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If John travels 90┬аkm at a speed of 30┬аkm/h and returns at a speed of 60┬аkm/h, his average speed is:
30 km/h
60 km/h
40 km/h
45 km/h
40 km/h
Use the harmonic mean formula for equal distances: Average Speed = (2 * product of speeds) / (sum of speeds). Here, (2 * 30 * 60) / (30 + 60) = 3600 / 90 = 40 km/h.
Distance traveled one way d = 90 km, Outbound speed v1тАЛ = 30 km/h, Return speed v2тАЛ = 60 km/h.
Average┬аSpeed=v1тАЛ+v2тАЛ2v1тАЛv2тАЛтАЛ
Use the harmonic mean formula for equal distances: Average Speed = (2 * product of speeds) / (sum of speeds). Here, (2 * 30 * 60) / (30 + 60) = 3600 / 90 = 40 km/h.
Many students incorrectly calculate the average speed by taking the arithmetic mean of the two speeds, i.e., (30 + 60) / 2 = 45 km/h, which is mathematically incorrect for distance-based problems.
Identify given variables
The distance d=90 km is the same for both legs of the journey. The speeds are v1тАЛ=30 km/h and v2тАЛ=60 km/h.
v1тАЛ=30┬аkm/h,v2тАЛ=60┬аkm/h
Apply average speed formula
For equal distances, the average speed VavgтАЛ is given by the harmonic mean of the speeds.
VavgтАЛ=v1тАЛ+v2тАЛ2├Чv1тАЛ├Чv2тАЛтАЛ
Calculate the result
Substitute the values into the formula: VavgтАЛ=30+602├Ч30├Ч60тАЛ=903600тАЛ.
VavgтАЛ=40┬аkm/h
C is correct because the calculated average speed using the harmonic mean formula for equal distances is 40 km/h.
This harmonic mean shortcut is extremely useful for 'time, speed, and distance' problems involving round trips, where the distance traveled in both directions is identical regardless of the actual length.
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