A survey shows that 63% of the Americans like cheese whereas 76% like apples. If the x% of the Americas like both cheese and apples , find the value of x.

Let A denotes, the set of Americans who like cheese andB denotes those who like apples. Let the population of America be 100, then n(A)=63andn(B)=76n(A)=63andn(B)=76Now, n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∪B)=63+76−n(A∩B)N(A∩B)=139−n(A∪B)N(A∩B)=139−n(A∪B)But n(A∪B)≤100n(A∪B)≤100So, 139−n(A∪B)≥139−100139−n(A∪B)≥139−100139−n(A∪B)≥39139−n(A∪B)≥39n(A∩B)≥39n(A∩B)≥39…………(i)Now,n(A∩B)≤n(A)and n(A∩B)≤n(B)n(A∩B)≤n(A)and n(A∩B)≤n(B)n(A∩B)≤63and n(A∩B)≤76n(A∩B)≤63and n(A∩B)≤76n(A∩B)≤63n(A∩B)≤63……………(ii)From Eqs. (i) and (ii),39≤n(A∩B)≤6339≤n(A∩B)≤6339≤x≤63

 

n(C U A) = n(A) + n(C) - n(C n A)

100 = 76 + 63 - X

100 = 139 - X

X = 139 - 100

X = 39

 

You can tell like that total % of union of set C and A can only be 100 that is why we take n(c U a) = 100

 

Dear student,

Please find the solution to your problem below.

 

Let us assume the no. of people surveyed to be 100

Let A denotes, the set of Americans who like cheese and B denote those who like apples, then

n(A) = 63  and   n(B)=76

Now,

n(A∪B) = n(A) + n(B) − n(A∩B)

n(A∪B) = 63 + 76 − n(A∩B)

n(A∩B) = 139 − n(A∪B)

 

 

n(A∪B) ≤ 100

So, 139 − n(A∪B) ≥ 139−100

139 − n(A∪B) ≥ 39

Hence, n(A∩B) ≥ 39 -----------(i)

Now, n(A∩B) ≤ n(A)     and     n(A∩B) ≤ n(B)

Hence, n(A∩B) ≤ 63    and     n(A∩B) ≤ 76

Hence, n(A∩B) ≤ 63 -----------(ii)

 

From Eqs. (i) and (ii),

39 ≤ n(A∩B) ≤ 63

39 ≤ x ≤ 63

 

Hope it helps.

Thanks and regards,

Kushagra