Two persons start running simultaneously around a circular track of length 600 meters from the same point. Their speeds are 25 km/h and 35 km/h respectively, and they run in the same direction. After how many seconds will they meet side by side for the first time anywhere on the track?

Introduction / Context:
This question is about two runners moving on a circular track at different speeds, starting from the same point and running in the same direction. The problem tests the concept of relative speed and how to compute the time taken for the faster runner to gain exactly one full lap on the slower runner. Such circular track problems are standard in time and distance topics for aptitude exams.

Given Data / Assumptions:

• Length of circular track = 600 meters.
• Speed of first runner = 25 km/h.
• Speed of second runner = 35 km/h.
• Both start together from the same point and move in the same direction.
• We need the time when they next come together side by side, anywhere on the track.

Concept / Approach:
When two runners start together in the same direction on a circular track, they will be side by side again when the faster runner has gained exactly one lap on the slower runner. The time to gain one lap is equal to the track length divided by the relative speed between them. We must convert speeds from km/h to m/s or track length from meters to kilometers so that the units are consistent.

Step-by-Step Solution:
Speed of first runner = 25 km/h. Speed of second runner = 35 km/h. Relative speed = 35 - 25 = 10 km/h. Convert relative speed to m/s: 10 km/h = (10 * 1000) / 3600 m/s. Relative speed = 10000 / 3600 m/s ≈ 2.7778 m/s. Track length = 600 meters. Time to gain one lap = distance / relative speed = 600 / 2.7778 seconds. Approximately, time ≈ 216 seconds.

Verification / Alternative check:
We can work directly in kilometers. Track length = 600 meters = 0.6 km. Relative speed = 10 km/h. Time in hours = distance / speed = 0.6 / 10 = 0.06 hours. Convert hours to seconds: 0.06 * 3600 = 216 seconds. This matches the previous calculation and confirms the result.

Why Other Options Are Wrong:
180 seconds, 144 seconds, 252 seconds, and 300 seconds do not satisfy the relationship between distance and relative speed. If we multiply the relative speed in m/s by any of these times, we do not get exactly 600 meters. Only 216 seconds leads to the correct lap distance, so the other options represent incorrect or approximate guesses.

Common Pitfalls:
One common mistake is to use the sum of the speeds instead of the difference when the runners are moving in the same direction. Another error is forgetting to convert units consistently, for example using km/h with meters, which gives wrong time values. Always ensure consistent units and remember that for same-direction movement, relative speed is the difference of speeds, not the sum.

Final Answer:
The two runners will meet side by side again for the first time after 216 seconds.