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.