You go from point A to point B in a straight line at 60 miles per hour and return along the same path at 40 miles per hour. What is your average speed?
Solution: The obvious (wrong) answer is 50 miles per hour. To see why it is wrong, imagine you never came back. Would your average speed be 30 miles per hour? No, your average speed would be zero. So, the answer cannot be the arithmetic mean of the speed going and speed coming back. Here is the correct calculation: Let d be the distance between A and B. Then the time it takes to go from A to B is d/60 hours and the return time is d/40 hours. The total distance covered is 2d and the total time taken is d/50 + d/70. Hence,
Average speed = 2d/(d/60 + d/40) miles per hour
Conveniently d cancels out and we get,
Average speed = 2/(1/60+1/40) = 48 miles per hour.
This type of average is called the “Harmonic” mean.