Let's say we need to do a comparison. Set
a = 34241432415/344425151233
b = 45034983295/453218433828
Or even more feindish, Set
a = 14488683657616/14488641242046
b = 10733594563328/10733563140768
By what algorithm would you do the computation, and could you guarantee me the same compute time as comparing 2/3 and 4/5?
a/b > x/y is the same as ay > xb
Assuming you don’t overflow your integer type.
There's your answer :)
It's far too easy to overflow your integer type by simply adding a bunch of rationals whose denominators happen to be coprime, or just by multiplying rationals. For this reason, the vast majority of rational implementations use arbitrary precision integers, and of course arithmetic on those isn't constant time.