Question

A survey shows that $70\%$ of the Indian like mango whereas $82\%$ like an apple. If $x\%$ of Indian like both mango and apples then

• A. $x=52$
• B. $52\le x\le 70$
• C. $\mathrm{x}=70$
• D. $70\le x\le 82$

Answer 1

The correct option is B

$52\le x\le 70$

Explanation for correct option:

Step:1 Let $100$ denote the population of India

The percentage of people who like mangoes, $n\left(A\right)=70$

The percentage of people who like apples , $n\left(B\right)=82$

Step:2 Let $n(A\cap B)$ be $x\%$ of Indian:

$\Rightarrow n(A\cap B)=152-n(A\cup B)$

but $n(A\cup B)\le 100$ $\mathit{n}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}\mathbf{=}\mathit{n}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{+}\mathit{n}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{-}\mathit{n}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}$

$\Rightarrow$ $n(A\cap B)\ge 52$

$\Rightarrow$ $n(A\cap B)\le 70$

$\Rightarrow$$52\le n(A\cap B)\le 70$

$\mathbf{\therefore }\mathbf{52}\mathbf{\le }\mathit{x}\mathbf{\le }\mathbf{70}$

Hence, Option ‘B’ is Correct.