Within a group of students, x students are taking English

Question

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

A) x + y - z
B) x + y + z
C) 2x + 2y - z
D) x + y - 2z 
E) x + y + 2z

FillFM · Answer

Option D.

x - English
y - Math
z - English and Math

(x-z) = just English
(y-z) = just Math

English or Math but not both = (x-z) + (y-z) = x + y - 2z

BrentGMATPrepNow · Answer

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:
 https://i.imgur.com/heYIMc2.png 
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:
 https://i.imgur.com/5COZBKU.png

GIVEN:  z students are taking both English and Math 
So, we can add this information to our diagram as follows:
 https://i.imgur.com/k4Waizn.png

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. 
 https://i.imgur.com/fYwOXjy.png

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.
 https://i.imgur.com/uF2BNsb.png

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:
 https://www.youtube.com/watch?v=jK-tiBrrf04

BvR · Answer

Hey, all!

Could anyone plase explain why the standard formula doesn't work in this question, i.e. Total = Set1+Set2 + Neither - Both

I understand that according to the text neither in this case = 0, but other parts of the formula do not lead to the right asnwer. So, total = Set1(x) + Set2 (y) - Both (z) = x+y-z.

Thank you in advance!

UPD: the mystery is solved. When we substract z only once, we still acoount it but avoid doubling. And when we substract z twice we find pure set of x and set of y.

CEdward · Answer

The key here is to recognize that x and y EACH include the number who are also taking English or math respectively. In other words, "Both English and Math" have been double-counted so if we want either English or Math, we have to remove z twice.

x + y - 2z

Answer is D.

x + y - 2z

Answer is D.

bumpbot · Answer

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Within a group of students, x students are taking English

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Within a group of students, x students are taking English

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