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.

							
n(C U A) = n(A) + n(C) - n(C n A)
100 = 76 + 63 - X
100 = 139 - X
X = 139 - 100
X = 39

sher mohammad
B.Tech, IIT Delhi



						

							
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
Cheers !!
 
						

							
 
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)
 
Here, we run into a problem. Since it hasn’t been mentioned whether there are people who like neither of apple nor cheese. We have to either assume it to be 0 or solve for an arbitary value. So if this question comes as an MCQ, take it to be 0, which will give n(A∪ B) = 100, then your answer will be 39. Otherwise in a subjective question, Proceed as follows:
 
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