Question

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
(i) how many can speak both Hindi and English:
(ii) how many can speak Hindi only;
(iii) how many can speak English only.

Answer 1

Let A & B denote the sets of the persons who like Hindi & English, respectively.
$$\begin{gathered} \mathrm{Given}: \\ n\left(A\right)=750 \\ n\left(B\right)=460 \\ n\left(A\cup B\right)=950 \\ \left(\mathrm{i}\right)\mathrm{We}\mathrm{know}: \\ n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right) \\ \Rightarrow 950=750+460-n\left(A\cap B\right) \\ \Rightarrow n\left(A\cap B\right)=260 \\ \mathrm{Thus},260\mathrm{persons}\mathrm{can}\mathrm{speak}\mathrm{both}\mathrm{Hindi}\mathrm{and}\mathrm{English}. \\ \left(\mathrm{ii}\right)n\left(A-B\right)=n\left(A\right)-n\left(A\cap B\right) \\ n\left(A-B\right)=750-260=490 \\ \mathrm{Thus},490\mathrm{persons}\mathrm{can}\mathrm{speak}\mathrm{only}\mathrm{Hindi}. \\ \left(\mathrm{iii}\right)n\left(B-A\right)=n\left(B\right)-n\left(A\cap B\right) \\ \Rightarrow n\left(B-A\right)=460-260 \\ =200 \\ \mathrm{Thus},200\mathrm{persons}\mathrm{can}\mathrm{speak}\mathrm{only}\mathrm{English}. \end{gathered}$$