PS-1

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.

so this tells us:

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

OR

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

so we know that the answer will be greater than 10

since y>x means that y/(x+y) will be less than 1.

So the answer is between 10 and 20

B. 12
C. 15
D. 18

if y and x are the same y/(x+y) will be 0.5 but and the answer will be 15 but since y > x the ratio is greater than 0.5

So the answer should be > 15

And D remains!

^ better explanation.

(x/(x+y) )(10) +(y/(x+y))(20) = k
It is the equaton of weighted avg. in these cases value of equation lies between multiple of two term (i.e. between 10 or 20) and displace towards higher term i.e. 20.

Now as the avg of 10 and 20 is 15, value will be between 15-10 or 15-20, as 20>10 then value will displace towards 20, the only answer between 15-20 is 18. Answers is D. 18.