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September 1st, 2018, 03:03 AM   #1
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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?
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September 1st, 2018, 09:45 AM   #2
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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.
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September 1st, 2018, 12:36 PM   #3
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Ganesh, did you at least google "circular track problems"?
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September 1st, 2018, 01:29 PM   #4
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Quote:
Originally Posted by Denis View Post
Ganesh, did you at least google "circular track problems"?
don't be absurd
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September 2nd, 2018, 03:52 AM   #5
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Quote:
Originally Posted by romsek View Post
basically you need to find $t$ such that

$3 t = 5t = 8t \pmod{100}$
can't you proceed without using modular arithmetic?
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September 2nd, 2018, 06:45 AM   #6
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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.
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September 2nd, 2018, 08:40 AM   #7
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Quote:
Originally Posted by skipjack View Post
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.
3x50 = 150 % 100 = 50

5x50 = 250 % 100 = 50

8x50 = 400 % 100 = 0

I think you are mistaken.
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September 2nd, 2018, 10:58 AM   #8
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I had replaced 8 with 7 to show that you got the correct answer, but by an incorrect method.
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September 2nd, 2018, 11:16 AM   #9
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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.

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September 2nd, 2018, 11:32 AM   #10
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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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