Lesson Draining-replacing mixture problems
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<H2>Draining-replacing mixture problems</H2> <H3>Problem 1</H3>The car radiator holds 20 liters. It contains a 25% antifreeze solution. Since the weather is getting colder, the mechanic should make the solution 40% antifreeze. How much of the 25% solution should be drained and replaced with pure antifreeze to give a 40% solution. <B>Solution</B> <pre> Let V be the volume to drain off from 20 liters of antifreeze. <U>Step 1: Draining</U>. After draining, you have 20-V liters of the 25% antifreeze. It contains 0.25*(20-V) of pure antifreeze. <U>Step 2: Replacing</U>. Then you add V liters of the pure antifreeze (the replacing step). After the replacing, you have the same total liquid volume of 20 liters. It contains 0.25(20-V) + V liters of pure antifreeze. So, the antifreeze concentration after replacement is {{{(0.25*(20-V)+V)/20}}}. It is the ratio of the pure antifreeze volume to the total volume. Therefore, your "concentration equation" is {{{(0.25*(20-V)+V)/20}}} = 0.4. (1) The setup is done and completed. To solve the equation (1), multiply both sides by 20. You will get 0.25*(20-V) + V= 0.4*20, 5 - 0.25V + V= 8, 0.75V = 8 - 5 = 3 ====> V = {{{3/0.75}}} = 4 liters. <U>Answer</U>. 4 liters of the 25% antifreeze must be drained and replaced by 4 liters of pure antifreeze. <U>Check</U>. {{{(0.25*(20-4)+4)/20}}} = 0.4. ! Correct ! </pre> <H3>Problem 2</H3>A tank holds 80 liters of a chemical solution. Currently, the solution has a strength of 30%. How much of this solution must be drained and replaced with a 70% solution to get a strength of 40%? <B>Solution</B> <pre> Let W be the volume to drain off from 80 liters of solution. <U>Step 1: Draining</U>. After draining, you have (80-W) liters of the 30% acid solution. <U>Step 2: Replacing</U>. Then you add W liters of the 70% solution (the replacing step). After the replacing, you have the same total liquid volume of 80 liters. It contains 0.3*(80-W) + 0.7*W of pure solvent. So, your "concentration equation" is {{{(0.30*(80-W) + 0.7*W)/80}}} = 0.4. (1) At this point, the setup is done and completed. Now you need to solve your basic equation (1). As the first step, multiply both sides by 80, and then simplify 0.30*(80-W) + 0.7*W = 0.4*80 24 - 0.3W + 0.7W = 32 0.4W = 32 - 24 = 8 W = {{{8/0.4}}} = 20. <U>ANSWER</U>. 20 liters of the original 30% solution should be drained and replaced by 20 liters of the 70% solutions. <U>CHECK</U>. {{{(0.3*(80-20) + 0.7*20)/80}}} = 0.4. ! Precisely correct ! </pre> <H3>Problem 3</H3>A 2.5 liters container has a mixture of 25% alcohol. How many liters of the mixture must be drained out and replaced with pure alcohol in order to obtain a mixture containing 40% alcohol? <B>Solution</B> <pre> Let W be the volume to drain off from 2.5 liters of the original mixture. <U>Step 1: Draining</U>. After draining, you have 5-W liters of the 25% mixture. <U>Step 2: Replacing</U>. Then you add W liters of the pure alcohol (the replacing step). After the replacing, you have the same total liquid volume of 2.5 liters. It contains 0.25(2.5-W) + W of the pure alcohol. So, your "concentration equation" is {{{(0.25*(2.5-W)+W)/2.5}}} = 0.4. (1) The setup is done and completed. Now you need to solve your basic equation (1). 0.25*(2.5-W) + W = 0.4*2.5 0.25*2.5 - 0.25W + W = 0.4*2.5 0.75W = 1 - 0.625 W = {{{(1-0.625)/0.75}}} = 0.5. <U>Answer</U>. 0.5 of a liter of the original mixture should be drained and replaced with the pure alcohol. </pre> My other lessons on word problems for mixtures in this site are - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Mixture-problems.lesson>Mixture problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/More-Mixture-problems.lesson>More Mixture problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Solving-typical-mixture-problems.lesson>Solving typical word problems on mixtures for solutions</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-antifreeze-solutions.lesson>Word problems on mixtures for antifreeze solutions</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Solving-typical-word-problems-on-mixtures-for-dry-substances-coffee-nuts.lesson>Word problems on mixtures for dry substances like coffee beans, nuts, cashew and peanuts</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-dry-substances-like-cookies-candies.lesson>Word problems on mixtures for dry substances like candies, dried fruits</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-dry-substances-like-soil-and-sand.lesson>Word problems on mixtures for dry substances like soil and sand</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-alloys.lesson>Word problems on mixtures for alloys</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Typical-word-problems-on-mixtures-from-the-archive.lesson>Typical word problems on mixtures from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Advanced-mixture-problems.lesson>Advanced mixture problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Advanced-mixture-problem-for-three-alloys.lesson>Advanced mixture problem for three alloys</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Unusual-word-problem-on-mixtures.lesson>Unusual word problem on mixtures</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Check-if-you-know-the-basics-of-mixtures-from-Science.lesson>Check if you know the basics of mixtures from Science</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Special-type-mixture-problems-on-DILUTION-adding-water.lesson>Special type mixture problems on DILUTION adding water</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Increasing-concentration-of-an-acid-solution-by-adding-pure-acid.lesson>Increasing concentration of an acid solution by adding pure acid</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Increasing-concentration-of-alcohol-in-solution-by-adding-pure-alcohol.lesson>Increasing concentration of alcohol solution by adding pure alcohol</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/How-many-kilograms-of-sand-must-be-added-to-a-mixture-of-sand-and-cement.lesson>How many kilograms of sand must be added to a mixture of sand and cement</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/How-much-water-should-be-evaporated.lesson>How much water must be evaporated</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Advanced-problems-on-draining-and-replacing.lesson>Advanced problems on draining and replacing</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Using-effective-methodology-to-solve-many-steps-dilution-problems.lesson>Using effective methodology to solve many-steps dilution problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Entertainment-problems-on-mixtures.lesson>Entertainment problems on mixtures</A> - <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/OVERVIEW-of-lesson-on-word-problems-for-mixtures.lesson>OVERVIEW of lessons on word problems for mixtures</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
