Two Trains with a Twist

Date: 9/11/96 at 22:41:27
From: Anonymous
Subject: Two Trains with a Twist

The local train heading for Westford at 60km/h and the express train 
heading for Eastford at 80 km/h pass each other at Centerville while 
heading in opposite directions along a straight roadbed.  Westford and 
Eastford are 195 km apart, and if the local train makes one additional 
15 minute stop after passing Centerville the express completes its run 
before the local does.  What is the maximum distance to the nearest 
kilometer between Westford and Centerville?

Date: 9/14/96 at 21:34:9
From: Doctor Ana
Subject: Re: Two Trains with a Twist


Westford                    Centerville                 Eastford
   W----------------------------C---------------------------E
   
   |----------------------------|
                 x              |---------------------------|
                                          195 - x

                                           [diagram by Dr. Chuck]

First of all, let's set up our variables. Let's call the distance 
between Westford and Centerville x. This will make the distance 
between Eastford and Centerville 195-x.

We know that distance equals rate times time, or d = rt. This means 
that time equals distance divided by rate, t = d/r. We can say that it 
took the express train (195-x)/80 hours to reach Eastford. We can also 
say that it took the local train x/60 + .25 hours to reach Westford. 
If the express train got to its destination first, then we can set up 
the inequality

   (195-x)/80 < x/60 +.25

This is true because the number of hours to reach Eastford is less 
than the number of hours to reach Westford. Let's multiply both sides 
of this inequality by 240 to make it look nicer.

   3(195-x) < 4x + 60

   585-3x < 4x + 60

Adding 3x to both sides we have:

   585 < 7x + 60

Subtracting 60 from both sides we have:

   525 < 7x

Dividing both sides by 7 we have:

   75 < x 

Remember that x stands for the distance between Centerville and 
Westford, which is the distance that the local train has to travel 
after the 2 trains pass at Centerville. This inequality states that 
x has to be more than 75 kilometers, which should make sense, because 
the express train goes faster than the local train, so it can go 
further and still beat the local train. 

The problem is that you are asked for the maximum distance between 
Westford and Centerville, and all we have is a minimum distance. First 
of all, why don't you read the problem again and be sure that they 
didn't ask for the distance between Eastford and Centerville? If they 
did ask for the distance between Westford and Centerville, then they 
could be as much as 195 kilometers apart, because the express train 
will certainly beat the local train, and that is the only requirement.

This is a complicated problem. I hope you understand my explanation 
and my answer. Try to do this problem again on your own now that 
you've read how I got my answer and be sure that you read the problem 
correctly. If you have any more questions please write back.

-Doctor Ana,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/