The answer by lhf is the
only real way (other than the elegant approach described by Bill Dubuque). You can use some ad hoc techniques to makes the numbers smaller. One possible trick is that for positive integers $a,b,c,d$ we always have that the mediant $(a+c)/(b+d)$ is between $a/b$ and $c/d$. Works here. The mediant of the first two is $260/520=1/2$, so to compare these two you only need to decide, which is larger than $1/2$, and that is easier with lhf's test (= the definition of the order of rationals).
A weighted mediant might work better sometimes, but occasionally you just cannot avoid getting some dirt on your hands.