A car drives 40 mph to a destination and 60 mph back. The average speed is not 50 mph. It is 48 mph. The error is intuitive and almost universal. The arithmetic mean weights each speed equally, but the car does not spend equal time at each speed. It covers equal distance. The outbound leg takes 50 per cent longer than the return, so the slower speed dominates the clock. Work it through for a 120-mile one-way trip: - Outbound: 120 / 40 = 3 hours - Return: 120 / 60 = 2 hours - Total: 240 miles in 5 hours = 48 mph The distance cancels entirely. For two equal-distance legs, average speed is 2ab / (a + b), the harmonic mean: the reciprocal of the average of the reciprocals. Speed is distance per time. When distance is held fixed, the reciprocal, hours per mile, is the quantity that actually adds linearly. This generalises into one of the cleanest inequalities in mathematics. For any set of positive numbers: - Arithmetic mean ≥ geometric mean ≥ harmonic mean - Equality holds only when ...