Three friends A, B and 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 and 40 secs and C takes 12 mins. One day they decided to find out if they set out together in the same direction from a point what would be the fastest time for all of them to meet at any point on the track. What did they conclude?
The second day C decides to run in the opposite direction from the starting point. When will they all meet? Will this be sooner than the time taken on day one? Where do they meet in both cases?
(In reply to
Puzzle Answer by K Sengupta)
(I) On the FIRST day, we observe that:
o A and B will respectively complete 4 laps and 3 laps in 32 minutes
o B and C will respectively complete 9 laps and 8 laps in 96 minutes.
Since 32 divides 96, we are in a position to assert that:
A, B, and, C will meet for the first time in 96 minutes - having respectively completed 12 laps, 9 laps, and 8 laps.
It is then TRIVIAL to observe all three will meet at the STARTING POINT.
(II) On the SECOND day, we observe that:
As C runs in the opposite direction:
A&C together will cover (1/8+1/12)= 5/24 of the lap every 1 minute.
=> They will meet every (24/5) minutes
B&C together will cover (3/32+1/12)= 17/96 of the lap every 1 minute.
=> They will meet every 96/17 minutes
Now, lcm(24/5, 96/17) = lcm(24, 96)/gcd(5,17) = 96
Consequently, all three will meet after 96 minutes for the first time.
Since each of 8, 32/3 and 12 divides 96, it follows that all three will meet at the starting point.
Explanation to Puzzle Answer
