Karthik walks to his destination at 5 km/h and reaches 8 minutes late. If he walks at 6 km/h, he reaches exactly on time. What is the distance he has to cover (in km)?

Introduction / Context:
This problem is very similar in style to other early and late arrival questions. Karthik travels the same distance at two different walking speeds, and his arrival time relative to the schedule changes. We use this information to find the distance to his destination, applying the basic relation between distance, speed, and time.

Given Data / Assumptions:

- Distance to the destination = D km (unknown).
- Scheduled (on time) travel time = T hours (unknown).
- At 5 km/h, Karthik is 8 minutes late, so time taken = T + 8/60 hours.
- At 6 km/h, he reaches on time, so time taken = T hours.
- Motion is along a straight path with constant speeds.

Concept / Approach:
We write two expressions for the distance D using the two different speeds and the corresponding travel times. Because the distance is the same in both cases, we equate the expressions and solve for the scheduled time T. Once we have T, we can compute the distance D using the speed at which he arrives on time.

Step-by-Step Solution:
Let D be the distance and T be the scheduled time. At 5 km/h: D = 5 * (T + 8/60). At 6 km/h: D = 6 * T. Equate the two distances: 5 (T + 8/60) = 6 T. Convert 8 minutes to hours: 8 / 60 = 2 / 15. So 5 (T + 2 / 15) = 6 T. Expand: 5T + 10 / 15 = 6 T. Simplify 10 / 15 = 2 / 3. So 5T + 2 / 3 = 6 T. Rearrange: 2 / 3 = 6 T - 5 T = T. Therefore T = 2 / 3 hours = 40 minutes. Distance D = 6 * T = 6 * (2 / 3) = 4 km.

Verification / Alternative check:
With D = 4 km: At 6 km/h, time taken = 4 / 6 hours ≈ 0.6667 hours, which is 40 minutes, so this matches the scheduled time. At 5 km/h, time taken = 4 / 5 hours = 0.8 hours, which is 48 minutes. The difference between 48 minutes and 40 minutes is 8 minutes, so he is 8 minutes late at 5 km/h. Both conditions are satisfied.

Why Other Options Are Wrong:
10 km, 8 km, and 6 km: For each, if you compute the travel times at 5 km/h and 6 km/h, you will not get the exact combination of 8 minutes late at 5 km/h and exactly on time at 6 km/h. Only 4 km fits both conditions at once.

Common Pitfalls:
A common error is to try to average the speeds or directly subtract times without writing any equations. Another mistake is forgetting that 8 minutes must be converted to 2 / 15 hours when working in km/h. Always set up the distance equality for the two speeds and carefully handle the unit conversions.

Final Answer:
Karthik covers a distance of 4 km to reach his destination on time at 6 km/h.