So, adding two rationals is the same as adding two such fractions, which will result in another fraction of this same form since integers are closed under addition and multiplication. Thus, adding two rational numbers produces another rational number.
a/b and c/d are rational numbers with a,b,c,d intergers
so with the same denominators ( equaling a+c over b) or different ones (equaling ab+cb over bd) they end up with the answers of
same denominator= b is not equal to 0
different denominators= d is not equal to 0
this may not make sense but i'm pretty sure i understood you right