A boatman can row 3 km against the stream in 20 minutes and return the same distance with the stream in 18 minutes. What is the rate of the current (in km/h)?

Introduction / Context:
This problem involves a boatman travelling the same distance upstream and downstream in different times. From these times and the known distance, we can determine the effective upstream and downstream speeds and then compute the rate of the current in the river.

Given Data / Assumptions:

Distance upstream = 3 km; distance downstream = 3 km (same path back).
Time upstream = 20 minutes = 20/60 = 1/3 hour.
Time downstream = 18 minutes = 18/60 = 3/10 hour.
Let boat speed in still water be b km/h.
Let speed of current be c km/h.
Upstream speed = (b - c) km/h; downstream speed = (b + c) km/h.

Concept / Approach:
We compute the effective speeds from distance and time. For upstream, speed = distance / time, and similarly for downstream. Then we use the standard relationships:
b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 Our target is c, the rate of the current.

Step-by-Step Solution:
Step 1: Compute upstream speed. Upstream speed v_u = distance / time = 3 / (1/3) = 9 km/h. Step 2: Compute downstream speed. Downstream time = 18/60 = 3/10 hour. Downstream speed v_d = 3 / (3/10) = 3 * (10/3) = 10 km/h. Step 3: Find the rate of current. c = (v_d - v_u) / 2 = (10 - 9) / 2 = 1/2 km/h.

Verification / Alternative check:
The still-water speed b = (v_d + v_u) / 2 = (10 + 9) / 2 = 19/2 = 9.5 km/h. Check upstream: b - c = 9.5 - 0.5 = 9 km/h, giving time 3/9 = 1/3 h = 20 min. Check downstream: b + c = 9.5 + 0.5 = 10 km/h, giving time 3/10 h = 0.3 h = 18 min. Both match the given data, confirming the correctness of c = 1/2 km/h.

Why Other Options Are Wrong:
1/3 km/h or 2/3 km/h would not yield the observed upstream and downstream times when combined with any consistent still-water speed.
1/4 km/h is too small, leading to almost equal upstream and downstream speeds, which is inconsistent with the clearly different times of 20 and 18 minutes.

Common Pitfalls:
Learners often forget to convert minutes into hours, causing incorrect speed calculations. Another common slip is to subtract upstream speed from downstream speed but forget to divide by 2 when computing current speed. Maintaining clear units and applying the formulas for b and c carefully will avoid these issues.

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