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### 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
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
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/
```
Associated Topics:
Middle School Algebra

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