You and I have to travel from Startville to Endville, but we have only one bicycle between us. So we decide to leapfrog: We’ll leave Startville at the same time, you walking and I riding. I’ll ride for 1 mile, and then I’ll leave the bicycle at the side of the road and continue on foot. When you reach the bike you’ll ride it for 1 mile, passing me at some point, then leave the bike and continue walking. And so on — we’ll continue in this way until we’ve both reached the destination.
Will this save any time? You say yes: Each of us is riding for part of the distance, and riding is faster than walking, so using the bike must increase our average speed.
I say no: One or the other of us is always walking; ultimately every inch of the distance between Startville and Endville is traversed by someone on foot. So the total time is unchanged — leapfrogging with the bike is no better than walking the whole distance on foot.
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You are. My argument would be sound if each of us simply stood by the bike after dismounting until the pedestrian caught up. But instead we’re investing that extra time in walking, which accounts for the faster progress.
Suppose the total trip is 2 miles, and each of us can walk at 4 mph and ride at 12 mph. I ride 1 mile in 5 minutes and leave the bike for you, walking the remaining 1 mile in 15 minutes. You take 15 minutes to reach the bike and then ride to Endville in 5 minutes. We both arrive at the destination in 20 minutes, where it would have taken 30 minutes if we’d walked the whole way.