{"id":5389,"date":"2014-12-29T09:00:02","date_gmt":"2014-12-29T17:00:02","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=5389"},"modified":"2020-01-15T10:48:23","modified_gmt":"2020-01-15T18:48:23","slug":"gmat-practice-problems-variables-in-the-answer-choices","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/","title":{"rendered":"GMAT Practice Problems: Variables in the Answer Choices"},"content":{"rendered":"

Variables in the Answer Choices<\/h2>\n

The GMAT loves these question, not least because so many real-world situations something of this feel: we have a few changing variables (number of items sold, or stock price, or commodity price, etc.) and we want to figure out something else that depends on these changing quantities.<\/p>\n

There are two basic approaches: an algebraic approach vs. a numerical approach<\/a>.\u00a0 Each one has advantages and disadvantages.\u00a0 Ultimately, you should practice both approaches on a number of these problems, to get a sense of what comes most naturally for you.<\/p>\n

If you know how to do the algebra, it is efficient and it always leads to an unambiguous answer choice.\u00a0 If you can do the algebra, that’s great, but the disadvantage is that it’s sometimes hard to know how to perform this approach.<\/p>\n

Picking numbers is inefficient but easier.\u00a0\u00a0 It important to know that there is an art to picking numbers<\/a>.\u00a0\u00a0 Of course, if I pick a set of numbers for the variables in the prompt, and an answer choice doesn’t produce the desired answer, I can eliminate it, but if these numbers do produce the desired answer, I can draw no conclusion from that.\u00a0 Picking numbers is, de facto<\/em>, a process of elimination approach.\u00a0\u00a0 We have to pick numbers until four answer choices have been clearly eliminated.\u00a0 Obviously, the test loves to design questions so that the most obvious choices for numbers eliminate few or no answers.\u00a0 Each round of picking numbers is simple and relatively quick, but when you begin, you have no idea how many rounds it will take before four answer choices have been eliminated.\u00a0 That’s the inefficient part.<\/p>\n

In the problems below, I will demonstrate both approaches for each problem, and in demonstrating the numerical approach, I will discuss in context some points of strategy for picking numbers.<\/p>\n

1) Jennifer can buy watches at a price of B dollars per watch, which she marks up by a certain percentage before selling. If she makes a total profit of T by selling N watches, then in terms of B and T and N, what is the percent of the markup from her buy price to her sell price?<\/p>\n

    \n\t(A) 100T\/(NB)
    \n\t(B) TB\/(100N)
    \n\t(C) 100TN\/B
    \n\t(D) ((T\/N) \u2013 B)\/(100B)
    \n\t(E) 100(T \u2013 NB)\/N\n<\/ol>\n

    2) At the store, Sam bought a shirt and a toaster.\u00a0 There was an 8% sales tax on each item, and with tax, Sam paid a total of K.\u00a0 If the price of the toaster before tax was T, what, in terms of K and T, is the price of the shirt before tax?<\/p>\n

      \n\t(A) 0.92(K \u2013 T)
      \n\t(B) 0.92K \u2013 T
      \n\t(C) 0.92(K \u2013 1.08T)
      \n\t(D) (K \u2013 T) \/1.08
      \n\t(E) (K\/1.08) \u2013 T\n<\/ol>\n

      3) Before January, the price of a dress was D and the price of a matching pair of shoes was H.\u00a0 In January, the price of the dress increased by 40% and the price of the shoes increased by 50%.\u00a0 In March, Roberta bought both items with a 30% discount.\u00a0 If D = 5H, which of the following represents the amount that Roberta paid?<\/p>\n

        \n\t(A) D + 40
        \n\t(B) D + H \u2013 1
        \n\t(C) D + 2H
        \n\t(D) 5.95H
        \n\t(E) 1.21D\n<\/ol>\n

        4) A marketer bought N crates of empty cardboard gift boxes.\u00a0 Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars.\u00a0 This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection.\u00a0 (Note: J is a divisor of Q.)\u00a0 The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction.\u00a0 What price P must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z.<\/p>\n

        \"gpp-vitac_img1\"<\/a><\/p>\n

        Solutions are given below.<\/p>\n

         <\/p>\n

        \"gpp-vitac_img6\"<\/a><\/p>\n

         <\/p>\n

        Practice problem explanations<\/h2>\n

        As you read these explanations, pay attention to which solution seems more natural and more doable for you.<\/p>\n

        1) Algebraic Solution<\/strong>: If she makes a total profit of T for N watches, then that must be a profit of T\/N for each watch.\u00a0 That must be the markup above cost on each watch, the amount of the increase.\u00a0 Well, percent increase = (amount of increase)\/(starting amount) x 100% = (T\/N)\/B *100 = 100T\/(NB)<\/p>\n

        Answer = (A)<\/strong><\/p>\n

        That Algebraic solution was elegant if you saw it, but if not, here’s a full solution with picking numbers.<\/p>\n

        Numerical Solution<\/strong>: The price of each watch, I will pick B = 7, a prime number.\u00a0\u00a0 For the percent increase, I will make this easy.\u00a0 Picking 100% is too easy, and too predictable, so I am going to pick 400% — 400% of 7 is 28, and the $7 watches are marked up $25, then the selling price is $35.\u00a0 The profit per watch is $28, and so if she sells N = 11 watches (another prime number), that would be a profit of T = 28*11 = $308.\u00a0 Leaving T in unmultiplied form will make it easier to cancel.<\/p>\n

        OK, now we will plug in T = 308 = 11*28, N = 11, and B = 7, and hope to get 400 as our answer.<\/p>\n

        (A) 100T\/(NB) = 100*11*28\/(11*4) = 100*28\/7 = 110*4 = 400 = works!<\/p>\n

        (B) TB\/(100N) = 11*28*7\/(100*11) = 28*7\/100 = doesn’t work!<\/p>\n

        (C) 100TN\/B = 100*11*28*11\/7 = 100*11*4*11 = doesn’t work!<\/p>\n

        (D) ((T\/N) \u2013 B)\/(100B) = [(11*28\/11) \u2013 7]\/(7*100) = (28 \u2013 7)\/(7*100) = 21\/(7*100) = 3\/100 = doesn’t work<\/p>\n

        (E) 100(T \u2013 NB)\/N = 100(11*28 \u2013 11*7)\/7 = 100*11*21\/7 = 100*11*3 = doesn’t work!<\/p>\n

        We were lucky here.\u00a0 With one choice of numbers, we were able to eliminate four answer choices, leaving (A)<\/strong> as the only possible answer.<\/p>\n

        2) Algebraic Solution<\/strong>: Call the price of the shirt S.\u00a0 The price of the toaster is T.\u00a0 The total before tax is (S + T).\u00a0 Then an 8% tax was applied.\u00a0 For an 8% increase, we use the multiplier <\/a>1.08.\u00a0 Thus, K = 1.08*(S + T). Solve this for S.<\/p>\n

        S + T = K\/1.08<\/p>\n

        S = K\/1.08 \u2013 T<\/p>\n

        Answer = (E)<\/strong><\/p>\n

        Numerical Solution<\/strong>: we make our lives much easier if we first recognize the percent trap.\u00a0 The multiplier 1.08 is the correct multiplier for an 8% increase, and it makes sense that this would appear somehow in the correct answer.\u00a0\u00a0 BUT, 0.92 is the multiplier for an 8% decrease.\u00a0 This is tricky: an 8% decrease is not the opposite of an 8% increase; it does not “undo” an 8% increase.\u00a0 In other words, an 8% increase followed by an 8% decrease does NOT get us back to the original starting point.\u00a0 That’s a major percent trap.\u00a0 There is a 8% increase in the problem, but no 8% decrease: this means that any answer in which the multiplier for an 8% decrease, 0.92, appears, it is automatically wrong.\u00a0\u00a0 We can immediately eliminate (A)<\/strong> & (B)<\/strong> & (C)<\/strong> without plugging in anything.<\/p>\n

        OK, let’s say T = 50, shirt = 150, so total before tax is $200. (Again, avoiding the clich\u00e9 price of $100 for a percent.)\u00a0 Now, 8% of $200 is $16, for a total of K = 216.\u00a0 Because dividing by a decimal isn’t fun, I am going to change the division by 1.08 into multiplication by the fraction 100\/108.\u00a0 We are looking for an answer that equals 150, the price of the shirt.<\/p>\n

        (D) (K \u2013 T) \/1.08 = (216 \u2013 50)\/1.08 = 156*100\/108 = 39*100\/27 = 13*100\/9<\/p>\n

        well, 13\/9 doesn’t equal 1.5, so this doesn’t work.<\/p>\n

        (E) (K\/1.08) \u2013 T = 216\/1.08 \u2013 50 = 216*(100\/108) \u2013 50 = 2*108*(100\/108) \u2013 50<\/p>\n

        = 2*100 \u2013 50 = 200 \u2013 50 = 150<\/p>\n

        That’s the price of the shirt!\u00a0 Choice (E)<\/strong> is the answer!<\/p>\n

        3) Algebraic Solution<\/strong>: dress increase to 1.4D, and the shoes increase to 1.5H.\u00a0 The multiplier <\/a>for a 30% decrease is 0.7, so the total amount could be written as 0.7*(1.4D + 1.5H).\u00a0 This does not appear as an answer.\u00a0 Since H is smaller, write D in terms of H: D = 5H.\u00a0 Then the total is 0.7*(1.4(5H) + 1.5H) = 0.7*(7H + 1.5H) = 4.9H + 1.05H = 5.95H. \u00a0That’s answer (D)<\/strong>.<\/p>\n

        Numerical Solution<\/strong>: What do you think every single drooling GMAT test taker is going to pick for the price of the dress?\u00a0 You guessed it: $100 dress and $20 shoes.\u00a0 It’s a spectacularly bad idea to make that choice.\u00a0 As will happen on the real GMAT, I was anticipating this choice, and composed trap answers that would work for this test.\u00a0 If you plug this in, I guarantee you will not be able to eliminate four answers at once.<\/p>\n

        Think about making a choice that is still relatively easy, but not everyone would make it.<\/p>\n

        Make the dress D = 300, so H = 60.\u00a0 D increased by 40% — 10% of 300 is 30, so 40% must be 4*30 = 120, and the new dress price is $420.\u00a0 The shoes increase by 50%, to $90.\u00a0 Together, after the increase, dress + shoes would cost 420 + 90 = $510.\u00a0 Now, Roberta gets a 30% decrease on this price.\u00a0 Well, 10% of 510 is 51, so 30% would be 3*51 = 153.\u00a0 510 \u2013 153 = 357.\u00a0 The correct answer must equal 357.<\/p>\n

        (A) D + 40 = 340 = not correct<\/p>\n

        (B) D + H \u2013 1 = 300 + 60 \u2013 1 = 359 = close, but no cigar!<\/p>\n

        (C) D + 2H = 300 + 2*60 = 420 = not even close<\/p>\n

        (D) 5.95H = 5.95*60 — hmmm — I am going to think of 60 as 10*2*3.\u00a0 First, 59.5*10 = 59.5. Then 59.5*1 = 119.\u00a0 Then 119*3 = 357.\u00a0 This works.<\/p>\n

        (E) 1.21D = 1.21*300 = (1 + 0.2 + 0.01)*300 = 300 + 60 + 3 = 363 = not correct<\/p>\n

        The only possible answer is (D)<\/strong>.<\/p>\n

        4) Algebraic Solution<\/strong>: So, the marketer initially paid W: that was his outlay, his cost.\u00a0\u00a0 There are Q boxes in each crate, and J boxes make a collection, so there are Q\/J collections in each crate, and NQ\/J collections in total.\u00a0 If he charges a price P, his revenue would be PNQ\/J.\u00a0 Now, profit equals revenue minus cost, so Z = PNQ\/J \u2013 W .\u00a0 Solve this for P<\/p>\n

        \"gpp-vitac_img7\"<\/a><\/p>\n

        Numerical Solution<\/strong>: Here, we have several choices to make, and it would be a good idea to pick numbers that aren’t the same, or aren\u2019t even divisible by each other if that’s not required. Picking some different prime numbers is good way eliminate more than one answer that comes out to the correct value.<\/p>\n

        I will say the price is 2, a nice round prime number.\u00a0\u00a0 Let’s say J = 3, so Q must be a multiple of that — say Q = 30.\u00a0 Let’s N = 7.\u00a0 This means he has 10 collections from every box, and 70 collections from the total lot.\u00a0 If he charges $2 per collection, that’s a revenue of $140.\u00a0 Let’s say his outlay was W = $95; then his profit is Z = $45.<\/p>\n

        OK, that’s an unusual enough set of number that we expect no more than one solution will work.\u00a0 We will plug in N = 7, Q = 30, W = 95, J = 3, and Z = 45, and we hope to get an output of P = 2.<\/p>\n

        \"gpp-vitac_img8\"<\/a><\/p>\n

        Thus, because we chose numbers well, the only possible answer is (B)<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

        Variables in the Answer Choices The GMAT loves these question, not least because so many real-world situations something of this feel: we have a few changing variables (number of items sold, or stock price, or commodity price, etc.) and we want to figure out something else that depends on these changing quantities. There are two […]<\/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-5389","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"\nGMAT Practice Problems: Variables in the Answer Choices - 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-practice-problems-variables-in-the-answer-choices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Practice Problems: Variables in the Answer Choices\" \/>\n<meta property=\"og:description\" content=\"Variables in the Answer Choices The GMAT loves these question, not least because so many real-world situations something of this feel: we have a few changing variables (number of items sold, or stock price, or commodity price, etc.) and we want to figure out something else that depends on these changing quantities. 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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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There are two […]","og_url":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/","og_site_name":"Magoosh Blog \u2014 GMAT\u00ae Exam","article_publisher":"https:\/\/www.facebook.com\/MagooshGMAT\/","article_published_time":"2014-12-29T17:00:02+00:00","article_modified_time":"2020-01-15T18:48:23+00:00","og_image":[{"url":"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/gpp-vitac_img1.png"}],"author":"Mike M\u1d9cGarry","twitter_card":"summary_large_image","twitter_creator":"@MagooshGMAT","twitter_site":"@MagooshGMAT","twitter_misc":{"Written by":"Mike M\u1d9cGarry","Est. reading time":"9 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/#article","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/"},"author":{"name":"Mike M\u1d9cGarry","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b"},"headline":"GMAT Practice Problems: Variables in the Answer Choices","datePublished":"2014-12-29T17:00:02+00:00","mainEntityOfPage":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/"},"wordCount":1661,"commentCount":2,"publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"articleSection":["GMAT Math"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/","url":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/","name":"GMAT Practice Problems: Variables in the Answer Choices - Magoosh Blog \u2014 GMAT\u00ae Exam","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/#website"},"datePublished":"2014-12-29T17:00:02+00:00","breadcrumb":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/magoosh.com\/gmat\/gmat-practice-problems-variables-in-the-answer-choices\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/magoosh.com\/gmat\/"},{"@type":"ListItem","position":2,"name":"GMAT Practice Problems: Variables in the Answer Choices"}]},{"@type":"WebSite","@id":"https:\/\/magoosh.com\/gmat\/#website","url":"https:\/\/magoosh.com\/gmat\/","name":"Magoosh Blog \u2014 GMAT\u00ae Exam","description":"Everything you need to know about the GMAT","publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/magoosh.com\/gmat\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/magoosh.com\/gmat\/#organization","name":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/","url":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","contentUrl":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"Magoosh"},"image":{"@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/MagooshGMAT\/","https:\/\/twitter.com\/MagooshGMAT"]},{"@type":"Person","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b","name":"Mike 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. 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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\/5389","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=5389"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/5389\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=5389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=5389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=5389"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=5389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}