A train takes 10 seconds to cross a pole and twice this time to cross a platform of length 200 m. What is the length of the train in metres?

Introduction / Context:
This problem combines two related situations: a train crossing a pole and the same train crossing a platform. It tests basic understanding of how distance, speed, and time interact, and how the effective distance changes depending on whether the train is passing a point object or an extended object like a platform.

Given Data / Assumptions:

• Time taken to cross a pole = 10 s.
• Time taken to cross a platform = 20 s (twice 10 s).
• Length of the platform = 200 m.
• The train moves with constant speed in both cases.
• The track is straight and there are no stops or changes in speed.

Concept / Approach:
When a train crosses a pole, the distance covered is just the length of the train. When it crosses a platform, the distance covered from the instant the front of the train reaches the platform until the rear leaves it is the sum of the train length and platform length. Since the time for the second scenario is double the first, we can relate the two distances using the same train speed.

Step-by-Step Solution:
Let L be the length of the train in metres. When crossing a pole, distance = L and time = 10 s. Speed of the train = L / 10 m/s. When crossing the platform, distance = L + 200 m and time = 20 s. Speed must be the same, so (L + 200) / 20 = L / 10. Multiply both sides by 20: L + 200 = 2L. Rearrange: 2L − L = 200, so L = 200 m.

Verification / Alternative check:
If the train is 200 m long, then its speed is 200 / 10 = 20 m/s when crossing the pole. When crossing the platform, distance = 200 + 200 = 400 m. At 20 m/s, time = 400 / 20 = 20 s, which matches the given condition of double the original time.

Why Other Options Are Wrong:
If the train were 180 m long, the crossing time for the platform would not be exactly twice 10 s. Similar mismatches occur for lengths 190 m and 210 m, because they do not satisfy the equation (L + 200) / 20 = L / 10. Only 200 m gives a consistent speed across both scenarios.

Common Pitfalls:
Some students mistakenly think that platform length does not matter or they confuse which distance corresponds to which time. Others forget that the same train speed applies in both situations, which is crucial for equating the two distance over time expressions. Carefully setting up the equation avoids these errors.

Final Answer:
The length of the train is 200 m.