Here, we have,
Suppose, the total number of people be n (P) = 950
The people who can speak English n (E) = 460
The people who can speak Hindi n (H) = 750
(i) How many people can speak both Hindi and English.
The people who can speak both Hindi and English = n (H ∩ E)
As we know,
n (P) = n (E) + n (H) – n (H ∩ E)
By 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 people can speak Hindi only.
As 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
∴ 490 people can speak only Hindi.
(iii) How many people can speak English only.
As 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) = 460 – 260 = 200
Hence, 200 people can speak only English.
