{"id":13485,"date":"2017-04-06T16:44:12","date_gmt":"2017-04-06T16:44:12","guid":{"rendered":"http:\/\/www.manhattanprep.com\/gmat\/?p=13485"},"modified":"2019-09-05T15:53:53","modified_gmt":"2019-09-05T15:53:53","slug":"how-to-handle-3-group-overlapping-sets-on-the-gmat","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gmat\/blog\/how-to-handle-3-group-overlapping-sets-on-the-gmat\/","title":{"rendered":"How to Handle 3-Group Overlapping Sets on the GMAT"},"content":{"rendered":"

\"Manhattan<\/p>\n

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! <\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n

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<\/i><\/b>Most overlapping sets on the GMAT have two distinct groups. Students take French and\/or Spanish (or neither), pianists play either classical and\/or jazz (or neither), people like either QDoba and\/or Baja Fresh (definitely neither. Chipotle, please)\u2014and for these situations, the familiar, double-set matrix approach works best.<\/span><\/p>\n

\"ManhattanI am very loyal to Chipotle. <\/span><\/p>\n

However, eventually the GMAT ratchets up the difficulty, and, instead of giving two groups overlapping, gives you three.<\/p>\n

Now, for those of you who haven\u2019t yet cracked 40 on the Quant, stop now. This is an advanced subject, and you shouldn\u2019t worry about it until you\u2019ve mastered the fundamentals of all the other question types. Fear not, this post will still be here. For those of you that have cracked 40, you\u2019ve probably suffered through one of these nauseating problems before. <\/span><\/p>\n

For three-group overlapping sets on the GMAT, our matrix-method has met its match. You could conceivably try a cube, if you wanted. But believe me, you don\u2019t. Some nerd already tried it. <\/span><\/p>\n

Okay, it was me.<\/span><\/p>\n

This is when the Venn Diagram, that old thing people had kind of forgotten about, gets called back into the game and finds itself significant again. Like John McCain.<\/span><\/p>\n

First, let\u2019s specify what it is that makes this question type challenging, because seeing what\u2019s difficult might help clarify what you need to do to work through it.<\/span><\/p>\n

What makes these problems difficult is figuring out <\/span>how much overlap there is. <\/span><\/i>Visually, that\u2019s the same as figuring out <\/span>how many times we\u2019ve counted each section of the Venn Diagram<\/span><\/i>. <\/span><\/p>\n

This becomes the fundamental skill in working through these problems: specifying how many times each section has been counted, and adjusting our equation by adding or subtracting sections until each part has only been counted once. <\/span><\/p>\n

This is best demonstrated with an example. Let\u2019s say there are 300 people at a film festival. 210 people watch Comedies, 115 watch Horror films, and 70 watch anything starring Kevin James. If 45 people watch both Comedies and Kevin James, 40 watch Horror and Kevin James, and 35 watch Comedy and Horror, and everyone watches at least one of these genres of film,\u00a0<\/span>how many people watch all three types of films?<\/p>\n

This is a pretty standard 3-group question. Let\u2019s build our diagram while making our equation.<\/span><\/p>\n

\"Manhattan<\/p>\n

First, we know that 210 watch Comedies, so let\u2019s put that in our equation and acknowledge we\u2019ve counted the Comedy circle once by shading it in.<\/span><\/p>\n

\"Manhattan<\/p>\n

Then bring in the \u2018Horror\u2019 section, add 115 in our equation, and shade the Horror section. <\/span><\/p>\n

\"ManhattanBut notice what happened. We\u2019ve counted the intersection of H and C twice now (for visualization, it\u2019s a shade darker). At some point, I have to get rid of that duplication in the equation I\u2019m building. <\/span><\/p>\n

Once I add the 70 for the \u2018Kevin\u2019 circle:<\/span><\/p>\n

\"Manhattan<\/p>\n

I see that I\u2019ve counted three overlaps twice and the very middle section <\/span>three times<\/span><\/i>. So now I need to find a way to get rid of these overlaps. I know the overlap of Comedy and Kevin is 45, so I can subtract that out of my equation. But, an important question, what sections does this apply to on the Venn Diagram? Is it the whole slice? Or just the image labeled \u2018A\u2019?<\/span><\/p>\n

\"Manhattan<\/p>\n

This is another reason why these problems are tough (and why Venn Diagrams are imperfect of labeling). In this case, 45 represents the<\/span> whole slice<\/span><\/i>. If it just represented A, the problem would have said \u201945 people watch <\/span>only<\/span><\/i> Comedy and Kevin James films.\u2019 Look out for the word \u2018only\u2019 or \u2018exactly\u2019 in these problems.<\/span> They are very important.<\/span><\/p>\n

(Aside: in labeling the Venn Diagram, you can create a system for yourself. If the number given for an overlap <\/span>includes<\/span><\/i> the center section, I write the numbers <\/span>close<\/span><\/i> to the center section. If the number I\u2019m given specifies it\u2019s \u2018only A and B,\u2019 I write it farther from the center section). <\/span><\/p>\n

So I subtract the 45, which takes out the whole slice. This means the intersection between <\/span>only<\/span><\/i> Kevin and Comedy has been counted once, as desired, <\/span>and<\/span><\/i> that I\u2019ve also taken out one of the three overlaps in the very center. It\u2019s now only been counted <\/span>twice<\/span><\/i>. <\/span><\/p>\n

\"Manhattan<\/p>\n

I then subtract the 40 for the overlap between Horror and Kevin. Now the overlap between <\/span>only<\/span><\/i> Horror and Kevin has been counted once, <\/span>and<\/span><\/i> the intersection of all three has been counted once.<\/span><\/p>\n

\"Manhattan<\/p>\n

I have to get rid of the last duplicated set, though, so I subtract the 35 who watch Horror and Comedy. Now, though, notice, I\u2019ve counted each of my \u2018only two groups\u2019 once, as desired, but I\u2019ve <\/span>entirely removed<\/span><\/i> the triple overlap:<\/span><\/p>\n

\"Manhattan<\/p>\n

So, I need to add that back in. I don\u2019t know the value. But that\u2019s okay\u2014when you don\u2019t know a value, assign a variable (especially when it\u2019s the value the question is asking about).<\/span><\/p>\n

So I add back x. And since it was specified that in this situation there are no people in the \u2018other\u2019 \u00a0category (because the problem specifies everyone watches at least one of these genres), I know my equation adds up to 300. <\/span><\/p>\n

\"Manhattan<\/p>\n

So I have the equation: 210 + 115 + 70 – 45 – 40 – 35 + x = 300. Solve for x and you\u2019re done.<\/span><\/p>\n

Unfortunately, you can\u2019t just memorize every step we just did and apply them to every 3-group problem, as the GMAT can give the information in different ways, and you won\u2019t be able to actually shade in and un-shade the circles as we have here. But the skill always boils down to the same thing: figure out how many times you\u2019ve counted each section of the Venn diagram, and set up an equation so that every part of it is only counted once. Some students like to draw little hash marks in the sections of the Venn diagram to show how many times its been counted, and then cross off them off as they subtract overlaps to get their answer. <\/span><\/p>\n

For practice, here are two other ways the GMAT might give the information:<\/span><\/p>\n

At a 300 person film festival, there are 210 people who watch Comedies, 115 who watch Horror, and 70 who watch Kevin James films. If 20 watch only Comedies and Kevin James, 10 watch only Comedies and Horror, and 40 watch only Horror and Kevin James, and every viewer watches at least one of these genres of movie, how many people watch all three? <\/span><\/p>\n

(These are the same numbers, but the presentation is different and the equation you build will be different. Go through each section and count how many times you\u2019ve put it in\/taken it out of your equation to solve). <\/span><\/p>\n

One thing you can do to improve at this question type is build your own sets. Once you understand how to come up with a three-group overlapping set, you\u2019ll be able to deconstruct the numbers better on the actual GMAT.<\/span><\/p>\n

Happy studying. I\u2019m going to pick up Chipotle and watch<\/span> Paul Blart 2<\/span><\/i>. ?<\/span><\/p>\n