Question

A survey show that 63% of the american likes cheese whereas 76% of the american like apples if x% of the american like both cheese and apples find the value of x.

Answer 1

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

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

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

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

So, $139-n(A\cap B)\le 100$

$n(A\cap B)\ge 39$.......(i)

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

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

$(63$ is smallest between $n\left(A\right)$ and $n\left(B\right))$