Laps and Least Common MultiplesDate: 05/28/2002 at 13:02:03 From: Pushkal Subject: Running around a track Hi Doctor, A, B, and C start at the same time, and in the same direction, to run around a circular course. A makes it in 252 seconds, B in 308 seconds, C in 198 seconds. At what time will three of them reach the starting point and how many courses would each have completed by that time? Pushkal Date: 05/28/2002 at 13:29:07 From: Doctor Ian Subject: Re: Running around a track Hi Pushkal, What makes this problem seem hard is the big, ugly numbers. But suppose we make it a smaller, tidier problem: A, B, and C start at the same time and in the same direction, to run around a circular course. A makes one circuit in 2 minutes, B in 3 minutes, and C in 5 minutes, and so on. When you look at it this way, it becomes clear that you need to find the smallest number of minutes that is evenly divisible by 2, 3, and 5. Does that make sense? Do you see how to apply the same reasoning to the original problem? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/28/2002 at 13:54:30 From: Pushkal Subject: (Continued) Dear Sir, Why should I find the L.C.M of the numbers? Date: 05/28/2002 at 14:11:36 From: Doctor Ian Subject: Re: (Continued) Hi Pushkal, If A can run one circuit in 2 minutes, he can run 1/2 circuit in 1 minute, right? Similarly, B can run 1/3 circuit in 1 minute, and C can run 1/5 circuit in 1 minute. So let's see where they are minute by minute: Minutes A B C ------- ----- ----- ----- 1 1/2 1/3 1/5 2 2/2* 2/3 2/5 3 3/2 3/3* 3/5 4 4/2* 4/3 4/5 5 5/2 5/3 5/5* and so on. The *'s mark completed laps, i.e., minutes after which the number of numerator is a multiple of the denominator. You want to get three *'s in the same row, which is to say, you want to find a numerator that is divisible by all three denominators simultaneously. Moreover, you want to find the _smallest_ such numerator. What would you call that number? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/28/2002 at 15:41:52 From: Pushkal Subject: Thank you Thank you Doctor Math. I really love your true dedication and way of explanation. Pushkal |
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