Set theory is no one’s favorite GMAT concept (unless you’re a masochist), but since nearly all test-takers will see at least one overlapping-sets question on the Quantitative section of the GMAT, it’s certainly important. And take solace in this – becoming confident with this challenging type of word problem can be as simple as learning how to rock a Venn diagram.
To refresh a bit on Venn diagrams, we have to recall two definitions: union and intersection. The union of sets is all elements from all sets. The intersection of sets is only those elements common to all sets. Let’s look at how we can take apart a complex question without any messy set theory formulas.
This year, x people won an Olympic medal for water competitions. One-third of the winners earned a medal for swimming and one-fourth of those who earned a medal for swimming also earned a medal for diving. How many people won an Olympic medal for water competitions but did not both receive a medal for swimming and a medal for diving?
Let’s work on a Venn diagram and fill in what we do know – an important first step as many of these problems come down in large part to “getting organized”. We know there will be some medal winners who swam but did not dive, some who dove but did not swim, and some who swam AND dove.
“x” here will be at the top of our Venn because it is the total for ALL PARTS of the Venn. That is, all three categories will sum to x. x/3 represents the 1/3 of the total (“x”) who swam, including those who swam only AND those who swam and dove.
We can make up a variable, let’s say “y,” to represent the total number of divers. The key to understanding this question lies in the last sentence and the phrase “not both.”
We need to know the people who ONLY swam but did NOT dive, and the people who ONLY dove but did NOT swim. I made up variables for these two groups: “a” and “z.”
Let’s use the answer choices to our advantage! Since they have the denominators of 12 and 7, let’s use one of those and work backwards! 12 appears more often, so we can start there.
If x = 12, there are 12/3 = 4 swimmers total, (12/3)/4 = 1 of whom swam and dove. That means a = 3. If 4 people swam, then 12-4 = 8 dove, so z = 8.
The two categories we’re looking for (a + z) are 3 + 8 = 11. We are looking for an answer choice that gives us 11 when x = 12.
The answer is (A).
The next thing you’ll need to understand is how 3 sets interact, but luckily the rules of the Venn diagram still apply! Look for another blog coming soon to cover 3 overlapping circles!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing tips and tricks to help students better prepare for the GMAT and the SAT.