{"id":3983,"date":"2013-09-25T09:00:49","date_gmt":"2013-09-25T16:00:49","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=3983"},"modified":"2020-01-15T10:48:55","modified_gmt":"2020-01-15T18:48:55","slug":"multiple-solutions-in-gmat-math","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/multiple-solutions-in-gmat-math\/","title":{"rendered":"Multiple Solutions in GMAT Math"},"content":{"rendered":"<p>First, a couple questions to start:<\/p>\n<p>1) A large research project was scheduled over the course of a 28-month period, and a budget of $84,000 was allotted for the course of the entire project.\u00a0 After seven months, the project had spent $19,800.\u00a0 Was the project at this point over budget or under budget, and by how much?<\/p>\n<ul>\n\t(A) under, $4200<br \/>\n\t(B) under, $1200<br \/>\n\t(C) under, $700<br \/>\n\t(D) over $1800<br \/>\n\t(E) over $5800\n<\/ul>\n<p>2) A special municipal payroll tax charges not tax on a payroll less than $200,000 and only 0.1% on a company&#8217;s payroll above $200,000.\u00a0 If Belfried Industries paid $200 in this special municipal payroll tax, then they must have had a payroll of<\/p>\n<ul>\n\t(A) $180,000<br \/>\n\t(B) $202,000<br \/>\n\t(C) $220,000<br \/>\n\t(D) $400,000<br \/>\n\t(E) $2,200,000\n<\/ul>\n<p>Solutions will come at the end of this article.<\/p>\n<p>&nbsp;<\/p>\n<h2>Multiple perspectives in math<\/h2>\n<p>Sometimes GMAT students, especially folks who are not very confident in math, think about GMAT math problems in a very binary framework &#8212; they get the problem either right or wrong.\u00a0 If they get the problem right, they assume they know &#8220;the way&#8221; to do the problem, and if they get the problem wrong, they read the solutions to determine what &#8220;the way&#8221; was.<\/p>\n<p>This, it turns out, is a quite na\u00efve approach to math.\u00a0 Many times, especially on GMAT problems, there is not a single &#8220;right way&#8221; to do the problem, but a variety of approaches.\u00a0 Sometimes one, a non-standard approach, is more efficient than the standard &#8220;approved by your Algebra Two teacher&#8221; approach.\u00a0 Sometimes two or more solutions are equally efficient, or perhaps one more than the other depending on the proclivities of the test taker.<\/p>\n<p>Folks who want to improve their GMAT Quant performance often over-emphasize the importance of practice a sheer quantity of problems and often vastly underestimate the profound importance of learning to see multiple solutions to a single problem.\u00a0\u00a0 These students often underestimate the importance of studying carefully the solution to a problem they got right: it may be that the solution shows a considerably more efficient approach than they devised, and studying and learning this new method of solution may well save them crucial time in the future.<\/p>\n<p>It is very much a poverty mentality in math for the student to be caught in a view of &#8220;<i>can I do <span style=\"text-decoration: underline\">anything<\/span> that might lead to a solution<\/i>?&#8221;\u00a0 I understand some folks might feel this initially, especially if they are returning to math after a long absence.\u00a0\u00a0 Recognize, though, that somewhere between the beginner stage and master, one had to learn to adjust one&#8217;s perspective from that question to &#8220;<i>of all the possible approaches, what would be the best, the most efficient, for this problem<\/i>?&#8221;\u00a0\u00a0 Mathematical mastery is about thinking in terms of options among possible solutions, and more quickly one can adopt this sort of perspective, the faster one will move toward mastery.<\/p>\n<p>&nbsp;<\/p>\n<h2>Some options<\/h2>\n<p>The foregoing discussion, well raising an important point, was somewhat abstract.\u00a0 What are some examples of some alternative approaches, beyond the way that your Algebra Two teacher told you that you were supposed to solve the problem?\u00a0 Here are some thoughts.<\/p>\n<p>a. thinking in terms of <a href=\"https:\/\/magoosh.com\/gmat\/gmat-quantitative-ratio-and-proportions\/\">ratios &amp; proportions<\/a> often simplifies problems, and <a href=\"https:\/\/magoosh.com\/gmat\/gmat-math-similar-shapes\/\">scale factors<\/a> are a powerful proportional reasoning trick to know and understand<\/p>\n<p>b. sometimes <a href=\"https:\/\/magoosh.com\/gmat\/a-geometric-and-visual-approach-to-gmat-math\/\">a visual solution<\/a> is very helpful, engaging the skills of the right brain; this can be especially important if algebraic information about x &amp; y can be moved into the x-y plan<\/p>\n<p>c. instead of following equations in a very lock-step, left-brain way, one can reason directly with the data, especially with percents &#8212; well if 5% is this, that must me 10% is that, and therefore 30% would be three times that.\u00a0 Arguably, this is a variant of proportional reasoning, specific to percents.<\/p>\n<p>d.\u00a0 Remember that <a href=\"https:\/\/magoosh.com\/gmat\/gmat-plugging-in-strategy-always-start-with-answer-choice-c\/\">backsolving<\/a> is an important option when all five answers are numbers.<\/p>\n<p>e.\u00a0 Remember that, especially if the answers are spread widely, <a href=\"https:\/\/magoosh.com\/gmat\/the-power-of-estimation-for-gmat-quant\/\">estimation<\/a> might be a valid solution, and much faster than a direct &amp; detailed calculation.<\/p>\n<p>Those are some examples, but nothing replaces the value of reading carefully the solution to each and every math problem you solve &#8212; most especially the solutions of problems you get right.\u00a0 It may be that, say, for 80% of the problems you get right, you did exactly what was in the solution, and that&#8217;s fine, but for that other 20%, what you could learn about mathematical reasoning in those solutions could activate the next level of mathematical potential for you.<\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>In the solution section below, I will show a few different alternates for solving each version of the problem.\u00a0\u00a0 Here&#8217;s another question from inside Magoosh for practice:<\/p>\n<p>3) <a href=\"http:\/\/gmat.magoosh.com\/questions\/829\">http:\/\/gmat.magoosh.com\/questions\/829<\/a><\/p>\n<p>If you have anything you would like to share after reading this, please let us know in the comments section below.<\/p>\n<p>&nbsp;<\/p>\n<h2>Practice problems: multiple solutions<\/h2>\n<p>1) <span style=\"text-decoration: underline\">METHOD ONE<\/span>: The straightforward methodical Algebra Two approach (not terribly efficient)<\/p>\n<p>First, figure out the cost of each month.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3984\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img1.png\" alt=\"msigm_img1\" width=\"351\" height=\"59\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img1.png 351w, https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img1-300x50.png 300w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><\/a><\/p>\n<p>Now, multiply by seven months: 7*($3000) = $21,000.\u00a0 That would be the budgeted amount, the amount theoretically supplied to the project.\u00a0 The real amount spent is less than this, so it is <span style=\"text-decoration: underline\">under budget<\/span>.<\/p>\n<p>$21,000 &#8211; #19,800 = $1200<\/p>\n<p>Answer = <b>(B)<\/b><\/p>\n<p><span style=\"text-decoration: underline\">METHOD TWO<\/span>: a slightly more efficient approach using ratios.<\/p>\n<p>Notice that seven months is exactly 1\/4 of 28-months.\u00a0 Therefore, after seven months, they should have spent 1\/4 of the total budget.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3985\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img2.png\" alt=\"msigm_img2\" width=\"320\" height=\"50\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img2.png 320w, https:\/\/magoosh.com\/gmat\/files\/2013\/09\/msigm_img2-300x46.png 300w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/a><\/p>\n<p>Again, this budgeted amount is more than what was spent, so what was spent was <span style=\"text-decoration: underline\">under budget<\/span>.<\/p>\n<p>$21,000 &#8211; #19,800 = $1200<\/p>\n<p>Answer = <b>(B)<\/b><\/p>\n<p>2) <span style=\"text-decoration: underline\">METHOD ONE<\/span>: the step-by-step algebraic inefficiency method:<\/p>\n<p>payroll tax = 0.1% of (payroll income \u2013 200,000)<\/p>\n<p>Remember that 0.1% as a decimal is 0.001.<\/p>\n<p>payroll tax = 0.001*(payroll income \u2013 200,000)<\/p>\n<p>200 = 0.001*(x \u2013 200,000)<\/p>\n<p>200\/0.001 = 200,000 = x \u2013 200,000<\/p>\n<p>200,000 + 200,000 = 400,000 = x<\/p>\n<p>Answer = <b>(D)<\/b><\/p>\n<p>NB: this method requires one to know a couple tricky ideas about percents &amp; decimals.<\/p>\n<p><span style=\"text-decoration: underline\">METHOD TWO<\/span>: thinking about percents.<\/p>\n<p>Well, 0.1% of something is $200.\u00a0 Multiply this by ten &#8212; then 1% of something is $2000.\u00a0 Now, multiply this by 100 &#8212;- 100% of something is $200,000.\u00a0 That&#8217;s the amount over the $200,000 baseline, so the total payroll must have been the sum, $400,000.\u00a0 Answer = <b>(D)<\/b><\/p>\n<p><span style=\"text-decoration: underline\">METHOD THREE<\/span>: backsolving<\/p>\n<p>One would use this either if one didn&#8217;t know how to do either of the first two approaches, or simply if one thought this would be faster.<\/p>\n<p>Start with <b>(C)<\/b>.<\/p>\n<p>Suppose payroll = $220,000.\u00a0 The first $200,000 is tax free, so only $20,000 is taxed at 0.1%.\u00a0 Well, 1% of $20,000 &#8212; knock off two zeros: 1% would be $200.\u00a0 Divide by ten now: 0.1% would be $20.\u00a0 That&#8217;s too little.\u00a0 Payroll must be more, so we can immediately eliminate choice <b>(A)<\/b> &amp; <b>(B)<\/b> &amp; <b>(C)<\/b>.\u00a0 Even if we had no more time to work on this problem, we could employ <a href=\"https:\/\/magoosh.com\/gmat\/when-to-guess-on-the-gmat\/\">solution behavior<\/a> very effectively at this point.<\/p>\n<p>You could try another answer, <b>(D)<\/b> or <b>(E)<\/b>, or you could just notice &#8212; with <b>(C)<\/b>, the taxable part was <i>ten times too small<\/i>, because the payroll tax was ten times too small.\u00a0\u00a0 We need a taxable portion that is ten times bigger &#8212; instead of $20,000, we need a taxable portion of $200,000.\u00a0 Add this to the baseline = $400,000.\u00a0 Answer = <b>(D)<\/b>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>First, a couple questions to start: 1) A large research project was scheduled over the course of a 28-month period, and a budget of $84,000 was allotted for the course of the entire project.\u00a0 After seven months, the project had spent $19,800.\u00a0 Was the project at this point over budget or under budget, and by [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13209],"class_list":["post-3983","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) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Multiple Solutions in GMAT Math<\/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\/multiple-solutions-in-gmat-math\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" 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M\u1d9cGarry","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/15a1e36ef1c2c3940179212433de141a","url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","caption":"Mike M\u1d9cGarry"},"description":"Mike holds an A.B. in Physics (graduating magna cum laude) 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 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\/3983","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=3983"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3983\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=3983"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=3983"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=3983"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=3983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}