A boy can swim in still water at 4.5 km/h, but it takes him twice as long to swim upstream as it takes to swim downstream over the same distance. What is the speed of the stream (in km/h)?

Introduction / Context:
Swimming problems in a river are analogous to boat and stream questions. The swimmer's speed in still water is known, and the current either helps or hinders him depending on direction. Here, we are told that the time taken upstream is twice the time taken downstream for the same distance, and we must find the speed of the stream.

Given Data / Assumptions:

Swimmer's speed in still water, b = 4.5 km/h.
Let the speed of the stream be c km/h.
Upstream speed = (b - c) km/h.
Downstream speed = (b + c) km/h.
For the same distance d, time upstream is twice time downstream.
Distance d cancels out in the ratio calculations.

Concept / Approach:
Using time = distance / speed, if the time upstream is twice the time downstream for the same distance d, we have:
d / (b - c) = 2 * [d / (b + c)]. The distance cancels, leaving an equation involving b and c. Substituting b = 4.5 and solving this equation gives c, the speed of the stream.

Step-by-Step Solution:
Step 1: Set up the time relation. d / (b - c) = 2d / (b + c). Cancel d: 1 / (b - c) = 2 / (b + c). Step 2: Cross-multiply. b + c = 2(b - c). b + c = 2b - 2c. Step 3: Solve for c. Rearrange: b + c = 2b - 2c ⇒ 3c = b ⇒ c = b / 3. Given b = 4.5 km/h, c = 4.5 / 3 = 1.5 km/h.

Verification / Alternative check:
With c = 1.5 km/h, downstream speed = 4.5 + 1.5 = 6 km/h, upstream speed = 4.5 - 1.5 = 3 km/h. For distance d, time downstream = d / 6, time upstream = d / 3. Since d / 3 = 2 * (d / 6), the upstream time is exactly twice the downstream time, which matches the condition perfectly.

Why Other Options Are Wrong:
1.8 km/h or 2 km/h as the stream speed would change the ratio of upstream to downstream times and would not give exactly a 2:1 ratio.
2.2 km/h is too high and would make upstream speed too slow relative to downstream, creating a different time ratio than specified in the problem.

Common Pitfalls:
A common mistake is to try to average times or speeds, or to use b ± c incorrectly (sometimes students swap them). It is essential to remember that upstream speed is b - c and downstream speed is b + c. Setting up the correct ratio of times using time = distance / speed and cancelling the common distance term leads to a simple and accurate solution.

Final Answer:
The speed of the stream is 1.5 km/h.