Shortcut Trick
Speed ratio Rohan : Soham = 2 : 1. In the same time, Rohan covers twice the laps of Soham.
For the first time meeting at the starting point, the slowest runner (Soham) must complete at least 1 full lap.
If Soham completes 1 lap in 10 min, Rohan completes 2 laps in 10 min.
Total Race = 8000 m = 4 laps (since 8000 ÷ 2000 = 4).
Time for Rohan for 4 laps = (10 min ÷ 2 laps) × 4 laps = 20 min.
∴ The correct answer is 20 min.
Alternate Method
Given:
Total Distance = 8000 m | Circumference = 2000 m | Speed of Rohan (VR) = 2 × Speed of Soham (VS)
Let the speed of Soham be v m/min. Then the speed of Rohan is 2v m/min.
Time to reach the starting point for Soham = (2000 × n1) ÷ v
Time to reach the starting point for Rohan = (2000 × n2) ÷ 2v
They meet at the starting point at 10 min for the first time:
⇒ 10 = (2000 × n1) ÷ v and 10 = (2000 × n2) ÷ 2v
⇒ v = 200n1 and 2v = 200n2
⇒ 2(200n1) = 200n2 ⇒ n2 = 2n1
For the first time after start, smallest integer n1 = 1, which means n2 = 2.
⇒ 10 = (2000 × 1) ÷ v ⇒ v = 200 m/min (Soham's speed)
⇒ Rohan's speed = 2 × 200 = 400 m/min
⇒ Time taken by Rohan to finish 8000 m = 8000 ÷ 400
⇒ Time = 20 min
∴ The correct answer is 20 min.
Additional Information
Meeting at Starting Point
The time when two or more runners meet for the first time at the starting point is the LCM of the time taken by each runner to complete one full lap.
Relative Speed in Circular Track
If two runners move in the same direction, their relative speed is (u − v). The time taken to meet for the first time anywhere on the track is Circumference ÷ (u − v).
Number of Meeting Points
If the ratio of speeds of two runners is a : b (in simplest form), the number of distinct meeting points on the track is |a − b| for same direction and (a + b) for opposite directions.