# 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

$\begin {array} {1 1} (A)\;39 \leq x < 63 & \quad (B)\; 39 \leq x \leq 63 \\ (C)\;39 < x < 63 & \quad (D)\;39 < x \leq 63 \end {array}$

Ans : (B)
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
$n(A)=63\: and\: n(B)=76$
Now, $n(A \cup B)=n(A) + n(B) - n(A \cap B)$
$N(A \cup B)=63+76 - n(A \cap B)$
$N(A \cap B)=139 - n(A \cup B)$
But $n(A \cup B) \leq 100$
So, $139 - n(A \cup B) \geq 139 - 100$
$139 - n(A \cup B) \geq 39$
$n(A \cap B) \geq 39$…………(i)
Now,$n(A \cap B) \leq n(A) \: and\: n(A \cap B) \leq n(B)$
$n(A \cap B) \leq 63\: and\: n(A \cap B) \leq 76$
$n(A \cap B) \leq 63$……………(ii)
From Eqs. (i) and (ii),
$39 \leq n(A \cap B) \leq 63$
$39 \leq x \leq 63$
answered Jan 23, 2014