A train overtakes two girls who are walking towards it at speeds of 3 km/h and 6 km/h and passes them completely in 36 seconds and 30 seconds respectively. What is the length of the train (in metres)?

Introduction / Context:
This time and distance question tests the concept of relative speed when a train overtakes people walking in the opposite direction, along with correct unit conversion between kilometres per hour and metres per second. The key idea is that the length of the train is the distance it travels relative to each girl while completely passing her.

Given Data / Assumptions:

- Speed of the train = V km/h (unknown).
- Speed of first girl = 3 km/h, walking towards the train.
- Speed of second girl = 6 km/h, also walking towards the train.
- Time taken to overtake first girl completely = 36 seconds.
- Time taken to overtake second girl completely = 30 seconds.
- Length of the train = L metres, the same in both situations.
- All motions are along the same straight track and speeds are constant.

Concept / Approach:
When two bodies move towards each other, their relative speed is the sum of their individual speeds. The distance covered while completely overtaking a walker is equal to the length of the train. We must convert km/h to m/s using 5/18 and then form two equations for L using the two given times. Equating these expressions for L lets us solve first for the speed of the train and then for its length in metres.

Step-by-Step Solution:
Let the speed of the train be V km/h and its length be L metres. Relative speed with the first girl = (V + 3) km/h. Convert to m/s: (V + 3) * 5 / 18. Distance to overtake = L = relative speed * time = (V + 3) * 5 / 18 * 36. So L = (V + 3) * 10. (Equation 1) Relative speed with the second girl = (V + 6) km/h. In m/s: (V + 6) * 5 / 18. Again L = (V + 6) * 5 / 18 * 30. So L = (V + 6) * 25 / 3. (Equation 2) Equate both expressions for L: (V + 3) * 10 = (V + 6) * 25 / 3. Multiply both sides by 3: 30(V + 3) = 25(V + 6). 30V + 90 = 25V + 150. 5V = 60, so V = 12 km/h. Substitute in Equation 1: L = (12 + 3) * 10 = 15 * 10 = 150 metres.

Verification / Alternative check:
If the train is 150 m long and moves at 12 km/h, its speed in m/s is 12 * 5 / 18 = 10/3 m/s. With the first girl, relative speed is (12 + 3) km/h = 15 km/h = 25/3 m/s. Time to cover 150 m is 150 / (25/3) = 18 seconds * 2 = 36 seconds, which matches the question. With the second girl, relative speed is (12 + 6) = 18 km/h = 5 m/s. Time = 150 / 5 = 30 seconds, which also matches. So the result is consistent with both sets of data.

Why Other Options Are Wrong:
120 m: Using 120 m as the length does not satisfy both overtaking times when you solve for the speed, so it contradicts the given numbers.
125 m: This length also fails to give 36 seconds and 30 seconds simultaneously for any single train speed.
None of these: There is a valid consistent length of 150 m, so this option is incorrect.

Common Pitfalls:
A common mistake is to subtract the walking speeds instead of adding them, even though the girls walk towards the train, not in the same direction. Another frequent error is to forget to convert km/h to m/s before using time in seconds, which produces inconsistent answers. Some learners also assume that the two given times must be averaged instead of building proper equations from the two situations.

Final Answer:
The length of the train is 150 m.