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

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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),

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

$39 \leq x \leq 63$

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