Within a group of students, x students are taking English, y students are taking Math, and z students are taking both English and Math. Which of the following represents the number of students who are taking English or Math but not both?

A) x + y - z
B) x + y + z
C) 2x + 2y - z
D) x + y - 2z
E) x + y + 2z
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We can solve this by using the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).

Here, we have a population of students, and the two characteristics are:
- taking English or not taking English
- taking Math or not taking Math

We want to find: (number of students taking English but not Math) + (number of students taking Math but not English)
So, we can set up our matrix as follows:

We need to find the number of students in the 2 boxes with stars in them.

GIVEN: x students are taking English, and y students are taking Math
So, we can add this information to our diagram as follows:


GIVEN: z students are taking both English and Math
So, we can add this information to our diagram as follows:


Let's first examine the top ROW.
If a total of x students are taking English, and z of them are also taking Math, then x-z of them are NOT taking Math.


Now examine the left-hand COLUMN.
If a total of y students are taking Math, and z of them are also taking English, then y-z of them are NOT taking English.


So, the number of students who are taking English or Math but not both = (x - z) + (y - z)
= x + y - 2z

Answer: D

NOTE: This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:

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