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### Running Laps and LCMs

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Date: 02/11/99 at 17:02:46
From: David Bernier
Subject: Math

Bill, Bob and John start at the same time at the start line. Bill runs
1/3rd lap per minute. Bob runs 1/5th lap per min., and John runs 1/6th
lap per min. How many laps need to be run by all in order to cross the
finish line at the same time?

I don't know how to set the problem up. Thank you for your help.
```

```
Date: 02/11/99 at 18:40:16
From: Doctor Micah
Subject: Re: Math

Hi David,

Thanks for writing to Dr. Math. Neat question. Running and math are two
of my favorite things to do.

The easiest way I know to solve this problem is to convert each
runner's rate to minutes/lap instead of laps/minute. The way to do this
is to take the reciprocal of each runner's rate. Taking the reciprocal
means putting the numerator (top number) of a fraction on the bottom
and the denominator (bottom number) on top. For example, the reciprocal
of 2/3 is 3/2, and the reciprocal of 1/4 is 4/1 = 4. So for each
runner:

Bill:   1/3 lap/min = 3/1 min/lap = 3 min/lap
Bob:    1/5 lap/min = 5/1 min/lap = 5 min/lap
John:   1/6 lap/min = 6/1 min/lap = 6 min/lap

So Bill takes 3 minutes to run 1 lap, Bob takes 5 minutes, and John
takes 6 minutes. This may seem a little bit like magic, but if you
think about it for a minute, it makes sense.

So, now we know how long it takes each of them to finish one lap. The
question is, when will they all cross the finish line at the same time?
Well, let's look at the time each of them crosses the finish line:

Bill crosses at 3 min, at 6 min, at 9 min, at 12 min, and so on.
Bob crosses at 5 min, at 10 min, at 15 min and so on.
John crosses at 6 min, at 12 min, at 18 min, and so on.

Notice that each of them crosses at every multiple of his given rate,
so if you can find the smallest number that is a multiple of 3, 5, and
6, (the least common multiple, or LCM) you'll have the answer.

Here are a couple of good archives on finding LCMs:

3 Methods for Finding Least Common Denominator
http://mathforum.org/dr.math/problems/rooney2.4.98.html

Least Common Multiple
http://mathforum.org/dr.math/problems/fields3.29.96.html

I hope this helps you solve the problem. If you still have trouble,
or if you don't know how to find the LCM, please feel free to write
back to Dr. Math for more help.

Good luck,

- Doctor Micah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Multiplication
Elementary Word Problems
Middle School Word Problems

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