A train with 120 wagons overtakes Arun, who is riding a horse in the same direction, and completely passes him in 36 seconds. From the instant it starts overtaking Arun, the train runs for 30 minutes before it starts overtaking Sriram, who is riding a horse from the opposite direction and is completely passed in 24 seconds. After the train has crossed Sriram, how many seconds later will Arun meet Sriram?

Introduction / Context:
This is a multi step time and distance puzzle involving a long train and two riders on horses. The train first overtakes Arun, then after running for half an hour it overtakes Sriram coming from the opposite direction. The question asks for the time between the instant when the train finishes overtaking Sriram and the moment when Arun and Sriram meet each other on the track. The key ideas are relative speed and careful use of a time line.

Given Data / Assumptions:

- Train speed = V m/s (unknown).
- Length of the train = L metres (unknown).
- Speed of Arun on his horse = vA m/s, same direction as the train.
- Speed of Sriram on his horse = vS m/s, opposite direction to the train.
- Train overtakes Arun completely in 36 seconds when moving in the same direction.
- Train overtakes Sriram completely in 24 seconds when moving in the opposite direction.
- From the instant it starts overtaking Arun to the instant it starts overtaking Sriram, 30 minutes = 1800 seconds elapse.
- All bodies move at constant speeds along a straight track.

Concept / Approach:
When the train overtakes someone moving in the same direction, the relative speed is V - vA. When it overtakes someone moving towards it, the relative speed is V + vS. In both cases, the distance covered relative to the rider equals the length of the train L. Using the two overtaking times, we build equations for L. Then we relate the positions of Arun and Sriram at the start of each overtaking event and find when they will meet each other. An important observation is that the final answer turns out to be independent of the actual numerical values of V, vA, vS, and L.

Step-by-Step Solution:
From overtaking Arun (same direction): (V - vA) * 36 = L. (1) From overtaking Sriram (opposite direction): (V + vS) * 24 = L. (2) Equate (1) and (2): (V - vA) * 36 = (V + vS) * 24. Divide by 12: 3(V - vA) = 2(V + vS). So 3V - 3 vA = 2V + 2 vS, which gives V = 3 vA + 2 vS. Take t = 0 at the instant the train starts overtaking Arun. Position of train front: xT(t) = x0 + V t. Position of Arun: xA(t) = x0 + vA t. At t = 1800 seconds, the train starts overtaking Sriram: xT(1800) = xS(1800). Let Sriram's position be xS(t) = s0 - vS t (coming from opposite direction). Then x0 + V * 1800 = s0 - vS * 1800, so s0 - x0 = 1800 (V + vS). Arun and Sriram meet when xA(t) = xS(t). So x0 + vA t = s0 - vS t, which gives (vA + vS) t = s0 - x0. Substitute s0 - x0 from above: (vA + vS) t = 1800 (V + vS). But V + vS = 3 vA + 3 vS = 3 (vA + vS). Therefore t = 1800 * 3 (vA + vS) / (vA + vS) = 5400 seconds. The train finishes overtaking Sriram at t = 1800 + 24 = 1824 seconds. Time after crossing Sriram when Arun meets Sriram = 5400 - 1824 = 3576 seconds.

Verification / Alternative check:
Notice that the final time 5400 seconds (for Arun and Sriram to meet) depends only on the 1800 second gap and the relationship V = 3 vA + 2 vS. The extra 24 seconds for overtaking Sriram simply shifts the time of crossing, and subtracting it from 5400 seconds gives 3576 seconds. No contradictory information arises, so the numerical answer is consistent and unique. Any numerical assignment of vA and vS satisfying V = 3 vA + 2 vS will reproduce the same time difference.

Why Other Options Are Wrong:
3560 sec and 3600 sec: These values do not match the exact algebraic difference 5400 - 1824 derived from the equations and would imply inconsistent speeds or distances.
Cannot be determined: Although the problem looks complicated, the information provided is sufficient, and unknown speeds cancel out, so the answer can be determined exactly.

Common Pitfalls:
The most common mistake is to try to find explicit speeds for the horses and the train unnecessarily, which can lead to algebraic mistakes. Another error is to misinterpret the 30 minutes as the time between the ends of overtaking rather than between the starts. Learners also sometimes forget that Arun and Sriram continue to move after all overtaking events, which is essential for finding when they meet each other.

Final Answer:
After the train has crossed Sriram, Arun and Sriram meet each other after 3576 seconds.