Running Laps and LCMs

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/