In Country X, 1/2 of the population can speak English, 1/3 can speak French, and 1/5 can speak both English and French.

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GMAT Prep

In Country X, 1/2 of the population can speak English, 1/3 can speak French, and 1/5 can speak both English and French. What fraction of the population of Country X can speak neither English nor French?

A. 2/15
B. 1/6
C. 3/10
D. 11/30
E. 13/30

OA D

Here, we have a population of citizens, and the two characteristics are:
- speaks English or doesn't speak English
- speaks French or doesn't speak French

Aside: We can also use Venn diagrams and formulae to solve overlapping sets questions. However, as difficulty levels increase, it becomes harder to apply those other approaches, whereas the Double Matrix Method works every time.

Since the question asks us to find the value of a FRACTION, that's assign a "nice" value to the population of Country X.
When I scan the other choices, I see that all of the denominators are factors of 30.
So let's let 30 equal the population of Country X.

Our initial diagram looks something like this:

I placed a star in the box representing citizens who speak neither English nor French, since this is the value we want to determine.

1/2 of the population can speak English, 1/3 can speak French, and 1/5 can speak both English and French
1/2 of 30 = 15, which means 15 people speak English, which also means the remaining 15 people do not speak English.
1/3 of 30 = 10, which means 10 people speak French, which also means the remaining 20 people do not speak French.
1/5 of 30 = 6, which means 6 people speak both English and French
When we add all of this to our diagram we get:

From here, we can fill in the remaining information to get:

What fraction of the population of Country X can speak neither English nor French?
Out of the population of 30 people, 11 speak neither English nor French
Fraction = 11/30

Answer: D

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

EXTRA PRACTICE QUESTION
https://www.youtube.com/watch?v=DsbPUSH_Wu4

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