A train 125 metres long passes a man who is running at 4 km/h in the same direction as the train. If the train takes 10 seconds to completely overtake the man, what is the speed of the train in km/h?

Introduction / Context:
This question combines relative speed and unit conversion from km/h to m/s, both of which are frequent in quantitative aptitude problems involving trains and moving persons. The train and the man are moving in the same direction, so the train only gains on the man at the relative speed. The time taken to completely cross the man depends on the train length and this relative speed.

Given Data / Assumptions:

• Length of the train = 125 metres.
• Speed of the man = 4 km/h.
• The man and the train move in the same direction.
• Time taken by the train to pass the man completely = 10 seconds.
• Speeds are uniform and motion is along a straight track.

Concept / Approach:
When two bodies move in the same direction, the effective speed that determines how fast one overtakes the other is the difference of their speeds. Here, the distance covered relative to the man is the full length of the train. So, relative speed = train speed minus man speed. We first find relative speed in metres per second from the length and time, then convert the final train speed to km/h.

Step-by-Step Solution:
Step 1: Relative speed in m/s = distance / time = 125 / 10 = 12.5 m/s. Step 2: Convert man speed to m/s: 4 km/h = 4 * 5/18 = 20/18 = 1.111... m/s. Step 3: Let train speed in m/s be v. Step 4: Relative speed v - 1.111... = 12.5. Step 5: So v = 12.5 + 1.111... ≈ 13.611... m/s. Step 6: Convert v back to km/h: v km/h = 13.611... * 18/5 ≈ 48.9996 km/h. Step 7: Round this to the nearest whole number: approximately 49 km/h.

Verification / Alternative check:
Take train speed as exactly 49 km/h. Convert to m/s: 49 * 5/18 ≈ 13.611 m/s. Relative speed with respect to the man is then 13.611 - 1.111 ≈ 12.5 m/s. Time to cross a 125 m distance at 12.5 m/s is 125 / 12.5 = 10 seconds, which matches the data perfectly. This confirms that 49 km/h is consistent.

Why Other Options Are Wrong:
Speeds 50, 51, or 52 km/h would give relative speeds slightly larger than 12.5 m/s and hence smaller crossing times, while 48 km/h would give a slower relative speed and a longer time than 10 seconds. Therefore, they do not match the given time exactly and must be rejected.

Common Pitfalls:
Typical mistakes include adding the speed of the man instead of subtracting it, forgetting to convert km/h to m/s, or treating the man as stationary. Some learners also incorrectly assume that time depends on both train and man lengths, but here only the train length matters because the man is treated as a point.

Final Answer:
The speed of the train is 49 kmph.