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which of the following could be k? [#permalink]
06 Apr 2009, 23:15

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Question Stats:

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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

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.

Re: which of the following could be k? [#permalink]
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.

Re: which of the following could be k? [#permalink]
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:

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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