Question

A survey conducted in a city reveals that $48$% children like cricket while $77$% children like football. Then the percentage of children who like both cricket and football can be

• A. $23$
• B. $31$
• C. $51$
• D. $65$

Answer 1

The correct option is B $31$

Let us assume that the number of children in city be $100.$
Let $C$ denote the set of children who like cricket,
and $F$ denote the set of children who like football.
$n\left(C\right)=48$ and $n\left(F\right)=77$
$n(C\cup F)\le 100$
$\Rightarrow n\left(C\right)+n\left(F\right)-n(C\cap F)\le 100$
$\Rightarrow 48+77-n(C\cap F)\le 100$
$\Rightarrow n(C\cap F)\ge 25\cdots \left(1\right)$

Also, $n(C\cap F)\le min\left\{n\right(C),n(F\left)\right\}$
$\Rightarrow n(C\cap F)\le 48\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$
$25\le n(C\cap F)\le 48$