A policeman is 400 metres behind a thief when he starts chasing. At that instant, the thief starts running at 5 km/h and the policeman runs at 9 km/h, both along the same straight road in the same direction. Assuming both maintain their speeds, how far (in metres) will the thief have run by the time he is caught by the policeman?

Introduction / Context:
This problem is a classic chase or pursuit question in time and distance. A faster person (the policeman) chases a slower person (the thief) who has a head start. The goal is to find how far the thief runs before being caught. It tests the understanding of relative speed when two bodies move in the same direction along a straight line.

Given Data / Assumptions:

Initial distance between the policeman and the thief = 400 metres.
Speed of the thief = 5 km/h.
Speed of the policeman = 9 km/h.
They run in the same direction along a straight road without changing speed.
We must find the distance run by the thief by the time he is caught.
1 kilometre = 1000 metres for conversion if needed.

Concept / Approach:
When two objects move in the same direction, the relative speed between the faster and the slower is the difference of their speeds. The time taken by the faster object to catch the slower is equal to the initial gap divided by the relative speed. After finding this time, we multiply it by the speed of the thief to get the distance he covers before being caught. All speeds should be in the same units when computing time and distance.

Step-by-Step Solution:
Step 1: Convert the head start distance into kilometres for consistency. Initial gap = 400 metres = 0.4 kilometres. Step 2: Compute the relative speed when both move in the same direction: relative speed = speed of policeman − speed of thief = 9 − 5 = 4 km/h. Step 3: Time taken for the policeman to catch the thief = initial gap / relative speed = 0.4 / 4 hours = 0.1 hour. Step 4: Distance run by the thief in that time = speed * time = 5 km/h * 0.1 hour = 0.5 kilometres. Step 5: Convert 0.5 kilometres back into metres: 0.5 * 1000 = 500 metres. Step 6: Hence, the thief runs 500 metres before being caught.

Verification / Alternative check:
We can check by computing how far the policeman runs in the same 0.1 hour. Distance covered by the policeman = 9 km/h * 0.1 hour = 0.9 kilometres = 900 metres. Since the thief has run 500 metres, the distance between their starting positions is 400 metres plus 500 metres for the thief, which is 900 metres, matching the policeman's distance. This confirms that they meet after the thief has run 500 metres.

Why Other Options Are Wrong:
If the thief ran only 400 metres, the policeman would still be behind because he would have to cover more than the initial gap. Distances like 300 or 450 metres would not satisfy the equality of distances travelled when checked with their respective speeds and time. A distance of 600 metres would require a longer time, which would give the policeman more than enough time to close the gap. Only 500 metres leads to consistent positions for both at the meeting point.

Common Pitfalls:
Learners sometimes add the speeds instead of subtracting them when both move in the same direction, which is incorrect. Another common mistake is to use metres for the head start and kilometres per hour for speed without conversion, leading to unit inconsistency. Carefully using relative speed as a difference and maintaining consistent units avoids these errors.

Final Answer:
The thief will have run 500 metres before being caught by the policeman.