Question

A survey shows that $63$% of Americans like cheese where as $76$% like apples. If $x$% of the american like both cheese and apples, then

• A. $x=39$
• B. $x=63$
• C. $39\le x\le 63$
• D. None of these

Answer 1

The correct option is C $39\le x\le 63$

Let $C$ and $A$ denotes the percentage of people likes cheese and apples respectively.

Given, $n\left(C\right)=63,n\left(A\right)=76$ and $n(C\cap A)=x$.
We know that,
$n(C\cup A)=n\left(C\right)+n\left(A\right)-n(C\cap A)$
$\Rightarrow n(C\cup A)=63+76-x$
$\Rightarrow x=139-n(C\cup A)$

Now, $n(C\cup A)\le 100$

$\Rightarrow 139-n(C\cup A)\ge 139-100=39$

$\Rightarrow x\ge 39$

Also, $n(C\cap A)\le n\left(C\right)$ and $n(C\cap A)\le n\left(A\right)$
$\Rightarrow x\le 63$
$\therefore 39\le x\le 63$.