Just to record a nice fact, related to the answers already given: if $r, s$ are distinct slopes on the torus, then their Farey distance is bounded above by $\log_2(i(r, s)) + 1$, where $i(r, s)$ is the geometric intersection number (which is also the unsigned algebraic intersection number). A moment's thought shows that this bound is not tight.
Nonetheless, writing the bound this way can be useful.