
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
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)(XY) 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: 19,882 Thanks: 1835 
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 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
I get inconsistent conditions: After first lap for A: DB1DC1=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: 19,882 Thanks: 1835 
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 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
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. bc=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: 19,882 Thanks: 1835 
That makes essentially the same mistake you made previously.

September 5th, 2018, 04:33 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
September 5th, 2018, 10:09 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,882 Thanks: 1835 
As explained in post #4.

September 5th, 2018, 10:18 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,479 Thanks: 950 
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)16B(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  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
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:
Soooooooo. skip AND Denis are wrong.  

Tags 
circular, problem, solve, track 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to solve this circular track problem?  Ganesh Ujwal  Calculus  12  September 3rd, 2018 02:01 PM 
Circular track race question  happy21  Elementary Math  7  June 16th, 2016 03:20 AM 
Circular Track  Aman Verma  Elementary Math  5  July 13th, 2014 09:24 PM 
Solve for Height of Circular Segment  kthr33  Algebra  2  January 8th, 2009 06:13 AM 
Circular Jogging Track  symmetry  Algebra  2  March 20th, 2007 01:02 PM 