Man Crossing a Bridge

Date: 09/27/2001 at 16:48:56
From: Eric Yang
Subject: Man crossing bridge problem

Dear Dr. Math,

A man is jogging across a bridge. When he is 3/8 of the way across, 
he hears a train coming from behind him. He calculates that if he 
keeps running, he will reach the end of the bridge at the same instant 
as the train. He also calculates that if he turns around and runs 
back, he will reach the beginning of the bridge at the same instant as 
the train. Say the man runs consistently at 8 mph. What is the speed 
of the train?

I tried two solutions and I get 2 mph.  Please help.

Date: 09/28/2001 at 01:18:10
From: Doctor Ian
Subject: Re: Man crossing bridge problem

Hi Eric,

Here is the man, 3/8 of the way across the bridge, with the train 
coming from behind, some distance D from the bridge. 
                           
  T              M
        |--|--|--|--|--|--|--|--|

  |-----|                          D
        |-----------------------|  B
        |--------|                 (3/8)B
                 |--------------|  (5/8)B

The train will cover distance D in the same time it takes the man to 
cover distance (3/8)B. The train will cover distance (D+B) in the same 
time it takes the man to cover distance (5/8)B.  

Recall that time = distance / rate.  We know that

                     (3/8)B         D
  time to run back = ------ = --------------
                     8 mph    speed of train

We also know that

                        (5/8)B       D+B
  time to run forward = ------ = --------------
                        8 mph    speed of train

Now, this looks bad, because we have two equations, but three 
unknowns. But let's rewrite the second equation to look like this:

      (3/8)B   (2/8)B         D                B
      ------ + ------ = -------------- + --------------
      8 mph    8 mph    speed of train   speed of train

This is the first equation, with something extra added to each side.  
Since the extras must be equal, we know that

     (2/8)B         B
     ------ = --------------
     8 mph    speed of train

Do you see why? 

Note that there is a way to solve the problem without equations, which 
may be one reason that it comes up so often.  Let's say the man chooses 
to run away from the train.  When the train reaches the bridge, he's moved 
3/8 of the length of the bridge, meaning he has 2/8, or 1/4, of the bridge 
left to run.  So the man can run 1/4 the length of the bridge in the time 
it takes the train to move the entire length of the bridge.  And we know 
the man runs at 8 mph.  So that tells us the speed of the train, right? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/