Sophie bikes s miles per hour. Her friend Rana is slower. Rana bikes r miles per hour. Rana and Sophie went for a 20-mile bike ride today. Each girl biked at her usual speed. How long did Sophie wait at the end of the route?
To solve this problem, we need to use the formula:
time = distance / speed
Let's start by finding the time it took for Sophie to complete the 20-mile bike ride:
time for Sophie = distance / speed = 20 / s
Now, let's find the time it took for Rana to complete the same distance:
time for Rana = distance / speed = 20 / r
Since Sophie is faster than Rana, she will finish the bike ride before Rana. Therefore, Sophie will have to wait for Rana at the end of the route. The amount of time she waits is equal to the difference between their times:
wait time = time for Rana - time for Sophie
wait time = (20 / r) - (20 / s)
So, the answer to the question is:
Sophie waited for (20 / r) - (20 / s) hours at the end of the route.