, they assume a <i>common raw material</i> of some kind. Specifically, the common raw material assumed throughout all of 72–78 is called <i>wedyet flour</i>, which is even implicated in the production of beer, so that beer can be exchanged for bread in the latter problems. 74's original statement also mentions "Upper Egyptian barley", but for our purposes this is cosmetic. What problems 72–78 say, then, is really this: equal amounts of raw material are used in two different production processes, to produce two different units of finished food, where each type has a different pefsu. One of the two finished food units is given. Find the other. This can be accomplished by dividing both units (known and unknown) by their respective pefsu, where the units of finished food vanish in dimensional analysis, and only the same raw material is considered. One can then easily solve for x. 72–78 therefore really require that x be given so that equal amounts of raw material are used in two different production processes. </td></tr> <tr> <td>73</td> <td>100 bread loaves of pefsu 10 are to be evenly exchanged for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> loaves of pefsu 15. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=150}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>150</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=150}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4ad5b857628f48da0e95e0494921d3a5d131e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.916ex; height:2.176ex;" alt="{\displaystyle x=150}"></span></td> <td>- </td></tr> <tr> <td>74</td> <td>1000 bread loaves of pefsu 5 are to be divided evenly into two heaps of 500 loaves each. Each heap is to be evenly exchanged for two other heaps, one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> loaves of pefsu 10, and the other of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> loaves of pefsu 20. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c9ad206da21f7d9fdfe515fade9f0feecc9909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.078ex; height:2.176ex;" alt="{\displaystyle x=1000}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a7d8628e464d8a3996c2fcaff9929449aecbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.904ex; height:2.509ex;" alt="{\displaystyle y=2000}"></span> </p> </td> <td>- </td></tr> <tr> <td>75</td> <td>155 bread loaves of pefsu 20 are to be evenly exchanged for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> loaves of pefsu 30. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=232+{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>232</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=232+{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f437162aa8bc4c26192f44a727a6f91d9acc26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.754ex; height:5.176ex;" alt="{\displaystyle x=232+{\frac {1}{2}}}"></span></td> <td>- </td></tr> <tr> <td>76</td> <td>1000 bread loaves of pefsu 10, one heap, will be evenly exchanged for two other heaps of loaves. The other two heaps each has an equal number of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> loaves, one being of pefsu 20, the other of pefsu 30. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1200}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec2c7db7b984ac71d02b5884d7db718d0625508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.078ex; height:2.176ex;" alt="{\displaystyle x=1200}"></span></td> <td>- </td></tr> <tr> <td>77</td> <td>10 des-measure of beer, of pefsu 2, are to be evenly exchanged for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> bread loaves, of pefsu 5. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbe492fdf59c36d880068344239762c8fd62451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=25}"></span></td> <td>- </td></tr> <tr> <td>78</td> <td>100 bread loaves of pefsu 10 are to be evenly exchanged for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> des-measures of beer of pefsu 2. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e665893a76846cafb19376187c796812eeaea02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=20}"></span></td> <td>- </td></tr> <tr> <td>79</td> <td>An estate's inventory consists of 7 houses, 49 cats, 343 mice, 2401 spelt plants (a type of wheat), and 16,807 units of heqat (of whatever substance—a type of grain, suppose). List the items in the estates' inventory as a table, and include their total.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}houses&7\\cats&49\\mice&343\\spelt&2401\\heqat&16807\\Total&19607\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>h</mi> <mi>o</mi> <mi>u</mi> <mi>s</mi> <mi>e</mi> <mi>s</mi> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mi>a</mi> <mi>t</mi> <mi>s</mi> </mtd> <mtd> <mn>49</mn> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mi>i</mi> <mi>c</mi> <mi>e</mi> </mtd> <mtd> <mn>343</mn> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mi>p</mi> <mi>e</mi> <mi>l</mi> <mi>t</mi> </mtd> <mtd> <mn>2401</mn> </mtd> </mtr> <mtr> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mn>16807</mn> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> </mtd> <mtd> <mn>19607</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}houses&7\\cats&49\\mice&343\\spelt&2401\\heqat&16807\\Total&19607\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05895c6065b6a8cd99b58089b7802eac8654fc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:19.048ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}houses&7\\cats&49\\mice&343\\spelt&2401\\heqat&16807\\Total&19607\\\end{bmatrix}}}"></span> </p> </td> <td>Problem 79 has been presented in its most literal interpretation. However, the problem is among the most interesting in the papyrus, as its setup and even method of solution suggests <a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a> (that is, geometric sequences), elementary understanding of finite <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>, as well as the <a href="/wiki/As_I_was_going_to_St_Ives" title="As I was going to St Ives">St. Ives problem</a>—even Chace cannot help interrupting his own narrative in order to compare problem 79 with the St. Ives nursery rhyme. He also indicates that a suspiciously familiar third instance of these types of problems is to be found in Fibonacci's <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a>. Chace suggests the interpretation that 79 is a kind of savings example, where a certain amount of grain is saved by keeping cats on hand to kill the mice which would otherwise eat the spelt used to make the grain. In the original document, the 2401 term is written as 2301 (an obvious mistake), while the other terms are given correctly; it is therefore corrected here. <p>Moreover, one of Ahmes' methods of solution for the sum suggests an understanding of finite <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>. Ahmes performs a direct sum, but he also presents a simple multiplication to get the same answer: "2801 x 7 = 19607". Chace explains that since the first term, the number of houses (7) is <i>equal</i> to the common ratio of multiplication (7), then the following holds (and can be generalized to any similar situation): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum \limits _{k=1}^{n}7^{k}=7{\bigg (}1+\sum \limits _{k=1}^{n-1}7^{k}{\bigg )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum \limits _{k=1}^{n}7^{k}=7{\bigg (}1+\sum \limits _{k=1}^{n-1}7^{k}{\bigg )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5c5010c3f9a3682359863b9b75d1e76644603f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.671ex; height:7.343ex;" alt="{\displaystyle \sum \limits _{k=1}^{n}7^{k}=7{\bigg (}1+\sum \limits _{k=1}^{n-1}7^{k}{\bigg )}}"></span> </p><p>That is, when the first term of a geometric sequence is equal to the common ratio, partial sums of geometric sequences, or finite geometric series, can be reduced to multiplications involving the finite series having one less term, which does prove convenient in this case. In this instance then, Ahmes simply adds the first four terms of the sequence (7 + 49 + 343 + 2401 = 2800) to produce a partial sum, adds one (2801), and then simply multiplies by 7 to produce the correct answer. </p> </td></tr> <tr> <td>80</td> <td>The hinu is a further unit of volume such that one heqat equals ten hinu. Consider the situations where one has a <a href="/wiki/Eye_of_Horus" title="Eye of Horus">Horus eye</a> fraction of heqats, and express their conversions to hinu in a table.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&heqat&=&10&hinu\\{\frac {1}{2}}&heqat&=&5&hinu\\{\frac {1}{4}}&heqat&=&(2+{\frac {1}{2}})&hinu\\{\frac {1}{8}}&heqat&=&(1+{\frac {1}{4}})&hinu\\{\frac {1}{16}}&heqat&=&({\frac {1}{2}}+{\frac {1}{8}})&hinu\\{\frac {1}{32}}&heqat&=&({\frac {1}{4}}+{\frac {1}{16}})&hinu\\{\frac {1}{64}}&heqat&=&({\frac {1}{8}}+{\frac {1}{32}})&hinu\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&heqat&=&10&hinu\\{\frac {1}{2}}&heqat&=&5&hinu\\{\frac {1}{4}}&heqat&=&(2+{\frac {1}{2}})&hinu\\{\frac {1}{8}}&heqat&=&(1+{\frac {1}{4}})&hinu\\{\frac {1}{16}}&heqat&=&({\frac {1}{2}}+{\frac {1}{8}})&hinu\\{\frac {1}{32}}&heqat&=&({\frac {1}{4}}+{\frac {1}{16}})&hinu\\{\frac {1}{64}}&heqat&=&({\frac {1}{8}}+{\frac {1}{32}})&hinu\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c254cc2f884191da61c84b5a2a423644839ea0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:36.646ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}1&heqat&=&10&hinu\\{\frac {1}{2}}&heqat&=&5&hinu\\{\frac {1}{4}}&heqat&=&(2+{\frac {1}{2}})&hinu\\{\frac {1}{8}}&heqat&=&(1+{\frac {1}{4}})&hinu\\{\frac {1}{16}}&heqat&=&({\frac {1}{2}}+{\frac {1}{8}})&hinu\\{\frac {1}{32}}&heqat&=&({\frac {1}{4}}+{\frac {1}{16}})&hinu\\{\frac {1}{64}}&heqat&=&({\frac {1}{8}}+{\frac {1}{32}})&hinu\\\end{bmatrix}}}"></span> </p> </td> <td>Compare problems 47 and 64 for other tabular information with repeated Horus eye fractions. </td></tr> <tr> <td>81</td> <td>Perform "another reckoning of the hinu." That is, express an assortment of Egyptian fractions, many terms of which are also Horus eye fractions, in various terms of heqats, hinu, and ro.</td> <td><figure class="mw-default-size mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:Rhind_Papyrus_Problem_81.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Rhind_Papyrus_Problem_81.png/220px-Rhind_Papyrus_Problem_81.png" decoding="async" width="220" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Rhind_Papyrus_Problem_81.png/330px-Rhind_Papyrus_Problem_81.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Rhind_Papyrus_Problem_81.png/440px-Rhind_Papyrus_Problem_81.png 2x" data-file-width="1164" data-file-height="1514" /></a><figcaption></figcaption></figure></td> <td>Problem 81's main section is a much larger conversion table of assorted Egyptian fractions, which expands on the idea of problem 80—indeed, it represents one of the largest tabular forms in the entire papyrus. The first part of problem 81 is an exact repetition of the table in problem 80, without the first row which states that 1 heqat = 10 hinu; it is therefore not repeated here. The second part of problem 81, or its "body", is the large table which is given here. The attentive reader will notice two things: several rows repeat identical information, and several forms (but not all) given in both of the "heqat" areas on either side of the table are in fact identical. There are two points worth mentioning, to explain why the table looks the way that it does. For one thing, Ahmes does in fact exactly repeat certain groups of information in different areas of the table, and they are accordingly repeated here. On the other hand, Ahmes also starts out with certain "left-hand" heqat forms, and makes some mistakes in his early calculations. However, in many cases he corrects these mistakes later in his writing of the table, producing a consistent result. Since the present information is simply a re-creation of Chace's translation and interpretation of the papyrus, and since Chace elected to interpret and correct Ahmes' mistakes by substituting the later correct information in certain earlier rows, thereby fixing Ahmes' mistakes and also therefore repeating information in the course of translation, this method of interpretation explains the duplication of information in certain rows. As for the duplication of information in certain columns (1/4 heqat = ... = 1/4 heqat, etc.), this seems simply to have been a convention that Ahmes filled in while considering certain important Horus-eye fractional ratios from both the standpoint of the hinu, and also of the heqat (and their conversions). In short, the various repetitions of information are the result of choices made by Ahmes, his potential source document, and the editorial choices of Chace, in order to present a mathematically consistent translation of the larger table in problem 81. </td></tr> <tr> <td>82</td> <td>Estimate in wedyet-flour, made into bread, the daily portion of feed for ten <a href="/wiki/Foie_gras" title="Foie gras">fattening</a> geese. To do this, perform the following calculations, expressing the quantities in Egyptian fractional terms of <i>hundreds</i> of heqats, heqats and ro, except where specified otherwise: <p>Begin with the statement that "10 fattening geese eat 2 + 1/2 heqats in one day". In other words, the daily rate of consumption (and initial condition) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is equal to 2 + 1/2. Determine the number of heqats which 10 fattening geese eat in 10 days, and in 40 days. Call these quantities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, respectively. </p><p>Multiply the above latter quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> by 5/3 to express the amount of "spelt", or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, required to be ground up. </p><p>Multiply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> by 2/3 to express the amount of "wheat", or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span>, required. </p><p>Divide <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> by 10 to express a "portion of wheat", or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, which is to be subtracted from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. </p><p>Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f-p=g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f-p=g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46d16930b4bf78d17fd7e0801afe5815de383b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.503ex; height:2.509ex;" alt="{\displaystyle f-p=g}"></span>. This is the amount of "grain", (or wedyet flour, it would seem), which is required to make the feed for geese, presumably on the interval of 40 days (which would seem to contradict the original statement of the problem, somewhat). Finally, express <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> again in terms of <i>hundreds of double heqats, double heqats and double ro</i>, where 1 hundred double heqat = 2 hundred heqat = 100 double heqat = 200 heqat = 32,000 double ro = 64,000 ro. Call this final quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"></span>. </p> </td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=25\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>25</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=25\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ac404e9228245bcbd7c89389c03485785cbc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.76ex; height:2.509ex;" alt="{\displaystyle t=25\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=100\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>100</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=100\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a600cf1f9e0494a5229428eaca79b8ad27d497b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.361ex; height:2.509ex;" alt="{\displaystyle f=100\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\bigg (}1+{\frac {1}{2}}{\bigg )}hundred\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>h</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\bigg (}1+{\frac {1}{2}}{\bigg )}hundred\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b02cc64657201b810b6c7a977496746ce2808b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.736ex; height:6.176ex;" alt="{\displaystyle s={\bigg (}1+{\frac {1}{2}}{\bigg )}hundred\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}16+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>16</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}16+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9093381dbacee9b95c1d7c7f9102a5ff6ab49be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.105ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}16+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\bigg (}{\frac {1}{3}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>h</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w={\bigg (}{\frac {1}{3}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b048822f13447303007a05617af41d173981459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.146ex; height:6.176ex;" alt="{\displaystyle w={\bigg (}{\frac {1}{3}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}8+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>8</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}8+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bb0cc96e9b230e2649eb8f81b5c1628a0a5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.105ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}8+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\bigg (}6+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\bigg (}6+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af5696e9efc6ec16040b329e06df67211ebd6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:48.492ex; height:6.176ex;" alt="{\displaystyle p={\bigg (}6+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;heqat+{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;ro}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\bigg (}{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>h</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\bigg (}{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9135cddfbe715c7308869bed38dd3fd0b4863b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.597ex; height:6.176ex;" alt="{\displaystyle g={\bigg (}{\frac {1}{2}}+{\frac {1}{4}}{\bigg )}hundred\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}18+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>18</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}18+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dceae67ed46fb41e58349cd46b8683f148aa64b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.268ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}18+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;heqat+{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;ro}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}={\bigg (}{\frac {1}{4}}{\bigg )}hundred\;\;\;double\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mi>h</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}={\bigg (}{\frac {1}{4}}{\bigg )}hundred\;\;\;double\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36b08bf01b472bb6ee5003206534bb2fe02814a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.189ex; height:6.176ex;" alt="{\displaystyle g_{2}={\bigg (}{\frac {1}{4}}{\bigg )}hundred\;\;\;double\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}21+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;double\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>21</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}21+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;double\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8f748a68a075b71ac39cac47f83852289e51de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.114ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}21+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}{\bigg )}\;\;\;double\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;double\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;double\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850a3726af462befe98e51bddec19c8c5013a5e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.726ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}3+{\frac {1}{3}}{\bigg )}\;\;\;double\;\;\;ro}"></span> </p> </td> <td>Beginning with problem 82, the papyrus becomes increasingly difficult to interpret (owing to mistakes and missing information), to the point of unintelligibility. However, it is yet possible to make some sense of 82. Simply put, there seem to exist established rules, or good estimates, for fractions to be taken of this-or-that food material in a cooking or production process. Ahmes' 82 simply gives expression to some of these quantities, in what is after all declared in the original document to be an "estimate", its somewhat contradictory and confused language notwithstanding. In addition to their strangeness, problems 82, 82B, 83 and 84 are also notable for continuing the "food" train of thought of the recent pefsu problems, this time considering how to feed animals instead of people. Both 82 and 82B make use of the "hundred heqat" unit with regard to t and f; these conventions are cosmetic, and not repeated here. Licence is also taken throughout these last problems (per Chace) to fix numerical mistakes of the original document, to attempt to present a coherent paraphrase. </td></tr> <tr> <td>82B</td> <td>Estimate the amount of feed for other geese. That is, consider a situation which is identical to problem 82, with the single exception that the initial condition, or daily rate of consumption, is exactly half as large. That is, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> = 1 + 1/4. Find <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and especially <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"></span> by using elementary algebra to skip the intermediate steps.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\bigg (}12+{\frac {1}{2}}{\bigg )}\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>12</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\bigg (}12+{\frac {1}{2}}{\bigg )}\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874824a438a008ac48897d85cc9e62cc9e821dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.02ex; height:6.176ex;" alt="{\displaystyle t={\bigg (}12+{\frac {1}{2}}{\bigg )}\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=50\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>50</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=50\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65b89f18263f2733c0d7aad3d7b426e85a98cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.199ex; height:2.509ex;" alt="{\displaystyle f=50\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}={\bigg (}23+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;double\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>23</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}={\bigg (}23+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;double\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2f859ab7b23658eccdd2ea40596783ff31e3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.73ex; height:6.176ex;" alt="{\displaystyle g_{2}={\bigg (}23+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}{\bigg )}\;\;\;double\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;double\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;double\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f5a617c35bcbc8b6feb86abedc2f0c7c2962a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.726ex; height:6.176ex;" alt="{\displaystyle +{\bigg (}1+{\frac {2}{3}}{\bigg )}\;\;\;double\;\;\;ro}"></span> </p> </td> <td>Problem 82B is presented in parallel with problem 82, and quickly considers the identical situation where the associated quantities are halved. In both cases, it appears that Ahmes' real goal is to find g_2. Now that he has a "procedure", he feels free to skip 82's onerous steps. One could simply observe that the division by two carries through the entire problem's work, so that g_2 is also exactly half as large as in problem 82. A slightly more thorough approach using elementary algebra would be to backtrack the relationships between the quantities in 82, make the essential observation that g = 14/15 x f, and then perform the unit conversions to transform g into g_2. </td></tr> <tr> <td>83</td> <td>Estimate the feed for various kinds of birds. This is a "problem" with multiple components, which can be interpreted as a series of remarks: <p>Suppose that four geese are cooped up, and their collective daily allowance of feed is equal to one hinu. Express one goose's daily allowance of feed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></span> in terms of heqats and ro. </p><p>Suppose that the daily feed for a goose "that goes into the pond" is equal to 1/16 + 1/32 heqats + 2 ro. Express this same daily allowance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}"></span> in terms of hinu. </p><p>Suppose that the daily allowance of feed for 10 geese is one heqat. Find the 10-day allowance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76557dbaa5a3859bd1b7180ea8b73b6c0d870df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.106ex; height:2.009ex;" alt="{\displaystyle a_{10}}"></span> and the 30-day, or one-month allowance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{30}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{30}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/941c1e2261d05f9537adfacd6d37c12af6a9bee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.106ex; height:2.009ex;" alt="{\displaystyle a_{30}}"></span> for the same group of animals, in heqats. </p><p>Finally a table will be presented, giving daily feed portions to fatten one animal of any of the indicated species. </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}={\frac {1}{64}}\;\;\;heqat+3\;\;\;ro}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mn>3</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}={\frac {1}{64}}\;\;\;heqat+3\;\;\;ro}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5271639955e3778dd89c59b56afd158bc39b89e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.155ex; height:5.343ex;" alt="{\displaystyle a_{1}={\frac {1}{64}}\;\;\;heqat+3\;\;\;ro}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}=1\;\;\;hinu}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}=1\;\;\;hinu}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7109a3be1ab2e22c5cce38edab0fe0f7b75eb0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.346ex; height:2.509ex;" alt="{\displaystyle a_{2}=1\;\;\;hinu}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{10}=10\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>=</mo> <mn>10</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{10}=10\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87c97122ac5f52315d73358dbdc08d8d7954f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.026ex; height:2.509ex;" alt="{\displaystyle a_{10}=10\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{30}=30\;\;\;heqat}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msub> <mo>=</mo> <mn>30</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{30}=30\;\;\;heqat}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a252f0f0bc803d7fe8635c8c933a53c1cf9f9536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.026ex; height:2.509ex;" alt="{\displaystyle a_{30}=30\;\;\;heqat}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\terp-goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\crane&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\set-duck&({\frac {1}{32}}+{\frac {1}{64}})&heqat&+&1&ro\\ser-goose&{\frac {1}{64}}&heqat&+&3&ro\\dove&&&&3&ro\\quail&&&&3&ro\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>g</mi> <mi>o</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mi>e</mi> <mi>r</mi> <mi>p</mi> <mo>−</mo> <mi>g</mi> <mi>o</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>e</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mi>e</mi> <mi>t</mi> <mo>−</mo> <mi>d</mi> <mi>u</mi> <mi>c</mi> <mi>k</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mi>e</mi> <mi>r</mi> <mo>−</mo> <mi>g</mi> <mi>o</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mrow> </mtd> <mtd> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>o</mi> <mi>v</mi> <mi>e</mi> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd> <mi>r</mi> <mi>o</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\terp-goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\crane&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\set-duck&({\frac {1}{32}}+{\frac {1}{64}})&heqat&+&1&ro\\ser-goose&{\frac {1}{64}}&heqat&+&3&ro\\dove&&&&3&ro\\quail&&&&3&ro\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f73e324390308c1209ab9a1f3ba872b93a09af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:54.618ex; height:26.509ex;" alt="{\displaystyle {\begin{bmatrix}goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\terp-goose&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\crane&({\frac {1}{8}}+{\frac {1}{32}})&heqat&+&(3+{\frac {1}{3}})&ro\\set-duck&({\frac {1}{32}}+{\frac {1}{64}})&heqat&+&1&ro\\ser-goose&{\frac {1}{64}}&heqat&+&3&ro\\dove&&&&3&ro\\quail&&&&3&ro\\\end{bmatrix}}}"></span> </p> </td> <td>Since problem 83's various items are concerned with unit conversions between heqats, ro and hinu, in the spirit of 80 and 81, it is natural to wonder what the table's items become when converted to hinu. The portion shared by the goose, terp-goose and crane is equal to 5/3 hinu, the set-ducks' portion is equal to 1/2 hinu, the ser-gooses' portion is equal to 1/4 hinu (compare the first item in the problem), and the portion shared by the dove and quail is equal to 1/16 + 1/32 hinu. The presence of various Horus eye fractions is familiar from the rest of the papyrus, and the table seems to consider feed estimates for birds, ranging from largest to smallest. The "5/3 hinu" portions at the top of the table, specifically its factor of 5/3, reminds one of the method for finding s in problem 82. Problem 83 makes mention of "Lower-Egyptian grain", or barley, and it also uses the "hundred-heqat" unit in one place; these are cosmetic, and left out of the present statement. </td></tr> <tr> <td>84</td> <td>Estimate the feed for a stable of oxen.</td> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}&Loaves&Common\;food\\4\;fine\;oxen&24\;heqat&2\;heqat\\2\;fine\;oxen&22\;heqat&6\;heqat\\3\;cattle&20\;heqat&2\;heqat\\1\;ox&20\;heqat&\\Total&86\;heqat&10\;heqat\\in\;spelt&9\;heqat&(7+{\frac {1}{2}})\;heqat\\10\;days&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&+15\;heqat&\\one\;month&200\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&&+15\;heqat\\double\;heqat&{\frac {1}{2}}\;c.\;heqat&{\frac {1}{4}}\;c.\;heqat\\&+(11+{\frac {1}{2}}+{\frac {1}{8}})\;heqat&+5\;heqat\\&+3\;ro&\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mi>L</mi> <mi>o</mi> <mi>a</mi> <mi>v</mi> <mi>e</mi> <mi>s</mi> </mtd> <mtd> <mi>C</mi> <mi>o</mi> <mi>m</mi> <mi>m</mi> <mi>o</mi> <mi>n</mi> <mspace width="thickmathspace" /> <mi>f</mi> <mi>o</mi> <mi>o</mi> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mspace width="thickmathspace" /> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mi>o</mi> <mi>x</mi> <mi>e</mi> <mi>n</mi> </mtd> <mtd> <mn>24</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mn>2</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mspace width="thickmathspace" /> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mi>o</mi> <mi>x</mi> <mi>e</mi> <mi>n</mi> </mtd> <mtd> <mn>22</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mn>6</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mspace width="thickmathspace" /> <mi>c</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>l</mi> <mi>e</mi> </mtd> <mtd> <mn>20</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mn>2</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mspace width="thickmathspace" /> <mi>o</mi> <mi>x</mi> </mtd> <mtd> <mn>20</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>T</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> </mtd> <mtd> <mn>86</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mn>10</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mi>n</mi> <mspace width="thickmathspace" /> <mi>s</mi> <mi>p</mi> <mi>e</mi> <mi>l</mi> <mi>t</mi> </mtd> <mtd> <mn>9</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>10</mn> <mspace width="thickmathspace" /> <mi>d</mi> <mi>a</mi> <mi>y</mi> <mi>s</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>+</mo> <mn>15</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>o</mi> <mi>n</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mi>m</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> </mtd> <mtd> <mn>200</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> <mn>15</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>o</mi> <mi>u</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>c</mi> <mo>.</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>+</mo> <mo stretchy="false">(</mo> <mn>11</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> <mtd> <mo>+</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mi>h</mi> <mi>e</mi> <mi>q</mi> <mi>a</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>+</mo> <mn>3</mn> <mspace width="thickmathspace" /> <mi>r</mi> <mi>o</mi> </mtd> <mtd /> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}&Loaves&Common\;food\\4\;fine\;oxen&24\;heqat&2\;heqat\\2\;fine\;oxen&22\;heqat&6\;heqat\\3\;cattle&20\;heqat&2\;heqat\\1\;ox&20\;heqat&\\Total&86\;heqat&10\;heqat\\in\;spelt&9\;heqat&(7+{\frac {1}{2}})\;heqat\\10\;days&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&+15\;heqat&\\one\;month&200\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&&+15\;heqat\\double\;heqat&{\frac {1}{2}}\;c.\;heqat&{\frac {1}{4}}\;c.\;heqat\\&+(11+{\frac {1}{2}}+{\frac {1}{8}})\;heqat&+5\;heqat\\&+3\;ro&\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72e322f031f27463de9b19860937422a5f29efd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.005ex; width:59.156ex; height:49.176ex;" alt="{\displaystyle {\begin{bmatrix}&Loaves&Common\;food\\4\;fine\;oxen&24\;heqat&2\;heqat\\2\;fine\;oxen&22\;heqat&6\;heqat\\3\;cattle&20\;heqat&2\;heqat\\1\;ox&20\;heqat&\\Total&86\;heqat&10\;heqat\\in\;spelt&9\;heqat&(7+{\frac {1}{2}})\;heqat\\10\;days&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&+15\;heqat&\\one\;month&200\;heqat&({\frac {1}{2}}+{\frac {1}{4}})\;c.\;heqat\\&&+15\;heqat\\double\;heqat&{\frac {1}{2}}\;c.\;heqat&{\frac {1}{4}}\;c.\;heqat\\&+(11+{\frac {1}{2}}+{\frac {1}{8}})\;heqat&+5\;heqat\\&+3\;ro&\\\end{bmatrix}}}"></span> </p> </td> <td>84 is the last problem, or number, comprising the mathematical content of the Rhind papyrus. With regard to 84 itself, Chace echoes Peet: "One can only agree with Peet that 'with this problem the papyrus reaches its limit of unintelligibility and inaccuracy.'"(Chace, V.2, Problem 84). Here, instances of the "hundred heqat" unit have been expressed by "c. heqat" in order to conserve space. The three "cattle" mentioned are described as "common" cattle, to differentiate them from the other animals, and the two headers concerning loaves and "common food" are with respect to heqats. The "fine oxen" at the table's beginning are described as Upper Egyptian oxen, a phrase also removed here for space reasons. <p>Problem 84 seems to suggest a procedure to estimate various food materials and allowances in similar terms as the previous three problems, but the extant information is deeply confused. Still, there are hints of consistency. The problem seems to start out like a conventional story problem, describing a stable with ten animals of four different types. It seems that the four types of animals consume feed, or "loaves" at different rates, and that there are corresponding amounts of "common" food. These two columns of information are correctly summed in the "total" row, however they are followed by two "spelt" items of dubious relationship to the above. These two spelt items are indeed each multiplied by ten to give the two entries in the "10 days" row, once unit conversions are accounted for. The "one month" row items do not seem to be consistent with the previous two, however. Finally, information in "double heqats" (read hundred double heqats, double heqats and double ro for these items) concludes the problem, in a manner reminiscent of 82 and 82B. The two items in the final row are in roughly, but not exactly, the same proportion to one another as the two items in the "one month" row. </p> </td></tr> <tr> <td>Number 85</td> <td>A small group of cursive hieroglyphic signs is written, which Chace suggests may represent the scribe "trying his pen." It appears to be a phrase or sentence of some kind, and two translations are suggested: 1) "Kill vermin, mice, fresh weeds, numerous spiders. Pray the god Re for warmth, wind and high water." 2) "Interpret this strange matter, which the scribe wrote ... according to what he knew."</td> <td> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:Rhind_Papyrus_Number_85.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Rhind_Papyrus_Number_85.png/220px-Rhind_Papyrus_Number_85.png" decoding="async" width="220" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Rhind_Papyrus_Number_85.png/330px-Rhind_Papyrus_Number_85.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Rhind_Papyrus_Number_85.png/440px-Rhind_Papyrus_Number_85.png 2x" data-file-width="851" data-file-height="901" /></a><figcaption></figcaption></figure> </td> <td>The remaining items 85, 86 and 87, being various errata that are not mathematical in nature, are therefore styled by Chace as "numbers" as opposed to problems. They are also located on areas of the papyrus that are well away from the body of the writing, which had just ended with Problem 84. Number 85, for example, is some distance away from Problem 84 on the verso—but not too far away. Its placement on the papyrus therefore suggests a kind of coda, in which case the latter translation, which Chace describes as an example of the "enigmatic writing" interpretation of ancient Egyptian documents, seems most appropriate to its context in the document. </td></tr> <tr> <td>Number 86</td> <td>Number 86 seems to be from some account, or memorandum, and lists an assortment of goods and quantities, using words familiar from the context of the rest of the papyrus itself. [The original text is a series of lines of writing, which are therefore numbered in the following.]</td> <td> <p>"1... living forever. List of the food in Hebenti... </p><p>2... his brother the steward Ka-mose... </p><p>3... of his year, silver, 50 pieces twice in the year... </p><p>4... cattle 2, in silver 3 pieces in the year... </p><p>5... one twice; that is, 1/6 and 1/6. Now as for one... </p><p>6... 12 hinu; that is, silver, 1/4 piece; one... </p><p>7... (gold or silver) 5 pieces, their price therefor; fish, 120, twice... </p><p>8... year, barley, in quadruple heqat, 1/2 + 1/4 of 100 heqat 15 heqat; spelt, 100 heqat... heqat... </p><p>9... barley, in quadruple heqat, 1/2 + 1/4 of 100 heqat 15 heqat; spelt, 1 + 1/2 + 1/4 times 100 heqat 17 heqat... </p><p>10... 146 + 1/2; barley, 1 + 1/2 + 1/4 times 100 heqat 10 heqat; spelt, 300 heqat... heqat... </p><p>11... 1/2, there was brought wine, 1 ass(load?)... </p><p>12... silver 1/2 piece; ... 4; that is, in silver... </p><p>13... 1 + 1/4; fat, 36 hinu; that is, in silver... </p><p>14... 1 + 1/2 + 1/4 times 100 heqat 21 heqat; spelt, in quadruple heqat, 400 heqat 10 heqat... </p><p>15-18 (These lines are repetitions of line 14.)" </p> </td> <td>Chace indicates that number 86 was pasted onto the far left side of the verso (opposite the later geometry problems on the recto), to strengthen the papyrus. Number 86 can therefore be interpreted as a piece of "scrap paper". </td></tr> <tr> <td>Number 87</td> <td>Number 87 is a brief account of certain events. Chace indicates an (admittedly now dated and possibly changed) scholarly consensus that 87 was added to the papyrus not long after the completion of its mathematical content. He goes on to indicate that the events described in it "took place during the period of the Hyksos domination."</td> <td>"Year 11, second month of the harvest season. Heliopolis was entered. <p>The first month of the inundation season, 23rd day, the commander (?) of the army (?) attacked (?) Zaru. </p><p>25th day, it was heard that Zaru was entered. </p><p>Year 11, first month of the inundation season, third day. Birth of Set; the majesty of this god caused his voice to be heard. </p><p>Birth of Isis, the heavens rained." </p> </td> <td>Number 87 is located toward the middle of the verso, surrounded by a large, blank, unused space. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has original text related to this article: <div lang="mul" style="margin-left: 10px;"><b><a href="https://wikisource.org/wiki/Rhind_Mathematical_Papyrus" class="extiw" title="oldwikisource:Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a></b> </div></div></div> </div> <ul><li><a href="/wiki/List_of_ancient_Egyptian_papyri" class="mw-redirect" title="List of ancient Egyptian papyri">List of ancient Egyptian papyri</a></li> <li><a href="/wiki/Akhmim_wooden_tablet" class="mw-redirect" title="Akhmim wooden tablet">Akhmim wooden tablet</a></li> <li><a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">Ancient Egyptian units of measurement</a></li> <li><a href="/wiki/As_I_was_going_to_St._Ives" class="mw-redirect" title="As I was going to St. Ives">As I was going to St. Ives</a></li> <li><a href="/wiki/Berlin_Papyrus_6619" title="Berlin Papyrus 6619">Berlin Papyrus 6619</a></li> <li><a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a></li> <li><a href="/wiki/Lahun_Mathematical_Papyri" title="Lahun Mathematical Papyri">Lahun Mathematical Papyri</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=12" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChace1927" class="citation book cs1"><a href="/wiki/Arnold_Buffum_Chace" title="Arnold Buffum Chace">Chace, Arnold Buffum</a>; et al. (1927). <a rel="nofollow" class="external text" href="https://archive.org/details/TheRhindPapyrusVolume1"><i>The Rhind Mathematical Papyrus</i></a>. Vol. 1. <a href="/wiki/Oberlin,_Ohio" title="Oberlin, Ohio">Oberlin, Ohio</a>: <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a> – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus&rft.place=Oberlin%2C+Ohio&rft.pub=Mathematical+Association+of+America&rft.date=1927&rft.aulast=Chace&rft.aufirst=Arnold+Buffum&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FTheRhindPapyrusVolume1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChace1929" class="citation book cs1">Chace, Arnold Buffum; et al. (1929). <a rel="nofollow" class="external text" href="https://archive.org/details/arnoldbuffumchaceludlowbullhenryparkermanningtherhindmathematicalpapyrus.volumei"><i>The Rhind Mathematical Papyrus</i></a>. Vol. 2. Oberlin, Ohio: Mathematical Association of America – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus&rft.place=Oberlin%2C+Ohio&rft.pub=Mathematical+Association+of+America&rft.date=1929&rft.aulast=Chace&rft.aufirst=Arnold+Buffum&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farnoldbuffumchaceludlowbullhenryparkermanningtherhindmathematicalpapyrus.volumei&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillings1972" class="citation book cs1">Gillings, Richard J. (1972). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsintim0000gill"><i>Mathematics in the Time of the Pharaohs</i></a></span> (Dover reprint ed.). MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-24315-X" title="Special:BookSources/0-486-24315-X"><bdi>0-486-24315-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+the+Time+of+the+Pharaohs&rft.edition=Dover+reprint&rft.pub=MIT+Press&rft.date=1972&rft.isbn=0-486-24315-X&rft.aulast=Gillings&rft.aufirst=Richard+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsintim0000gill&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinsShute1987" class="citation book cs1"><a href="/wiki/Gay_Robins" title="Gay Robins">Robins, Gay</a>; <a href="/wiki/Charles_Shute_(academic)" title="Charles Shute (academic)">Shute, Charles</a> (1987). <i>The Rhind Mathematical Papyrus: an Ancient Egyptian Text</i>. London: British Museum Publications Limited. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7141-0944-4" title="Special:BookSources/0-7141-0944-4"><bdi>0-7141-0944-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus%3A+an+Ancient+Egyptian+Text&rft.place=London&rft.pub=British+Museum+Publications+Limited&rft.date=1987&rft.isbn=0-7141-0944-4&rft.aulast=Robins&rft.aufirst=Gay&rft.au=Shute%2C+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Spalinger-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Spalinger_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Spalinger_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Spalinger_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Spalinger_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Spalinger_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Spalinger_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpalinger1990" class="citation journal cs1">Spalinger, Anthony (1990). "The Rhind Mathematical Papyrus as a Historical Document". <i>Studien zur Altägyptischen Kultur</i>. <b>17</b>. Helmut Buske Verlag: 295–337. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25150159">25150159</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studien+zur+Alt%C3%A4gyptischen+Kultur&rft.atitle=The+Rhind+Mathematical+Papyrus+as+a+Historical+Document&rft.volume=17&rft.pages=295-337&rft.date=1990&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25150159%23id-name%3DJSTOR&rft.aulast=Spalinger&rft.aufirst=Anthony&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-Clagett-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Clagett_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Clagett_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Clagett_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Clagett_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Clagett_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Clagett_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClagett1999" class="citation book cs1 cs1-prop-long-vol">Clagett, Marshall (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/ancientegyptians03clag"><i>Ancient Egyptian Science, A Source Book</i></a></span>. Memoirs of the American Philosophical Society. Vol. Three: Ancient Egyptian Mathematics. American Philosophical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87169-232-0" title="Special:BookSources/978-0-87169-232-0"><bdi>978-0-87169-232-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ancient+Egyptian+Science%2C+A+Source+Book&rft.series=Memoirs+of+the+American+Philosophical+Society&rft.pub=American+Philosophical+Society&rft.date=1999&rft.isbn=978-0-87169-232-0&rft.aulast=Clagett&rft.aufirst=Marshall&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fancientegyptians03clag&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-peet-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-peet_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-peet_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeet1923" class="citation book cs1"><a href="/wiki/T._Eric_Peet" title="T. Eric Peet">Peet, T. Eric</a> (1923). <i>The Rhind Mathematical Papyrus: British Museum 10057 and 10058</i>. University Press of Liverpool.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus%3A+British+Museum+10057+and+10058&rft.pub=University+Press+of+Liverpool&rft.date=1923&rft.aulast=Peet&rft.aufirst=T.+Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span> For the location where the papyrus was found see <a rel="nofollow" class="external text" href="https://archive.org/details/Peet_1923/page/n5">page 2</a>.</span> </li> <li id="cite_note-Chace,_Arnold_Buffum_1929-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chace,_Arnold_Buffum_1929_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chace,_Arnold_Buffum_1929_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChace1979" class="citation book cs1 cs1-prop-long-vol">Chace, Arnold Buffum (1979) [1927–29]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/rhindmathematica0000unse"><i>The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations</i></a></span>. Classics in Mathematics Education. Vol. 8. 2 vols (Reston: National Council of Teachers of Mathematics Reprinted ed.). Oberlin: Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87353-133-7" title="Special:BookSources/0-87353-133-7"><bdi>0-87353-133-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rhind+Mathematical+Papyrus%3A+Free+Translation+and+Commentary+with+Selected+Photographs%2C+Translations%2C+Transliterations+and+Literal+Translations&rft.place=Oberlin&rft.series=Classics+in+Mathematics+Education&rft.edition=Reston%3A+National+Council+of+Teachers+of+Mathematics+Reprinted&rft.pub=Mathematical+Association+of+America&rft.date=1979&rft.isbn=0-87353-133-7&rft.aulast=Chace&rft.aufirst=Arnold+Buffum&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frhindmathematica0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britishmuseum.org/collection/object/Y_EA10057">"The Rhind Mathematical Papyrus"</a>. <i>The British Museum</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-12-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+British+Museum&rft.atitle=The+Rhind+Mathematical+Papyrus&rft_id=https%3A%2F%2Fwww.britishmuseum.org%2Fcollection%2Fobject%2FY_EA10057&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-Schneider-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Schneider_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Schneider_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">cf. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchneider2006" class="citation book cs1">Schneider, Thomas (2006). "The Relative Chronology of the Middle Kingdom and the Hyksos Period (Dyns. 12–17)". In Hornung, Erik; Krauss, Rolf; Warburton, David (eds.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/ancientegyptianc00horn_842"><i>Ancient Egyptian Chronology</i></a></span>. Handbook of Oriental Studies. Brill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/ancientegyptianc00horn_842/page/n206">194</a>–195. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789004113855" title="Special:BookSources/9789004113855"><bdi>9789004113855</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Relative+Chronology+of+the+Middle+Kingdom+and+the+Hyksos+Period+%28Dyns.+12%E2%80%9317%29&rft.btitle=Ancient+Egyptian+Chronology&rft.series=Handbook+of+Oriental+Studies&rft.pages=194-195&rft.pub=Brill&rft.date=2006&rft.isbn=9789004113855&rft.aulast=Schneider&rft.aufirst=Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fancientegyptianc00horn_842&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_mathematics/">"Egyptian mathematics"</a>. <i>Maths History</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Maths+History&rft.atitle=Egyptian+mathematics&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FEgyptian_mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuggenbühl1964" class="citation journal cs1"><a href="/wiki/Laura_Guggenb%C3%BChl" title="Laura Guggenbühl">Guggenbühl, Laura</a> (October 1964). Eves, Howard (ed.). "The New York fragments of the Rhind Mathematical Papyrus". Historically Speaking. <i>The Mathematics Teacher</i>. <b>57</b> (6): 406–410. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fmt.57.6.0406">10.5951/mt.57.6.0406</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27957095">27957095</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Teacher&rft.atitle=The+New+York+fragments+of+the+Rhind+Mathematical+Papyrus&rft.volume=57&rft.issue=6&rft.pages=406-410&rft.date=1964-10&rft_id=info%3Adoi%2F10.5951%2Fmt.57.6.0406&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27957095%23id-name%3DJSTOR&rft.aulast=Guggenb%C3%BChl&rft.aufirst=Laura&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.brooklynmuseum.org/opencollection/objects/118304/Fragments_of_Rhind_Mathematical_Papyrus">"Collections: Egyptian, Classical, Ancient Near Eastern Art: Fragments of Rhind Mathematical Papyrus"</a>. Brooklyn Museum<span class="reference-accessdate">. Retrieved <span class="nowrap">November 1,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Collections%3A+Egyptian%2C+Classical%2C+Ancient+Near+Eastern+Art%3A+Fragments+of+Rhind+Mathematical+Papyrus&rft.pub=Brooklyn+Museum&rft_id=http%3A%2F%2Fwww.brooklynmuseum.org%2Fopencollection%2Fobjects%2F118304%2FFragments_of_Rhind_Mathematical_Papyrus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> <li id="cite_note-Maor-7-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Maor-7_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Maor-7_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaor1998" class="citation book cs1">Maor, Eli (1998). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/trigonometricdel00maor_316"><i>Trigonometric Delights</i></a></span>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/trigonometricdel00maor_316/page/n28">20</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-09541-8" title="Special:BookSources/0-691-09541-8"><bdi>0-691-09541-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometric+Delights&rft.pages=20&rft.pub=Princeton+University+Press&rft.date=1998&rft.isbn=0-691-09541-8&rft.aulast=Maor&rft.aufirst=Eli&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrigonometricdel00maor_316&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rhind_Mathematical_Papyrus&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Rhind_Mathematical_Papyrus" class="extiw" title="commons:Category:Rhind Mathematical Papyrus"><span style="font-style:italic; font-weight:bold;">Rhind Mathematical Papyrus</span></a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.britishmuseum.org/collection/object/Y_EA10057">British Museum webpage on the first section of the Papyrus</a></li> <li><a rel="nofollow" class="external text" href="https://www.britishmuseum.org/collection/object/Y_EA10058">British Museum webpage on the second section of the Papyrus</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150629192919if_/https://www.britishmuseum.org/explore/highlights/highlight_objects/aes/r/rhind_mathematical_papyrus.aspx">British Museum webpage on the Papyrus</a> at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (archived June 29, 2015).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html">"Mathematics in Egyptian Papyri"</a>. <i><a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor+History+of+Mathematics+archive&rft.atitle=Mathematics+in+Egyptian+Papyri&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2Fhistory%2FHistTopics%2FEgyptian_papyri.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Rhind_Papyrus"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RhindPapyrus.html">"Rhind Papyrus"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Rhind+Papyrus&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRhindPapyrus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARhind+Mathematical+Papyrus" class="Z3988"></span></span></li> <li>Williams, Scott W. <a rel="nofollow" class="external text" href="http://www.math.buffalo.edu/mad/index.html"><i>Mathematicians of the African Diaspora</i></a>, containing a page on <a rel="nofollow" class="external text" href="http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html"><i>Egyptian Mathematics Papyri</i></a>.</li></ul> <table class="wikitable succession-box noprint" style="margin:0.5em auto; font-size:small;clear:both;"> <tbody><tr style="text-align:center;"> <td style="width:30%;" rowspan="1">Preceded by<div style="font-weight: bold">16: <a href="/wiki/Flood_tablet" class="mw-redirect" title="Flood tablet">Flood tablet</a></div> </td> <td style="width: 40%; text-align: center;" rowspan="1"><b> <a href="/wiki/A_History_of_the_World_in_100_Objects" title="A History of the World in 100 Objects">A History of the World in 100 Objects</a><br /><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/ahistoryoftheworld/about/transcripts/episode17/">Object 17</a> </b> </td> <td style="width: 30%; text-align: center;" rowspan="1">Succeeded by<div style="font-weight: bold">18: <a href="/wiki/Minoan_Bull-leaper" title="Minoan Bull-leaper">Minoan Bull-leaper</a></div> </td></tr> </tbody></table> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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<ul><li><a href="/wiki/Akan_Drum" title="Akan Drum">Akan Drum</a></li> <li><a href="/wiki/Aubin_Codex" title="Aubin Codex">Aubin Codex</a></li> <li><a href="/wiki/Benin_Bronzes" title="Benin Bronzes">Benin Bronzes</a></li> <li><a href="/wiki/Briggs_Enigma" title="Briggs Enigma">Briggs Enigma</a></li> <li><a href="/wiki/Bronze_Head_from_Ife" title="Bronze Head from Ife">Bronze Head from Ife</a></li> <li><a href="/wiki/Codex_Kingsborough" title="Codex Kingsborough">Codex Kingsborough</a></li> <li><a href="/wiki/Double-headed_serpent" title="Double-headed serpent">Double-headed serpent</a></li> <li><a href="/wiki/Hoa_Hakananai%27a" title="Hoa Hakananai'a">Hoa Hakananai'a</a></li> <li><i><a href="/wiki/Throne_of_Weapons" title="Throne of Weapons">Throne of Weapons</a></i></li> <li><a href="/wiki/Kayung_totem_pole" title="Kayung totem pole">Kayung totem pole</a></li> <li><i><a href="/wiki/Tree_of_Life_(Kester)" title="Tree of Life (Kester)">Tree of Life</a></i></li> <li><a href="/wiki/Yaxchilan_Lintel_24" title="Yaxchilan Lintel 24">Yaxchilan Lintel 24</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_Ancient_Egypt_and_Sudan" title="British Museum Department of Ancient Egypt and Sudan">Ancient Egypt<br />and Sudan</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Battlefield_Palette" title="Battlefield Palette">Battlefield Palette</a></li> <li><a href="/wiki/Colossal_quartzite_statue_of_Amenhotep_III" title="Colossal quartzite statue of Amenhotep III">Colossal quartzite statue of Amenhotep III</a></li> <li><a href="/wiki/Colossal_red_granite_statue_of_Amenhotep_III" title="Colossal red granite statue of Amenhotep III">Colossal head of Amenhotep III</a></li> <li><a href="/wiki/El-Amra_clay_model_of_cattle" title="El-Amra clay model of cattle">El-Amra clay model of cattle</a></li> <li><a href="/wiki/Gebelein_predynastic_mummies" title="Gebelein predynastic mummies">Gebelein predynastic mummies</a></li> <li><a href="/wiki/Greenfield_papyrus" class="mw-redirect" title="Greenfield papyrus">Greenfield papyrus</a></li> <li><a href="/wiki/Hornedjitef" title="Hornedjitef">Hornedjitef</a></li> <li><a href="/wiki/Hunters_Palette" title="Hunters Palette">Hunters Palette</a></li> <li><a href="/wiki/MacGregor_plaque" title="MacGregor plaque">MacGregor plaque</a></li> <li><a href="/wiki/Min_Palette" title="Min Palette">Min Palette</a></li> <li><a href="/wiki/Papyrus_of_Ani" title="Papyrus of Ani">Papyrus of Ani</a></li> <li><a href="/wiki/Prudhoe_Lions" title="Prudhoe Lions">Prudhoe Lions</a></li> <li><a class="mw-selflink selflink">Rhind Mathematical Papyrus</a></li> <li><a href="/wiki/Rosetta_Stone" title="Rosetta Stone">Rosetta Stone</a></li> <li><a href="/wiki/Sphinx_of_Taharqo" title="Sphinx of Taharqo">Sphinx of Taharqo</a></li> <li><a href="/wiki/Statues_of_Amun_in_the_form_of_a_ram_protecting_King_Taharqa" title="Statues of Amun in the form of a ram protecting King Taharqa">Amun in the form of a ram protecting King Taharqa</a></li> <li><a href="/wiki/Younger_Memnon" title="Younger Memnon">Younger Memnon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_Asia" title="British Museum Department of Asia">Asia</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Admonitions_Scroll" title="Admonitions Scroll">Admonitions Scroll</a></li> <li><a href="/wiki/Aphsad_inscription_of_%C4%80dityasena" title="Aphsad inscription of Ādityasena">Aphsad inscription of Ādityasena</a></li> <li><a href="/wiki/Amaravati_Marbles" title="Amaravati Marbles">Amaravati Marbles</a></li> <li><a href="/wiki/Ambika_Statue_from_Dhar" title="Ambika Statue from Dhar">Ambika Statue from Dhar</a></li> <li><a href="/wiki/Amit%C4%81bha_Buddha_from_Hancui" title="Amitābha Buddha from Hancui">Amitābha Buddha from Hancui</a></li> <li><a href="/wiki/Bimaran_casket" title="Bimaran casket">Bimaran casket</a></li> <li><a href="/wiki/Buddhapad_Hoard" title="Buddhapad Hoard">Buddhapad Hoard</a></li> <li><a href="/wiki/Mogao_Christian_painting" title="Mogao Christian painting">Mogao Christian painting</a></li> <li><a href="/wiki/David_Vases" title="David Vases">David Vases</a></li> <li><a href="/wiki/Dhaneswar_Khera_Buddha_image_inscription" title="Dhaneswar Khera Buddha image inscription">Dhaneswar Khera Buddha image inscription</a></li> <li><a href="/wiki/Hephthalite_silver_bowl" title="Hephthalite silver bowl">Hephthalite silver bowl</a></li> <li><a href="/wiki/Huixian_Bronze_Hu" title="Huixian Bronze Hu">Huixian Bronze Hu</a></li> <li><a href="/wiki/Jade_terrapin_from_Allahabad" title="Jade terrapin from Allahabad">Jade terrapin from Allahabad</a></li> <li><a href="/wiki/Kakiemon_elephants_(British_Museum)" title="Kakiemon elephants (British Museum)">Kakiemon elephants</a></li> <li><a href="/wiki/Kang_Hou_gui" title="Kang Hou gui">Kang Hou gui</a></li> <li><a href="/wiki/Kanishka_casket" class="mw-redirect" title="Kanishka casket">Kanishka casket</a></li> <li><a href="/wiki/Klang_Bell" title="Klang Bell">Klang Bell</a></li> <li><a href="/wiki/Kulu_Vase" title="Kulu Vase">Kulu Vase</a></li> <li><a href="/wiki/Mathura_lion_capital" title="Mathura lion capital">Mathura lion capital</a></li> <li><a href="/wiki/Percival_David_Foundation_of_Chinese_Art" title="Percival David Foundation of Chinese Art">Percival David Foundation of Chinese Art</a></li> <li><a href="/wiki/Seated_Buddha_from_Gandhara" title="Seated Buddha from Gandhara">Seated Buddha from Gandhara</a></li> <li><a href="/wiki/Stamp_seal_(BM_119999)" title="Stamp seal (BM 119999)">Stamp seal (BM 119999)</a></li> <li><a href="/wiki/Statue_of_Tara" title="Statue of Tara">Statue of Tara</a></li> <li><i><a href="/wiki/The_Great_Wave_off_Kanagawa" title="The Great Wave off Kanagawa">The Great Wave off Kanagawa</a></i></li> <li><a href="/wiki/Vishnu_Nicolo_Seal" class="mw-redirect" title="Vishnu Nicolo Seal">Vishnu Nicolo Seal</a></li> <li><a href="/wiki/Wardak_Vase" title="Wardak Vase">Wardak Vase</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_Greece_and_Rome" class="mw-redirect" title="British Museum Department of Greece and Rome">Greece<br />and Rome</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aegina_Treasure" title="Aegina Treasure">Aegina Treasure</a></li> <li><a href="/wiki/Aineta_aryballos" title="Aineta aryballos">Aineta aryballos</a></li> <li><a href="/wiki/Apollo_of_Cyrene" title="Apollo of Cyrene">Apollo of Cyrene</a></li> <li><a href="/wiki/Archangel_ivory" title="Archangel ivory">Archangel ivory</a></li> <li><a href="/wiki/Arcisate_Treasure" title="Arcisate Treasure">Arcisate Treasure</a></li> <li><a href="/wiki/Armento_Rider" title="Armento Rider">Armento Rider</a></li> <li><a href="/wiki/Arundel_Head" title="Arundel Head">Arundel Head</a></li> <li><a href="/wiki/Asclepius_of_Milos" title="Asclepius of Milos">Asclepius of Milos</a></li> <li><a href="/wiki/Barber_Cup_and_Crawford_Cup" title="Barber Cup and Crawford Cup">Barber Cup and Crawford Cup</a></li> <li><a href="/wiki/Bassae_Frieze" title="Bassae Frieze">Bassae Frieze</a></li> <li><a href="/wiki/Beaurains_Treasure" title="Beaurains Treasure">Beaurains Treasure</a></li> <li><a href="/wiki/Blacas_Cameo" title="Blacas Cameo">Blacas Cameo</a></li> <li><a href="/wiki/Boscoreale_Treasure" title="Boscoreale Treasure">Boscoreale Treasure</a></li> <li><a href="/wiki/Boy_with_Thorn" title="Boy with Thorn">Boy with Thorn</a></li> <li><a href="/wiki/Braganza_Brooch" title="Braganza Brooch">Braganza Brooch</a></li> <li><a href="/wiki/Bronze_Head_of_Hypnos_from_Civitella_d%27Arna" title="Bronze Head of Hypnos from Civitella d'Arna">Bronze head of Hypnos</a></li> <li><a href="/wiki/Burgon_vase" title="Burgon vase">Burgon vase</a></li> <li><a href="/wiki/Bursa_Treasure" title="Bursa Treasure">Bursa Treasure</a></li> <li><a href="/wiki/Campo_Iemini_Venus" title="Campo Iemini Venus">Campo Iemini Venus</a></li> <li><a href="/wiki/Carthage_Treasure" title="Carthage Treasure">Carthage Treasure</a></li> <li><a href="/wiki/Caubiac_Treasure" title="Caubiac Treasure">Caubiac Treasure</a></li> <li><a href="/wiki/Chaourse_Treasure" title="Chaourse Treasure">Chaourse Treasure</a></li> <li><a href="/wiki/Chatsworth_Head" title="Chatsworth Head">Chatsworth Head</a></li> <li><a href="/wiki/Chatuzange_Treasure" title="Chatuzange Treasure">Chatuzange Treasure</a></li> <li><a href="/wiki/Choiseul-Gouffier_Apollo" title="Choiseul-Gouffier Apollo">Choiseul-Gouffier Apollo</a></li> <li><a href="/wiki/Demeter_of_Knidos" title="Demeter of Knidos">Demeter of Knidos</a></li> <li><a href="/wiki/Dionysus_Sardanapalus" title="Dionysus Sardanapalus">Dionysus Sardanapalus</a></li> <li><a href="/wiki/Elgin_Amphora" title="Elgin Amphora">Elgin Amphora</a></li> <li><a href="/wiki/Elgin_Marbles" title="Elgin Marbles">Elgin Marbles</a></li> <li><a href="/wiki/Esquiline_Treasure" title="Esquiline Treasure">Esquiline Treasure</a></li> <li><a href="/wiki/Euphorbos_plate" title="Euphorbos plate">Euphorbos plate</a></li> <li><a href="/wiki/Farnese_Diadumenos" title="Farnese Diadumenos">Farnese Diadumenos</a></li> <li><a href="/wiki/Guilford_Puteal" title="Guilford Puteal">Guilford Puteal</a></li> <li><a href="/wiki/Harpy_Tomb" title="Harpy Tomb">Harpy Tomb</a></li> <li><a href="/wiki/Herculean_Sarcophagus_of_Genzano" title="Herculean Sarcophagus of Genzano">Herculean Sarcophagus of Genzano</a></li> <li><a href="/wiki/Isis_Tomb,_Vulci" title="Isis Tomb, Vulci">Isis Tomb</a></li> <li><a href="/wiki/Jennings_Dog" title="Jennings Dog">Jennings Dog</a></li> <li><a href="/wiki/Lion_of_Knidos" title="Lion of Knidos">Lion of Knidos</a></li> <li><a href="/wiki/Lycurgus_Cup" title="Lycurgus Cup">Lycurgus Cup</a></li> <li><a href="/wiki/Macmillan_aryballos" title="Macmillan aryballos">Macmillan aryballos</a></li> <li><a href="/wiki/M%C3%A2con_Treasure" title="Mâcon Treasure">Mâcon Treasure</a></li> <li><a href="/wiki/Mainz_Gladius" title="Mainz Gladius">Mainz Gladius</a></li> <li><a href="/wiki/Mero%C3%AB_Head" title="Meroë Head">Meroë Head</a></li> <li><a href="/wiki/Minoan_Bull-leaper" title="Minoan Bull-leaper">Minoan Bull-leaper</a></li> <li><a href="/wiki/Nereid_Monument" title="Nereid Monument">Nereid Monument</a></li> <li><a href="/wiki/Oscan_Tablet" title="Oscan Tablet">Oscan Tablet</a></li> <li><a href="/wiki/Oxyrhynchus_Papyri" title="Oxyrhynchus Papyri">Papyri of Oxyrhynchus</a> <ul><li><a href="/wiki/Papyrus_Oxyrhynchus_84" title="Papyrus Oxyrhynchus 84">84</a></li> <li><a href="/wiki/Papyrus_Oxyrhynchus_85" title="Papyrus Oxyrhynchus 85">85</a></li> <li><a href="/wiki/Papyrus_Oxyrhynchus_102" title="Papyrus Oxyrhynchus 102">102</a></li> <li><a href="/wiki/Papyrus_Oxyrhynchus_103" title="Papyrus Oxyrhynchus 103">103</a></li></ul></li> <li><a href="/wiki/Paramythia_Hoard" title="Paramythia Hoard">Paramythia Hoard</a></li> <li><a href="/wiki/Pericles_with_the_Corinthian_helmet" title="Pericles with the Corinthian helmet">Pericles bust</a></li> <li><a href="/wiki/Petelia_Gold_Tablet" title="Petelia Gold Tablet">Petelia Gold Tablet</a></li> <li><a href="/wiki/Piranesi_Vase" title="Piranesi Vase">Piranesi Vase</a></li> <li><a href="/wiki/Portland_Vase" title="Portland Vase">Portland Vase</a></li> <li><a href="/wiki/Priene_inscription_of_Alexander_the_Great" title="Priene inscription of Alexander the Great">Priene dedicatory inscription</a></li> <li><a href="/wiki/Alexander_the_Great%27s_edict_to_Priene" title="Alexander the Great's edict to Priene">Priene edict inscription</a></li> <li><a href="/wiki/San_Sosti_Axe-Head" title="San Sosti Axe-Head">San Sosti Axe-Head</a></li> <li><a href="/wiki/Sant%27Angelo_Muxaro_Patera" title="Sant'Angelo Muxaro Patera">Sant'Angelo Muxaro Patera</a></li> <li><a href="/wiki/Sarcophagus_of_Seianti_Hanunia_Tlesnasa" title="Sarcophagus of Seianti Hanunia Tlesnasa">Sarcophagus of Seianti Hanunia Tlesnasa</a></li> <li><a href="/wiki/Satala_Aphrodite" title="Satala Aphrodite">Satala Aphrodite</a></li> <li><a href="/wiki/Stony_Stratford_Hoard" class="mw-redirect" title="Stony Stratford Hoard">Stony Stratford Hoard</a></li> <li><a href="/wiki/Strangford_Apollo" title="Strangford Apollo">Strangford Apollo</a></li> <li><a href="/wiki/Tomb_of_Payava" title="Tomb of Payava">Tomb of Payava</a></li> <li>Townley collection <ul><li><a href="/wiki/Townley_Antinous" title="Townley Antinous">Antinous</a></li> <li><a href="/wiki/Townley_Caryatid" title="Townley Caryatid">Caryatid</a></li> <li><a href="/wiki/Townley_Discobolus" class="mw-redirect" title="Townley Discobolus">Discobolus</a></li> <li><a href="/wiki/Townley_Hadrian" title="Townley Hadrian">Hadrian</a></li> <li><a href="/wiki/Townley_Vase" title="Townley Vase">Vase</a></li> <li><a href="/wiki/Townley_Venus" title="Townley Venus">Venus</a></li></ul></li> <li><a href="/wiki/Uerdingen_Hoard" title="Uerdingen Hoard">Uerdingen Hoard</a></li> <li><a href="/wiki/Vaison_Diadumenos" title="Vaison Diadumenos">Vaison Diadumenos</a></li> <li><a href="/wiki/Warren_Cup" title="Warren Cup">Warren Cup</a></li> <li><a href="/wiki/Xanten_Horse-Phalerae" title="Xanten Horse-Phalerae">Xanten Horse-Phalerae</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_the_Middle_East" title="British Museum Department of the Middle East">Middle East</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ain_Sakhri_figurine" title="Ain Sakhri figurine">Ain Sakhri figurine</a></li> <li><a href="/wiki/Assyrian_lion_weights" title="Assyrian lion weights">Assyrian lion weights</a></li> <li><a href="/wiki/Balawat_Gates" title="Balawat Gates">Balawat Gates</a></li> <li><a href="/wiki/Blacas_ewer" title="Blacas ewer">Blacas ewer</a></li> <li><a href="/wiki/Uruk_Trough" title="Uruk Trough">Uruk Trough</a></li> <li><a href="/wiki/Blau_Monuments" title="Blau Monuments">Blau Monuments</a></li> <li><a href="/wiki/Palmer_Cup" title="Palmer Cup">Palmer Cup</a></li> <li><a href="/wiki/Standard_of_Ur" title="Standard of Ur">Standard of Ur</a></li> <li><a href="/wiki/Ram_in_a_Thicket" title="Ram in a Thicket">Ram in a Thicket</a></li> <li><a href="/wiki/Lyres_of_Ur" title="Lyres of Ur">Lyres of Ur</a></li> <li><a href="/wiki/Tell_al-%27Ubaid_Copper_Lintel" title="Tell al-'Ubaid Copper Lintel">Tell al-'Ubaid Copper Lintel</a></li> <li><a href="/wiki/White_Obelisk" title="White Obelisk">White Obelisk</a></li> <li><a href="/wiki/Black_Obelisk_of_Shalmaneser_III" title="Black Obelisk of Shalmaneser III">Black Obelisk of Shalmaneser III</a></li> <li><a href="/wiki/Kition_Necropolis_Phoenician_inscriptions" title="Kition Necropolis Phoenician inscriptions">Kition Necropolis Phoenician inscriptions</a></li> <li><a href="/wiki/Burney_Relief" title="Burney Relief">Burney Relief</a></li> <li><a href="/wiki/Complaint_tablet_to_Ea-n%C4%81%E1%B9%A3ir" title="Complaint tablet to Ea-nāṣir">Complaint tablet to Ea-nāṣir</a></li> <li><a href="/wiki/Tablet_of_Shamash" title="Tablet of Shamash">Tablet of Shamash</a></li> <li><a href="/wiki/Statue_of_Idrimi" title="Statue of Idrimi">Statue of Idrimi</a></li> <li><a href="/wiki/Babylonian_Map_of_the_World" title="Babylonian Map of the World">Babylonian Map of the World</a></li> <li><a href="/wiki/Rassam_cylinder" title="Rassam cylinder">Rassam cylinder</a></li> <li><a href="/wiki/Cylinders_of_Nabonidus" title="Cylinders of Nabonidus">Cylinders of Nabonidus</a></li> <li><a href="/wiki/Cyrus_Cylinder" title="Cyrus Cylinder">Cyrus Cylinder</a></li> <li><a href="/wiki/Gilgamesh_flood_myth" title="Gilgamesh flood myth">Flood tablet (Gilgamesh)</a></li> <li><a href="/wiki/Jar_of_Xerxes_I" title="Jar of Xerxes I">Jar of Xerxes I</a></li> <li><a href="/wiki/Library_of_Ashurbanipal" title="Library of Ashurbanipal">Library of Ashurbanipal</a></li> <li><i><a href="/wiki/Lion_Hunt_of_Ashurbanipal" title="Lion Hunt of Ashurbanipal">Lion Hunt of Ashurbanipal</a></i></li> <li><a href="/wiki/Oxus_Treasure" title="Oxus Treasure">Oxus Treasure</a></li> <li><a href="/wiki/Sennacherib%27s_Annals" title="Sennacherib's Annals">Taylor Prism</a></li> <li><a href="/wiki/Lachish_reliefs" title="Lachish reliefs">Lachish reliefs</a></li> <li><a href="/wiki/Babylonian_Chronicles" title="Babylonian Chronicles">Babylonian Chronicles</a></li> <li><a href="/wiki/Ur_Box_inscription" title="Ur Box inscription">Ur Box inscription</a></li> <li><a href="/wiki/Kurkh_Monoliths" title="Kurkh Monoliths">Kurkh Monoliths</a></li> <li><a href="/wiki/Antiochus_cylinder" title="Antiochus cylinder">Antiochus cylinder</a></li> <li><a href="/wiki/Nimrud_ivories" title="Nimrud ivories">Nimrud ivories</a></li> <li><a href="/wiki/Phoenician_metal_bowls" title="Phoenician metal bowls">Phoenician metal bowls</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_Prehistory_and_Europe" class="mw-redirect" title="British Museum Department of Prehistory and Europe">Prehistory<br />and Europe</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Portable_altar_(WB.232)" class="mw-redirect" title="Portable altar (WB.232)">Boxwood altar</a></li> <li><a href="/wiki/Franks_Casket" title="Franks Casket">Franks Casket</a></li> <li><a href="/wiki/Holy_Thorn_Reliquary" title="Holy Thorn Reliquary">Holy Thorn Reliquary</a></li> <li><a href="/wiki/Hoxne_Hoard" title="Hoxne Hoard">Hoxne Hoard</a></li> <li><a href="/wiki/Lewis_chessmen" title="Lewis chessmen">Lewis chessmen</a></li> <li><a href="/wiki/Lindow_Man" title="Lindow Man">Lindow Man</a></li> <li><a href="/wiki/Mildenhall_Treasure" title="Mildenhall Treasure">Mildenhall Treasure</a></li> <li><a href="/wiki/Ringlemere_Cup" title="Ringlemere Cup">Ringlemere Cup</a></li> <li><a href="/wiki/Royal_Gold_Cup" title="Royal Gold Cup">Royal Gold Cup</a></li> <li><a href="/wiki/Seax_of_Beagnoth" title="Seax of Beagnoth">Seax of Beagnoth</a></li> <li><a href="/wiki/Sutton_Hoo" title="Sutton Hoo">Sutton Hoo</a></li> <li><a href="/wiki/Sutton_Hoo_helmet" title="Sutton Hoo helmet">Sutton Hoo helmet</a></li> <li><a href="/wiki/Sutton_Hoo_purse-lid" title="Sutton Hoo purse-lid">Sutton Hoo purse-lid</a></li> <li><i><a href="/wiki/Swimming_Reindeer" title="Swimming Reindeer">Swimming Reindeer</a></i></li> <li><a href="/wiki/Hedwig_glass" title="Hedwig glass">Hedwig glass</a></li> <li><a href="/wiki/Vindolanda_tablets" title="Vindolanda tablets">Vindolanda Tablets</a></li> <li><a href="/wiki/Lampsacus_Treasure" title="Lampsacus Treasure">Lampsacus Treasure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;"><a href="/wiki/British_Museum_Department_of_Prints_and_Drawings" class="mw-redirect" title="British Museum Department of Prints and Drawings">Prints and<br />Drawings</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/D%C3%BCrer%27s_Rhinoceros" title="Dürer's Rhinoceros">Dürer's Rhinoceros</a></i></li> <li><i><a href="/wiki/Epifania_(Michelangelo_drawing)" class="mw-redirect" title="Epifania (Michelangelo drawing)">Epifania</a></i></li> <li><i><a href="/wiki/I_Modi" title="I Modi">I Modi</a></i></li> <li><i><a href="/wiki/Isabella_Brant_(drawing)" title="Isabella Brant (drawing)">Isabella Brant</a></i></li> <li><i><a href="/wiki/The_Ancient_of_Days" title="The Ancient of Days">The Ancient of Days</a></i></li> <li><i><a href="/wiki/The_Disasters_of_War" title="The Disasters of War">The Disasters of War</a></i></li> <li><i><a href="/wiki/The_Tale_of_The_Flopsy_Bunnies#Illustrations" class="mw-redirect" title="The Tale of The Flopsy Bunnies">The Tale of the Flopsy Bunnies</a></i></li> <li><i><a href="/wiki/Triumphal_Arch_(woodcut)" title="Triumphal Arch (woodcut)">Triumphal Arch</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;background-color: #eee;">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/British_Museum_Department_of_Coins_and_Medals" title="British Museum Department of Coins and Medals">Coins and Medals</a></li> <li><a href="/wiki/British_Museum_Department_of_Conservation_and_Scientific_Research" class="mw-redirect" title="British Museum Department of Conservation and Scientific Research">Conservation and Scientific Research</a></li> <li><a href="/wiki/British_Museum_Department_of_Libraries_and_Archives" class="mw-redirect" title="British Museum Department of Libraries and Archives">Libraries and Archives</a></li> <li><a href="/wiki/Portable_Antiquities_Scheme" title="Portable Antiquities Scheme">Portable Antiquities and Treasure</a></li> <li><a href="/wiki/Rondanini_Faun" title="Rondanini Faun">Rondanini Faun</a></li> <li><i><a href="/wiki/A_History_of_the_World_in_100_Objects" title="A History of the World in 100 Objects">A History of the World in 100 Objects</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: left;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blythe_House" title="Blythe House">Blythe House</a></li> <li><a href="/wiki/British_Museum_Act" title="British Museum Act">British Museum Acts</a></li> <li><a href="/wiki/Dingwall_Beloe_Lecture_Series" title="Dingwall Beloe Lecture Series">Dingwall Beloe Lectures</a></li> <li><a href="/wiki/List_of_films_shot_at_the_British_Museum" title="List of films shot at the British Museum">Films shot at the British Museum</a></li> <li><a href="/wiki/Private_Case" title="Private Case">Private Case</a></li> <li><a href="/wiki/British_Museum#Contested_artefacts" title="British Museum">Repatriation controversy</a></li> <li><a href="/wiki/Secretum_(British_Museum)" title="Secretum (British Museum)">Secretum</a></li> <li>Staff <ul><li><a href="/wiki/List_of_directors_of_the_British_Museum" title="List of directors of the British Museum">Directors</a></li> <li><a href="/wiki/The_British_Museum_Friends" title="The British Museum Friends">Friends</a></li> <li><a href="/wiki/List_of_keepers_of_the_British_Museum" title="List of keepers of the British Museum">Keepers</a></li> <li><a href="/wiki/Royal_Commission_on_the_British_Museum" title="Royal Commission on the British Museum">Royal Commission on the British Museum</a></li> <li><a href="/wiki/List_of_trustees_of_the_British_Museum" title="List of trustees of the British Museum">Trustees</a></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:British_Museum" title="Category:British Museum">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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Rhind Mathematical Papyrus - Wikipedia