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Circular Jogging Track Problem 3 (Posted on 2006-04-14) Difficulty: 4 of 5
Continuing with the original problem. Three friends A B & C regularly jog in circular jogging track every morning. The track is 1000 m in circumference.

A takes 8 mins to complete one lap, B takes 10 mins & 40 secs & C takes 12 mins. They all start from a point on the track but C decides to run in the opposite direction from the same point.

As they set out B suddenly has cramps (not at the starting point) and starts walking in the same direction as before. His walking speed takes him 32 mins to complete a lap. He decides to continue walking till they all meet. At what point should he have cramps such that they all meet at any point in the fastest time?

See The Solution Submitted by Salil    
Rating: 3.0000 (1 votes)

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Solution solution | Comment 1 of 2

A and C meet each other every 288 seconds, as their relative angular speed is 1/480+1/720 = 1/288.

When they meet the first time, at t=288 seconds, A has traveled 3/5 of the way around the track, while C has traveled 2/5 in the other direction.  As B runs slower than A, no matter what time he could slow down, he couldn't make this first meeting.

After another 288 seconds, t=576, A has traveled 6/5 of the way around; that is, he's 200 m into his second lap. In any combination of B's walking and running paces (1/640 rev/sec and 1/1920 rev/sec), he would be between 300 and 900 m along the track at this time, so he can't make this meeting.

The next meeting between A and C, at t=864, will be where A has traveled 9/5 of the way around the track: he's 800 m into his second lap. At an all-walking pace, B would be at the 450 m mark; running, he'd be 350 m into his second lap, so some combination of running and walking will leave him at the 800 m mark.  Let the time he has cramps be called x.

x/640 + (864 - x) / 1920 = 800/1000
x/640 - x/1920 = 4/5 - 864/1920
x/960 = 7/20
x = 336

So if B gets the cramps at t=336 seconds (5 min, 36 sec), when he's 525 m along the track, he will make this meeting at t=864 (14 min, 24 sec), 800 m along the track.

  Posted by Charlie on 2006-04-15 00:33:07
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