{"id":2790,"date":"2012-09-27T09:00:01","date_gmt":"2012-09-27T16:00:01","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2790"},"modified":"2020-01-15T10:50:27","modified_gmt":"2020-01-15T18:50:27","slug":"gmat-solution-and-mixing-problems","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-solution-and-mixing-problems\/","title":{"rendered":"GMAT Solution and Mixing Problems"},"content":{"rendered":"<p>Attention, mad scientists out there!\u00a0 Consider these two practice questions.<\/p>\n<p>&nbsp;<\/p>\n<p>1) A scientist has 400 units of a 6% phosphoric acid solution, and an unlimited supply of 12% phosphoric acid solution.\u00a0 How many units of the latter must she add to the former to produce a 10% phosphoric acid solution?<\/p>\n<p>(A) 200<\/p>\n<p>(B) 400<\/p>\n<p>(C) 500<\/p>\n<p>(D) 600<\/p>\n<p>(E) 800<\/p>\n<p>&nbsp;<\/p>\n<p>2) A chef mixes P ounces of 60% sugar solution with Q ounces of a 10% sugar solution to produce R ounces of a 25% sugar solution.\u00a0 What is the value of P?<\/p>\n<p>[Statement #1] Q = 455 mL<\/p>\n<p>[Statement #2] R = 660 mL<\/p>\n<p>&nbsp;<\/p>\n<h2>Mixtures, solutions, and concentration<\/h2>\n<p>First of all, keep in mind: this is relatively rare kind of GMAT Math problem.\u00a0 There&#8217;s a good chance you could take five GMATs in row and never see a mixture problem.\u00a0 Nevertheless, they do arise occasionally, and it&#8217;s good to be familiar with them.<\/p>\n<p>Let&#8217;s think for a moment about &#8220;concentration&#8221; &#8212; not the mental quality, but in the chemical solution sense.\u00a0 What does it mean to say we have a &#8220;400 units of a 6% phosphoric acid solution&#8221;?\u00a0 Whatever those units are, the total amount of mixture, of solution, we have is 400 units, and of that, 6% is pure phosphoric acid.\u00a0 Well, 6% of 400 is 24, so we know we have 24 units of pure phosphoric acid.\u00a0 That is the amount of concentrate we have in our solution.\u00a0 (Those &#8220;units&#8221; could be units of volume or units of mass, but that&#8217;s far more detail that you need to know on the GMAT).<\/p>\n<p>I bet &#8220;<a href=\"http:\/\/en.wikipedia.org\/wiki\/Phosphoric_acid\" target=\"_blank\" rel=\"noopener noreferrer\">phosphoric acid<\/a>&#8221; sounds highly technical and abstruse: it&#8217;s actually one of the ingredients listed on every can of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Coca-Cola_formula\" target=\"_blank\" rel=\"noopener noreferrer\">Coca Cola<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<h2>The two equations<\/h2>\n<p>The secret to any mixture or concentration problem is to use the two equations.\u00a0 First is the amount of stuff, the total volume.\u00a0 This is called the volume equation.\u00a0 The basic idea is:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2791\" title=\"samp_img1\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img1.png\" alt=\"\" width=\"369\" height=\"82\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img1.png 369w, https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img1-300x66.png 300w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><\/a><\/p>\n<p>This makes sense when you think about it: the volume of the resultant solution had to come from the volumes of the two things we mixed.<\/p>\n<p>The second equation is similar.\u00a0 This concerns, specifically, the amount of concentrate, of whatever the chemical or substances is of which we have a solution.\u00a0 In problem #1, the concentrate is phosphoric acid, and in problem #2, sugar.\u00a0\u00a0 The amount of concentrate that winds up in the resultant solution must come from somewhere.\u00a0 It must come from the amount of concentrate in the two solutions mixed.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2792\" title=\"samp_img2\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img2.png\" alt=\"\" width=\"503\" height=\"83\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img2.png 503w, https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img2-300x49.png 300w\" sizes=\"(max-width: 503px) 100vw, 503px\" \/><\/a><\/p>\n<p>As in the section above, these amounts of concentrate will always be (the concentration percentage) <span style=\"text-decoration: underline\">times<\/span> (the total volume of the solution concerned).<\/p>\n<p>If you had trouble with the problems above before, go back to them, and see if you can set up both of these equations and solve.\u00a0 Give them another try before reading the explanations below.<\/p>\n<p>Also, here&#8217;s another practice question:<\/p>\n<p>3) <a href=\"http:\/\/gmat.magoosh.com\/questions\/29\">http:\/\/gmat.magoosh.com\/questions\/29<\/a><\/p>\n<p>&nbsp;<\/p>\n<h2>Practice problems explanations<\/h2>\n<p>1) We could backsolve from the numerical answer choices, but let&#8217;s use a straight algebra approach.\u00a0 Let X equal the units of 12% phosphoric acid solution we use, and let Y be the units of 10% sulfuric acid solution that result.<\/p>\n<p>The <strong>volume equation<\/strong> is:<\/p>\n<p>400 + X = Y<\/p>\n<p>In the first solution, we have 6% of 400, or 24 units of phosphoric acid.<\/p>\n<p>In the second solution, we have 12% of X = 0.12*X of phosphoric acid.<\/p>\n<p>In the resultant solution, we have 10% of Y = 0.10*Y of phosphoric acid.<\/p>\n<p>The <strong>concentration equation<\/strong> is:<\/p>\n<p>24 + 0.12*X = 0.10*Y<\/p>\n<p>Multiply this by 100, to clear the decimals:<\/p>\n<p>2400 + 12X = 10Y<\/p>\n<p>Everything is even, so divide by 2 to simplify:<\/p>\n<p>1200 + 6X = 5Y<\/p>\n<p>We want X, so let&#8217;s multiply the volume equation by -5 and add that to this equation we just got:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img3.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2793\" title=\"samp_img3\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/09\/samp_img3.png\" alt=\"\" width=\"167\" height=\"112\" \/><\/a><\/p>\n<p>Answer = <strong>E<\/strong><\/p>\n<p>2) This one is very elegant.\u00a0 We\u00a0 have three variables &#8212; the amount of 60% sugar solution, the amount of 15% sugar solution, and the amount of the resultant 25% sugar solution.\u00a0 Three variables.\u00a0 We have two equations: the volume equation and the concentration equation.\u00a0 Right now, three variables and two equations: we can&#8217;t solve.<\/p>\n<p>Now, look at the statements.\u00a0 Each statement gives us the value of one of the variables.\u00a0\u00a0 If we get the value of one variable, that&#8217;s no longer a variable, and thus we are down to <a href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-solve-two-equations-with-two-variables\/\">two variables with two equations<\/a>: that&#8217;s a situation in which we can find a full solution.\u00a0 Thus, given the value of either Q or R, we enter a situation in which we can solve for everything, and thus we would know P.\u00a0 Therefore, each statement, by itself, is sufficient.\u00a0 Answer = <strong>D<\/strong><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Attention, mad scientists out there!\u00a0 Consider these two practice questions. &nbsp; 1) A scientist has 400 units of a 6% phosphoric acid solution, and an unlimited supply of 12% phosphoric acid solution.\u00a0 How many units of the latter must she add to the former to produce a 10% phosphoric acid solution? (A) 200 (B) 400 [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[193],"tags":[],"ppma_author":[13209],"class_list":["post-2790","post","type-post","status-publish","format-standard","hentry","category-word-problems"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Solution and Mixing Problems - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/gmat-solution-and-mixing-problems\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Solution and Mixing Problems\" \/>\n<meta property=\"og:description\" content=\"Attention, mad scientists out there!\u00a0 Consider these two practice questions. &nbsp; 1) A scientist has 400 units of a 6% phosphoric acid solution, and an unlimited supply of 12% phosphoric acid solution.\u00a0 How many units of the latter must she add to the former to produce a 10% phosphoric acid solution? 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=2790"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2790\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=2790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=2790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=2790"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=2790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}