A man can row at 8 km/h in still water. If the river current flows at 2 km/h, it takes him 4 hours more to row a certain distance upstream than to row the same distance downstream. What is that distance (in km)?

Introduction / Context:
This is a typical boats and streams time-difference problem. The boat's speed in still water and the current speed are known, and you are told that rowing upstream takes 4 hours more than rowing downstream for the same distance. From this information, you must determine the distance.

Given Data / Assumptions:

Boat speed in still water, b = 8 km/h.
River current speed, c = 2 km/h.
Upstream speed = b - c = 8 - 2 = 6 km/h.
Downstream speed = b + c = 8 + 2 = 10 km/h.
Let the one-way distance be d km.
Time upstream is 4 hours more than time downstream.

Concept / Approach:
Time taken to cover a distance equals distance divided by speed. For distance d, upstream time is d/6 hours and downstream time is d/10 hours. The condition that upstream time is 4 hours more leads to the equation:
d/6 = d/10 + 4. Solving this linear equation in d gives the required distance.

Step-by-Step Solution:
Step 1: Write the time relation. Upstream time - downstream time = 4. So d/6 - d/10 = 4. Step 2: Combine the fractions on the left-hand side. Find LCM of 6 and 10, which is 30. d/6 - d/10 = (5d - 3d) / 30 = 2d / 30 = d / 15. So d / 15 = 4. Step 3: Solve for d. d = 4 * 15 = 60 km.

Verification / Alternative check:
Check times for d = 60 km. Upstream time = 60 / 6 = 10 hours. Downstream time = 60 / 10 = 6 hours. Difference = 10 - 6 = 4 hours, exactly as stated in the problem. This confirms that 60 km is the correct distance.

Why Other Options Are Wrong:
For 54 km, upstream time = 54/6 = 9 hours and downstream time = 54/10 = 5.4 hours, difference = 3.6 hours, not 4.
For 32 km or 45 km, similar calculations show time differences different from 4 hours, so they cannot satisfy the given condition.

Common Pitfalls:
A frequent mistake is reversing the time difference (writing d/10 = d/6 + 4 instead of d/6 = d/10 + 4) or mixing up which speed is upstream and which is downstream. Some students also incorrectly average speeds or distances rather than using the time-difference equation. Carefully identifying upstream and downstream speeds and then applying time = distance / speed with the given difference prevents these errors.

Final Answer:
The required distance is 60 km.