Up the hill at 10, down at 30

Question

A cyclist rides up a hill at a steady 10 km/h and comes back down the same road at a steady 30 km/h. What is her average speed for the whole round trip? Explain why it is not 20 km/h, and give the general formula.

Accepted Answer

15 km/h. Take a 30 km hill. Up: 30 km at 10 km/h takes 3 hours. Down: 30 km at 30 km/h takes 1 hour. Total: 60 km in 4 hours, so 15 km/h. Any hill length gives the same answer, since both distance and time scale with it. Why not 20. The average speed is total distance over total time, and the cyclist spends three quarters of the time on the slow climb. Averaging the two speeds equally, as if she spent equal time at each, overweights the descent. She spends equal distances at each speed, and for equal distances the right average is the harmonic mean: 2 × 10 × 30 / (10 + 30) = 600 / 40 = 15. Th…

Up the hill at 10, down at 30

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