jmc

The first approach that comes to mind is probably also the best one. So we cross-multiply to obtain $(x-4)(x-9)=36$. Multiplying out the left-hand side and cancelling the 36 yields $x^2-13x=0$, or $x(x-13)=0$. This equation has two solutions, $x=0$ and 13. Since we are looking for positive answers, we take $x=\boxed{13}$.