A survey shows that 63% of the Americans like cheese whereas 76% like apples. If  x% of the Americans like both cheese and apples, then a value of x can be

Let A denote the set of Americans who like cheese and let B denote the set of Americans who like apples. Let population of America be 100.  Then n(A)=63,n(B)=76 Now, n(A∪B)=n(A)+n(B)−n(A∩B)=63+76−n(A∩B)⇒ n(A∩B)=139−n(A∪B) But n(A∪B)≤100∴ 139−n(A∪B)≥139−100=39∴ n(A∩B)≥39 i.e., 39≤n(A∩B) Again, A∩B⊆A,A∩B⊆B∴ n(A∩B)≤n(A)=63 and n(A∩B)≤n(B)=76∴ n(A∩B)≤63 Then, 39≤n(A∩B)≤n(B)⇒39≤x≤63