Two trains are moving in the same direction at speeds of 72 km/h and 36 km/h. The faster train completely crosses a girl who is sitting at a window seat in the slower train in 32 seconds. What is the length of the faster train in metres?

Introduction / Context:
In many aptitude exams, train problems test the idea of relative speed when two objects move in the same direction or opposite directions. In this question, two trains move in the same direction, and the faster train overtakes a girl sitting in the slower train. We are asked to find the length of the faster train using the time taken to completely pass the girl and the relative speed between the trains.

Given Data / Assumptions:

• Speed of the faster train = 72 km/h.
• Speed of the slower train = 36 km/h.
• The girl is sitting in the slower train, treated as a point object.
• Time taken by the faster train to completely cross the girl = 32 seconds.
• We assume uniform speeds and straight tracks.

Concept / Approach:
When two bodies move in the same direction, the effective or relative speed is the difference between their individual speeds. Here, the faster train must cover its own length relative to the girl in the slower train. Once we calculate the relative speed in metres per second, we multiply it by the time taken to cross in order to get the length of the faster train.

Step-by-Step Solution:
Step 1: Relative speed in km/h = 72 - 36 = 36 km/h. Step 2: Convert this into m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed in m/s = 36 * 5/18 = 10 m/s. Step 4: Time taken to cross the girl = 32 s. Step 5: Length of the faster train = relative speed * time = 10 * 32 = 320 m.

Verification / Alternative check:
We can verify the calculation quickly. If the faster train is 320 m long and the relative speed is 10 m/s, then time to cross the girl = distance / speed = 320 / 10 = 32 s, which matches the given time. The use of relative speed is correct because both trains are moving in the same direction and the girl shares the motion of the slower train.

Why Other Options Are Wrong:
170 m, 100 m, and 270 m are all smaller than the correct length and would give times less than 32 seconds at 10 m/s. For example, 270 m at 10 m/s would give 27 s. The distractor 220 m also leads to 22 s, which does not match the given crossing time. Therefore, none of these satisfy the condition of 32 seconds.

Common Pitfalls:
A common mistake is to add the two speeds instead of subtracting them when the motion is in the same direction. Another error is to forget to convert from km/h to m/s before multiplying by time in seconds. Some learners also incorrectly treat the entire slower train length, but only the length of the faster train matters when overtaking a single point like a sitting passenger.

Final Answer:
The length of the faster train is 320 m.