Kinesys settles, wobbling, on a mountaintop on Dynamos. It’s the only place on the planet where you can almost stand still, and sure enough it’s packed with cyclists catching their breath before hurling themselves downhill again.
Liz takes the chance to stretch her legs with a few dance steps, to the cyclists’ amazement. Meanwhile a local cyclist reaches the summit, drenched in sweat, and checks his bike computer. Then he turns his bike around and races down the valley.
“Ten miles an hour going up, thirty coming down,” M00N reads off the display. “So the average is easy: twenty!” Dord stiffens as if he’d heard heresy: “Easy? Please, M00N. Nothing is ever that easy.”
The Riddle
A cyclist climbs a mountain at an average of 10 mph and then, once at the top, turns his bike around and rides back down the same road at an average of 30 mph. What is the cyclist’s overall average speed for the whole trip?
Hint
Beware the gut answer: the cyclist doesn’t spend the same amount of time going up as coming down. Pick a convenient length for the climb and work out the times.
Solution
The correct answer is 15 mph.
The answer most people give is 20 mph, based on the fact that the cyclist covered exactly half the road at an average of 10 mph and exactly the other half at an average of 30 mph. But that answer is wrong: since he goes downhill three times as fast as uphill, the cyclist spent three times as long riding at 10 mph as he did at 30 mph.
So the average has to be weighted by time: v = (3 × 10 + 1 × 30) / 4 = 15 mph. For example, if the road is 15 miles long, the climb takes 1½ hours and the descent half an hour: 30 miles in 2 hours, or 15 mph.
Source: Giuliano Spirito, “Matematica dell’incertezza” (Mathematics of Uncertainty)