{"id":2564,"date":"2024-06-03T09:00:33","date_gmt":"2024-06-03T16:00:33","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2564"},"modified":"2019-04-20T01:39:09","modified_gmt":"2019-04-20T08:39:09","slug":"gmat-sets-venn-diagrams","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-sets-venn-diagrams\/","title":{"rendered":"GMAT Sets: Venn Diagrams"},"content":{"rendered":"<p><strong>Learn this technique to master set questions of GMAT Math word problems.\u00a0<\/strong><\/p>\n<p><iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Sc8WA_6kPrE\" style=\"border: 0; margin: 0 auto; display: flex\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Practice questions<\/h2>\n<p>First, try these challenging practice questions.<\/p>\n<p>1) Of the 80 houses in a development, 50 have a two-car garage, 40 have an in-the-ground swimming pool, and 35 have both a two-car garage and an in-the-ground swimming pool.\u00a0 How many houses in the development have neither a two-car garage nor an in-the-ground swimming pool?<\/p>\n<p>A. 10<br \/>\nB. 15<br \/>\nC. 20<br \/>\nD. 25<br \/>\nE. 30<\/p>\n<p>2) A certain school has three performing arts extracurricular activities: Band, Chorus, or Drama.\u00a0 Students must participate in at least one, and may participate in two or even in all three.\u00a0 There are 120 students in the school.\u00a0 There are 70 students in Band, 73 in the Chorus, and 45 in the Drama.\u00a0 Furthermore, 37 students are in both the Band and Chorus, 20 are in both the Band and the Drama, and 8 students are in all three groups.\u00a0 Twenty-five students are just in the chorus, not in anything else.\u00a0 \u00a0How many students participate in only the drama?<\/p>\n<p>A. 11<br \/>\nB. 12<br \/>\nC. 14<br \/>\nD. 17<br \/>\nE. 21<\/p>\n<p>&nbsp;<\/p>\n<h2>Introduction to sets<\/h2>\n<p>The idea of a set is, in some sense, the most fundamental idea in all of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Set_theory\" target=\"_blank\" rel=\"noopener noreferrer\">mathematics<\/a>.\u00a0 Nonetheless, it&#8217;s a very simple idea.\u00a0 A set is simply a collection of objects or elements.\u00a0 When the members of a set are all numbers, then we use &#8220;set notation&#8221;, which consists of brackets.\u00a0 For example:<\/p>\n<p>A = {2, 3, 3, 5, 7}<\/p>\n<p>That notation denotes set A with five numerical members.\u00a0 When all the members of the set are numbers, typical questions involve computations like the mean or the median: here, the mean (or average) of set A is 4, and the median of A is 3.\u00a0 You can read more about those calculations at <a href=\"https:\/\/magoosh.com\/gmat\/common-gmat-topic-descriptive-statistics\/\" target=\"_self\" rel=\"noopener noreferrer\">this post<\/a>.<\/p>\n<h2>Overlapping sets<\/h2>\n<p>Numerical sets can be handled with statistical calculations.\u00a0 Non-numerical sets, sets in which the members are people or cars or companies, are the stuff of tricky word problems.\u00a0 Ordinarily, there&#8217;s nothing particular challenging if there&#8217;s only one set: some of the people are in that set, whatever it is, and the rest aren&#8217;t.\u00a0 No challenge.<\/p>\n<p>Things get more interesting if there are two or more overlapping sets.\u00a0 For example, in the SF Bay Area, many adult residents were born out of state &#8212; many, but not all; many adult residents have a college education &#8212; many, but not all; and many adult residents are SF Giants fans &#8212; many, but not all.\u00a0 Those are three overlapping sets &#8212;- any particular adult resident of the SF Bay Area many be a member of none, one, two, or all three of those three categories.\u00a0 As it happens, I am a member of exactly two of those categories.\u00a0 This is exactly the situation of the practice questions posed above.<\/p>\n<p>&nbsp;<\/p>\n<h2>Venn diagrams<\/h2>\n<p>Venn diagrams are the best method for untangling overlapping sets.\u00a0 If you have two overlapping sets, you need a two-circle Venn diagram:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2565\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img1.png\" alt=\"\" \/><\/a><\/p>\n<p>This diagram contains three discrete regions:<\/p>\n<p>A = those elements in just the left circle<\/p>\n<p>B = those element in both categories, in the overlap<\/p>\n<p>C = those elements in just the right circle<\/p>\n<p>There may also be a fourth discrete region, those elements that are not members of any set.\u00a0 Typically, the problem will only give us information about totals &#8212; the total number of elements altogether, the total number in each circle, and the overlap.\u00a0 If you are told there are 70 members in the right circle, and 20 members in the overlap, then you would know B + C = 70 and B = 20, so from that you could deduce C = 50, the number of elements that are just in the portion of the circle labeled C.\u00a0 In general, you work from the center outward, figuring out one discrete region after another.\u00a0 Once you know the value of each individual discrete region, you will be able to answer any question about the number in any particular grouping.<\/p>\n<p>If there are three overlapping categories, we use a three-circle Venn diagram:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2566\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img2.png\" alt=\"\" width=\"621\" height=\"567\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img2.png 621w, https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img2-300x273.png 300w\" sizes=\"(max-width: 621px) 100vw, 621px\" \/><\/a><\/p>\n<p>This diagram has at least seven discrete regions<\/p>\n<p>A = members of all three circles<\/p>\n<p>B = members of the green and blue circles, but not the red circle<\/p>\n<p>C = members of the green and red circles, but not the blue circle<\/p>\n<p>D = members of the blue and red circles, but not the green circle<\/p>\n<p>E = members of the green circle but of neither the blue nor the red circles<\/p>\n<p>F = members of the blue circle but of neither the green nor the red circles<\/p>\n<p>G = members of the red circle but of neither the green nor the blue circles<\/p>\n<p>Depending on context, there may also be a eighth discrete region, those elements that are not members of any of the three set.\u00a0 Typically, the problem will only give us information about totals.\u00a0 This gets very tricky.\u00a0 If we are told the total in any one circle, that includes four discrete regions; for example, the green circle includes A + B + C + E.\u00a0 Similarly, the overlap of two circles contains two discrete regions: for example, the overlap of the blue and red circles includes A + D.\u00a0 The problem will always tell you how many elements are in the central region (A), and will often tell you how many are in each circle, and how many in each overlap of two circles.\u00a0 In general, you work from the center outward, figuring out one discrete region after another.\u00a0 Once you know the value of each individual discrete region, you will be able to answer any question about the number in any particular grouping.<\/p>\n<p>With these strategies, you may find the practice problems at the beginning somewhat more approachable.\u00a0 Try them again, before reading the explanations below.<\/p>\n<p>In the next post, we will look at set problems in which each element is categorized according to two different variables at once.<\/p>\n<p>&nbsp;<\/p>\n<h2>Practice question explanations<\/h2>\n<p>1) Here, we have two categories: (a) with or without two-car garage, and (b) with or without an in-the-ground pool.\u00a0 Houses can be members of either, both, or neither category.\u00a0 We will use a two circle Venn diagram:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img3.png\"><img decoding=\"async\" class=\"alignnone  wp-image-2567\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img3.png\" alt=\"\" width=\"646\" height=\"331\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img3.png 807w, https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img3-300x153.png 300w\" sizes=\"(max-width: 646px) 100vw, 646px\" \/><\/a><\/p>\n<p>We know the total of the group is 80 &#8212;&#8211; A + B + C + D = 80.\u00a0 We know the green circle, two-car garages, has 50 members, so A + B = 50.\u00a0 We know the blue circle, in-the-ground pool, has 40 members, so B + C = 40.\u00a0 We also know the crucial overlap region, B = 35.\u00a0 If B = 35, in the green circle, we can deduce that A = 15, and in the blue circle, we can deduce that C = 5.\u00a0 Then<\/p>\n<p>A + B + C + D = 15 + 35 + 5 + D = 80<\/p>\n<p>D = 25<\/p>\n<p>Thus, 25 houses in this development have neither a two-car garage nor an in-the-ground swimming pool. \u00a0Answer = <strong>D<\/strong>.<\/p>\n<p>2) Here, we have three categories, so we need three circles.\u00a0 Every student must take at least one of these three performing arts extracurricular activities, so there will be no one outside the three circles.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img4.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-2568\" src=\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/set_img4.png\" alt=\"\" \/><\/a><\/p>\n<p>The sum of all seven = 120 (we never use this number in this question)<\/p>\n<p>The totals for the band (70), the chorus (73), and the drama (45) each involve the sum of four discrete regions.\u00a0 We will have to find other information before we can employ them.<\/p>\n<p>&#8220;8 students are in all three groups&#8221;<\/p>\n<p><strong>N = 8<\/strong>.<\/p>\n<p>&#8220;37 students are in both the Band and Chorus&#8221;<\/p>\n<p>37 = K + N = K + 8 &#8212;&gt;\u00a0<strong>K = 29<\/strong><\/p>\n<p>&#8220;20 are in both the Band and the Drama&#8221;<\/p>\n<p>20 = M + N = M + 8 \u00a0&#8212;&gt;\u00a0<strong>M = 12<\/strong><\/p>\n<p>&#8220;twenty-five students are just in the chorus, not in anything else&#8221;<\/p>\n<p><strong>L = 25<\/strong><\/p>\n<p>We now have identified three of the regions in the Chorus circle, so we can solve for P.<\/p>\n<p>chorus = 73 = K + L + N + P<\/p>\n<p>73 = 29 + 25 + 8 + P<\/p>\n<p><strong>P = 11<\/strong><\/p>\n<p>Now, we have identified three of the regions in the Drama circle, so we can solve for Q.<\/p>\n<p>drama = 45 = M + N + P + Q<\/p>\n<p>45 = 12 + 8 + 11 + Q<\/p>\n<p><strong>Q = 14<\/strong><\/p>\n<p>This is precisely what the question was asking: how many students are only in drama? There are 14 students who take only drama.<\/p>\n<p>Answer = <strong>C<\/strong><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Learn this technique to master set questions of GMAT Math word problems.\u00a0 &nbsp; Practice questions First, try these challenging practice questions. 1) Of the 80 houses in a development, 50 have a two-car garage, 40 have an in-the-ground swimming pool, and 35 have both a two-car garage and an in-the-ground swimming pool.\u00a0 How many houses [&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-2564","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 Sets: Venn Diagrams - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"Venn diagram GMAT problems might throw you for a loop--but not if you know what to expect. 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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. 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