Solving Mixture Problems | Brilliant Math & Science Wiki

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For example, we might want to know how much water to add to dilute a saline solution, or we might want to determine the percentage of concentrate in a jug of orange juice. We can use fractions, ratios, or percentages to describe quantities in mixtures. A B C Reveal the answer If each can of …" > <meta property="og:type" content="website"> <meta property="og:title" content="Solving Mixture Problems | Brilliant Math & Science Wiki"> <meta property="og:description" content="Mixture problems involve combining two or more things and determining some characteristic of either the ingredients or the resulting mixture. For example, we might want to know how much water to add to dilute a saline solution, or we might want to determine the percentage of concentrate in a jug of orange juice. We can use fractions, ratios, or percentages to describe quantities in mixtures. 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href="https://brilliant.org/account/signup/?signup=true&next=/wiki/solving-mixture-problems/" class="btn signup-email ax-click" data-ax-id="clicked_signup_modal_email" data-ax-type="button"> Sign up manually</a> </div> <div class="alternative"> <p> Already have an account? <a href="https://brilliant.org/account/login/?next=/wiki/solving-mixture-problems/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link"> Log in here. </a> </p> </div> </div> </div> <div class="wiki-top-editors" id="cmp_wiki_top_editors_id"> <b>Janae Pritchett</b>, <b>Anton Kriksunov</b>, and <b>Pi Han</b> contributed </div> <div id="wiki-main" data-controller="app/newsfeed:feed"> <div class="summary-container" id="cmp_wiki_canonical_page_id"> <div class="summary wiki-content" data-controller="app/wiki:summary,app/zoomable:images" data-cmp-url="/wiki/solving-mixture-problems/" data-page-key="wiki_canonical_page" data-cmp-key="wiki_canonical_page"> <div class="section collapsed" id="section-pre-header-section"><div class="section-container"><p><strong>Mixture problems</strong> involve combining two or more things and determining some characteristic of either the ingredients or the resulting mixture. For example, we might want to know how much water to add to dilute a saline solution, or we might want to determine the percentage of concentrate in a jug of orange juice.</p> </div> </div> <div class="toc wiki-toc"> <h4>Contents</h4> <ul class="unstyled"> <li> <a href="#introduction-to-mixtures">Introduction to Mixtures</a> </li> <li> <a href="#using-a-table-to-problem-solve">Using a Table to Problem Solve</a> </li> <li> <a href="#practice-problems">Practice Problems</a> </li> </ul> </div> <div id="introduction-to-mixtures" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Introduction to Mixtures</h2> </header> </div> <div class="section collapsed" id="section-introduction-to-mixtures"><div class="section-container"> <p>We can use fractions, ratios, or percentages to describe quantities in mixtures. </p> <blockquote class="example"> <h3>If 200 grams of a saline solution has 40 grams of salt, what percentage of the solution is salt?</h3> <hr /> <p><div class="show-hide-btn-container" data-controller="util/ui:showAndHideContent"> <button class="btn ax-click" data-ax-type="button" data-ax-id="clicked_show_here_button"> Answer </button> </div> </p><blockquote class="hidden"> <p>\(\frac{40}{200} = 0.20 = 20\%\) of the solution is salt.</p> </blockquote><p> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-mixture-problems-1"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> A </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> B </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> C </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>If each can of orange juice concentrate contains the same amount of concentrate, which recipe will make the drink that is most orangey? <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/iDu8hFql1n-group-90.svg?width=300" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/iDu8hFql1n-group-90.svg?width=300 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/iDu8hFql1n-group-90.svg?width=600 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/iDu8hFql1n-group-90.svg?width=900 3x" alt="" style="width:300px;max-width:100%;" /> </span></p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>A</strong> <div> <div class="solv-nux-solution row"> <p>To determine which drink is most orangey, we need to figure out what percentage of each drink consists of concentrate. We can arrange the given data in a table as follows:</p> <div class="table-wrapper"> <table class="table"> <tbody> <tr><td>Amount of Concentrate</td><td>Amount of Water</td><td>Total Amount of Drink</td><td>% Concentrate</td></tr> <tr><td>2</td><td>3</td><td>5</td><td>\(\frac{2}{5} = 40\%\)</td></tr> <tr><td>3</td><td>5</td><td>8</td><td>\(\frac{3}{8} =37.5\%\)</td></tr> <tr><td>5</td><td>10</td><td>15</td><td>\(\frac{5}{15} \approx 33\%\)</td></tr> </tbody> </table> </div> <p>Since drink A contains the highest percentage of concentrate, it will be the most orangey.</p> </div> </div> </div> </div> </div> </div> </p> </div> </div> <div id="using-a-table-to-problem-solve" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Using a Table to Problem Solve</h2> </header> </div> <div class="section collapsed" id="section-using-a-table-to-problem-solve"><div class="section-container"> <p>There is a general strategy for solving these mixture problems that uses simple algebra organized with a chart.</p> <blockquote class="example"><p> </p> <p>How much 40% rubbing alcohol do we need to add to 90% rubbing alcohol to make a 50% solution of rubbing alcohol?</p> <hr /> <p>We could organize the data we are given in the following chart:</p> <div class="table-wrapper"> <table class="table"> <tbody> <tr><td>Solution Type</td><td>Concentration</td><td>Amount of Solution</td><td>Amount of Pure Alcohol</td></tr> <tr><td>40% rubbing alcohol</td><td>0.4</td><td>? liters</td><td>? liters</td></tr> <tr><td>90% rubbing alcohol</td><td>0.9</td><td>? liters</td><td>? liters</td></tr> <tr><td>50% rubbing alcohol</td><td>0.5</td><td>10 liters</td><td>\(0.5(10) = 5\) liters</td></tr> </tbody> </table> </div> <p>In general, the rows of the chart are the mixture types that you have. The columns describe the amount of each compound you have and the concentration of that component in each mixture (represented as a decimal). </p> <p><strong>When you don't have some of the information needed to fill in the cart, use a variable instead.</strong></p> <div class="table-wrapper"> <table class="table"> <tbody> <tr><td>Solution Type</td><td>Concentration</td><td>Amount of Solution</td><td>Amount of Pure Compound</td></tr> <tr><td>40% rubbing alcohol</td><td>0.4</td><td>\(x\)</td><td>\(0.4(x)\)</td></tr> <tr><td>90% rubbing alcohol</td><td>0.9</td><td>\(10-x\)</td><td>\(0.9(10-x)\)</td></tr> <tr><td>50% rubbing alcohol</td><td>0.5</td><td>10</td><td>\(0.5(10) = 5\)</td></tr> </tbody> </table> </div> <p>The amount of 40% solution that we'll need is unknown (so make it \(x\)). The amount of 90% solution that we'll need is also unknown, but must be \(10-x\) liters so that we'll, in total, make 10 liters of the final solution. </p> <p>The amount of alcohol that each part of the mixture adds to the final result is equal to the amount of each solution mixed in, times the fraction of alcohol that that solution is made from.</p> <p><strong>To use this chart to solve the problem, we will use the fourth column as an equation to solve for \(x.\)</strong> </p> <p>The 10 liters of our final mixture must have a total volume of 5 liters of alcohol in it in order to be 50% alcohol. Those 5 liters must come from a combination of the amount of 40% solution we mix in and the amount of 90% solution. If the volume (in liters) of 40% solution that we mix in is \(x,\) then \(0.4x\) will be the volume (in liters) of the amount of alcohol contributed. Similarly, \(0.9(10-x)\) will be the amount of alcohol contributed by the \(10-x\) liters of 90% alcohol solution that we add. Therefore, in total, \(0.4x + 0.9(10-x)\) must be equal to the 5 total liters we'll need in the final solution to make it 50% alcohol.</p> <p>Solving the equation, \[\begin{align} 0.4x + (9 - 0.9x) &= 5\\ -0.5x + 9 &= 5\\ 0.5x &= 4\\ \Rightarrow x &= \frac{4}{0.5}= 8. \end{align}\] Therefore, we need \(x = 8\) liters of the 40% solution.</p> </blockquote> <blockquote class="example"> <h3>How many grams of pure water must be added to 40 grams of a 10% saline solution to make a saline solution that is 5% salt?</h3> <hr /> <p><div class="show-hide-btn-container" data-controller="util/ui:showAndHideContent"> <button class="btn ax-click" data-ax-type="button" data-ax-id="clicked_show_here_button"> Answer </button> </div> <blockquote class="hidden"> Let's set up a table to solve this problem, using \(x\) to represent the number of grams of water that must be added. </p> <div class="table-wrapper"> <table class="table"> <tbody> <tr><td>Solution Type</td><td>Concentration</td><td>Amount of Solution</td><td>Total Amount of Salt</td></tr> <tr><td>water</td><td>0</td><td>\(x\)</td><td>0</td></tr> <tr><td>10% solution</td><td>0.1</td><td>40</td><td>\((0.1)(40)\)</td></tr> <tr><td>5% solution</td><td>0.05</td><td>40+\(x\)</td><td>\((0.05)(40+x)\)</td></tr> </tbody> </table> </div> <p>Because the total amount of salt remains the same after we add the water, we can set up and solve the following equation: \[\begin{align} (0.1)(40)&=(0.05)(40+x) \\ 4 &= 2 + 0.05x \\ x&=40.\end{align}\]</p> <p>We need to add \(40 \) grams of pure water to create the solution that is \(5\%\) salt.</p> </blockquote><p> </p></blockquote> <p><div class="problem-modal-container anchor" id="problem-a-lot-of-milk"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> More than 5 ml </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> Exactly 5 ml </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> Less than 5 ml </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>I have a 100 ml mixture that is 20% isopropyl alcohol (80 ml of water and 20 ml of isopropyl alcohol).</p> <p>How much more alcohol do I need to add to make the mixture 25% alcohol?</p> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/WnhRipWOXV-alcohol-problem.svg?width=400" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/WnhRipWOXV-alcohol-problem.svg?width=400 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/WnhRipWOXV-alcohol-problem.svg?width=800 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/WnhRipWOXV-alcohol-problem.svg?width=1200 3x" alt="" style="width:400px;max-width:100%;" /> </span></p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>More than 5 ml</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-mixture-water-and-milk"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>There is a 40 litre solution of milk and water in which the concentration of milk is 72%. How much water must be added to this solution to make it a solution in which the concentration of milk is 60% ?</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>8</strong> </div> </div> </div> </div> </p> </div> </div> <div id="practice-problems" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Practice Problems</h2> </header> </div> <div class="section collapsed" id="section-practice-problems"><div class="section-container"> <p>Let's practice using the strategies from above on a variety of problems. </p> <blockquote class="example"> <h3>Jill mixes 100 liters of A-beverage that contains 45% juice with 200 liters of B-beverage. The resulting C-beverage is 30% fruit juice. What is the percent of fruit juice in the 200 liters of the B-beverage?</h3> <p><div class="show-hide-btn-container" data-controller="util/ui:showAndHideContent"> <button class="btn ax-click" data-ax-type="button" data-ax-id="clicked_show_here_button"> Answer </button> </div> <blockquote class="hidden"> Let's begin by making a table to show what we know.</p> <div class="table-wrapper"> <table class="table"> <tbody> <tr><td>Beverage</td><td>Liters of Beverage</td><td>Concentration</td><td>Liters of Juice</td></tr> <tr><td>Beverage A</td><td>100</td><td>0.45</td><td>\((0.45)(100) = 45\)</td></tr> <tr><td>Beverage B</td><td>200</td><td>\(x\)</td><td>\(200x\)</td></tr> <tr><td>Beverage C</td><td>300</td><td>0.30</td><td>\((0.30)(300)=90\)</td></tr> </tbody> </table> </div> <p>The total number of liters of juice in Beverages A and B must equal 90, so \[\begin{align} 45 + 200x &= 90 \\ 200x &= 45 \\ 0.225 &= x.\end{align}\]</p> <p>The concentration of juice in Beverage B is 22.5%.</p> </blockquote> </blockquote> <p><div class="problem-modal-container anchor" id="problem-acid-mixture"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>Unequal amounts of 40% and 10% acid solutions were mixed and the resulting mixture was 30% concentrated. However, the required concentration is 25%, so the Chemist added 300 cubic meters of 20% acid solution in order to get the required concentration. What was the original amount of 40% acid solution?</p> <p>Write only the quantity without the units (cubic meters).</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>200</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-who-wants-strawberries"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 0.49 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 0.33 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 0.59 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 0.67 </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/59d81409c6.c0b51c46e4.q06ybR.jpg?width=250" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/59d81409c6.c0b51c46e4.q06ybR.jpg?width=250 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/59d81409c6.c0b51c46e4.q06ybR.jpg?width=500 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/59d81409c6.c0b51c46e4.q06ybR.jpg?width=750 3x" alt="" style="width:250px;max-width:100%;" /> </span> Legend: wt = weight</p> <p>Strawberries contain about 15 wt% solids and 85 wt% water. To make strawberry jam, crushed strawberries and sugar are mixed in a 45:55 mass ratio, and the mixture is heated to evaporate water until the residue contains one-third water by mass.</p> <p>Question: 1.) Calculate how many pounds of strawberries are needed to make a pound of jam.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>0.49</strong> </div> </div> </div> </div> </p> <p><div class="problem-modal-container anchor" id="problem-milk-water-ratio-problem"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <div class="solv-mcq-wrapper row"> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 40 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 20 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 60 </span> </span> <span class="btn-wiki-mcq " > <span class="bg"></span> <span> 30 </span> </span> </div> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>In a mixture of 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity of water to be further added is:</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>60</strong> </div> </div> </div> </div> </p> </div> </div> </div> </div> </div> <div class="wiki-self-citation" data-controller="app/wiki:getCitationTime"> <strong>Cite as:</strong> Solving Mixture Problems. <em>Brilliant.org</em>. 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