A train is travelling at 48 km/h. It crosses another train, having half of its own length and travelling in the opposite direction at 42 km/h, in 12 seconds. It also passes completely over a railway platform in 45 seconds. What is the length of the platform in metres?

Introduction / Context:
This question is structurally similar to a typical two part trains problem, where information about two trains crossing each other is used to deduce the length of one train, and then that length is used with a platform crossing time to find the length of the platform. It tests multi step reasoning, unit conversion, and consistent application of relative speed concepts.

Given Data / Assumptions:

• Speed of the main train = 48 km/h.
• Speed of the second train = 42 km/h.
• Length of the second train = half the length of the main train.
• The trains move in opposite directions and cross each other in 12 seconds.
• The main train crosses a platform completely in 45 seconds at 48 km/h.
• We must find the length of the platform in metres.

Concept / Approach:
Let the length of the main train be L metres, so the other train has length L / 2 metres. First, when two trains run in opposite directions, their relative speed is the sum of their speeds and the total distance covered during crossing is L + L / 2. From this we compute L. Next, when the main train crosses a platform, the distance travelled is L plus the platform length. Using its speed and crossing time, we solve for the platform length.

Step-by-Step Solution:
Step 1: Convert speeds to metres per second: 48 km/h = 48 * 5 / 18 = 40 / 3 metres per second; 42 km/h = 42 * 5 / 18 = 35 / 3 metres per second.Step 2: Relative speed in opposite directions = 40 / 3 + 35 / 3 = 75 / 3 = 25 metres per second.Step 3: Total distance when they cross each other = L + L / 2 = 3L / 2 metres.Step 4: Time for crossing is 12 seconds, so 3L / 2 = 25 * 12 = 300, giving L = 200 metres.Step 5: Now consider the main train crossing the platform. Let the platform length be P metres.Step 6: Total distance while crossing the platform = L + P = 200 + P.Step 7: Speed of the train = 40 / 3 metres per second, and time taken = 45 seconds, so 200 + P = (40 / 3) * 45 = 600.Step 8: Therefore P = 600 - 200 = 400 metres.

Verification / Alternative check:
Check the crossing of trains: with L = 200 and the other train 100 metres long, total distance is 300 metres. At relative speed 25 metres per second, time = 300 / 25 = 12 seconds, matching the data.Check platform crossing: train length 200 metres plus platform 400 metres gives 600 metres. At 40 / 3 metres per second, time = 600 / (40 / 3) = 45 seconds, also matching the given information.

Why Other Options Are Wrong:
Platform lengths such as 320 metres, 360 metres, 480 metres or 500 metres do not satisfy both the crossing and platform conditions when checked with the same speeds and times. Only 400 metres ensures both the 12 second crossing and the 45 second platform passage work out correctly.

Common Pitfalls:
Common errors include forgetting that the other train has only half the length of the main train or failing to add both train lengths when calculating the crossing distance. Some candidates also confuse km/h with metres per second or mishandle the arithmetic when solving 3L / 2 = 300. Separating the problem clearly into two stages and checking each numerically helps avoid these mistakes.

Final Answer:
The length of the platform is 400 metres.