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# Each correctional facility in a certain state has the same

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Senior Manager
Joined: 22 Sep 2005
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Each correctional facility in a certain state has the same [#permalink]  23 Jan 2008, 13:22
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Each correctional facility in a certain state has the same number of interns, and each facility has interns of each sex. If the correctional has 22% of the female interns in the state, does it have less than 20% of the male interns?

1) The Correctional has between 18% and 22% of the state's male interns.
2) The Correctional has more female interns than does any other facility in the state.
Director
Joined: 01 Jan 2008
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Re: DS: correctional facility [#permalink]  23 Jan 2008, 13:45
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1 is sufficient, let's denote x - total number of female interns in the state, y - total number of male interns in the state, d - percent of the male interns in the considered facility. We know d is between 18% and 22%.

Total number of facilities is (x+y)/(22%*x+d*y) = n, it's easy to show that 4 < n < 6

(x+y)/(22%*x+d*y) <= (x+y)/(22%*x + 18%*d) < (x+y)/(18%*x+18%*y) = 1/18% < 6
(x+y)/(22%*x + d*y) >= (x+y)/(22%*x + 22%*y) = 1/22% > 4

Therefore, n = 5, then d < 20%

2 is sufficient -> there are at least 4 other facilities or 5 total (22%*4 = 88% < 100%), in each facility there are more male interns than in the considered facility. Therefore percentage of male interns in the considered facility is below 20% of the male interns in the state
CEO
Joined: 17 Nov 2007
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Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
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Kudos [?]: 2244 [0], given: 359

Re: DS: correctional facility [#permalink]  23 Jan 2008, 22:12
Expert's post
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Director
Joined: 03 Sep 2006
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Re: DS: correctional facility [#permalink]  24 Jan 2008, 00:06
total males =M
total females =F
total interns = (M+F)
total number of correctional facilities = n
Each has no. of interns = (M+F)/n
Each has no. of females = (22/100)*F
Each has male interns = x such that => (18/100)*M < x < (22/100)*M
(M+F)/n = (22/100)*F + x

x= (M+F)/n - (22/100)*F

(18/100)*M < (M+F)/n - (22/100)*F < (22/100)*M

how do you get n = 5?

in this case each correctional facility has (M+F)/n interns, but that does not mean each has M/n and F/n interns.

M can be 100, F can be 5 and vice versa!

Could you please explain in detail?
Director
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Re: DS: correctional facility [#permalink]  24 Jan 2008, 06:20
LM wrote:
Each has no. of interns = (M+F)/n
Each has no. of females = (22/100)*F
Each has male interns = x such that => (18/100)*M < x < (22/100)*M
Could you please explain in detail?

LM, it's not "each" but only one correctional facility. One particular correctional facility has 22% of female interns (22/100)*F and has d*M males -> total number of interns in the facility is (22/100)*F + d*M. Since each facility has the same total number of interns (F+M)/((22/100)*F + d*M) has to be a positive integer. Using inequalities I showed that the integer is greater than 4 and less than 6. It has to be equal to 5. Since there are more than 1/5 female interns, d < 1/5, otherwise the facility will have more than 1/5 of all interns which contradicts the condition that each facility has the same number of interns.
Senior Manager
Joined: 22 Sep 2005
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Re: DS: correctional facility [#permalink]  24 Jan 2008, 10:39
You are right.

OA is D
Re: DS: correctional facility   [#permalink] 24 Jan 2008, 10:39
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