In a survey, 63% of Americans like cheese, whereas 76% like apples. If x% of the Americans like both cheese and apples, then find the range of x.
Answer: A
Let C be the set of people who like cheese and A be the set of people who like apples.
n(C) = 63, n(A) = 76. We are looking for n(C ∩ A) = x.
The value of x (the intersection) must be less than or equal to the smaller of the two individual percentages. So, x ≤ 63.
We also know n(C ∪ A) = n(C) + n(A) - n(C ∩ A). The maximum value for the union is 100% (everyone). So, \(100 \ge 63 + 76 - x \Rightarrow 100 \ge 139 - x \Rightarrow x \ge 39\).
Combining the two inequalities, we get 39 ≤ x ≤ 63.