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 September 3rd, 2018, 11:30 PM #1 Senior Member   Joined: Aug 2014 From: India Posts: 343 Thanks: 1 How to solve this circular track problem? A, B, C are running clockwise around a circular track. When A completes the first round, the distance between B and C is 2/3 the distance between A and B. When A completes the 2nd round, C reaches the same position where B was at the time A finished the Ist round. If A is the fastest and C is the slowest among them. What is the ratio of speeds of A, B and C? I tried: Round 1: A is at X, B is at Y, and C is at (2/3)(X-Y) Round 2: A is at 2X, B is at 2Y, and C is at Y. From here, how to proceed?
 September 4th, 2018, 04:19 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1967 After round 2, C's total distance run has doubled to be Y, so C runs half as fast as B. Hence Y/2 = (2/3)(X - Y), and so Y/X = 4/7. Hence the desired ratio is 1:4/7:2/7, which is 7:4:2. Note that although A has returned to the start position after round 1, the distance between A and B must then be X - Y rather than Y.
 September 4th, 2018, 08:01 AM #3 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 I get inconsistent conditions: After first lap for A: DB1-DC1=2/3 DB1 -> DC1=1/3 DB1 After second lap for A, ie, after same period of time has elapsed DC2=2DC1=DB1 - > DC1=1/2 DB1 EDIT: So my assumptions were wrong. Either B, or B and C must have completed a lap. I'll assume skipjack's proof is correct. . Last edited by zylo; September 4th, 2018 at 08:15 AM.
 September 4th, 2018, 11:40 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1967 The problem states that C is the slowest runner, so B must complete more than a lap overall. In round 1, B covered 4/7 of a lap, reaching a point Y that was 3/7 of a lap from the starting point, so the distance that the problem refers to as "the distance between A and B" is 3/7 of a lap, not 4/7 of a lap.
 September 4th, 2018, 02:17 PM #5 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 The problem is not solvable. Assume they all start together and A has run a lap ccw. If you draw a circle, A is then at the top of the circle, and ccw from A is C and then B. First condition: Let b, c be respective distances of B, C from A after one lap of A. b-c=2/3 b, c=1/3b In the time A runs a second lap, C moves a distance c again to 2/3 b, and so can't reach B. After the third lap of A, C reaches the initial position of b. Last edited by skipjack; September 4th, 2018 at 03:07 PM.
 September 4th, 2018, 03:07 PM #6 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1967 That makes essentially the same mistake you made previously.
September 5th, 2018, 04:33 AM   #7
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Quote:
 Originally Posted by skipjack That makes essentially the same mistake you made previously.
Which is?

 September 5th, 2018, 10:09 AM #8 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1967 As explained in post #4.
 September 5th, 2018, 10:18 AM #9 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,958 Thanks: 991 Assume track is straight road = 28 Assume Skip's ratio: 7:4:2. Make up simple scenario, like speeds A=28, B=16, C=8 (LCM(7,4,2): A...............28................>(28)--------------28------------->A(56) B..........16.........>(16)--------16--------B(32) C...8...>(8)---8--->C(16) "When A completes the first round, the distance between B and C is 2/3 the distance between A and B." B - C = 8 = 2/3(12) "When A completes the 2nd round, C reaches the same position where B was at the time A finished the 1st round." B - C = 2/3(A - B) 8 = 2/3(12) 8 = 8 Soooooo: Skip is correct! Last edited by Denis; September 5th, 2018 at 10:34 AM.
September 5th, 2018, 12:59 PM   #10
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Reference previous post.

After first lap A is at beginning, B is at 16 and C is at 8.
Distance between B and C is 8 and distance between A and B is 16.

Quote:
 Originally Posted by Ganesh Ujwal A, B, C are running clockwise around a circular track. When A completes the first round, the distance between B and C is 2/3 the distance between A and B.....
8 is not 2/3 of 16.

Soooooooo. skip AND Denis are wrong.

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