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Sets & Relations: Survey Shows Americans Like Cheese Whereas Like Apples

JEE Maths question with a full step-by-step solution.

Question
A survey shows that 63%63\% of the Americans like cheese whereas 76%76\% like apples. If x%x\% of the Americans like both cheese and apples, then:
Ax=39x = 39
Bx=63x = 63
C39≤x≤6339 \le x \le 63correct
Dx=50x = 50
Solution
Let n(A)=63n(A) = 63, n(B)=76n(B) = 76. By inclusion-exclusion,
n(A∩B)=n(A)+n(B)−n(A∪B)=139−n(A∪B).n(A \cap B) = n(A) + n(B) - n(A \cup B) = 139 - n(A \cup B).
 Step 1: n(A∪B)≤100⇒n(A∩B)≥139−100=39.n(A \cup B) \le 100 \Rightarrow n(A \cap B) \ge 139 - 100 = 39. Step 2: n(A∩B)≤min⁡(63,76)=63.n(A \cap B) \le \min(63, 76) = 63.
∴39≤x≤63.\therefore 39 \le x \le 63.
 Correct answer: (3)
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