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

Introduction / Context:
This composite train problem first uses relative speed in opposite directions to determine the length of a train, and then uses that length with the train's speed and the time to cross a bridge to find the bridge length. It tests multi step reasoning, unit conversion, and careful algebra.

Given Data / Assumptions:

• Speed of first train = 48 km/h.
• Speed of second train = 42 km/h.
• Length of second train = half the length of the first train.
• The trains cross each other in 12 seconds while travelling in opposite directions.
• The first train passes a bridge in 45 seconds at 48 km/h.
• We must find the length of the bridge in metres.

Concept / Approach:
Let the length of the first train be L metres, so the second train has length L / 2 metres. In opposite directions, relative speed is the sum of their speeds, and the total distance during crossing is L + L / 2. We first determine L from this information. Then, treating the bridge as a fixed object, the distance covered while passing the bridge is L plus the bridge length. Using the constant speed of 48 km/h and the 45 second crossing time, we solve for the bridge length.

Step-by-Step Solution:
Step 1: Convert speeds to metres per second: 48 km/h = 48 * 5 / 18 = 40 / 3 metres per second, and 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 trains cross each other = L + L / 2 = 3L / 2 metres.Step 4: Time for crossing = 12 seconds, so 3L / 2 = 25 * 12 = 300, which gives L = 200 metres.Step 5: Now consider the first train passing a bridge. Distance covered = L + bridge length = 200 + B, speed = 48 km/h = 40 / 3 metres per second, time = 45 seconds.Step 6: Use distance = speed * time: 200 + B = (40 / 3) * 45 = 40 * 15 = 600, giving B = 600 - 200 = 400 metres.

Verification / Alternative check:
Check crossing of trains: with L = 200, second train length = 100 metres. Total distance = 300 metres, relative speed = 25 metres per second, time = 300 / 25 = 12 seconds as required.Check bridge crossing: with bridge length 400 metres, total distance = 600 metres and speed = 40 / 3 metres per second, time = 600 / (40 / 3) = 600 * 3 / 40 = 45 seconds.

Why Other Options Are Wrong:
Bridge lengths such as 250 metres, 320 metres, 390 metres or 280 metres do not satisfy both the crossing and bridge conditions when verified. Some values correspond to using wrong relative speeds or incorrect time conversions. Only 400 metres is consistent with all parts of the question.

Common Pitfalls:
Learners may forget that the second train is only half as long as the first, leading to an incorrect total distance during the crossing. Another common error is to use the difference of speeds instead of the sum for opposite directions. Mistakes can also occur when converting km/h to metres per second or when solving the equation 3L / 2 = 300. Organising the problem in two clear stages helps to avoid confusion.

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