A man can row at 9 km/h in still water. It takes him twice as long to row a certain distance upstream as it takes to row the same distance downstream. What is the speed of the stream?

Introduction / Context:
This is a classic boats and streams question involving a comparison of the time taken to travel upstream and downstream. The man's speed in still water is given, and you know that it takes him twice as long to row upstream as downstream over the same distance. From this, you must deduce the speed of the stream. Such problems help you practice equations involving relative speed and time.

Given Data / Assumptions:

• Speed of man (boat) in still water b = 9 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 the same distance.

Concept / Approach:
If the distance in each direction is D, then:

• Time downstream = D / (b + c).
• Time upstream = D / (b − c).

The condition given is that time upstream is twice time downstream, so D / (b − c) = 2 * D / (b + c). We cancel D and solve the resulting equation for c. Once c is known, we have the speed of the stream directly.

Step-by-Step Solution:
Step 1: Write the time condition: D / (b − c) = 2 * D / (b + c).Step 2: Cancel D from both sides, giving 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: Bring like terms together: b + c − 2b + 2c = 0 ⇒ −b + 3c = 0.Step 6: So 3c = b ⇒ c = b / 3.Step 7: Since b = 9 km/h, c = 9 / 3 = 3 km/h.Step 8: Therefore, the speed of the stream is 3 km/h.

Verification / Alternative check:
With b = 9 km/h and c = 3 km/h, downstream speed = 9 + 3 = 12 km/h and upstream speed = 9 − 3 = 6 km/h. For any distance D, time downstream is D/12 and time upstream is D/6. The ratio of times is (D/6) : (D/12) = 2 : 1, which matches the condition that upstream time is twice downstream time.

Why Other Options Are Wrong:
If c were 2, 2.5, 3.5 or 4 km/h, substituting into upstream and downstream speeds would not produce the required 2:1 ratio of times. For example, if c = 2, downstream speed = 11 and upstream speed = 7, and the time ratio is not exactly 2. Only c = 3 yields upstream and downstream speeds (6 and 12) with the correct time relationship.

Common Pitfalls:

• Reversing the relationship and taking downstream time as twice upstream time.
• Miswriting speeds as (b − 2c) or (b + 2c) by mistake.
• Forgetting to cancel the distance D when setting up the time ratio.
• Attempting to plug in options directly without forming the equation, which can be slower and error-prone.

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