Same two speeds, but equal times, which mean now?

Question

You drive for 2 hours at 60 km/h, then for 2 hours at 40 km/h. Solve in your head. What is your average speed over the four hours? Aim: under 10 seconds.

Accepted Answer

Average speed is still total distance over total time, but now the weighting is by time, and the times are equal. That makes it the plain arithmetic mean: \bar v = \frac{60 + 40}{2} = 50 	ext{ km/h}. Check directly: 2 h at 60 covers 120 km, 2 h at 40 covers 80 km, so 200 km in 4 h = 50 km/h. Contrast this with the equal-distance version of the same two speeds, where the average would be the harmonic mean 	frac{2 	imes 60 	imes 40}{100} = 48 km/h. Equal time gives 50; equal distance gives 48. The slow leg drags the average down only when it also eats up more of the trip, which happens unde…

Same two speeds, but equal times, which mean now?

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