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

Let the total numbers be 100 

given that $\mathrm{n}\left(\mathrm{A}\right)=63,\mathrm{n}\left(\mathrm{B}\right)=76$ 
$\mathrm{n}(\mathrm{A}\cap \mathrm{B})=\mathrm{x}$

 we know that $\mathrm{n}(\mathrm{A}\cap \mathrm{B})\le$ minimum of ( n(A), n(B) )
$\therefore \mathrm{n}(\mathrm{A}\cap \mathrm{B})=\mathrm{x}\le 63$ 

we know that

 $$\begin{gathered} \mathrm{n}(\mathrm{A}\cup \mathrm{B})\le 100 \\ \Rightarrow n\left(A\right)+n\left(B\right)-n(A\cap B)\le 100\Rightarrow 63+76-\mathrm{x}\le 100\Rightarrow \mathrm{x}\ge 39 \\ \Rightarrow 39\le \mathrm{x}\le 63 \end{gathered}$$