Walking and Riding to Town, Is It Faster?

Apr 26, 2004

This puzzler was sent in by Doug Burnside. Of course, I did have to obfuscate and de-clarify.

Doug writes, "John and Fred have a dilemma. They need to get back to town as quickly as possible, but they only have one bicycle between them. And, of course, you can travel faster on the bike than you can by walking.

John says, "We'll leave at the same time. You start walking, and I'll ride the bike for a mile. Then I'll leave it by the side of the road, while you keep walking. When you get to the bike, you get on it, and you ride for a mile, and then you leave it by the side of the road. When I get to it, I'll ride another mile. We'll keep leap frogging like this until we get to town.

"Since we'll be riding part of the way and walking part of the way, each of us will have a higher average speed than if we walked, so we'll get there faster.

Fred says, "That isn't going to do any good, you dope! Think about it. Between the two of us, one or the other is going to walk every inch of the way from here to town. So, we can't possibly get there any faster than if we just walked all the way and didn't use the bike at all."

Who's right? Or, like us, are they both wrong?


RAY: Between the two of them, one or the other is going to walk every inch of the way to town.

TOM: That almost sounds credible, doesn't it?

RAY: It sounds credible to say that someone is walking all the time, therefore John's plan is not any faster.

That would be true, if the guy riding the bike stopped and waited for the walker to catch up, but he doesn't. Instead, he leaves the bike and he walks.

As it turns out, John is correct, and I can give a little example.

Let's say the trip to town is two miles, and let's say that you can walk at four miles an hour, and you can ride the bike at twenty miles an hour.

If you walk at four miles an hour you can do a mile in 15 minutes. And because you can ride your bicycle five times faster, you can do that same mile in three minutes.

So, let's say they take off from the starting point and John rides the bike the first mile, which takes him three minutes.

He leaves the bike. He begins to walk to town. From that point to town is going to take him another 15 minutes. The guy who was walking has to walk for 15 minutes before he gets to the bike. When he gets to the bike he rides to town. It takes him three minutes to go a mile.

They both arrive in town in 18 minutes. If they had walked it would have taken them 30 minutes.

Do we have a winner?

TOM: Yeah. The winner is Cammy Bates from Centennial, Colorado. And for having her answer selected at random from among all the correct answers that we got, Cammy gets a $26 gift certificate to the Shameless Commerce Division of cartalk.com.

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