A thief drives away with a Maruti car at a constant speed of 40 km/h. The theft is discovered after 30 minutes, and the owner immediately starts chasing on a bike at 50 km/h. After how much time from the moment the thief left will the owner overtake the thief?

Introduction / Context:
This is a typical chase or pursuit problem in time and distance. One vehicle starts earlier at a lower speed, and another starts later at a higher speed. The key idea is to understand the concept of head start and use relative speed to determine when the faster chaser will catch the slower vehicle that started earlier.

Given Data / Assumptions:

• Speed of the thief in the car = 40 km/h.
• The theft is discovered 30 minutes (0.5 hour) later.
• Speed of the owner on the bike = 50 km/h.
• The owner starts the chase exactly 0.5 hour after the thief starts.
• Both travel along the same straight route in the same direction.
• We are asked for the time from the thief's start until the owner catches him.

Concept / Approach:
When a faster vehicle is chasing a slower one, the catch up time depends on the initial lead and the difference in speeds. The lead is the distance already covered by the thief before the owner starts. After that, the relative speed between them is the difference of their speeds. Time to catch up is equal to lead divided by relative speed. Then we can convert that time back to the total time from the thief's start.

Step-by-Step Solution:
Let t be the time in hours from the thief's start until the catch up. Owner starts 0.5 hour later, so owner travel time = t - 0.5 hours. Distance traveled by thief = 40 * t km. Distance traveled by owner = 50 * (t - 0.5) km. At catch up, distances are equal: 40 * t = 50 * (t - 0.5). 40t = 50t - 25 10t = 25, so t = 25 / 10 = 2.5 hours. 2.5 hours = 2 hours 30 minutes.

Verification / Alternative check:
The thief has a 0.5 hour head start at 40 km/h, so the lead distance is 40 * 0.5 = 20 km. The relative speed between the owner and the thief is 50 - 40 = 10 km/h. Time to close a 20 km gap at 10 km/h is 20 / 10 = 2 hours. These 2 hours are measured from the moment the owner starts. Therefore, total time from the thief's start is 0.5 hour of head start plus 2 hours of chase, which again gives 2.5 hours or 2 hours 30 minutes.

Why Other Options Are Wrong:
2 hours and 2 hours 5 minutes are both too short because they ignore or underestimate the head start and the relative speed. 2 hours 10 minutes is still smaller than the correct 2.5 hours and does not satisfy the distance equality. The value 1 hour 50 minutes is much too small and clearly inconsistent with the head start and speed difference. Only 2 hours 30 minutes satisfies the correct equations.

Common Pitfalls:
Students often forget that the owner travels for less time than the thief and incorrectly set the travel times equal. Another frequent error is to add the speeds instead of subtracting them when calculating relative speed in a same-direction chase. Careful attention to who starts first, how long each travels, and whether the directions are the same or opposite is crucial for these problems.

Final Answer:
The owner will overtake the thief 2 hours 30 minutes after the thief starts driving away.