You can take advantage of the answer choices and some math "logic" to get to the correct answer. Let's TEST THE ANSWERS....
We're told that X, Y and K are POSITIVE and that X < Y.
We're also told that 10X/(X+Y) + 20Y/(X+Y) = K. We're asked what COULD be the value of K, which means that there's more than one possible answer. Since the answers are NUMBERS, one of the them MUST be a possible answer.
We can manipulate the given equation into:
10X + 20Y = K(X+Y)
Since all of the variables are POSITIVE, K MUST be between 10 and 20. Here's the proof:
IF.....K=10, then the equation becomes...
10X + 20Y = 10X + 10Y
20Y = 10Y
Since Y is positive, 20Y = 10Y is NOT possible.
Eliminate Answer A
In that same way, K can't be 20 (or greater) because the end equation would be an impossibility.
With the remaining 3 answers, we can TEST the possibilities...
IF...K = 12, then the equation becomes...
10X + 20Y = 12X + 12Y
8Y = 2X
4Y = X
In this scenario, X > Y which is the OPPOSITE of what we were told. This is NOT the answer.
IF....K=15, then the equation becomes...
10X + 20Y = 15X + 15Y
5Y = 5X
Y = X
In this scenario, X = Y, which is NOT a match for what we were told. This is NOT the answer.
IF....K=18, then the equation becomes....
10X + 20Y = 18X + 18Y
2Y = 8X
Y = 4X
Here, X < Y, which IS a match for what we were told. This IS a POSSIBLE answer.
GMAT assassins aren't born, they're made,