A person can row at 8.5 km/h in still water. He finds that 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 (in km/h)?

Introduction / Context:
This question explores the relationship between upstream and downstream times when the boat speed in still water is known. If the time taken upstream is double the time taken downstream for the same distance, there is a specific relationship between the boat speed and stream speed that can be derived using the time formula.

Given Data / Assumptions:
- Speed of rowing 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 (km/h). - Upstream speed = b - c, downstream speed = b + c. - Let the distance each way be L km.

Concept / Approach:
Time = distance / speed. For the same distance L, upstream time is L / (b - c) and downstream time is L / (b + c). The problem states that upstream time is twice downstream time, which gives an equation relating b and c. Substituting b = 8.5 into this relation allows us to solve for c and thus find the stream speed.

Step-by-Step Solution:
Step 1: Upstream time T up = L / (b - c). Step 2: Downstream time T down = L / (b + c). Step 3: Given T up = 2 * T down. Step 4: So L / (b - c) = 2L / (b + c). Step 5: Cancel L from both sides: 1 / (b - c) = 2 / (b + c). Step 6: Cross multiply: b + c = 2(b - c). Step 7: Expand right side: b + c = 2b - 2c. Step 8: Rearrange: b + c - 2b + 2c = 0 gives -b + 3c = 0. Step 9: So 3c = b and c = b / 3. Step 10: Substitute b = 8.5 km/h: c = 8.5 / 3 km/h. Step 11: 8.5 / 3 ≈ 2.833 km/h, which is about 2.83 km/h.

Verification / Alternative check:
With b = 8.5 and c ≈ 2.83, downstream speed ≈ 11.33 km/h and upstream speed ≈ 5.67 km/h. For a distance of, say, 11.33 km, downstream time ≈ 1 hour. Upstream time for the same distance is 11.33 / 5.67 ≈ 2 hours, which is twice the downstream time as required. This confirms that the computed stream speed is correct.

Why Other Options Are Wrong:
- 1.78, 2.35, and 3.15 km/h lead to time ratios that are not exactly 2:1 between upstream and downstream. - 4 km/h would make upstream speed very low and the ratio between times much larger than 2.

Common Pitfalls:
A common mistake is to try to average speeds or to assume that the stream speed is some simple fraction of 8.5 without using the time relationship. Others may set up the equation with the ratio inverted. Always be clear which time is longer and by what factor, and translate that precisely into a mathematical equation.

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