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Two Shortcut Formulas for Set Problems

Posted in Test Prep on July 28, 2011 by

If you’re one of those people who goes crazy over overlapping sets problems, this post is for you. Oftentimes, Venn Diagrams are the way to go on these problems, and I highly recommend that you master them in our GMAT course. However, if you’re more formula-oriented, there’s a handy shortcut you can use.

Let’s say we consider all the athletes at a school. We consider those on the football team and those on the hockey team. Some athletes are on both teams, and some athletes are on neither team. Let’s assign the following variables:

F = number of athletes on the football team (including those who also play hockey)
H = number of athletes on the hockey team (including those who also play football)
B = number of athletes who play both football and hockey
N = number of athletes who play neither football nor hockey
T = total number of athletes at the school.

The following formula would apply:

F + H – B + N = T

This applies to any pair of overlapping sets. You add the total number in each group (including the overlap in both cases), then you subtract the number in the overlap, add the number in the neither group, and that gets you the overall total.

In general, you could think of it as:

(Total in first group) + (Total in second group) – (overlap) + neither = Overall total

Now, it’s important to keep in mind that the total for each group includes the overlap and not just those elements that are only football or only hockey.

“All that is great,” you might say. But then you’d add, “What about the really annoying problems that involve not two but three overlapping sets?” Well, there’s even a formula for that situation. In general, if you have three overlapping groups, the formula would be:

(Total in Group 1) + (Total in Group 2) + (Total in Group 3) – (Overlap of 1 and 2) – (Overlap of 1 and 3) – (Overlap of 2 and 3) – [2 * (Overlap of 1, 2, and 3)] + (total in none of the groups) = Overall total

Very important reminder: When we say “Total in Group 1”, we mean everything in Group 1, including the elements in the overlaps. It does not mean those elements in Group 1 only!

If you’re curious about why we subtract the overlaps, it’s essentially because in adding up the totals of each group, we count the overlaps more than once. For example, in adding the totals of Groups 1 and 2, the overlap between them gets counted twice, so we must subtract out one of them. When we add the totals of all three groups, the overlap of all three groups gets counted 3 times, whereas it should only be counted once; that’s why we subtract that overlap twice.

Now, the formula is a great tool, but as you probably suspected, having it memorized is just a first step. The difficulty of GMAT problems lies not in complex formulas and calculations, but in the quirkiness of their constructions and setups. Try your hand at these two problems to see what I mean. The first is an Official Guide problem, and the second is one of my own. As usual, feel free to post your step-by-step solution in the comments. Good luck!

1. Of the 200 students at college T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

A) 20 to 50
B) 40 to 70
C) 50 to 130
D) 110 to 130
E) 110 to 150

2. In a particular neighborhood of 150 households, some households have no electronic devices. The rest have some combination of televisions, laptops, and stereos. 75 households have televisions, 50 have laptops, and 20 have all three devices. If 45 households have exactly two of the three devices, and the number of households that have stereos is four times the number of households that have none of the three devices, how many households have stereos?

A) 84
B) 88
C) 92
D) 96
E) 100