{"id":884,"date":"2012-04-02T12:15:47","date_gmt":"2012-04-02T19:15:47","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=884"},"modified":"2015-05-22T17:02:52","modified_gmt":"2015-05-23T00:02:52","slug":"gmat-og-case-study-problem-solving-practice-problem-163","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-og-case-study-problem-solving-practice-problem-163\/","title":{"rendered":"GMAT OG Case Study: Problem Solving Practice Problem #163"},"content":{"rendered":"<p>Here is the problem that appears in the OG, 12th Edition:<\/p>\n<h2>Problem Solving Practice Problem #163<\/h2>\n<p>This year Henry will save a certain amount of his income, and he will spend the rest.\u00a0 Next year Henry will have no income, but for each dollar he saves this year, he will have 1 + r dollars available to spend next year.\u00a0 In terms of r, what fraction of his income should Henry save this year so that next year the amount he has to spend will be equal to half the amount he spends this year?<\/p>\n<ul>\n\t(A) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_05cf80cb1198e433be7eca536de8c3b4.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"1\/(r + 2)\" title=\"1\/(r + 2)\"\/><br \/>\n\t(B) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_6f589605263cbd63094f7d0e07a7a767.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"\u00a01\/(2r + 2)\" title=\"\u00a01\/(2r + 2)\"\/><br \/>\n\t(C) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_a7ca3eeac7105d61c1357d1c3fe0937a.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"1\/(3r + 2)\" title=\"1\/(3r + 2)\"\/><br \/>\n\t(D) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_540abc4018d96038812268f54af654ee.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"1\/(r + 3)\" title=\"1\/(r + 3)\"\/><br \/>\n\t(E) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_38a78dbd3f4daae32d296003a8d10985.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"1\/(2r + 3)\" title=\"1\/(2r + 3)\"\/>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>This is a doozy of a question!\u00a0 It involves a complicated verbal scenario, thinking about proportions, and all the answer choices are in terms of variables.\u00a0 Holy schnikies!<\/p>\n<p>Notice that r is something like interest on an investment, and here, r is a decimal between 0 and 1.\u00a0 We know that because we got the multiplier (1 + r) simply by adding one to r.\u00a0 If r were, say, a percent between 0 and 100, we would have to divide r by 100 before adding one when we constructed the multiplier.<\/p>\n<p>I will show two completely different methods to solve this: the first, a more traditional algebraic approach, and the second, a numerical approach.<\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"text-decoration: underline;\">Method #1: Algebraic Solution<\/span><\/h2>\n<p>Let&#8217;s say that x is what the question is asking &#8212; x is the proportion of income that Henry saves this year.\u00a0 We will assume that x is a number between 0 and 1.<\/p>\n<p>We will also use the variable T for the dollar amount of the total income that Henry makes this year.\u00a0 We expect this variable to cancel out at some point.\u00a0 Given these two variables, what do we know?<\/p>\n<p>Well, the dollar that Henry saves<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_9a5a53b7fcec9eae10a36b1afd5007d9.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"saves = xT\" title=\"saves = xT\"\/><\/p>\n<p>Therefore, the dollar amount he spends this year is:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_cec132ad994961942391d77e809bdafe.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"spends this year = T - xT = T(1 - x)\" title=\"spends this year = T - xT = T(1 - x)\"\/><\/p>\n<p>If he saves xT this year, that means the dollar amount he will have available to spend next year is:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_bd9c4c3b12306ab206e177fa5adf8513.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"spends next year = xT(1 + r)\" title=\"spends next year = xT(1 + r)\"\/><\/p>\n<p>Well, the condition imposed by the last sentence in the question is:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_991_413f3c92a4723dfa38550e5068ff612f.png\" style=\"vertical-align:-9px; display: inline-block ;\" alt=\"(spends next year) = 0.5*(spends this year)\" title=\"(spends next year) = 0.5*(spends this year)\"\/><\/p>\n<p>Plug in algebraic expressions into this word equation:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_3b37f033d26032df12d7b05163f4e54c.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"xT(1 + r) = 0.5*T(1 - x)\" title=\"xT(1 + r) = 0.5*T(1 - x)\"\/><\/p>\n<p>Multiply by 2 to clear the fractions\/decimals:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_dbf4ab1c7da690a67cc95f8ed193665e.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"2xT(1 + r) = T(1 - x)\" title=\"2xT(1 + r) = T(1 - x)\"\/><\/p>\n<p>As promised, the T cancels.\u00a0 Divide by T.<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_0f31e6a4c95dbe586bd84831091eea8d.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"2x(1 + r) = (1 - x)\" title=\"2x(1 + r) = (1 - x)\"\/><\/p>\n<p>Now, we need to solve for x.\u00a0 Distribute on the left side.<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_e862aafcd22cccd324619d5808352621.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"2x + 2xr = 1 - x\" title=\"2x + 2xr = 1 - x\"\/><\/p>\n<p>Add x to both sides, so that all the x&#8217;s are on the left side of the equation:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_50302093702645bdd39e32582b6bb17f.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"3x + 2xr = 1\" title=\"3x + 2xr = 1\"\/><\/p>\n<p>Now, the death-defying Algebra One move: factor out a factor of x:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_5558f044033136f083fa1e884556aae1.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"x(3 + 2r) = 1\" title=\"x(3 + 2r) = 1\"\/><\/p>\n<p>To complete the process of isolating x, divide by the parentheses:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_979_2e6c4998138e177378f36b7b21e23003.png\" style=\"vertical-align:-21px; display: inline-block ;\" alt=\"x = 1\/(3 + 2r) = 1\/(2r + 3)\" title=\"x = 1\/(3 + 2r) = 1\/(2r + 3)\"\/><\/p>\n<p>This expression is equal to Answer Choice <strong>E<\/strong>, so this is the correct answer.<\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"text-decoration: underline;\">Method Two: Plugging in Clever Numbers<\/span><\/h2>\n<p>&nbsp;<\/p>\n<p>Let&#8217;s pick some ridiculous easy answers, and see if we can eliminate our way to the answer.<\/p>\n<p>First of all, let&#8217;s consider the extreme case where r = 0.\u00a0 Henry gets no interest at all: he might as well have buried the saved portion in an old shoe under the back porch.\u00a0 What he has to spend is exactly equal to what he saved.<\/p>\n<p>Well, let x bethe fraction of his income that he savesnow &amp; spends next year; then (1 \u2013 x) is the fraction he spends now.\u00a0 Givenwhat he saves &amp; spends next year is 1\/2 of what he spends this year, we have:<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_ab84632a6eb84ca1916d0faff3dbe7ad.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"0.5*(1 - x) = x\" title=\"0.5*(1 - x) = x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_0932773f2478197c773aca2c87cad039.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(1 - x) = 2x\" title=\"(1 - x) = 2x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_b61ba87fe091f99e3b853f8c1d36699b.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"1 = 3x\" title=\"1 = 3x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_ec80a1ce262cb11c65a8253ff5a80ae3.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"1\/3 = x\" title=\"1\/3 = x\"\/><\/p>\n<p>&nbsp;<\/p>\n<p>Therefore, he must spend 2\/3 of his income and save 1\/3 of his income.\u00a0 Thus, when r = 0, the ratio Henry has to save is 1\/3.<\/p>\n<p>Well, if we plug zero into the answer choices, how many of them equal 1\/3?\u00a0 (A) &amp; (B) &amp; (C) do not, so we can immediately eliminate them.\u00a0 One easy choice, and three potential answers are gone, leaving only (D) &amp; (E) as viable options.<\/p>\n<p>Now, an equally extreme answer: assume that r = 1.\u00a0 Whatever Henry saves is doubled in one year (a pretty sweet deal!\u00a0 I&#8217;d love to find <em>that<\/em> bank!)\u00a0 Let x equal the fraction of his income that he saves this year.\u00a0 This means he spends (1 \u2013 x) this year.\u00a0 It also means, his savings doubles, so next year he spends 2x.\u00a0 Again, what he spends next year is 1\/2 of what he spends this year<\/p>\n<p>&nbsp;<\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_72838a861e40bd8deb37c9ef5bb610c1.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"0.5*(1 - x) = 2x\" title=\"0.5*(1 - x) = 2x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_d77abfe6431543cf5620d8ccdf1fce7b.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"1-x = 4x\" title=\"1-x = 4x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_8a28fee55bb44e38a4b6ac1dae21b636.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"1 = 5x\" title=\"1 = 5x\"\/><\/p>\n<p align=\"center\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_ebb416e7592ecacc45dfe5bc0802cc7f.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"1\/5 = x\" title=\"1\/5 = x\"\/><\/p>\n<p>&nbsp;<\/p>\n<p>Therefore, he must save 1\/5 of his income and spend 4\/5 of it.\u00a0 If r = 1, the ratio Henry should save is 1\/5.<\/p>\n<p>Plug r = 1 into remaining choices (D) &amp; (E).\u00a0 (D) doesn&#8217;t work.\u00a0 (E) works.\u00a0 Bingo.\u00a0 We&#8217;re done, and we know the answer must be (E).<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is the problem that appears in the OG, 12th Edition: Problem Solving Practice Problem #163 This year Henry will save a certain amount of his income, and he will spend the rest.\u00a0 Next year Henry will have no income, but for each dollar he saves this year, he will have 1 + r dollars [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13209],"class_list":["post-884","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - 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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. 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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\/884","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=884"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/884\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=884"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=884"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=884"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=884"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}