Three equal legs at three speeds

Question

You cover three equal-distance legs of a route at 20 km/h, 30 km/h, and 60 km/h. Solve in your head. What is your average speed for the whole trip? Aim: under 15 seconds.

Accepted Answer

For n equal-distance legs, average speed is the harmonic mean: \bar v = \frac{n}{\frac{1}{v1} + \frac{1}{v2} + \frac{1}{v3}} = \frac{3}{\frac{1}{20} + \frac{1}{30} + \frac{1}{60}}. Put the reciprocals over a common denominator of 60: 	frac{1}{20} = 	frac{3}{60}, 	frac{1}{30} = 	frac{2}{60}, 	frac{1}{60} = 	frac{1}{60}. They sum to 	frac{6}{60} = 	frac{1}{10}. So \bar v = \frac{3}{1/10} = 30 	ext{ km/h}. Sanity check with 60 km per leg: 60/20 = 3 h, 60/30 = 2 h, 60/60 = 1 h, so 180 km in 6 h = 30 km/h. Takeaway. The harmonic mean generalizes to any number of equal-distance legs: n div…

Three equal legs at three speeds

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