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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.

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.

Grade:11

6 Answers

Sher Mohammad IIT Delhi
askIITians Faculty 174 Points
7 years ago

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



Ramesh sharma
30 Points
3 years ago
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
Alok kumar
19 Points
3 years ago
Let A denotes, the set of Americans who like cheese andB denotes those who like apples. Let the population of America be 100, thenn(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)andn(A∩B)≤n(B)n(A∩B)≤n(A)andn(A∩B)≤n(B)n(A∩B)≤63andn(A∩B)≤76n(A∩B)≤63andn(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
Krish Gupta
askIITians Faculty 82 Points
one year ago
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 !!
 
Krish Gupta
askIITians Faculty 82 Points
one year ago
 

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

 

Kushagra Madhukar
askIITians Faculty 629 Points
one year ago
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

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