The Mathematics Education Department
EMT 725, Lisa F. Garey
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Intuitively, 45 mph is the arithmetic mean of 30 and 60 mph. However, these are rates of change and cannot be treated as numbers. Average speed involves the total distance and the total time taken to cover that distance.
The total distance travelled is 20 + 20 = 40. At 60 mph, 20 miles takes 1/3 of an hour. Driving at 30 mph, 20 miles takes 2/3 of an hour. Hence, the entire trip of 40 miles took 1 hour. Thus the average speed is 40 mph.
This agrees with the harmonic mean formula. Average speed = 
=
40 mph.
What if the distances are different? 
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