Shortcut Trick
When running in the same direction, the first meeting time anywhere is the LCM of their pair-wise relative meeting times.
Time for M and N to meet = Distance ÷ Relative Speed = 150 ÷ (20 − 15) = 30 sec.
Time for N and P to meet = Distance ÷ Relative Speed = 150 ÷ (15 − 12) = 50 sec.
Time to meet for all three = LCM(30, 50) = 150 sec.
∴ The correct answer is 150 sec.
Alternate Method
Centre
Track Length = 150 m
Start Point
M = 20 m/s
N = 15 m/s
P = 12 m/s
Meeting Time = LCM of Individual Lap Times
Given:
⇒ Length of circular track = 150 m
⇒ Speed of M = 20 m/sec
⇒ Speed of N = 15 m/sec
⇒ Speed of P = 12 m/sec
Formula Used:
⇒ Time taken to complete one lap = Distance ÷ Speed
⇒ Time to meet at starting point = LCM of individual lap times
⇒ LCM of fractions = LCM of Numerators ÷ HCF of Denominators
Calculation:
⇒ Lap time for M = 150 ÷ 20 = 15/2 sec
⇒ Lap time for N = 150 ÷ 15 = 10/1 sec
⇒ Lap time for P = 150 ÷ 12 = 25/2 sec
⇒ Time to meet = LCM(15/2, 10/1, 25/2)
⇒ LCM of Numerators (15, 10, 25) = 150
⇒ HCF of Denominators (2, 1, 2) = 1
⇒ Meeting time = 150 ÷ 1 = 150 sec
(Note: For these specific speeds, the time to meet for the first time anywhere on the track is identically equal to the time to meet at the starting point.)
∴ The correct answer is 150 sec.
Additional Information
First Meeting at Starting Point
Regardless of the direction of running, the time taken for all runners to meet again at the starting point is always the LCM of their individual lap times: Time = LCM(T1, T2, T3).
First Meeting Anywhere on Track (Same Direction)
When running in the same direction, the time to meet for the first time is the LCM of the times taken to meet in pairs using relative speed: Time = LCM [ D ÷ (S1 − S2) , D ÷ (S2 − S3) ].
First Meeting Anywhere on Track (Opposite Direction)
When two persons run in opposite directions, their relative speeds are added. The time taken to meet for the first time anywhere on the track is: Time = D ÷ (S1 + S2).
LCM of Fractions Rule
To calculate the LCM of fractions, always use the rule: LCM(a/b, c/d) = LCM(a, c) ÷ HCF(b, d).