While my personal preference is to use a Venn diagram when dealing with most Sets questions, there are some questions in which a double-matrix is necessary (and much more powerful than a wimpy Venn). This little guy will make even the scariest-looking Sets question into a simple set of rows and columns, and its ability to help us determine whether a statement is sufficient in DS is unmatched!
To make a double-matrix, simply create a chart with rows and columns. The rows are assigned to one variable, and the columns to another. At the bottom of each column and at the far-right of each row, place a box for the column-total and row-total. Let’s check out a sample GMAT question to see what this might look like!
33 out of the 47 students in an advanced degree program have a higher than average GPA. How many students in the program are receiving some form of academic scholarship?
(1) More students do not have a scholarship than have a scholarship.
(2) The same number of students have a higher than average GPA and are receiving some form of academic scholarship as have neither a higher than average GPA nor an academic scholarship.
From the question-stem, we know 33 students have a high GPA, while 14 do not. We need more information about which of these students have scholarships to be able to answer this value question. Statement (1) is insufficient because it does not give us information to find the exact numerical value of the students receiving some form of scholarship.
Statement (2) tells us that the number of students who fit “both” is equal to the number of students who fit “neither.” Let’s set up a chart to visualize the four possible categories for the students. Since “both” = “neither,” let’s fill in “x” for those boxes.
Since each column and row must total, if there are “x” students receiving no scholarship and not a higher GPA, and the total students who don’t have a higher GPA is 14, then 14-x students must not have a higher GPA and have a scholarship. The total we are looking for is represented by the red “?,” and we can set up an equation to solve: x + (14 – x) = 14. Sufficient.
The correct response is (B).
To wrap up, keep in mind that it’s possible (although highly unlikely) that you might see a Sets question on the GMAT involving three variables instead of two. In that case, you’d have 5 columns instead of 4, and 5 rows instead of 4, but the same rules of totaling apply!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing advice to help students better prepare for the GMAT and the SAT.