A survey shows that 63% of the people watch a News channel whereas 76% watch another channel.
Question:

A survey shows that 63% of the people watch a News channel whereas 76% watch another channel. If x % of the people watch both channel, then

(a) x = 35

(b) x = 63

(c) 39 ≤ x ≤ 63

(d) x = 39

Solution:

Let A denote the percentage of people watching a news channel

Let B denote the percentage of people watching other channel

i.e

n(A) = 63

n(B) = 76

Let n(A∩B) = x

then n(A⋃B) = n(A) + n(B) – n(A∩B) = 63 + 76 – x

n(A⋃B) = 139 – x

i.e x = 139 – n (A⋃B)

Since n(A⋃B) ≤ 100

i.e. 139 – n(A⋃B) ≥ 139 – 100 = 39

i.e. x ≥ 39.

also, n(A∩B) ≤ n(A) and n(A∩B) ≤ n(B)

⇒ n(A∩B) ≤ 63

⇒ 39 ≤ n(A∩B) = x ≤ 63

Hence, the correct answer is option C.

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