In Country X, 1/2 of the population can speak English, 1/3 can speak

One approach is to use the Double Matrix Method. This technique can be used for questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
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:


EXTRA PRACTICE QUESTION


More questions to practice with:
EASY: https://gmatclub.com/forum/of-the-120-p ... 15386.html
MEDIUM: https://gmatclub.com/forum/in-a-certain ... 21716.html
HARD: https://gmatclub.com/forum/a-group-of-2 ... 24888.html
KILLER: https://gmatclub.com/forum/a-certain-hi ... 32899.html

2x2 matrix
---french---n f----total
Eng--6-----9-----15
NE --4------11---15
total--10----20---30
11/30
OPTION D

Let the population of country X be 30

Those who can speak only English will be A
Those who can speak only French be C
Those who can speak both be B

Now,
1/2 of total can speak English.
That will be A+B = 30*1/2=15
(Since only English is not stated. So both and only English will be included)

1/3 of total can speak French.
That will be C+B = 30*1/3=10
(Since only French is not stated. So both and only French will be included)

1/5 of total can speak both:
B = 1/5*30 = 6

Therefore A will be 15-6=9
And C will be 10-6=4

Those who can't speak English or French will be 30-9+4+6 = 11
The fraction who can't speak either of the language will be 11/30

Therefore D is the answer

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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?

Neither = Total - (English+ French - Both) = 1 - (1/2 + 1/3 - 1/5) = 1 - (15 + 10 - 6)/30 = 1 - 19/30 = 11/30

IMO D

We can let the total number of people = 60 and create the equation:

60 = 30 + 20 - 12 + N

22 = N

Thus, 22/60 = 11/30 of the people speak neither language.

Answer: D
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