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/
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