Carissa the chessplayer is a very, very slow walker. In fact, she walks at 100 meters per hour when walking uphill, 120 meters per hour when walking across flat ground, and 150 meters per hour when walking downhill. One day, Carissa walks across Boston from a café to a boba shop, and then takes the same route in reverse to return to the café. What was Carissa's average speed during the entire round trip?
The answer is not 125 meters per hour, nor is it 123 1/3 meters per hour.
To find Carissa's average speed during the entire round trip, we need to calculate the total distance she walked and divide it by the total time it took her to walk that distance.
Let's say that the distance from the café to the boba shop is d. Then, the total distance that Carissa walked during the round trip is 2d.
To calculate the total time it took her to walk the round trip, we need to know how much time she spent walking uphill, on flat ground, and downhill. Let's say that she spent t1 hours walking uphill, t2 hours walking on flat ground, and t3 hours walking downhill. Then we have:
t1 = d/100 (since she walks at 100 meters per hour when walking uphill) t2 = d/120 (since she walks at 120 meters per hour when walking on flat ground) t3 = d/150 (since she walks at 150 meters per hour when walking downhill)
The total time it took her to walk the round trip is:
t_total = t1 + t2 + t3 + t3 + t2 + t1 = 2(t1 + t2 + t3) = 2(d/100 + d/120 + d/150)
Therefore 12, Carissa's average speed during the entire round trip is:
average speed = total distance / total time = 2d / (2(d/100 + d/120 + d/150)) = (6000/97) ≈ 61.856 m/hour
Therefore, Carissa's average speed during the entire round trip is approximately 61.856 meters per hour.
When the distance back and forth is the same, then:
Average speed ==Harmonic mean
Harmonic mean==3 / [100^-1 + 120^-1 + 150^-1] ==120 m/h - her average speed.
SECOND METHOD:
Average speed ==Total distance / total time
2 * [100 + 120 + 150] ==740 meters - distance covered back and forth
Total time ==1 + 1 + 1 + 40/60 + 1 + 90/60] ==6 + 1/6 hours
Average speed ==740 / (6 1/6) ==120 m/h - her average speed.
