If x, y, and k are positive numbers such that ((x)/(x+y))(10) + ((y)/(x+y))(20) = k and if x < y

If x, y, and k are positive numbers such that ((x)/(x+y))(10) + ((y)/(x+y))(20) = k and if x < y, which of the following could be the value of k?
A. 10
B. 12
C. 15
D. 18
E. 30




The trick is to recognize that the equation just gives a weighted average of x and y. The equation above is the same as you'd use if you were asked "If x pounds of peanuts and y pounds of cashews are mixed together, and peanuts cost $10/pound and cashews cost $20/pound, what is the price per pound of the resulting mixture?" The answer must be between 10 and 20, and if y > x, the answer must be closer to 20 than to 10. So 18 is the only possible answer.

Thu Jan 28, 2010 1:23 pm
rahul.s wrote: Ian, In such a problem, where we need to experiment with numbers, how would we know which are the right numbers to choose? Well, you don't need to experiment with numbers in the question above; I didn't in my post above. That said, if you don't see a conceptual or algebraic solution fairly quickly, plugging in numbers is a good fallback option. It would be a bit lucky to find numbers that give the exact answer to this question, but if you plug in a few very simple sets of numbers (you don't want to waste any time on complicated numbers), making sure that x is less than y, you'll always find that k is between 15 and 20, which may lead you to the correct answer here. There are also algebraic solutions to the question: (10x + 20y)/(x+y) = k 10x + 20y = kx + ky 20y - ky = kx - 10x y(20 - k) = x(k - 10) (20 - k)/(k - 10) = x/y and since 0 < x < y, then 0 < x/y < 1, and it must be that 0 < (20 - k) / (k - 10) < 1. From the answer choices we can be sure k - 10 isn't negative, so we can multiply through this inequality by k-10 to find that 0 < 20 - k < k - 10, or that 15 < k < 20.