The Round-Trip Average

Question

You drive 60 km to a client at 30 km/h and drive the same 60 km back at 60 km/h. What is your average speed for the round trip?

Accepted Answer

Total distance 120 km. Times: frac6030 = 2 h out, frac6060 = 1 h back, 3 hours total: bar v = frac1203 = mathbf40 text km/h. Not 45. Average speed weights by time, and you spend two of the three hours at 30 km/h. For equal distances, speeds combine by the harmonic mean: bar v = frac2aba+b = frac2 cdot 30 cdot 6090 = 40, which is always below the arithmetic mean (AM–HM inequality), the slow leg dominates. Why trading firms ask this: rate-like quantities (speeds, P/E ratios, FX quotes) harmonically average when you fix the numerator-side exposure, arithmetically when you fix the denominator side. Knowing which mean applies, and that the naive average is biased in a predictable direction, is the same reflex as not averaging two bond yields when you actually hold equal notionals of price. Instant answer for speed rounds: harmonic mean of 30 and 60 is 40.

The Round-Trip Average

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