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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.

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6 years ago IIT Delhi
174 Points
```							n(C U A) = n(A) + n(C) - n(C n A)100 = 76 + 63 - X100 = 139 - XX = 139 - 100X = 39sher mohammadB.Tech, IIT Delhi
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6 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
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2 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
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2 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, 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≤63Cheers !!
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5 months ago
```							 n(C U A) = n(A) + n(C) - n(C n A)100 = 76 + 63 - X100 = 139 - XX = 139 - 100X = 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
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5 months ago 605 Points
```							Dear student,Please find the solution to your problem below. Let us assume the no. of people surveyed to be 100Let A denotes, the set of Americans who like cheese and B denote those who like apples, thenn(A) = 63  and   n(B)=76Now,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) ≤ 100So, 139 − n(A∪B) ≥ 139−100139 − n(A∪B) ≥ 39Hence, 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) ≤ 76Hence, n(A∩B) ≤ 63 -----------(ii) From Eqs. (i) and (ii),39 ≤ n(A∩B) ≤ 6339 ≤ x ≤ 63 Hope it helps.Thanks and regards,Kushagra
```
4 months ago
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