Question

A survey shows that 63% of Americans like cheese whereas 76% like apples. If x% of the American like both cheese and apples, then
A. x = 39
B. x = 63
C. \[39 \leqslant x \leqslant 63\]
D. None of these

Answer 1

Hint: Here, quantities are given in percentage, so total people surveyed are 100. We will use venn rules and formulas to find the value of x. Given that 63% like cheese and 76% like apple but some people who like cheese also like apples and some people who like apples also like cheese.

Complete step-by-step answer:
 Let C denote the percentage of people who like cheese and and A denote the percentage of people who like apples.
Given, n(C) = 63, n (A) = 76 and n (C ∩ A) = x.
We know that,
$n (C ∪ A) = n (C) + n (A) – n (C ∩ A)$
$\Rightarrow n (C ∪ A) = 63 + 76 − x $
$\Rightarrow x = 139 – n (C ∪ A) $ …(i)
Now, $n (C ∪ A) ≤ 100 $ [Since, total people surveyed is 100]
$\Rightarrow 139 – n (C ∪ A) ≥ 139 – 100 = 39 $ [From (i)]
$\Rightarrow x ≥ 39 $
Also, $ n (C ∩ A) ≤ n(C) $ and $ n (C ∩ A) ≤ n (A) $
$ \Rightarrow x ≤ 63 $
$ \therefore 39 ≤ x ≤ 63 $.
Thus, the value of x lies between 39 and 63 also; the value of x can be 39 or 63.
So, the correct answer is “Option C”.

Note: In these types of questions, we use the concept of union and intersection. From the total people surveyed 63% like cheese some of these people also included in liking apples and from 76% people liking apples also includes people liking cheese.