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

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