A man reaches his office 20 minutes late if he walks from his home at 3 km/h, and he reaches 30 minutes early if he walks at 4 km/h. What is the distance between his home and his office?

Introduction / Context:
This time and distance question involves early and late arrival based on two different walking speeds. The key idea is that there is a fixed scheduled time to reach the office, and changing the walking speed changes the arrival time relative to that schedule. We use this information to find the actual distance between home and office.

Given Data / Assumptions:

- Distance between home and office = D km (unknown).
- Usual scheduled travel time = T hours (unknown).
- If the man walks at 3 km/h, he is 20 minutes late, so time taken = T + 20/60 hours.
- If he walks at 4 km/h, he is 30 minutes early, so time taken = T - 30/60 hours.
- Motion is straight and speeds are constant.

Concept / Approach:
Distance is equal to speed multiplied by time. Since the distance from home to office is fixed, we can write two expressions for that distance using the two different speeds and arrival conditions. These give two equations in the two unknowns D and T. Solving the system of equations yields the distance D directly.

Step-by-Step Solution:
Let the distance be D km and the scheduled time be T hours. At 3 km/h: D = 3 * (T + 20/60). At 4 km/h: D = 4 * (T - 30/60). So we have: 3 (T + 1/3) = D. (1) 4 (T - 1/2) = D. (2) Set right sides equal: 3 (T + 1/3) = 4 (T - 1/2). Expand: 3T + 1 = 4T - 2. Rearrange: 1 + 2 = 4T - 3T, so 3 = T. Now substitute T = 3 in (1): D = 3 (3 + 1/3). D = 3 * (10/3) = 10 km.

Verification / Alternative check:
If D = 10 km, then at 3 km/h the time taken is 10 / 3 hours, which is 3 hours 20 minutes. The scheduled time T is 3 hours, so he is 20 minutes late, which matches the statement. At 4 km/h, the time taken is 10 / 4 = 2.5 hours, which is 2 hours 30 minutes. This is 30 minutes earlier than the scheduled 3 hours. Both conditions are satisfied, so D = 10 km is correct.

Why Other Options Are Wrong:
20 km, 16 km, 14 km: If you plug any of these distances into the time calculations with 3 km/h and 4 km/h, you will not get arrival times that differ from the schedule by exactly 20 minutes late and 30 minutes early. Each fails at least one of the conditions given in the problem.

Common Pitfalls:
Some learners mistakenly add or subtract both 20 minutes and 30 minutes from the same travel time, which is incorrect. Others forget to convert minutes to hours, leading to inconsistent equations. Always express all times in hours when speeds are given in km/h, and clearly write two separate equations for the two situations.

Final Answer:
The distance between his home and office is 10 km.