Solution to the bird and trains problem

Answer to the math problem from Posner (1973, Cognition: An introduction)

The problem:

Two train stations are 50 miles apart. At 2pm one Saturday afternoon two trains start toward each other, one frm each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies to the front of the second train. When it reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour, and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet?

How not to represent the problem:

If the problem is represented in terms of the flight path of the bird, the solution will be very difficult. Using this method of representing the problem, you might try to calculate how far the bird flew on its first flight, then on its second flight, and so on, and then add them all together.

A less obvious, but more productive way to represent the problem:

How much time does the bird spend in flight? If you represent the problem in terms of the amount of time the bird is flying, then a solution is fairly straightforward. The two trains are 50 miles apart, and are approaching each other at a relative speed of 50 mph (25 + 25), so it will take one hour for them to meet. If the bird spends one hour flying at 100 mph, then it will traverse 100 miles before the trains meet.