Question

A survey shows that $63\%$ of the Americans like cheese whereas $76\%$ like apples.

If $x\%$ of the Americans 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

$39\le x\le 63$

Explanation for the correct option:

Find the value of $x$:

Let $C$ and $A$ indicates the percentage of people who like cheese and apples respectively.

Given, $n\left(A\right)=76$, $n\left(C\right)=63$ and $n(C\cap A)=x$

As we know,

$\Rightarrow$$n(C\cup A)=63+76-x$ $\mathbf{\left[}\mathit{n}\mathbf{\right(}\mathit{C}\mathbf{\cup }\mathit{A}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathbf{(}\mathit{C}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathit{n}\mathbf{(}\mathit{A}\mathbf{)}\mathbf{}\mathbf{-}\mathbf{}\mathit{n}\mathbf{(}\mathit{C}\mathbf{\cap }\mathit{A}\mathbf{\left)}\mathbf{\right]}$

$\Rightarrow$ $x=139-n(C\cup A)$

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$

$\mathbf{\therefore }\mathbf{39}\mathbf{}\mathbf{\le }\mathbf{}\mathit{x}\mathbf{}\mathbf{\le }\mathbf{}\mathbf{63}$

Hence, Option ‘C’ is Correct.