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which of the following could be k?

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Director
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which of the following could be k? [#permalink] New post 06 Apr 2009, 23:15
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If x, y, k are positive numbers such that 10x/(x+y)+20y/((x+y)=k and x<y, which of the following could be the value of k?
A. 10
B. 12
C. 15
D. 18
E. 30
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Re: which of the following could be k? [#permalink] New post 07 Apr 2009, 04:21
I posted two different solutions to this question on another forum:

www.beatthegmat.com/two-questions-from- ... 34185.html
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Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

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Manager
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Re: which of the following could be k? [#permalink] New post 07 Apr 2009, 07:16
I think Ian's common sense approach to everything is phenomenal! Unfortunately, I am not among those who can think in such a way during exam conditions. I went for the simple algebra:

Got: 10(x + 2y) = k (x+y)
Use the answer choices now and with x<y and both positive, you will see only 18 makes sense.

Answer:D
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Re: which of the following could be k? [#permalink] New post 07 Apr 2009, 10:53
Thank you Ian - your solutions are eye opening!
OA D.
For others, I have copied Ian's reply below:

***************************************************
Say you were asked the following question: "There are x men and y women at a certain company. The average wage of the men is $10 per hour, and the average wage of the women is $20 per hour. If k is the average wage of all employees at the company, what is k?" Here, you have x+y employees in total, the men combined earn 10x in total, and the women combined earn 20y in total, so

k = (10x + 20y)/(x + y)

That's the equation in the question -- it's just a weighted average. If the average of one group is 10, and the other is 20, the weighted average must be between 10 and 20, and if y > x (we have more women than men) then the average must be closer to 20 than to 10, so 18 is the only possible value of k among the answer choices.

Alternatively, you could do the question algebraically, as follows. Cross multiply to get:

10x + 20y = kx + ky
20y - ky = kx - 10x
y(20 - k) = x(k - 10)
y/x = (k - 10)/(20 - k)

We know y > x > 0, so the left side is bigger than 1. Notice that the right side is negative unless 10 <= k < 20, and that the numerator will only be larger than the denominator if k > 15. So 15 < k < 20.
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Re: which of the following could be k? [#permalink] New post 07 Apr 2009, 17:56
Awesome problem.

Thanks a lot to Ian - great approaches one can learn from! :wave
Re: which of the following could be k?   [#permalink] 07 Apr 2009, 17:56
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