I'm not sure if this is the quickest way, but here's how I got the answer:
We can rewrite the problem to this:
Factor out 10.
10((x/(x+y) + 2(y/(x+y)) = k
Now combine the two fractions because they have the same denominator:
10((x+2y)/(x+y)) = k
So 10 times this fraction is k.
If k is 10, that would mean that the fraction must be 1 because 10*1 = 10. But ((x+2y)/(x+y)) can never be 1 because the numerator always multiplies y by 2 while the denominator is just y and we know that y is a positive number. Thus the numerator will always be bigger than the denominator, or always greater than 1.
If k is 12, then we have the equation 10*((x+2y)/(x+y)) = 12. Then ((x+2y)/(x+y)) must be 12/10 or 6/5. Then we have two equations. x+2y = 6 and x+y = 5. Solving this will be x=4, y=1. But, it was given that x<y so this cannot be the answer.
If k is 15, then we have the equation 10*((x+2y)/(x+y)) = 15. Then ((x+2y)/(x+y)) must be 15/10, or 3/2. Then we have two equations. x+2y = 3 and x+y = 2. Solving this will be x = 1, and y = 1. But x must be less than y so this cannot be it.
If k is 18, then we have the equation 10*((x+2y)/(x+y)) = 18. Then ((x+2y)/(x+y)) must be 18/10, or 9/5. Then we have two equations x+2y = 9 and x+y = 5. Solving this will be x = 1, y = 4. Since 1 is less than 4, this is the solution.
So the answer is D.
Now do this in 2 minutes!
I'm sure there's an easier way, I just can't figure it out at the moment.
