$\frac{10x+20y}{x+y}$ =k

10+ $\frac{10y}{x+y}$ = k

$\frac{10y}{x+y}$ = (k-10)

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

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

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

we have x < y ... so : the factor are reversed

(20-K) < (k-10)

30 < 2k

k > 15

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.

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.