Question

Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English? How many speak only French? How many of them speak English and French both?

Answer 1

Let A be the set of persons speaking English and B be the set of persons speaking French.

So, n(A) = 72; n(B) = 43; n(A ∪ B) = 100

Now,

n(A) + n(B) = n(A ∪ B) + n(A ∩ B)

⇒ n(A ∩ B) = 72 + 43 - 100

⇒ n(A ∩ B) = 15 

So, the number of person who speak French and English both is 15.

Also,

n(A) = n(A - B) + n(A ∩ B)

⇒ n(A - B) = 72 - 15

⇒ n(A - B) = 57 

So, the number of person who speak only English is 57.

And, 

n(B) = n(B - A) + n(A ∩ B)

⇒ n(B - A) = 43 - 15

⇒ n(B - A) = 28

So, the number of person who speak only French is 28.