Two athletes at the same position on a circular 4-mile race track start running. If they run in opposite directions, they meet in 12 minutes. How ever, it takes one hour for the faster runner to gain a lap if they run in the same direction. How fast, in miles per hour, does the faster athlete run?
This is the explanation to the answer provided by the book:
Let $x$ and $y$ be the speed of the faster athlete and slower athlete in mules per hour, respectively. Two athletes meet in 12 minutes or ($\frac{1}{5}$ hr) is 4 miles. This can be written as
$$\frac{x}{5}+\frac{y}{5}=4\Rightarrow x+y=20$$
It takes one hour for the faster athlete to gain a lap if both athletes run n the same directions, which means that the difference of the distances that they run is 4 miles. This can be written as
$$x-y=4$$
The explanation continues to solving the system and eventually get $x=12$. I don't understand the part "...which means the difference of the distances they run is 4 miles...". Why is that? and why the same variables when it says those represent the speed of the two athletes?
The faster runner runs 4 miles further than the slower runner because the problem states that the faster runner has gained a lap on the slower runner (i.e. that the runner has completed an enter extra [4 mile] lap more than the slower runner). They "use the same variables" because it takes 1 hour for that to happen, so $1x-1y=x-y$.