The set of rationals is closed under addition and nonzero division. Thus there is no way that the final expression you gave could be irrational.
I have the following problem, but I don't know what to do. I have been told by someone to think about gcd(a+c, b+d), but I still don't know how to prove this. Thanks for your help!
"Prove that for any positive integers a, b, c, d satisfying
no cancellation is possible in the fraction
, where the result is a rational number."