Question

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find :

(i) how manycan speak both Hindi and English

(ii) how many can speak Hindi only

(iii) how many can speak English only.

Answer 1

(i) Let,

n(P) denote the total number of persons,

n(H) denote the number of persons who speak Hindi and

n(E) denote the number of persons who speak English.

Then,

n(P) = 950, n (H) = 750, n(E) = 460

To find : $n(A\cap E)$

$n\left(P\right)=n\left(H\right)+n\left(E\right)-n(H\cap E)$

$\Rightarrow 950=750+460n(H\cap E)$

$\Rightarrow 950=2110-n(A\cap E)$

$\Rightarrow n(H\cap E)=2110-950$

= 260

Hence, 260 persons can speak both Hindi and English.

(ii) Clearly H is the disjoint union of H - E and $H\cap E$

i.e. $H=(H-E)\cup (H\cap E)$

$\therefore n\left(H\right)=n(H-E)+n(H\cap E)$ [$\because$ if A and B are disjoint then $n(A\cap B)$

= n(A)+n (B)

$\Rightarrow 750=n(H-E)+260$

$\Rightarrow n(H-E)=750-260$

= 490

Hence, 490 persons can speak Hindi only.

(iii) On s similiar lines we have

$E=(E-H)\cup (H\cap E)$

i.e. E is the disjoint union of E- H ans $H\cap E$

$\therefore n\left(E\right)=n(E-H)+n(H\cap E)$

$\Rightarrow 460=n(E-H)+260$

$\Rightarrow n(E-H)=460-260$

= 200

Hence, 200 persons can speak English only.