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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.

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Let , Total number of people be n(P) = 950 

People who can speak English n(E) = 460 

People who can speak Hindi n(H) = 750 

i. How many can speak both Hindi and English. 

People who can speak both Hindi and English = n (H ∩ E) 

We know, 

n (P) = n(E) + n(H) – n (H ∩ E) 

Substituting the values we get 

950 = 460+750 – n (H ∩ E) 

950= 1210 – n (H ∩ E) 

n (H ∩ E) = 260. 

Number of people who can speak both English and Hindi are 260. 

ii. How many can speak Hindi only. 

We can see that H is disjoint union of n(H–E) and n (H ∩ E). 

(If A and B are disjoint then n (A ∪ B) = n(A) + n(B)) 

∴ H = n(H–E) ∪ n (H ∩ E). 

⇒ n(H) = n(H–E) + n (H ∩ E). 

⇒ 750 = n (H – E)+ 260 

⇒ n(H–E) = 490. 

Only, 490 people speak Hindi. 

iii. how many can speak English only. 

We can see that E is disjoint union of n(E–H) and n (H ∩ E). 

(If A and B are disjoint then n (A ∪ B) = n(A) + n(B)) 

∴ E = n(E–H) ∪ n (H ∩ E). 

⇒ n(E) = n(E–H) + n (H ∩ E). 

⇒ 460 = n (H – E)+ 260 

⇒ n(H–E) = 200. 

Only, 200 people speak English.

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