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.

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Solution

## (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∩E) n(P)=n(H)+n(E)−n(H∩E) ⇒950=750+460n(H∩E) ⇒950=2110−n(A∩E) ⇒n(H∩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∩E i.e. H=(H−E)∪(H∩E) ∴n(H)=n(H−E)+n(H∩E) [∵ if A and B are disjoint then n(A∩B) = n(A)+n (B) ⇒750=n(H−E)+260 ⇒n(H−E)=750−260 = 490 Hence, 490 persons can speak Hindi only. (iii) On s similiar lines we have E=(E−H)∪(H∩E) i.e. E is the disjoint union of E- H ans H∩E ∴n(E)=n(E−H)+n(H∩E) ⇒460=n(E−H)+260 ⇒n(E−H)=460−260 = 200 Hence, 200 persons can speak English only.

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