Problem

While skiing, Josh rides a chair lift moving at 3 miles per hour up a hill. After he has skied down the hill, his average velocity for the round trip is 5 miles per hour. How fast did he ski?

Solution

We assume that the distance up the hill (and therefore down the hill also) is x miles. Using the formula distance = rate x time, we try to find r_down, his speed on the way down the hill.

First find the time it takes Josh to complete each part of his trip. The amount of time it takes to go up the hill, t_up, is

The amount of time it takes to go down the hill, t_down, is

The total time Josh spends on the hill is

This equation can be solved for the unknown r_down.

Therefore, Josh must ski down the hill at a speed of 15 miles per hour. Note that we did not need to know how long the hill was to solve this problem.