Two cyclists, k kilometres apart, and starting at the same time, would be together in r hours if they travelled in the same direction, but would pass each other in t hours if they travelled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is :

Let the speed of the faster and slower cyclists be x km/hr and y km/hr respectively
Then,
$$\eqalign{ & \frac{k}{{x - y}} = r \cr & \left( {x - y} \right)r = k.....(i) \cr & {\text{And,}} \cr & \frac{k}{{x + y}} = t \cr & \left( {x + y} \right)t = k.....(ii) \cr} $$
From (i) and (ii), we have :
$$\eqalign{ & \Rightarrow \left( {x - y} \right)r = \left( {x + y} \right)t \cr & \Rightarrow xr - yr = xt + yt \cr & \Rightarrow xr - xt = yr + yt \cr & \Rightarrow x\left( {r - t} \right) = y\left( {r + t} \right) \cr & \Rightarrow \frac{x}{y} = \frac{{r + t}}{{r - t}} \cr} $$