Two persons started running on a circular track simultaneously with speeds of 20 m/s and 30 m/s in opposite directions. If the circumference of the circular track is 100 m, then find at how many distinct points they will cross each other? 

Given : 

Speed of 1st person = 20 m/s

Speed of the 2nd person = 30 m/s

Circumference of the Track = 100 m

Concept used :

Speed = Distance covered / Time taken

Calculation :
The time taken by the first person to complete a full lap is 100 m / 20 m/s = 5 seconds. The time taken by the second person to complete a full lap is 100 m / 30 m/s = 10/3 seconds.

​The LCM of 5 seconds and 10/3 seconds is 10 seconds, which means they will end up in the same initial positions after 10 seconds.

​Let the two persons cross each other for the first time after x seconds.

Then, total distance covered by 1st person = 20x and the 2nd person = 30x

Now, the total distance covered by both the person = Circumference of the track

⇒ 30x + 20x = 100m ⇒ 50x = 100m ⇒ x = 2 s

∴ The two persons will cross each other for the first time after 2 seconds.

The two persons will cross each other again after 2 seconds, and then again after 2 seconds, and so on.

So in 10 seconds they will meet = 10/2 = 5 times

Therefore, the two persons will cross each other at 5 distinct points on the track.
Shortcut Trick

S₁ = 20 m/s and S₂ = 30 m/s

S₁/S₂ = 2/3

They are running in opposite directions so they will meet each other at distinct points = 2 + 3 = 5