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

i. Let U be the set of all the persons, E be the set of persons who speak English and F be the set of persons who speak French. 

∴ n(E) = 72, n(F) = 43 Since, each one out of 100 persons speak at least one language ∴ n(U) = n(E ∪ F)= 100, 

ii. n (E ∪ F) = n (E) + n (F) – n(E ∩ F) 

100 = 72 + 43 – n (E ∩ F) 

n (E ∩ F) = 72 + 43 – 100 

∴ n(E ∩ F) = 15 

Number of people who speak English and French = 15

iii. Number of people who speak only English = n(E) – n(E ∩ F) = 72 – 15 = 57 

iv. Number of people who speak only French = n(F) – n(E ∩ F) = 43 – 15 = 28 

Alternate Method: 

Let U be the set of all the persons, E be the set of persons who speak English, 

F be the set of persons who speak French and x people speak both the languages.

Since, each one out of 100 persons speak at least one language. 

∴ n(U) = n(E ∪ F) = 100 

∴ 72 – x + x + 43 – x = 100 

∴ 115 – x = 100 

∴ x = 115 – 100 = 15. 

Number of people who speak English and French = 15 

Number of people who speak only English = 72 – x = 72 – 15 = 57 

Number of people who speak only French = 43 – x = 43 – 15 = 28