Train Problems: Which Distance to Use for Each Crossing

Every train problem uses the ordinary relation time = distance ÷ speed. What makes these questions feel different is that a train is long. The front of the train has to travel further than the gap between two points before the whole train is clear, and the extra distance is exactly the train's own length.

So the only real decision in a train question is which distance to use. Once you have that and a speed in matching units, the arithmetic is a single division. This page sets out the four cases you will meet, then shows how to work backwards from times to lengths and speeds.

Lengths are usually in metres and times in seconds, while speeds are given in km/h. Convert before dividing. One kilometre is 1,000 metres and one hour is 3,600 seconds, so 1 km/h = 1,000/3,600 = 5/18 m/s.

• km/h to m/s: multiply by 5/18. For example, 72 km/h = 20 m/s and 54 km/h = 15 m/s.
• m/s to km/h: multiply by 18/5. For example, 25 m/s = 90 km/h.

Speeds that are multiples of 18 km/h convert to whole numbers of metres per second, which is why question setters favour 36, 54, 72 and 90 km/h.

When two trains move, use relative speed: imagine standing on one train and watching the other. Coming towards you, it seems to move at the sum of the two speeds; moving the same way, it seems to move at the difference. In both cases the distance is both trains' lengths, because the trains are clear of each other only when their rear ends have passed.

Many questions give two crossing times and ask for the train's length or speed. Compare the two situations. The difference in distance divided by the difference in time gives the speed, because the train's own length cancels out.

In km/h, 15 m/s is 15 × 18/5 = 54 km/h. Keep the answer in the unit the options use.

A moving person is a short train of zero length. A man walking in the same direction as the train reduces the relative speed; a man walking towards it increases it.

• ✘ Forgetting to add the platform or the second train's length.
• ✘ Mixing km/h with metres and seconds. Convert first.
• ✘ Adding speeds for trains moving the same way. Add only for opposite directions.
• ✘ Answering in m/s when the options are in km/h, or the reverse.

Train questions are a special case of relative motion. The same logic appears in boats and streams, where the current adds to or subtracts from the boat's speed, and in average speed questions about the same journey. For a one-page formula list, see the Quant cheat code.

When do I add the platform length to the train length?

Whenever the object being crossed has a length of its own, such as a platform, bridge, tunnel or another train. The train has crossed it only when its last coach clears the far end, so the front travels the train length plus the object's length.

Why is a person or pole treated as having zero length?

A pole, signal or standing person is narrow compared with a train, so exam questions treat it as a point. The train passes it once its whole length has gone by, so the distance is just the train's length.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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