# Two cyclists approach each other [duplicate]

Two cyclists, 120 miles apart, approach each other, each pedaling at 10 miles per hour. A fly starts at one cyclist and flies back and forth between the cyclists at 15 miles per hour. When the cyclists come together (and squash the fly between them), how far has the fly flown?

I know that I am supposed to find some sort of series here, but I don't really know how to approach this. How do I find the terms?

## marked as duplicate by Ilmari Karonen, Lee David Chung Lin, zz20s, verret, Theo BenditMar 26 at 4:14

• You actually don't need series at all! – Bolton Bailey Sep 26 '16 at 1:07
• You don't need a series. It's a well known problem I think it would be a shame to reveal the trick to solve it easily. There is a story about John von Neumann being given this problem (or one very much like it) and immediately answering it. The asker says "oh wow you spotted the trick? Most people try to sum the series" to which von Neumann replied "what other way is there to solve it?" – Dan Robertson Sep 26 '16 at 1:10

The cyclists approach each other at a combined speed of $20$ miles per hour, so they meet after $6$ hours. During all of this time the fly was travelling at a speed of $15$ miles per hour. Hence it travelled $90$ miles during these $6$ hours.

• That was the von Neumann answer !!!. – Felix Marin Sep 26 '16 at 3:44

This is actually a famous question which can be answered in two ways. One way is a series the other way is by the above answer.

video of what's called the train-bird problem