How to Handle 3-Group Overlapping Sets on the GMAT
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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)—and for these situations, the familiar, double-set matrix approach works best.
I am very loyal to Chipotle.
However, eventually the GMAT ratchets up the difficulty, and, instead of giving two groups overlapping, gives you three.
Now, for those of you who haven’t yet cracked 40 on the Quant, stop now. This is an advanced subject, and you shouldn’t worry about it until you’ve 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’ve probably suffered through one of these nauseating problems before.
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’t. Some nerd already tried it.
Okay, it was me.
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.
First, let’s specify what it is that makes this question type challenging, because seeing what’s difficult might help clarify what you need to do to work through it.
What makes these problems difficult is figuring out how much overlap there is. Visually, that’s the same as figuring out how many times we’ve counted each section of the Venn Diagram.
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.
This is best demonstrated with an example. Let’s 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, how many people watch all three types of films?
This is a pretty standard 3-group question. Let’s build our diagram while making our equation.
First, we know that 210 watch Comedies, so let’s put that in our equation and acknowledge we’ve counted the Comedy circle once by shading it in.
Then bring in the ‘Horror’ section, add 115 in our equation, and shade the Horror section.
But notice what happened. We’ve counted the intersection of H and C twice now (for visualization, it’s a shade darker). At some point, I have to get rid of that duplication in the equation I’m building.
Once I add the 70 for the ‘Kevin’ circle:
I see that I’ve counted three overlaps twice and the very middle section three times. 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 ‘A’?
This is another reason why these problems are tough (and why Venn Diagrams are imperfect of labeling). In this case, 45 represents the whole slice. If it just represented A, the problem would have said ’45 people watch only Comedy and Kevin James films.’ Look out for the word ‘only’ or ‘exactly’ in these problems. They are very important.
(Aside: in labeling the Venn Diagram, you can create a system for yourself. If the number given for an overlap includes the center section, I write the numbers close to the center section. If the number I’m given specifies it’s ‘only A and B,’ I write it farther from the center section).
So I subtract the 45, which takes out the whole slice. This means the intersection between only Kevin and Comedy has been counted once, as desired, and that I’ve also taken out one of the three overlaps in the very center. It’s now only been counted twice.
I then subtract the 40 for the overlap between Horror and Kevin. Now the overlap between only Horror and Kevin has been counted once, and the intersection of all three has been counted once.
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’ve counted each of my ‘only two groups’ once, as desired, but I’ve entirely removed the triple overlap:
So, I need to add that back in. I don’t know the value. But that’s okay—when you don’t know a value, assign a variable (especially when it’s the value the question is asking about).
So I add back x. And since it was specified that in this situation there are no people in the ‘other’ category (because the problem specifies everyone watches at least one of these genres), I know my equation adds up to 300.
So I have the equation: 210 + 115 + 70 – 45 – 40 – 35 + x = 300. Solve for x and you’re done.
Unfortunately, you can’t 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’t 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’ve 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.
For practice, here are two other ways the GMAT might give the information:
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?
(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’ve put it in/taken it out of your equation to solve).
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’ll be able to deconstruct the numbers better on the actual GMAT.
Happy studying. I’m going to pick up Chipotle and watch Paul Blart 2. ?
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Reed Arnold
is a Manhattan Prep instructor based in New York, NY. He has a B.A. in economics, philosophy, and mathematics and an M.S. in commerce, both from the University of Virginia. He enjoys writing, acting, Chipotle burritos, and teaching the GMAT. Check out Reed’s upcoming GMAT courses here.