A survey shows that $73$% of the persons working in an office like coffee, whereas $65$% like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be :
(A)
$63$
(B)
$36$
(C)
$54$
(D)
$38$
MEDIUM
Correct Answer:B
Explanation
Let C be the set of persons who like coffee and T be the set of persons who like tea. Given: $n(C) = 73$ and $n(T) = 65$. Also, $n(C \cup T) \le 100$. Using the formula for the union of two sets: $n(C \cup T) = n(C) + n(T) - n(C \cap T)$. Let $x = n(C \cap T)$ be the percentage of people who like both coffee and tea. So, $n(C \cup T) = 73 + 65 - x \le 100$. This implies $138 - x \le 100$, which gives $x \ge 38$. Also, $x$ must be less than or equal to both $73$ and $65$ since $x$ represents the intersection. So, $x \le 73$ and $x \le 65$. Therefore, $38 \le x \le 65$. Since $x$ must be in the range $[38, 65]$, the value $x=36$ is not possible.