A man can row a boat at 9 km/h in still water. For a given distance on a river, it takes him twice as long to row upstream as it takes to row downstream. What is the rate of flow of the stream (in km/h)?

Introduction / Context:
This boat and stream problem focuses on the relationship between speeds upstream and downstream and the time taken to travel a fixed distance. When the time upstream is a simple multiple of the time downstream, you can use that ratio to find the speed of the current in terms of the speed of the boat in still water.

Given Data / Assumptions:

Speed of the boat in still water, b = 9 km/h.
Let the speed of the stream be c km/h.
For a fixed distance d, time upstream is twice the time downstream.
Upstream speed = (b - c) km/h.
Downstream speed = (b + c) km/h.
Distance d is the same in both directions and cancels out in ratios.

Concept / Approach:
Time is equal to distance divided by speed. For the same distance, if the time upstream is twice the time downstream, then:
d / (b - c) = 2 * [d / (b + c)]. The distance d cancels, leaving a relation involving b and c only. We then solve this equation for c using the known value of b = 9 km/h. Finally, we interpret c as the rate of the stream.

Step-by-Step Solution:
Step 1: Set up the time ratio equation. d / (b - c) = 2d / (b + c). Cancel d from both sides to get 1 / (b - c) = 2 / (b + c). Step 2: Cross-multiply to eliminate denominators. b + c = 2(b - c). b + c = 2b - 2c. Step 3: Group like terms. Bring terms together: 3c = b. Thus c = b / 3. Step 4: Substitute the value of b. c = 9 / 3 = 3 km/h.

Verification / Alternative check:
If c = 3 km/h, then downstream speed = 9 + 3 = 12 km/h and upstream speed = 9 - 3 = 6 km/h. For distance d, time downstream = d / 12, time upstream = d / 6 = 2 * (d / 12). This matches the condition that upstream time is twice downstream time, confirming that c = 3 km/h is correct.

Why Other Options Are Wrong:
2 km/h or 1.5 km/h would give a time ratio that is not exactly 2:1 for upstream versus downstream for the same distance.
4 km/h or 5 km/h would make the current very strong relative to the boat, altering the times so that the upstream journey would take more than twice the downstream time or might even become too slow to be reasonable for b = 9 km/h.

Common Pitfalls:
Learners sometimes try to average speeds or assume that time differences translate directly into additive adjustments of speed. The correct approach is always to use time = distance / speed and set up equations using the given ratio of times. Also, ensure that you keep track of which speed is upstream (b - c) and which is downstream (b + c).

Final Answer:
The rate of flow of the stream is 3 km/h.