There are different cable channels, namely Airtel, TataSky and Dish TV. In a survey, it was found that 85% of the viewers respond to Dish TV, 20% to TataSky, and 30% to Airtel. 20% of the viewers respond to exactly two channels and 5% to none. What percentage of the viewers responded to all three channels?
Answers
Answer:
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Solution :-
Let us assume that, the total viewers are 100 .
So,
→ Number of viewers respond to Dish TV = n(D) = 85% of 100 = 85
→ Number of viewers respond to Tata sky = n(T) = 20% of 100 = 20
→ Number of viewers respond to Airtel = n(A) = 30% of 100 = 30
and,
→ Number of viewers respond to none = 5% of 100 = 5 .
So,
→ n (D u T u A) = 100 - 5 = 95 ------- Eqn.(1)
and,
→ Number of viewers respond to exactly two channels = n(D n T) + n(T n A) + n(A n D) - 3•n(D n T n A) = 20% of 100 = 20 -------- Eqn.(2)
now, we know that,
- n(A u B u C) = n(A) + n(B) + n(C) - n(A n B) - n(B n C) - n(C n A) + n(A n B n C)
then,
→ n(D u T u A) = n(D) + n(T) + n(A) - n(D n T) - n(T n A) - n(A n D) + n(D n T n A)
adding and subtracting 3•n(D n T n A) in RHS side we get,
→ n(D u T u A) = n(D) + n(T) + n(A) - n(D n T) - n(T n A) - n(A n D) + n(D n T n A) + 3•n(D n T n A) - 3•n(D n T n A)
→ n(D u T u A) = n(D) + n(T) + n(A) - n(D n T) - n(T n A) - n(A n D) + 3•n(D n T n A) + {n(D n T n A) - 3•n(D n T n A)}
→ n(D u T u A) = n(D) + n(T) + n(A) - [n(D n T) + n(T n A) + n(A n D) - 3•n(D n T n A)] + {n(D n T n A) - 3•n(D n T n A)}
→ n(D u T u A) = n(D) + n(T) + n(A) - [n(D n T) + n(T n A) + n(A n D) - 3•n(D n T n A)] - 2•n(D n T n A)}
putting values from above and Eqn.(1) and Eqn.(2) we get,
→ 95 = 85 + 20 + 30 - 20 - 2 * n(D n T n A)}
→ 95 = 85 + 30 - 2 * n(D n T n A)}
→ 95 = 115 - 2 * n(D n T n A)}
→ 2 * n(D n T n A)} = 115 - 95
→ 2 * n(D n T n A)} = 20
→ n(D n T n A)} = 10 .
therefore,
→ Required % = (10 * 100)/100 = 10% (Ans.)
Hence, 10% of the viewers responded to all three channels .
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