18. A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist's average rate for the round trip in miles per hour?
Question
- A cyclist bikes x distance at 10 miles per hour and returns over the same path at 8 miles per hour. What is the cyclist's average rate for the round trip in miles per hour?
Solution
The average speed for a round trip is not simply the average of the two speeds. Instead, it is the total distance divided by the total time.
Step 1: Calculate the total distance The total distance for the round trip is 2x (x distance going and x distance returning).
Step 2: Calculate the total time The time taken to bike x distance at 10 mph is x/10 hours. The time taken to return over the same path at 8 mph is x/8 hours. So, the total time is x/10 + x/8 = 9x/40 hours.
Step 3: Calculate the average speed The average speed is the total distance divided by the total time, which is (2x) / (9x/40) = 80/9 mph.
So, the cyclist's average rate for the round trip is 80/9 miles per hour.
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