On a circular track of length 1200 m, P, Q and R start from the same point simultaneously with speeds of 18 kmph, 27 kmph and 36 kmph respectively. Find the minimum time after which they will meet if they are running in the same direction.
Question
On a circular track of length 1200 m, P, Q and R start from the same point simultaneously with speeds of 18 kmph, 27 kmph and 36 kmph respectively. Find the minimum time after which they will meet if they are running in the same direction.
Solution
To solve this problem, we need to find the time it takes for all three runners to meet at the same point on the track. This will happen when they have all run a distance that is a multiple of the track's length.
Step 1: Convert the speeds from km/h to m/s.
- Speed of P = 18 kmph = 18 * (1000/3600) m/s = 5 m/s
- Speed of Q = 27 kmph = 27 * (1000/3600) m/s = 7.5 m/s
- Speed of R = 36 kmph = 36 * (1000/3600) m/s = 10 m/s
Step 2: Find the relative speeds between the runners. Since they are running in the same direction, we subtract the slower speed from the faster speed.
- Relative speed of P and Q = 7.5 - 5 = 2.5 m/s
- Relative speed of Q and R = 10 - 7.5 = 2.5 m/s
- Relative speed of P and R = 10 - 5 = 5 m/s
Step 3: Find the time it takes for each pair of runners to meet. This is the track length divided by their relative speed.
- Time for P and Q to meet = 1200 / 2.5 = 480 seconds
- Time for Q and R to meet = 1200 / 2.5 = 480 seconds
- Time for P and R to meet = 1200 / 5 = 240 seconds
Step 4: The minimum time after which all three will meet is the least common multiple (LCM) of these times. The LCM of 480, 480, and 240 is 480 seconds.
So, the minimum time after which they will meet if they are running in the same direction is 480 seconds.
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