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set theory

by *arch* » Sun Aug 17, 2008 3:42 am
60% of the males of a community have black eyes, 70% have curly hair and 85% have dark complexion. What is the minimum percent of the males who will have all three characteristics?

(A) 20%
(B) 45%
(C) 80%
(D) 60%
(E) None of these
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by parallel_chase » Sun Aug 17, 2008 9:28 am
IMO 45% whats the OA?
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by pepeprepa » Sun Aug 17, 2008 9:43 am
I would say E
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by 4meonly » Sun Aug 17, 2008 1:12 pm
I think E - there really cannot be any person with all these criteria
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by pranavc » Sun Aug 17, 2008 3:19 pm
I am going with E. What is the OA?
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by Ian Stewart » Sun Aug 17, 2008 3:20 pm
The answer should be E (it should be 15%). Suppose 60% have black eyes (I'll just write B), and 40% do not (I'll write xB). We know 70% have C. We could have:

30% B, C
30% B, xC
40% xB, C

That is, if all of the people who *do not* have B *do* have C, we get the smallest percentage who have *both* B and C. We can continue:

15% B, C, D
15% B, C, xD
30% B, xC, D
40% xB, C, D

Now we have 15% with all three, 60% with B, 70% with C, and 85% with D. So the answer should be E.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by pepeprepa » Mon Aug 18, 2008 1:17 am
I write mine if it helps even if it seems long that's not when you are in.

First, I don't consider percents but a population of 100 guys, I don't think it is an error.

60: black eyes
70: curly hair
85: dark complexion

Minimum of guys with Black eyes and Curly hair is: 70+60-100=30
Minimum of guys with Black eyes and Dark complexion is: 60+85-100=45
Minimum of guys with Dark Complexion and Curly hair is: 70+85-100=55
I think everyone agrees for that, we have the minimum for junction of features.

If you draw the three circles graph and with the figures we found above you can write 3 equations.
Let's consider:
L: guys with only black eyes
K: guys with only curly hair
J: guys with only dark complexion
60=L + 30 + 45 - x
70=K+30+55 - x
85=J+45+55 -x

-15+x=L
-15+x=K
-15+x=J

Given L, K and J cannot be negative, but they can be 0, x must be at least 15.

So the answer is 15 or 15% of course.
Happy to share if you have thoughts about what I did.

Anyway thanks again to Ian, opposite way for this kind of question seems nice.
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