Question

A survey shows that 63% of the Indians like Paneer whereas 76% like Tofu. If x% of the Indians like both Paneer and Tofu, then which of the following is/are correct?

• A. Maximum value of x is 63.
• B. Maximum value of x is 76.
• C. Minimum value of x is 39.
• D. Minimum value of x is 0.

Answer 1

Answer: (A), (C)

Minimum value of x is 39.

Let X denote the set of Indians who like Paneer and let Y denote the set of Indians who like Tofu.
Let the population of India be 100. Then, n(A) = 63, n(B) = 76.
We know that,
$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)\Rightarrow n(A\cup B)=63+76-n(A\cap B)\Rightarrow n(A\cup B)=139-x\mathrm{Since}\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{Indians}\mathrm{who}\mathrm{like}\mathrm{both}\mathrm{Paneer}\mathrm{and}\mathrm{Tofu}\mathrm{will}\mathrm{be}\mathrm{less}\mathrm{than}\mathrm{or}\mathrm{equal}\mathrm{to}100.\therefore n(A\cup B)\le 100\Rightarrow 139-x\le 100\Rightarrow x\ge 39.\mathrm{The}\mathrm{minimum}\mathrm{value}\mathrm{of}x\mathrm{is}39.\mathrm{We}\mathrm{know}\mathrm{that},A\cap B\subset A\mathrm{and}A\cap B\subset B\therefore n(A\cap B)\le n\left(A\right)\mathrm{and}n(A\cap B)\le n\left(B\right)\Rightarrow n(A\cap B)\le 63\mathrm{and}n(A\cap B)\le 76\Rightarrow n(A\cap B)\le 63\Rightarrow x\le 63\mathrm{Maximum}\mathrm{value}\mathrm{of}x\mathrm{is}63$