Question

A survey shows that $63\%$ of the Americans like cheese whereas $76\%$ like apples. The statement,$x\%$ of Americans like both cheese and apples” is true for

• A. $39\le x<63$
• B. $39\le x\le 63$
• C. $39<x<63$
• D. $39<x\le 63$

Answer 1

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

Let $A$ denote, the set of Americans who like cheese and

$B$ denotes those who like apples.

Let the population of America be $100$, then

$n\left(A\right)=63$ and $n\left(B\right)=76$

Now, $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

$n(A\cup B)=63+76-n(A\cap B)$

$n(A\cap B)=139-n(A\cup B)$

But $n(A\cup B)\le 100$

So, $139-n(A\cup B)\ge 139-100$

$139-n(A\cup B)\ge 39$

$n(A\cap B)\ge 39$ …………$\left(i\right)$

Now,$n(A\cap B)\le n\left(A\right)$ and $n(A\cap B)\le n\left(B\right)$

$n(A\cap B)\le 63$ and $n(A\cap B)\le 76$

$n(A\cap B)\le 63$ ……………$\left(ii\right)$

From Eqs. $\left(i\right)$ and $\left(ii\right)$,

$39\le n(A\cap B)\le 63$

$\therefore 39\le x\le 63$