How Far Does the Fly Fly?

Here’s another one of those problems that is tricky, only in that it seems to be difficult, yet it is easy as pie if you think about it in a certain way.  You don’t have to be a whiz in algebra to work this problem, but you do have to think clearly.  Like the bayou problem, I don’t recall where I first saw it.  Try to work it before you read my solution.

Two trains are heading toward each other on the same track, bound for a head-on collision.  When the trains are exactly two miles apart, a fly on the headlight of one train, which we will designate as train A, starts flying toward the other train, which we will designate as train B.  When the fly reaches train B, he turns around and races back to train A.  When he gets back to train A, he turns around and heads back to the train B again.  Back and forth he goes, flying ever shorter distances on each pass as the trains get closer and closer, until the trains collide.  The trains are going 30 mph, and the fly is going 60 mph.  The question is, how far does the fly fly before he meets his maker?

A typical engineer would probably make this problem much more difficult than it is.  For example, he might work out the distance-equals-rate-times-time thing and come up with a series of smaller and smaller distances and then take the sum of the series – not an easy thing to do.  However, I am not a typical engineer, so I will take an altogether different, and far simpler, approach.

Since there are 60 minutes in an hour, 60 mph is a mile a minute, and 30 mph is a mile in two minutes.  Since the trains are two miles apart at the beginning of our consideration, and since they are traveling toward each other at identical speeds, it is apparent that they will collide at the midpoint between them.  Therefore, each train travels one mile before colliding, and at 30 mph, it takes them two minutes to travel that distance.  Now, the fly flies for the same amount of time as the trains do before they collide; that is to say that the fly also flies for two minutes before the collision occurs.  Since the fly flies for two minutes at a speed of one mile per minute, the fly flies two miles.  Eureka!