A boy can swim at 4.5 km/h in still water. It takes him twice as long to swim a certain distance upstream as it takes to swim the same distance downstream. What is the speed of the stream?

Introduction / Context:
This swimming-in-a-stream question is another relative speed problem. The swimmer's speed in still water is known and you are told that the time taken to swim upstream is twice the time taken downstream for the same distance. From this information, you must find the speed of the stream. The structure closely resembles standard boats and streams problems, just with a swimmer instead of a boat.

Given Data / Assumptions:

• Speed of boy in still water b = 4.5 km/h.
• Let speed of stream be c km/h.
• Downstream speed = b + c km/h.
• Upstream speed = b − c km/h.
• Time upstream is twice the time downstream for a fixed distance.

Concept / Approach:
Let the distance in each direction be D. Then:

• Downstream time t_d = D / (b + c).
• Upstream time t_u = D / (b − c).

The condition t_u = 2 t_d leads to an equation in b and c. Since b is known, this equation allows us to solve for c.

Step-by-Step Solution:
Step 1: Write the time relation: D / (b − c) = 2 * D / (b + c).Step 2: Cancel D from both sides: 1 / (b − c) = 2 / (b + c).Step 3: Cross multiply: b + c = 2(b − c).Step 4: Expand right side: b + c = 2b − 2c.Step 5: Rearranging: b + c − 2b + 2c = 0 ⇒ −b + 3c = 0.Step 6: Therefore, 3c = b ⇒ c = b / 3.Step 7: Substitute b = 4.5 km/h: c = 4.5 / 3 = 1.5 km/h.Step 8: Thus, the speed of the stream is 1.5 km/h.

Verification / Alternative check:
With c = 1.5 km/h, downstream speed = 4.5 + 1.5 = 6 km/h and upstream speed = 4.5 − 1.5 = 3 km/h. For any distance D, downstream time is D/6 and upstream time is D/3. The ratio D/3 : D/6 simplifies to 2 : 1, which exactly matches the condition that upstream time is twice downstream time.

Why Other Options Are Wrong:
Speeds like 1.8, 2, 2.2 or 2.5 km/h would produce downstream and upstream speeds that do not satisfy the exact 2:1 ratio of times. Only when c is 1.5 km/h do the effective speeds (6 and 3) yield an upstream time that is double the downstream time for any fixed distance.

Common Pitfalls:

• Reversing the roles of upstream and downstream times, leading to the wrong equation.
• Taking c = b/2 instead of b/3, perhaps by misreading the condition.
• Neglecting to cancel the distance D before manipulating the equation.
• Confusing the swimmer's speed in still water with downstream speed.

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