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Delivering a Message

Date: 09/02/2000 at 19:19:37
From: Lucas Marshall
Subject: Pre-Calculus Word Problem, but it seems like Algebra

My problem is a word problem. You've answered a problem kind of like 
this already, but mine has different information and different 
nuances. I have an answer, but I think my logic is too simple. Here's 
the problem.

A column of soldiers 25 miles long marches 25 miles a day. One 
morning, just as the day's march began, a messenger started at the 
rear of the column with a message for the man at the front of the 
column. During the day he marched forward, delivered the message to 
the first man in the column and returned to his position just as the 
day's march ended. How far did the messenger march?

My answer is 50 miles, but it seems too easy. I find it odd that there 
was no speed given for either the column or the messenger. How would 
you solve this?

Thanks a lot.

Date: 09/02/2000 at 23:14:35
From: Doctor Peterson
Subject: Re: Pre-Calculus Word Problem, but it seems like Algebra

Hi, Lucas.

Often a simplistic line of reasoning can solve a problem that is hard 
to solve by more advanced reasoning. If you'd told me your reasoning, 
you might have convinced me, though I have a feeling you are in fact 
wrong. Let's see how I would do it.

I like to picture this sort of thing in a graph of position vs. time. 
The front and rear of the column will be represented by lines with the 
same slope; the messenger must be going faster, starting at the rear 
line, meeting the front line, and then returning (I presume at the 
same speed) to the rear line:

     50+              * front
       |          +*
       |        */ \
       |     *  /   \ messenger
       |  *   /      \   *
     25+     /        + rear
       |    /      *
       |  /     *
       | /   *
       |/ *
       0              1

The first thing I see is that the messenger wasn't going twice as fast 
as the column (which is going 25 miles per day - its speed IS given, 
though in rather vague terms with regard to time); if he were, 
he would have gone 50 miles in a day, and would have met the front 
just as they stopped, with no time to get back. I also see that he 
would have gone 50 miles if he had met them halfway along their march, 
going 37.5 miles forward and 12.5 miles back. But then he clearly 
wouldn't have gone the same speed both ways. So your answer is wrong.

I'll just suggest where I would go from here. Suppose he meets them at 
time x (measuring time as a fraction of a "day," however long their 
march takes). Where will he meet the front of the line? What is his 
speed going forward? What is it going back? Set those expressions 
equal and solve for x.

This is a nice problem, with an interesting solution. If you need more 
help, write back and show me how far you got.

- Doctor Peterson, The Math Forum
Associated Topics:
High School Basic Algebra

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