September 1st, 2018, 04:03 AM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  How to solve this circular track problem?
On a circular track of length 100 metres A,B and C start running clockwise from the same point, with speeds of 3m/sec, 5m/sec and 8m/sec respectively. At what distance from the starting position would they meet for the first time anywhere on the track?

September 1st, 2018, 10:45 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,311 Thanks: 1223 
basically you need to find $t$ such that $3 t = 5t = 8t \pmod{100}$ Note that $3,~5,~8$ have no factors in common. So it must be that $t=100k$ for this equality to hold. Obviously the first such occurrence other than at the start is $k=1$ which corresponds to $t=100s$ This corresponds to a distance of $0m$ from the starting line. 
September 1st, 2018, 01:36 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,958 Thanks: 991 
Ganesh, did you at least google "circular track problems"?

September 1st, 2018, 02:29 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,311 Thanks: 1223  
September 2nd, 2018, 04:52 AM  #5 
Senior Member Joined: Aug 2014 From: India Posts: 343 Thanks: 1  
September 2nd, 2018, 07:45 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,285 Thanks: 1967 
Every 20 seconds, C laps A. Every 50 seconds, B laps A. Hence A is first lapped at the same time by B and C after 100 seconds. The reasoning romsek gave wasn't quite right. For the equation 3t ≡ 5t ≡ 7t (mod 100), the coefficients would also have no common factor (other than 1), but the solution would then be t = 50, not t = 100. 
September 2nd, 2018, 09:40 AM  #7  
Senior Member Joined: Sep 2015 From: USA Posts: 2,311 Thanks: 1223  Quote:
5x50 = 250 % 100 = 50 8x50 = 400 % 100 = 0 I think you are mistaken.  
September 2nd, 2018, 11:58 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,285 Thanks: 1967 
I had replaced 8 with 7 to show that you got the correct answer, but by an incorrect method.

September 2nd, 2018, 12:16 PM  #9 
Newbie Joined: Dec 2016 From: Austin Posts: 16 Thanks: 1 
There are a few different ways of solving this. Could you let us know what type of course this question was asked in? For example, if this is for a Physics course, I might suggest an angular speed equation. If it is Geometry based, then a circumference approach. If it were a challenge problem in an introductory Algebra course, then we might use the Least Common Multiple and a couple of problem solving techniques. Thanks! 
September 2nd, 2018, 12:32 PM  #10 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,958 Thanks: 991 
Can someone provide an example where the 3 runners meet not at the starting point? Thanks. I must have goofed in my program... 

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