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href="#Number_of_terms_and_use_of_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Number of terms and use of fractions</span> </div> </a> <ul id="toc-Number_of_terms_and_use_of_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proportions_and_percentage_ratios" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proportions_and_percentage_ratios"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Proportions and percentage ratios</span> </div> </a> <ul id="toc-Proportions_and_percentage_ratios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reduction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Reduction</span> </div> </a> <ul id="toc-Reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irrational_ratios" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Irrational_ratios"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Irrational ratios</span> </div> </a> <ul id="toc-Irrational_ratios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Odds"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Odds</span> </div> </a> <ul id="toc-Odds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Units</span> </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangular_coordinates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Triangular_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Triangular coordinates</span> </div> </a> <ul id="toc-Triangular_coordinates-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">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">13</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" title="Table of Contents" > <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">Ratio</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 61 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-61" 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">61 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Verhouding" title="Verhouding – Afrikaans" lang="af" hreflang="af" data-title="Verhouding" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%8D%E1%8B%B5%E1%88%AD" title="ውድር – Amharic" lang="am" hreflang="am" data-title="ውድር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="نسبة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="نسبة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%AA%E0%A6%BE%E0%A6%A4" title="অনুপাত – Assamese" lang="as" hreflang="as" data-title="অনুপাত" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Nisb%C9%99t" title="Nisbət – Azerbaijani" lang="az" hreflang="az" data-title="Nisbət" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Rasyo" title="Rasyo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Rasyo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%8A%D0%BE%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5" title="Съотношение – Bulgarian" lang="bg" hreflang="bg" data-title="Съотношение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/R%C3%A0tio" title="Ràtio – Catalan" lang="ca" hreflang="ca" data-title="Ràtio" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D0%B0%D0%B9%D0%BB%D0%B0%D1%88%C4%83%D0%BD%D0%BD%C4%83%D0%BB%C4%83%D1%85" title="Шайлашăннăлăх – Chuvash" lang="cv" hreflang="cv" data-title="Шайлашăннăлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pom%C4%9Br" title="Poměr – Czech" lang="cs" hreflang="cs" data-title="Poměr" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Rheshiyo" title="Rheshiyo – Shona" lang="sn" hreflang="sn" data-title="Rheshiyo" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cymhareb" title="Cymhareb – Welsh" lang="cy" hreflang="cy" data-title="Cymhareb" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Forholdstal" title="Forholdstal – Danish" lang="da" hreflang="da" data-title="Forholdstal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%B1%D8%A7%D8%B5%D9%8A%D9%88" title="راصيو – Moroccan Arabic" lang="ary" hreflang="ary" data-title="راصيو" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Verh%C3%A4ltnis_(Mathematik)" title="Verhältnis (Mathematik) – German" lang="de" hreflang="de" data-title="Verhältnis (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CF%8C%CE%B3%CE%BF%CF%82_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Λόγος (μαθηματικά) – Greek" lang="el" hreflang="el" data-title="Λόγος (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Raz%C3%B3n_(matem%C3%A1tica)" title="Razón (matemática) – Spanish" lang="es" hreflang="es" data-title="Razón (matemática)" 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/Rilatumo" title="Rilatumo – Esperanto" lang="eo" hreflang="eo" data-title="Rilatumo" 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/Arrazoi_(matematika)" title="Arrazoi (matematika) – Basque" lang="eu" hreflang="eu" data-title="Arrazoi (matematika)" 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%86%D8%B3%D8%A8%D8%AA_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="نسبت (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="نسبت (ریاضیات)" 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/Rapport_(math%C3%A9matiques)" title="Rapport (mathématiques) – French" lang="fr" hreflang="fr" data-title="Rapport (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Ferh%C3%A2lding_(wiskunde)" title="Ferhâlding (wiskunde) – Western Frisian" lang="fy" hreflang="fy" data-title="Ferhâlding (wiskunde)" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Raz%C3%B3n_(matem%C3%A1ticas)" title="Razón (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Razón (matemáticas)" 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%B9%84_(%EC%88%98%ED%95%99)" title="비 (수학) – Korean" lang="ko" hreflang="ko" data-title="비 (수학)" 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%80%D5%A1%D6%80%D5%A1%D5%A2%D5%A5%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Հարաբերություններ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Հարաբերություններ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AA%E0%A4%BE%E0%A4%A4" title="अनुपात – Hindi" lang="hi" hreflang="hi" data-title="अनुपात" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Omjer" title="Omjer – Croatian" lang="hr" hreflang="hr" data-title="Omjer" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rasio" title="Rasio – Indonesian" lang="id" hreflang="id" data-title="Rasio" 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/Rapporto_(matematica)" title="Rapporto (matematica) – Italian" lang="it" hreflang="it" data-title="Rapporto (matematica)" 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%99%D7%97%D7%A1_(%D7%91%D7%99%D7%9F_%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D)" title="יחס (בין מספרים) – Hebrew" lang="he" hreflang="he" data-title="יחס (בין מספרים)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D1%8B%D1%88" title="Катыш – Kyrgyz" lang="ky" hreflang="ky" data-title="Катыш" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Santykis" title="Santykis – Lithuanian" lang="lt" hreflang="lt" data-title="Santykis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ar%C3%A1ny" title="Arány – Hungarian" lang="hu" hreflang="hu" data-title="Arány" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%82%E0%B4%B6%E0%B4%AC%E0%B4%A8%E0%B5%8D%E0%B4%A7%E0%B4%82" title="അംശബന്ധം – Malayalam" lang="ml" hreflang="ml" data-title="അംശബന്ധം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A5%8B%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%B0" title="गुणोत्तर – Marathi" lang="mr" hreflang="mr" data-title="गुणोत्तर" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nisbah" title="Nisbah – Malay" lang="ms" hreflang="ms" data-title="Nisbah" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verhouding_(wiskunde)" title="Verhouding (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Verhouding (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AA%E0%A4%BE%E0%A4%A4" title="अनुपात – Newari" lang="new" hreflang="new" data-title="अनुपात" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AF%94" title="比 – Japanese" lang="ja" hreflang="ja" data-title="比" 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/Forhold" title="Forhold – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Forhold" 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/Forhold" title="Forhold – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Forhold" 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-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Hira" title="Hira – Oromo" lang="om" hreflang="om" data-title="Hira" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Stosunek_(matematyka)" title="Stosunek (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Stosunek (matematyka)" 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/Raz%C3%A3o_(matem%C3%A1tica)" title="Razão (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Razão (matemática)" 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/Raport" title="Raport – Romanian" lang="ro" hreflang="ro" data-title="Raport" 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%A1%D0%BE%D0%BE%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5" title="Соотношение – Russian" lang="ru" hreflang="ru" data-title="Соотношение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ratio" title="Ratio – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ratio" 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/Pomer_(matematika)" title="Pomer (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Pomer (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Saami_qumman" title="Saami qumman – Somali" lang="so" hreflang="so" data-title="Saami qumman" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%DB%8E%DA%98%DB%95" title="ڕێژە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕێژە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Размера (математика) – Serbian" lang="sr" hreflang="sr" data-title="Размера (математика)" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Suhde" title="Suhde – Finnish" lang="fi" hreflang="fi" data-title="Suhde" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Rasyo" title="Rasyo – Tagalog" lang="tl" hreflang="tl" data-title="Rasyo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BF%E0%AE%95%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="விகிதம் – Tamil" lang="ta" hreflang="ta" data-title="விகிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A8%E0%B0%BF%E0%B0%B7%E0%B1%8D%E0%B0%AA%E0%B0%A4%E0%B1%8D%E0%B0%A4%E0%B0%BF" title="నిష్పత్తి – Telugu" lang="te" hreflang="te" data-title="నిష్పత్తి" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%B1%E0%B8%95%E0%B8%A3%E0%B8%B2%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99" title="อัตราส่วน – Thai" lang="th" hreflang="th" data-title="อัตราส่วน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Rasyo" title="Rasyo – Turkish" lang="tr" hreflang="tr" data-title="Rasyo" 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%A1%D0%BF%D1%96%D0%B2%D0%B2%D1%96%D0%B4%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%BD%D1%8F" title="Співвідношення – Ukrainian" lang="uk" hreflang="uk" data-title="Співвідношення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%B7_l%E1%BB%87" title="Tỷ lệ – Vietnamese" lang="vi" hreflang="vi" data-title="Tỷ lệ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Rasyo" title="Rasyo – Waray" lang="war" hreflang="war" data-title="Rasyo" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AF%94" title="比 – Chinese" lang="zh" hreflang="zh" data-title="比" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Relationship between two numbers of the same kind</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">For other uses, see <a href="/wiki/Ratio_(disambiguation)" class="mw-disambig" title="Ratio (disambiguation)">Ratio (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"is to" redirects here. For the grammatical construction, see <a href="/wiki/Am_to" class="mw-redirect" title="Am to">am to</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Aspect-ratio-4x3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/220px-Aspect-ratio-4x3.svg.png" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/330px-Aspect-ratio-4x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/440px-Aspect-ratio-4x3.svg.png 2x" data-file-width="139" data-file-height="106" /></a><figcaption>The ratio of width to height of <a href="/wiki/Standard-definition_television" title="Standard-definition television">standard-definition television</a></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>ratio</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'r' in 'rye'">r</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="/ʃ/: 'sh' in 'shy'">ʃ</span></span>(<span style="border-bottom:1px dotted"><span title="/i/: 'y' in 'happy'">i</span></span>)<span style="border-bottom:1px dotted"><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>/</a></span></span>) shows how many times one <a href="/wiki/Number" title="Number">number</a> contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). </p><p>The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive</a>. </p><p>A ratio may be specified either by giving both constituting numbers, written as "<i>a</i> to <i>b</i>" or "<i>a:b</i>", or by giving just the value of their <a href="/wiki/Quotient" title="Quotient">quotient</a> <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a <i><b>proportion</b></i>. </p><p>Consequently, a ratio may be considered as an ordered pair of numbers, a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, are <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, and may sometimes be natural numbers. </p><p>A more specific definition adopted in <a href="/wiki/Physical_sciences" class="mw-redirect" title="Physical sciences">physical sciences</a> (especially in <a href="/wiki/Metrology" title="Metrology">metrology</a>) for <i>ratio</i> is the <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> quotient between two <a href="/wiki/Physical_quantities" class="mw-redirect" title="Physical quantities">physical quantities</a> measured with the same <a href="/wiki/Measurement_unit" class="mw-redirect" title="Measurement unit">unit</a>.<sup id="cite_ref-ISO_80000-1_4-0" class="reference"><a href="#cite_note-ISO_80000-1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A quotient of two quantities that are measured with <em>different</em> units may be called a <a href="/wiki/Rate_(mathematics)" title="Rate (mathematics)"><i>rate</i></a>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation_and_terminology">Notation and terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=1" title="Edit section: Notation and terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ratio of numbers <i>A</i> and <i>B</i> can be expressed as:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <ul><li>the ratio of <i>A</i> to <i>B</i></li> <li><i>A:B</i></li> <li><i>A</i> is to <i>B</i> (when followed by "as <i>C</i> is to <i>D</i> "; see below)</li> <li>a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> with <i>A</i> as numerator and <i>B</i> as denominator that represents the quotient (i.e., <i>A</i> divided by <i>B, or</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 {\tfrac {A}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>A</mi> <mi>B</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {A}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0284d6d3707f6972edd6b5797aa405b91080ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.083ex; height:3.676ex;" alt="{\displaystyle {\tfrac {A}{B}}}" /></span>). This can be expressed as a simple or a decimal fraction, or as a percentage, etc.<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></li></ul> <p>When a ratio is written in the form <i>A</i>:<i>B</i>, the two-dot character is sometimes the <a href="/wiki/Colon_(punctuation)" title="Colon (punctuation)">colon</a> punctuation mark.<sup id="cite_ref-MathWorld-colon_8-0" class="reference"><a href="#cite_note-MathWorld-colon-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Unicode" title="Unicode">Unicode</a>, this is <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+003A</span> </span><span style="font-size:125%;line-height:1em">:</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">COLON</span>, although Unicode also provides a dedicated ratio character, <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">U+2236</span> </span><span style="font-size:125%;line-height:1em">∶</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">RATIO</span>.<sup id="cite_ref-Unicode_9-0" class="reference"><a href="#cite_note-Unicode-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The numbers <i>A</i> and <i>B</i> are sometimes called <i>terms of the ratio</i>, with <i>A</i> being the <i><a href="/wiki/Antecedent_(grammar)" title="Antecedent (grammar)">antecedent</a></i> and <i>B</i> being the <i><a href="/wiki/Consequent" title="Consequent">consequent</a></i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>A statement expressing the equality of two ratios <i>A</i>:<i>B</i> and <i>C</i>:<i>D</i> is called a <b>proportion</b>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> written as <i>A</i>:<i>B</i> = <i>C</i>:<i>D</i> or <i>A</i>:<i>B</i>∷<i>C</i>:<i>D</i>. This latter form, when spoken or written in the English language, is often expressed as </p> <dl><dd>(<i>A</i> is to <i>B</i>) as (<i>C</i> is to <i>D</i>).</dd></dl> <p><i>A</i>, <i>B</i>, <i>C</i> and <i>D</i> are called the terms of the proportion. <i>A</i> and <i>D</i> are called its <i>extremes</i>, and <i>B</i> and <i>C</i> are called its <i>means</i>. The equality of three or more ratios, like <i>A</i>:<i>B</i> = <i>C</i>:<i>D</i> = <i>E</i>:<i>F</i>, is called a <b>continued proportion</b>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Ratios are sometimes used with three or even more terms, e.g., the proportion for the edge lengths of a "<a href="/wiki/Dimensional_lumber" class="mw-redirect" title="Dimensional lumber">two by four</a>" that is ten inches long is therefore </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 {\text{thickness : width : length }}=2:4:10;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>thickness : width : length </mtext> </mrow> <mo>=</mo> <mn>2</mn> <mo>:</mo> <mn>4</mn> <mo>:</mo> <mn>10</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{thickness : width : length }}=2:4:10;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d292dcb1855d6edfdc4cfa81ec2ce283338cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.874ex; height:2.509ex;" alt="{\displaystyle {\text{thickness : width : length }}=2:4:10;}" /></span></dd> <dd>(unplaned measurements; the first two numbers are reduced slightly when the wood is planed smooth)</dd></dl> <p>a good concrete mix (in volume units) is sometimes quoted as </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 {\text{cement : sand : gravel }}=1:2:4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>cement : sand : gravel </mtext> </mrow> <mo>=</mo> <mn>1</mn> <mo>:</mo> <mn>2</mn> <mo>:</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{cement : sand : gravel }}=1:2:4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79e4fb8dc3a37a9bb0a7c7b8e06cac6cfaec187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.341ex; height:2.509ex;" alt="{\displaystyle {\text{cement : sand : gravel }}=1:2:4.}" /></span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></dd></dl> <p>For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement. </p><p>The meaning of such a proportion of ratios with more than two terms is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side. </p> <div class="mw-heading mw-heading2"><h2 id="History_and_etymology">History and etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=2" title="Edit section: History and etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to trace the origin of the word "ratio" to the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> <span title="Ancient Greek (to 1453)-language text"><span lang="grc">λόγος</span></span> (<i><a href="/wiki/Logos" title="Logos">logos</a></i>). Early translators rendered this into <a href="/wiki/Latin" title="Latin">Latin</a> as <i><span title="Latin-language text"><i lang="la"><a href="https://en.wiktionary.org/wiki/ratio#Latin" class="extiw" title="wikt:ratio">ratio</a></i></span></i> ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Medieval writers used the word <i><span title="Latin-language text"><i lang="la">proportio</i></span></i> ("proportion") to indicate ratio and <i><span title="Latin-language text"><i lang="la">proportionalitas</i></span></i> ("proportionality") for the equality of ratios.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Euclid collected the results appearing in the Elements from earlier sources. The <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreans</a> developed a theory of ratio and proportion as applied to numbers.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a>. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The existence of multiple theories seems unnecessarily complex since ratios are, to a large extent, identified with quotients and their prospective values. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Euclid's_definitions"><span id="Euclid.27s_definitions"></span>Euclid's definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=3" title="Edit section: Euclid's definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Book V of <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a> has 18 definitions, all of which relate to ratios.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a <i>part</i> of a quantity is another quantity that "measures" it and conversely, a <i>multiple</i> of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning <a href="/wiki/Aliquot_part" class="mw-redirect" title="Aliquot part">aliquot part</a>) is a part that, when multiplied by an integer greater than one, gives the quantity. </p><p>Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity <i>measures</i> the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII. </p><p>Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Euclid defines a ratio as between two quantities <i>of the same type</i>, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities <i>p</i> and <i>q</i>, if there exist integers <i>m</i> and <i>n</i> such that <i>mp</i>><i>q</i> and <i>nq</i>><i>p</i>. This condition is known as the <a href="/wiki/Archimedes_property" class="mw-redirect" title="Archimedes property">Archimedes property</a>. </p><p>Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities <i>p</i>, <i>q</i>, <i>r</i> and <i>s</i>, <i>p</i>:<i>q</i>∷<i>r</i> :<i>s</i> if and only if, for any positive integers <i>m</i> and <i>n</i>, <i>np</i><<i>mq</i>, <i>np</i>=<i>mq</i>, or <i>np</i>><i>mq</i> according as <i>nr</i><<i>ms</i>, <i>nr</i>=<i>ms</i>, or <i>nr</i>><i>ms</i>, respectively.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This definition has affinities with <a href="/wiki/Dedekind_cuts" class="mw-redirect" title="Dedekind cuts">Dedekind cuts</a> as, with <i>n</i> and <i>q</i> both positive, <i>np</i> stands to <i>mq</i> as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>p</i></span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span> stands to the rational number <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>m</i></span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span> (dividing both terms by <i>nq</i>).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Definition 6 says that quantities that have the same ratio are <i>proportional</i> or <i>in proportion</i>. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog". </p><p>Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities <i>p</i>, <i>q</i>, <i>r</i> and <i>s</i>, <i>p</i>:<i>q</i>><i>r</i>:<i>s</i> if there are positive integers <i>m</i> and <i>n</i> so that <i>np</i>><i>mq</i> and <i>nr</i>≤<i>ms</i>. </p><p><span class="anchor" id="EuclidDef8"></span>As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms <i>p</i>, <i>q</i> and <i>r</i> to be in proportion when <i>p</i>:<i>q</i>∷<i>q</i>:<i>r</i>. This is extended to four terms <i>p</i>, <i>q</i>, <i>r</i> and <i>s</i> as <i>p</i>:<i>q</i>∷<i>q</i>:<i>r</i>∷<i>r</i>:<i>s</i>, and so on. Sequences that have the property that the ratios of consecutive terms are equal are called <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progressions</a>. Definitions 9 and 10 apply this, saying that if <i>p</i>, <i>q</i> and <i>r</i> are in proportion then <i>p</i>:<i>r</i> is the <i>duplicate ratio</i> of <i>p</i>:<i>q</i> and if <i>p</i>, <i>q</i>, <i>r</i> and <i>s</i> are in proportion then <i>p</i>:<i>s</i> is the <i>triplicate ratio</i> of <i>p</i>:<i>q</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Number_of_terms_and_use_of_fractions">Number of terms and use of fractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=4" title="Edit section: Number of terms and use of fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, a comparison of the quantities of a two-entity ratio can be expressed as a <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is <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 {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}" /></span> that of the second entity. </p><p>If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason. </p><p>Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare the quantities of any two of the entities covered by the ratio: for example, from a ratio of 2:3:7 we can infer that the quantity of the second entity is <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 {\tfrac {3}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f2b824decf224d9a3143d4271666c7fba7ac83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{7}}}" /></span> that of the third entity. </p> <div class="mw-heading mw-heading2"><h2 id="Proportions_and_percentage_ratios">Proportions and percentage ratios<span class="anchor" id="Proportions"></span><span class="anchor" id="Percentage_ratios"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=5" title="Edit section: Proportions and percentage ratios"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the <a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">lowest common denominator</a>, or to express them in parts per hundred (<a href="/wiki/Percent" class="mw-redirect" title="Percent">percent</a>). </p><p>If a mixture contains substances A, B, C and D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, we have converted to <a href="/wiki/Percentages" class="mw-redirect" title="Percentages">percentages</a>: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10). </p><p>If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, <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 {\tfrac {2}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb22be2c480d6bb96c97cc2b6a1a796f8396489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{5}}}" /></span>, or 40% of the whole is apples 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 {\tfrac {3}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f5356bc6141848e00fa630e4e1443f5d6fd2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3}{5}}}" /></span>, or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion. </p><p>If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older <a href="/wiki/Television" title="Television">televisions</a> have a 4:3 <i><a href="/wiki/Display_aspect_ratio" title="Display aspect ratio">aspect ratio</a></i>, which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparison works only when values being compared are consistent, like always expressing width in relation to height. </p> <div class="mw-heading mw-heading2"><h2 id="Reduction">Reduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=6" title="Edit section: Reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ratios can be <a href="/wiki/Reduction_(mathematics)" title="Reduction (mathematics)">reduced</a> (as fractions are) by dividing each quantity by the common factors of all the quantities. As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers. </p><p>Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3. The verbal equivalent is "40 is to 60 as 2 is to 3." </p><p>A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">simplest form</a> or lowest terms. </p><p>Sometimes it is useful to write a ratio in the form 1:<i>x</i> or <i>x</i>:1, where <i>x</i> is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4) Alternatively, it can be written as 0.8:1 (dividing both sides by 5). </p><p>Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it a <a href="/wiki/Divisor" title="Divisor">factor</a> or <a href="/wiki/Multiplication" title="Multiplication">multiplier</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Irrational_ratios">Irrational ratios</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=7" title="Edit section: Irrational ratios"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ratios may also be established between <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a> quantities (quantities whose ratio, as value of a fraction, amounts to an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>). The earliest discovered example, found by the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a>, is the ratio of the length of the diagonal <span class="texhtml mvar" style="font-style:italic;">d</span> to the length of a side <span class="texhtml mvar" style="font-style:italic;">s</span> of a <a href="/wiki/Square" title="Square">square</a>, which is the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, formally <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:d=1:{\sqrt {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:d=1:{\sqrt {2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c9c8b0787fffe4bb3a27294785bcc7548b36cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.326ex; height:3.009ex;" alt="{\displaystyle a:d=1:{\sqrt {2}}.}" /></span> Another example is the ratio of a <a href="/wiki/Circle" title="Circle">circle</a>'s circumference to its diameter, which is called <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>, and is not just an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, but a <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>. </p><p>Also well known is the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> of two (mostly) lengths <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>, which is defined by the proportion </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 a:b=(a+b):a\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>a</mi> <mspace width="1em"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:b=(a+b):a\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b46ef8cf589510b404c696927c27d59f56e5dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.63ex; height:2.843ex;" alt="{\displaystyle a:b=(a+b):a\quad }" /></span> or, equivalently <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 \quad a:b=(1+b/a):1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em"></mspace> <mi>a</mi> <mo>:</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad a:b=(1+b/a):1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f17c19eee996c2ff03cbbbe62bcff2071f3b3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.534ex; height:2.843ex;" alt="{\displaystyle \quad a:b=(1+b/a):1.}" /></span></dd></dl> <p>Taking the ratios as fractions 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 a:b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3149b4f815ad9e8b3e8cdd29adcd02a42c22e5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.165ex; height:2.176ex;" alt="{\displaystyle a:b}" /></span> as having the value <span class="texhtml mvar" style="font-style:italic;">x</span>, yields the equation </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 x=1+{\tfrac {1}{x}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> <mspace width="1em"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1+{\tfrac {1}{x}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed756207f6c598df128a9827b01860dfa995038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.53ex; height:3.343ex;" alt="{\displaystyle x=1+{\tfrac {1}{x}}\quad }" /></span> 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 \quad x^{2}-x-1=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em"></mspace> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad x^{2}-x-1=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5a7006359da449fd255ba18fcb5399f51316e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.787ex; height:3.009ex;" alt="{\displaystyle \quad x^{2}-x-1=0,}" /></span></dd></dl> <p>which has the positive, irrational solution <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={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594d6a70808bd6407eae75b9bfc96a9098dd8c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.007ex; height:4.343ex;" alt="{\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.}" /></span> Thus at least one of <i>a</i> and <i>b</i> has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a>: even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio. </p><p>Similarly, the <a href="/wiki/Silver_ratio" title="Silver ratio">silver ratio</a> of <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> is defined by the proportion </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 a:b=(2a+b):a\quad (=(2+b/a):1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>a</mi> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:b=(2a+b):a\quad (=(2+b/a):1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329f5e036d9784c487b86facd67babb955d06a67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.003ex; height:2.843ex;" alt="{\displaystyle a:b=(2a+b):a\quad (=(2+b/a):1),}" /></span> corresponding to <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^{2}-2x-1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-2x-1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366b8357e27139666cdd6f9bb5df8c14d6fa6bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.627ex; height:2.843ex;" alt="{\displaystyle x^{2}-2x-1=0.}" /></span></dd></dl> <p>This equation has the positive, irrational solution <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={\tfrac {a}{b}}=1+{\sqrt {2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e653692eafd97589466bdc8a8ce6e5a2680193d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.98ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},}" /></span> so again at least one of the two quantities <i>a</i> and <i>b</i> in the silver ratio must be irrational. </p> <div class="mw-heading mw-heading2"><h2 id="Odds">Odds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=8" title="Edit section: Odds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Odds" title="Odds">Odds</a></div> <p><i>Odds</i> (as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. The probability of success is 30%. In every ten trials, there are expected to be three wins and seven losses. </p> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=9" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ratios may be <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">unitless</a>, as in the case they relate quantities in units of the same <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a>, even if their <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> are initially different. For example, the ratio <span class="nowrap">one minute : 40 seconds</span> can be reduced by changing the first value to 60 seconds, so the ratio becomes <span class="nowrap">60 seconds : 40 seconds</span>. Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2. </p><p>On the other hand, there are non-dimensionless quotients, also known as <a href="/wiki/Rate_(mathematics)" title="Rate (mathematics)"><i>rates</i></a> (sometimes also as ratios).<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In chemistry, <a href="/wiki/Mass_concentration_(chemistry)" title="Mass concentration (chemistry)">mass concentration</a> ratios are usually expressed as weight/volume fractions. For example, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions. </p> <div class="mw-heading mw-heading2"><h2 id="Triangular_coordinates">Triangular coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=10" title="Edit section: Triangular coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The locations of points relative to a triangle with <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> <i>A</i>, <i>B</i>, and <i>C</i> and sides <i>AB</i>, <i>BC</i>, and <i>CA</i> are often expressed in extended ratio form as <i>triangular coordinates</i>. </p><p>In <a href="/wiki/Barycentric_coordinates_(mathematics)" class="mw-redirect" title="Barycentric coordinates (mathematics)">barycentric coordinates</a>, a point with coordinates <i>α, β, γ</i> is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at <i>A</i> and <i>B</i> being <i>α</i> : <i>β</i>, the ratio of the weights at <i>B</i> and <i>C</i> being <i>β</i> : <i>γ</i>, and therefore the ratio of weights at <i>A</i> and <i>C</i> being <i>α</i> : <i>γ</i>. </p><p>In <a href="/wiki/Trilinear_coordinates" title="Trilinear coordinates">trilinear coordinates</a>, a point with coordinates <i>x</i> :<i>y</i> :<i>z</i> has <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> distances to side <i>BC</i> (across from vertex <i>A</i>) and side <i>CA</i> (across from vertex <i>B</i>) in the ratio <i>x</i> :<i>y</i>, distances to side <i>CA</i> and side <i>AB</i> (across from <i>C</i>) in the ratio <i>y</i> :<i>z</i>, and therefore distances to sides <i>BC</i> and <i>AB</i> in the ratio <i>x</i> :<i>z</i>. </p><p>Since all information is expressed in terms of ratios (the individual numbers denoted by <i>α, β, γ, x, y,</i> and <i>z</i> have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle. </p> <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=Ratio&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cross_ratio" class="mw-redirect" title="Cross ratio">Cross ratio</a></li> <li><a href="/wiki/Dilution_ratio" title="Dilution ratio">Dilution ratio</a></li> <li><a href="/wiki/Displacement%E2%80%93length_ratio" title="Displacement–length ratio">Displacement–length ratio</a></li> <li><a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">Dimensionless quantity</a></li> <li><a href="/wiki/Financial_ratio" title="Financial ratio">Financial ratio</a></li> <li><a href="/wiki/Fold_change" title="Fold change">Fold change</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">Interval (music)</a></li> <li><a href="/wiki/Odds_ratio" title="Odds ratio">Odds ratio</a></li> <li><a href="/wiki/Parts-per_notation" title="Parts-per notation">Parts-per notation</a></li> <li><a href="/wiki/Price%E2%80%93performance_ratio" title="Price–performance ratio">Price–performance ratio</a></li> <li><a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">Proportionality (mathematics)</a></li> <li><a href="/wiki/Ratio_distribution" title="Ratio distribution">Ratio distribution</a></li> <li><a href="/wiki/Ratio_estimator" title="Ratio estimator">Ratio estimator</a></li> <li><a href="/wiki/Rate_(mathematics)" title="Rate (mathematics)">Rate (mathematics)</a></li> <li><a href="/wiki/Twitter_usage#Ratio" title="Twitter usage">Ratio (Twitter)</a></li> <li><a href="/wiki/Rate_ratio" title="Rate ratio">Rate ratio</a></li> <li><a href="/wiki/Relative_risk" title="Relative risk">Relative risk</a></li> <li><a href="/wiki/Rule_of_three_(mathematics)" class="mw-redirect" title="Rule of three (mathematics)">Rule of three (mathematics)</a></li> <li><a href="/wiki/Scale_(map)" title="Scale (map)">Scale (map)</a></li> <li><a href="/wiki/Scale_(ratio)" title="Scale (ratio)">Scale (ratio)</a></li> <li><a href="/wiki/Sex_ratio" title="Sex ratio">Sex ratio</a></li> <li><a href="/wiki/Superparticular_ratio" title="Superparticular ratio">Superparticular ratio</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></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=Ratio&action=edit&section=12" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">New International Encyclopedia</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/numbers/ratio.html">"Ratios"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Ratios&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fnumbers%2Fratio.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStapel" class="citation web cs1">Stapel, Elizabeth. <a rel="nofollow" class="external text" href="https://www.purplemath.com/modules/ratio.htm">"Ratios"</a>. <i>Purplemath</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Purplemath&rft.atitle=Ratios&rft.aulast=Stapel&rft.aufirst=Elizabeth&rft_id=https%3A%2F%2Fwww.purplemath.com%2Fmodules%2Fratio.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span></span> </li> <li id="cite_note-ISO_80000-1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-ISO_80000-1_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFiso.org" class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en">"ISO 80000-1:2022(en) Quantities and units — Part 1: General"</a>. <i>iso.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-07-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=iso.org&rft.atitle=ISO+80000-1%3A2022%28en%29+Quantities+and+units+%E2%80%94+Part+1%3A+General&rft_id=https%3A%2F%2Fwww.iso.org%2Fobp%2Fui%2F%23iso%3Astd%3Aiso%3A80000%3A-1%3Aed-2%3Av1%3Aen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" 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"><i>"The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)"</i>, "The Mathematics Dictionary" <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=UyIfgBIwLMQC&dq=dictionary+ratio&pg=PA349">[1]</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">New International Encyclopedia</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">Decimal fractions are frequently used in technological areas where ratio comparisons are important, such as aspect ratios (imaging), compression ratios (engines or data storage), etc.</span> </li> <li id="cite_note-MathWorld-colon-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-MathWorld-colon_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein2022" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> (2022-11-04). <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Colon.html">"Colon"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-26</span></span>.</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=Colon&rft.date=2022-11-04&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FColon.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span></span> </li> <li id="cite_note-Unicode-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Unicode_9-0">^</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.unicode.org/charts/PDF/U0000.pdf">"ASCII Punctuation"</a> <span class="cs1-format">(PDF)</span>. <i>The Unicode Standard, Version 15.0</i>. Unicode, Inc. 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-26</span></span>. <q>[003A is] also used to denote division or scale; for that mathematical use 2236 ∶ is preferred</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Unicode+Standard%2C+Version+15.0&rft.atitle=ASCII+Punctuation&rft.date=2022&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU0000.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.britannica.com/topic/ratio">from the Encyclopædia Britannica</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Heath, p. 126</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">New International Encyclopedia</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.bellegroup.com/es/support/mixingHints.html">Belle Group concrete mixing hints</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Penny Cyclopædia, p. 307</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Smith, p. 478</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Heath, p. 112</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Heath, p. 113</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Smith, p. 480</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Heath, reference for section</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">"Geometry, Euclidean" <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica Eleventh Edition</a></i> p682.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Heath p.114</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Heath p. 125</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavid_Ben-ChaimYaffa_KeretBat-Sheva_Ilany2012" class="citation book cs1">David Ben-Chaim; Yaffa Keret; Bat-Sheva Ilany (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eawKLY71xvkC&q=perspective&pg=PA25"><i>Ratio and Proportion: Research and Teaching in Mathematics Teachers</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789460917844" title="Special:BookSources/9789460917844"><bdi>9789460917844</bdi></a>. <q><span class="cs1-kern-left"></span>"Velocity" can be defined as the ratio... "Population density" is the ratio... "Gasoline consumption" is measure as the ratio...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ratio+and+Proportion%3A+Research+and+Teaching+in+Mathematics+Teachers&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=9789460917844&rft.au=David+Ben-Chaim&rft.au=Yaffa+Keret&rft.au=Bat-Sheva+Ilany&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeawKLY71xvkC%26q%3Dperspective%26pg%3DPA25&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><i>"</i>Ratio as a Rate<i>. The first type [of ratio] defined by <a href="/wiki/Freudenthal" title="Freudenthal">Freudenthal</a>, above, is known as rate, and illustrates a comparison between two variables with difference units. (...) A ratio of this sort produces a unique, new concept with its own entity, and this new concept is usually not considered a ratio, per se, but a rate or density."</i>, "Ratio and Proportion: Research and Teaching in Mathematics Teachers" <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=eawKLY71xvkC&q=rate&pg=PA29">[2]</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ratio&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZqsrAAAAYAAJ&pg=PA307">"Ratio" <i>The Penny Cyclopædia</i> vol. 19</a>, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff</li> <li><a rel="nofollow" class="external text" href="https://books.google.com/books?id=qgAoAAAAYAAJ&pg=PA270">"Proportion" <i>New International Encyclopedia, Vol. 19</i> 2nd ed. (1916) Dodd Mead & Co. pp270-271</a></li> <li><a rel="nofollow" class="external text" href="https://books.google.com/books?id=sqMXAAAAYAAJ&pg=PA55">"Ratio and Proportion" <i>Fundamentals of practical mathematics</i>, George Wentworth, David Eugene Smith, Herbert Druery Harper (1922) Ginn and Co. pp. 55ff</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_lxkPAAAAIAAJ"><i>The thirteen books of Euclid's Elements, vol 2</i></a>. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. 1908. pp. 112ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+thirteen+books+of+Euclid%27s+Elements%2C+vol+2&rft.pages=112ff&rft.pub=Cambridge+Univ.+Press&rft.date=1908&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_lxkPAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARatio" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</span></li> <li>D.E. Smith, <i>History of Mathematics, vol 2</i> Ginn and Company (1925) pp. 477ff. Reprinted 1958 by Dover Publications.</li></ul> <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=Ratio&action=edit&section=14" title="Edit section: External links"><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 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li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/120px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/ratio" class="extiw" title="wiktionary:Special:Search/ratio">ratio</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <div class="navbox-styles"><style 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.navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Fractions_and_ratios" title="Template:Fractions and ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fractions_and_ratios" title="Template talk:Fractions and ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fractions_and_ratios" title="Special:EditPage/Template:Fractions and ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fractions_and_ratios54" style="font-size:114%;margin:0 4em"><a href="/wiki/Fraction" title="Fraction">Fractions</a> and <a class="mw-selflink selflink">ratios</a></div></th></tr><tr><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 2px 0 0;min-width: 60px"><div><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Unicode_0x0025.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/50px-Unicode_0x0025.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/75px-Unicode_0x0025.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/100px-Unicode_0x0025.svg.png 2x" data-file-width="16" data-file-height="16" /></a></span></div></td><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Division and ratio</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Dividend</a> ÷ <a href="/wiki/Divisor" title="Divisor">Divisor</a> = <a href="/wiki/Quotient" title="Quotient">Quotient</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File"><span><img alt="The ratio of width to height of standard-definition television." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/120px-Aspect-ratio-4x3.svg.png" decoding="async" width="66" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/131px-Aspect-ratio-4x3.svg.png 2x" data-file-width="139" data-file-height="106" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Fraction</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span style="font-size:120%"><span class="sfrac">⁠<span class="tion"><span class="num">Numerator</span><span class="sr-only">/</span><span class="den">Denominator</span></span>⁠</span></span> = Quotient</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_fraction" title="Algebraic fraction">Algebraic</a></li> <li><a href="/wiki/Aspect_ratio" title="Aspect ratio">Aspect</a></li> <li><a href="/wiki/Binary_number" title="Binary number">Binary</a></li> <li><a href="/wiki/Continued_fraction" title="Continued fraction">Continued</a></li> <li><a href="/wiki/Decimal#Decimal_fractions" title="Decimal">Decimal</a></li> <li><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic</a></li> <li><a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden</a> <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver</a></li></ul></li> <li><a href="/wiki/Integer" title="Integer">Integer</a></li> <li><a href="/wiki/Irreducible_fraction" title="Irreducible fraction">Irreducible</a> <ul><li><a href="/wiki/Reduction_(mathematics)" title="Reduction (mathematics)">Reduction</a></li></ul></li> <li><a href="/wiki/Just_intonation" title="Just intonation">Just intonation</a></li> <li><a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">LCD</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">Musical interval</a></li> <li><a href="/wiki/Paper_size" title="Paper size">Paper size</a></li> <li><a href="/wiki/Percentage" title="Percentage">Percentage</a></li> <li><a href="/wiki/Unit_fraction" title="Unit fraction">Unit</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; 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