A person can row at 8.5 km/h in still water and finds that it takes twice as long to row upstream as it does to row downstream over the same distance. What is the speed of the stream (in km/h)?

Introduction / Context:
This question relates the ratio of times taken upstream and downstream to the speeds of a boat in still water and the stream. The person’s speed in still water is given, and we know that rowing upstream takes twice as long as rowing downstream for the same distance. Using the relationship between time and speed, we can set up an equation and solve for the stream speed.

Given Data / Assumptions:

• Speed in still water b = 8.5 km/h.
• Time taken upstream is twice the time taken downstream for the same distance.
• Let c be the speed of the stream.
• Upstream speed = b - c, downstream speed = b + c.
• Distances for upstream and downstream are equal.

Concept / Approach:
If the distance is D, then:

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

We are told that upstream time is twice downstream time, so D / (b - c) = 2 * D / (b + c). Distance cancels, and we get an equation only in b and c. Solving this equation gives c. The stream speed can then be compared with the options.

Step-by-Step Solution:
Let b = 8.5 km/h and c be the stream speed. Upstream time = D / (b - c), downstream time = D / (b + c). Given that upstream time is twice downstream time: D / (b - c) = 2 * D / (b + c). Cancel D from both sides to get 1 / (b - c) = 2 / (b + c). Cross multiply: b + c = 2(b - c). So b + c = 2b - 2c. Rearrange: b + c - 2b + 2c = 0 which simplifies to -b + 3c = 0. Thus 3c = b so c = b / 3. Substitute b = 8.5 to get c = 8.5 / 3 ≈ 2.833 km/h.

Verification / Alternative check:
If c ≈ 2.83 km/h, then downstream speed is about 8.5 + 2.83 = 11.33 km/h and upstream speed is about 8.5 - 2.83 = 5.67 km/h. For any fixed distance, say D = 11.33 km, downstream time is 11.33 / 11.33 = 1 hour and upstream time is 11.33 / 5.67 ≈ 2 hours, which is twice as long. This confirms the relationship described in the question.

Why Other Options Are Wrong:
If the stream speed were 1.78 km/h, the time ratio would not be exactly 2:1. Similar mismatches occur with 2.35 km/h and 3.15 km/h. Only a stream speed close to 2.83 km/h makes the upstream time exactly double the downstream time for the same distance when the still water speed is 8.5 km/h.

Common Pitfalls:
A common misunderstanding is to invert the given ratio and set downstream time to be double instead of upstream. Another error is to relate speed ratio directly to time ratio without considering the reciprocal relationship. The correct approach is to express times as distance divided by speed and only then apply the time ratio condition.

Final Answer:
The speed of the stream is approximately 2.83 km/h.