I saw the question below in the 12th edition of the official study guide. I need further explanation of why it was solved the way it was solved in the official guide.

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

We transform k = x/(x + y) × (10) + y/(x + y) × (20) = x/(x + y) × (10) + y/(x + y) × (10) + y/( x+y ) × (10) = (x/( x+y ) + y/( x+y )) × (10) + y/(x + y) × (10) = (x + y)/(x + y) × (10) + y/(x+y) × (10) = 10 + y/(x+y) × (10)

k = 10 + y/(x+y) × (10)

y/(x+y) < 1 because y < (x + y)

Since x < y then y/(x + y) > y/(y + y) = 1/2

So 1/2 < y/(x + y) < 1
multiply all sides by 10:
1/2 × 10 < y/(x + y) × 10 < 1 × 10
add 10
10 + 1/2 × 10 < 10 + y/(x + y) × 10 < 10 + 1 × 10
We see that we have the expression for k in the middle, therefore:

15 < k < 20

The only answer choice that fits is (D).

The main ideas we use are following:
- proper transformation of the expression for k
- evaluation of y/(x + y) based on fact that 0 < x < y.