Three Overlapping Sets

I wrote a few weeks ago that Overlapping Sets are a hot GMAT topic. They don't have to be particularly hard questions, but they fit well into the criteria of what the GMAC is trying to test: mathematical reasoning, without a lot of nuts-and-bolts arithmetic. Some of these questions require nothing more than addition and subtraction.

Two Overlapping Sets

You've probably see plenty of problems with two overlapping sets. Something like: "A class has 40 students, 25 of whom are boys and 12 of whom play basketball. If 8 of the boys play basketball, how many girls do not play basketball?"

There are three distinct ways of handling these questions, each of which I cover in more detail in Total GMAT Math:

• A Venn Diagram
• A table
• A formula

The Overlapping Sets Formula

The formula I use is this:

Total = Group1 + Group2 - Both + Neither

In the example above, 40 is Total, 25 is Group1, 12 is Group2, and 8 is Both. (Group1 and Group2 are interchangeable.) Basically, G1+G2 is the sum of all of the people who do one or the other, but that sum double-counts the number who do both. That's why we subtract both.

Three Overlapping Sets

My focus today, though, is on questions with three overlapping sets. They aren't common on the GMAT, but at the same time, they don't require much more thinking that do the two-set questions.

Here's what one of those might look like:

Of the shoe stores in City X, 30 carry Brand A shoes, 40 carry Brand B shoes, and 25 carry Brand C shoes. If each store carries at least one of the brands, 32 of the stores carry two of the three shoe brands, and none of the stores carry all three, how many shoe stores are there in City X?

We can't use a table for this, because it would have to be three-dimensional. A Venn Diagram is a possibility, but there's no place to put 32. We'll have to look at an expansion of the formula.

The Three Overlapping Sets Formula

Here it is:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

This looks more complicated, but the concept is the same. G1+G2+G3 is the sum of all the stores that carry these brands, only we've double-counted some of them. Since 32 of the stores carry two brands, we've double-counted exactly 32, so we subtract 32, the "sum of 2-group overlaps."

The tricky part is the second-to-last term: "2*(all three)." It doesn't come up very often, as when it does, it's almost always on Data Sufficiency questions. However, it's worth thinking about.

If one of the stores in City X carried all three shoe brands, that store is being counted three times--once in the 30 A's, once in the 40 B's, and once in the 25 C's. But it's still only one store. Thus, if it's represented three times, we need to subtract it twice. In this case, the question tells us that the "all three" term is equal to zero.

So, to solve this example:

Total = 30 + 40 + 25 - 32 - 2*0 + 0 = 63

(Neither equals 0 because every shoe store in City X carries at least one of the brands.)

Problems with three overlapping sets can be daunting, but if you can master the concepts in this article, it'll be yet another question type you can attack with confidence when you take your GMAT.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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