# Time and Distance Tutorial II: Relative Speed, Linear & Circular Races, Meeting Points

## Circular race: When there are three runners around the circular track

Runners A, B and C running around a circular track with respective speeds a, b and c such that a > b > c. They start simultaneously and if the length of the race is 'L'.

Same Direction: Time taken to meet them for the first time along anywhere on the track = "LCM" [L/(a - b), L/(b - c)]
Time taken to meet them for the first time at the starting point is = "LCM" [L/a, L/b, L/c]
Opposite direction: If A is running clockwise while B and C are running anticlockwise;
Time taken to meet them for the first time along anywhere on the track = "LCM" [L/(a + b), L/(a + c)]
Time taken to meet them for the first time at the starting point is = "LCM" [L/a, L/b, L/c]

### Concept review questions

Avinash, Bhanu and Suresh run around a circular track of length 400 meters ¸ starting from the same point simultaneously in the same direction at respective speeds of 4 kmph, 6 kmph and 8 kmph. When they all will meet at the starting point for the first time after they started?
From the above shortcut, time taken to meet at the starting point for the first time after they started = "LCM"[400/4 xx 18/5, 400/6 xx 18/5, 400/8 xx 18/5]
= "LCM" [360, 240, 180] = 720 sec
Joy and Ajo simultaneously start running from two diametrically opposite points A and B respectively of a circular track in opposite direction. Joy is running at 10 kmph and the speed of Ajo is 15 kmph. If they took 12.5 seconds to meet at first. How long from the start they will have to be on diametrically opposite positions again?
Ratio of speeds of Joy and Ajo = 10 : 15 = 2 : 3
Divide the circular track into 10 equal distances.

Let the length of the track is 10 units.
When Joy cover 2 units counterclockwise and Ajo covered 3 units clockwise (as per diagram) they will meet at first (at point C)
For the meeting they took 12.5 seconds.
From the point C if Joy covers 2 units more on counter clockwise direction and Ajo covers 3 units more on clockwise direction then they will be in diametrically opposite positions.
Ie, for this additional movement, they require another 12.5 seconds
Therefore the total time required to become diametrically opposite is 12.5 + 12.5 = 25 sec
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