A boatman can row 3 km upstream against the current in 20 minutes and then return the same 3 km downstream in 18 minutes. What is the speed of the river current (in km/h)?

Introduction / Context:
In this boats and streams problem, we are given the time taken by a boatman to cover the same distance upstream and downstream. The question asks for the speed of the current. Such questions test understanding of relative speed concepts, where the effective speed of a boat changes depending on whether it is moving with or against the current. By working with distance, time, and speed relations, we can isolate and calculate the speed of the stream in km/h.

Given Data / Assumptions:

• The distance between the two points is 3 km each way.
• Time taken upstream (against the current) is 20 minutes.
• Time taken downstream (with the current) is 18 minutes.
• We assume uniform speed of the boat in still water and a uniform stream speed.
• We must express time in hours to find speeds in km/h.

Concept / Approach:
The core idea is that:

• Upstream speed = boat speed in still water - stream speed.
• Downstream speed = boat speed in still water + stream speed.

Once we compute both effective speeds from distance and time, we can use simple algebra. The average of upstream and downstream speeds gives the boat speed in still water, and half the difference of those speeds gives the stream speed.

Step-by-Step Solution:
Distance each way = 3 km. Upstream time = 20 minutes = 20 / 60 hours = 1/3 hours. Downstream time = 18 minutes = 18 / 60 hours = 3/10 hours. Upstream speed = distance / time = 3 / (1/3) = 9 km/h. Downstream speed = distance / time = 3 / (3/10) = 10 km/h. Let b = speed of boat in still water, c = speed of current. Then b - c = 9 and b + c = 10. Add the equations: 2b = 19 so b = 19 / 2 = 9.5 km/h. Subtract the equations: 2c = 1 so c = 1/2 = 0.5 km/h.

Verification / Alternative check:
If the stream speed is 0.5 km/h, then upstream speed is 9.5 - 0.5 = 9 km/h and downstream speed is 9.5 + 0.5 = 10 km/h. Time upstream is 3 / 9 = 1/3 hours, which is 20 minutes. Time downstream is 3 / 10 hours, which is 0.3 hours or 18 minutes. These match the given data, so the calculated stream speed is consistent and correct.

Why Other Options Are Wrong:
Option 1/3 km/h underestimates the current because it would not create the observed difference between 20 minutes and 18 minutes travel times. Option 2/3 km/h is too large and would make upstream travel significantly slower than given. Option 1/4 km/h is also too small and would not adequately explain the 2 minute time difference. Only 1/2 km/h reproduces both the upstream and downstream times correctly.

Common Pitfalls:
Many learners forget to convert minutes into hours before calculating speeds. Others incorrectly average the times instead of the speeds, which leads to wrong values for the boat and stream speeds. Another frequent mistake is to treat the boat speed in still water as the average of distance and time directly, without separating upstream and downstream situations. Careful unit handling and correct use of the upstream and downstream formulas are essential.

Final Answer:
The speed of the current is 1/2 km/h.