(3*a - 3*b)/(6*a - 6*b) = 1/2

Indeed, your answer key is correct.

I can see why you think it is 0. You are thinking the problem is:

(3*a)/(6*b) - (3*a)/(6*b) = 0.

However, this problem is:

(3*a)/(6*a - 6*b) - (3*b)/(6*a - 6*b)

Do you see they each have a common denominator. Now you can cancel to get:

(3*a)/(6*a - 6*b) = a/[2(a - b)] and:

(3*b)/(6*a - 6*b) = b/[2(a - b)]

Thus, a/[2(a - b)] - b/[2(a - b)] = (a - b)/[2(a - b)]

Cancel the (a - b) from the numerator and denominator. Note that there is a "1 times (a - b)" at the top; thus, you are left with 1/2.

Do you see why it isn't 0?