Picking numbers would actually be the most efficient strategy in my opinion. You could however reason this out but you have to be wary of this method. First, see that when you add the same thing to both the denominator and the numerator than the fraction will increase in value if the numerator is smaller than the denominator and the number you add is positive. Now if these conditions are not met, then the reverse will happen.

there is a rule,
given a,b > 0 and if another num x>0 is added
if a>b then a+x/b+x<a/b
if a<b then a+x/b+x> a/b
if a=b then a+x/b+x=a/b