A thief is 350 m ahead of a policeman when the policeman starts chasing. The thief also starts running at that moment. If the thief's speed is 5 km/h and the policeman's speed is 7 km/h, how far (in metres) will the thief have run before he is overtaken?

Introduction / Context:
This is another chase problem involving relative speed. A thief has a 350 m head start when the policeman begins to chase him. We know both speeds and must determine how far the thief runs before he is caught. Handle such situations by focusing on the difference in speeds when the motion is in the same direction, and treat the initial head start as the gap that must be closed.

Given Data / Assumptions:

• Initial lead of the thief = 350 m.
• Thief's speed = 5 km/h.
• Policeman's speed = 7 km/h.
• Both run in the same direction with constant speeds.
• The policeman starts chasing at the instant we start counting time.

Concept / Approach:
Relative speed when two bodies move in the same direction is the difference of their speeds. Once we know the relative speed, we can compute the time needed for the policeman to close the initial gap. That time, when multiplied by the thief's speed, gives the distance run by the thief.
relative speed = 7 - 5 = 2 km/h

Step-by-Step Solution:
Step 1: Convert the initial gap to kilometres. Initial gap = 350 m = 350 / 1000 km = 0.35 km. Step 2: Compute the time to close the gap. Relative speed = 2 km/h. time (hours) = gap / relative speed = 0.35 / 2 = 0.175 hours. Step 3: Compute the distance run by the thief. Thief's speed = 5 km/h. distance_thief = speed * time = 5 * 0.175 km. 5 * 0.175 = 0.875 km. Convert to metres: 0.875 * 1000 = 875 m. Therefore, the thief runs 875 m before he is overtaken.
Verification / Alternative check:
We can also compute the distance covered by the policeman in the same time. In 0.175 hours at 7 km/h, the policeman covers:
distance_policeman = 7 * 0.175 km = 1.225 km = 1225 m. The difference in their distances at the capture moment should equal the initial lead of 350 m:
1225 m - 875 m = 350 m. This matches the original lead, so our solution is consistent.

Why Other Options Are Wrong:
700 m implies less time than required to close the 350 m gap at a relative speed of 2 km/h.
1050 m implies more time than necessary, which would give a greater initial lead than 350 m when combined with the policeman's distance.
525 m and 650 m similarly fail to maintain the required 350 m initial difference when we reconstruct the situation.
Only 875 m fits both the head start and the speed conditions correctly.

Common Pitfalls:
Typical errors include adding the speeds instead of subtracting them when both are running in the same direction. Learners sometimes forget to convert metres to kilometres, causing mismatched units in the formula. Others may compute the distance covered by the policeman and mistakenly present it as the thief's distance. Focusing on relative speed and carefully tracking whose distance is being asked helps avoid these mistakes.

Final Answer:
The thief will have run 875 m before the policeman overtakes him.