KA-0174Rate, time, distance
4 points, difficulty 2 of 3Level 11–12about 135sA driver travels from one town to another at an average of 60 km/h and returns along the same road at an average of 40 km/h. What is the average speed for the whole journey?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Average speed is never the average of two speeds.
2The strategy
Pick a convenient distance, say 120 km each way, and work out the two times.
3The full solution
120/60 = 2 hours out and 120/40 = 3 hours back, so 240 km in 5 hours, which is 48 km/h.
Solution
The reliable way
Average speed means total distance divided by total time, and here the two legs take different times, so the two speeds cannot simply be averaged. Choose a convenient one-way distance - 120 km works because both speeds divide it - giving 120/60 = 2 hours out and 120/40 = 3 hours back. That is 240 km in 5 hours, so the average is 240/5 = 48 km/h. Notice it is below the midpoint of 50, because more of the journey's TIME is spent at the slower speed. Any distance gives the same answer. The transferable idea: average speed is total distance over total time, and equal distances weight the slower leg more heavily.
The elegant way
Equal distances at two speeds give their harmonic mean: 2 x 60 x 40 / (60 + 40) = 4800/100 = 48.
Why this is on the test: The plain average is the trap, and knowing the answer must fall below it eliminates two choices instantly.