Problem Solving

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

is just the equation for a weighted average. It's exactly the equation you would write down if you were asked the following question:

At Company A, the x men employees make an average of $10/hour. The y women employees make an average of $20/hour. What is the average wage of all employees at Company A?

If y > x, the answer could be anything between 15 and 20. D.

ildude02 wrote:How did you certainly conclude that it should be between 15 and 20. I can see that the average wage should be closer towards the Y's wage since y is greater then x, but not sure how we could say that it should be inbetween 15 and 20. Is it just because, if we have choose 14, it will be closer to X's wage then Y and that would mean , X is more than Y?

Yes, that's exactly right- because 15 is halfway between 10 and 20, if you have more men, the average wage will be between 10 and 15, and if you have more women, the average wage will be between 15 and 20.

If you can recognize that the given equation represents a weighted average, you can answer this question very quickly. If not, you can try choosing numbers, of course, but good luck finding a combination that gives you the exact answer (18) here- you'd need luck to get there. Or you can try the problem algebraically, though I don't like this solution much:

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

10x + 20y = kx + ky

(20 - k)y = (k - 10)x

Now we know that x < y, so it must be true that
k - 10 > 20 - k
2k > 30
k > 15

But we also know that y and x are positive. Since k > 15, k-10 must be positive, and because (20 - k)y = (k - 10)x, then 20-k must also be positive, so k < 20. k = 18 is the only possible answer choice.

There will surely be other ways to see the answer, and I'm sure there are algebraic ways more elegant than the above, but if you can recognize the equation as a weighted average, then you get the answer in ten seconds. Weighted averages are not uncommon on the GMAT, so it's worthwhile learning to recognize them.