Two bicyclists travel in opposite directions. One travels $5$ miles per hour faster than the other. In $2$ hours they are $50$ miles apart. What is the rate of the faster bicyclist?

Let the speed of the slower bicyclist be $x$ miles per hour. Then the speed of faster bicyclist would be $x+5$ miles per hour.

Using the concept of relative motion here, let’s keep the slower bicyclist at rest. Then the faster bicyclist would be traveling towards the slower bicyclist with a speed of $x+5-(-x)=2x+5$ miles per hour.

Now, the faster bicyclist travels $50$ miles in $2$ hours.

$\therefore$ $2=\frac{50}{2x+5}$ ($\because$ time $=$ distance/ speed)

$\Rightarrow$ $x=10$ miles per hour.

So, the speed of the faster bicyclist is, $10+5=15$ miles per hour.