Discussion
An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?
Answer: A
The average speed is 72 km/h.
For equal distances at two speeds, the average speed is the harmonic mean, not the arithmetic mean, because more time is spent at the lower speed:
\[ \begin{aligned} \bar{v} &= \frac{2v_{1}v_{2}}{v_{1} + v_{2}} \\ &= \frac{2(60)(90)}{150} = 72 \end{aligned} \]
Check with a distance of 180 km each way: 3 h out and 2 h back, 360 km in 5 h, 72 km/h. 75 is the arithmetic mean of 60 and 90, the usual trap; 73 and 74 match neither calculation.
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