A man covered a certain distance at a constant speed. If he had moved 3 km/h faster, he would have taken 40 minutes less. If he had moved 2 km/h slower, he would have taken 40 minutes more. What is the distance covered (in km)?

Introduction / Context:
This is a classic time, speed, and distance problem where the same distance is travelled at three different hypothetical speeds. The question asks you to use the information about changes in speed and corresponding changes in time to determine the unknown distance. Such questions test algebraic formulation of time and distance relations and are very common in aptitude exams.

Given Data / Assumptions:
- Let the original speed of the man be v km/h and the distance be d km.
- If he moves at (v + 3) km/h, his time decreases by 40 minutes (which is 2/3 hour).
- If he moves at (v - 2) km/h, his time increases by 40 minutes (2/3 hour).
- The distance d remains constant in all cases.
- Motion is along a straight path with uniform speed, and there are no stops.

Concept / Approach:
We use time = distance / speed. The original time is d / v. When the speed is changed, the time also changes. The given statements about 40 minutes less or more allow us to set up two algebraic equations involving d and v. Solving these simultaneous equations gives the values of v and d. This approach is a standard technique for handling problems with changed speed and changed time for a fixed distance.

Step-by-Step Solution:
Step 1: Let the original time be T = d / v hours.Step 2: At speed (v + 3) km/h, time taken is d / (v + 3). This is 2/3 hour less than T.So, d / v - d / (v + 3) = 2/3.Step 3: At speed (v - 2) km/h, time taken is d / (v - 2). This is 2/3 hour more than T.So, d / (v - 2) - d / v = 2/3.Step 4: Now we solve the system of equations:(i) d / v - d / (v + 3) = 2/3.(ii) d / (v - 2) - d / v = 2/3.Step 5: Solving these equations (for example, by algebraic manipulation) gives v = 12 km/h and d = 40 km.

Verification / Alternative check:
With v = 12 km/h and d = 40 km, original time T = 40 / 12 = 10/3 hours (3 hours 20 minutes). At speed 15 km/h (12 + 3), time = 40 / 15 = 8/3 hours (2 hours 40 minutes), which is 40 minutes less than 3 hours 20 minutes. At speed 10 km/h (12 - 2), time = 40 / 10 = 4 hours, which is 40 minutes more than 3 hours 20 minutes. Thus both conditions are satisfied, confirming that the distance is 40 km.

Why Other Options Are Wrong:
- 30 km, 36 km, and 42 km do not simultaneously satisfy the two conditions when you set up and solve the equations with the given speed changes.
- Only 40 km gives a consistent set of times that differ by exactly 40 minutes for the given increases and decreases in speed.

Common Pitfalls:
Many candidates incorrectly try to average speeds or directly subtract distances instead of using algebraic equations. Another frequent mistake is to convert 40 minutes incorrectly or to forget that the distance is the same in all three scenarios. Always translate the word statements into precise time equations using time = distance / speed and then solve the system step by step.

Final Answer:
The distance covered by the man is 40 km.