Question

Writing to Learn A driver averaged 30\(\mathrm { mph }\) on a 150 -mile trip and then returned over the same 150 miles at the rate of 50\(\mathrm { mph }\) . He figured that his average speed was 40\(\mathrm { mph }\) for the entire trip. (a) What was his total distance traveled? (b) What was his total time spent for the trip? (c) What was his average speed for the trip? (d) Explain the error in the driver's reasoning.

Short answer

a) The total distance traveled was 300 miles. b) The total time spent for the trip was 8 hours. c) The average speed for the trip was 37.5 mph. d) The driver's error was in thinking that the average speed was the simple average of his two speeds (30mph and 50mph), while in fact the average speed is calculated by total distance traveled divided by total time, which gives 37.5 mph.

Step-by-step solution

Total Distance Travelled

To calculate the total distance traveled, add the distance covered to the destination and the distance covered on the return trip. Here, the driver traveled 150 miles each way, thus the total distance traveled = 150 miles (to destination) + 150 miles (return) = 300 miles.

Total Time Spent

To find the total time spent, divide the distance covered by the speed for each leg of the trip and add them together. This gives time = distance/speed. Thus, the time to the destination = 150 miles ÷ 30 mph = 5 hours and the return time = 150 miles ÷ 50 mph = 3 hours. Therefore, the total time spent = 5 hours (to destination) + 3 hours (return) = 8 hours.

Average Speed

The average speed for the trip is computed by dividing the total distance by the total time. Therefore, average speed = total distance ÷ total time = 300 miles ÷ 8 hours = 37.5 mph.

Error in Reasoning

The error in the driver's reasoning is that he computed the intended average speed by simply finding the average of the two speeds he traveled at (30mph and 50mph) which gives 40 mph. However, average speed for a trip is calculated by dividing total distance by total time, which in this case gives 37.5mph, not 40mph. The two segments of the trip were travelled for unequal time periods, so the overall average cannot be simply the average of the two speeds. This is a common mistake in reasoning about average speeds over round trips.