A motorboat can go at 8 km/h in still water. It takes three times as long to cover a certain distance against the current as it takes to cover the same distance with the current. What is the speed of the current (in km/h)?

Introduction / Context:
This problem uses the relationship between upstream and downstream times when the speed of a boat in still water is known. You are told that travelling against the current takes three times as long as travelling with the current over the same distance, and you must infer the stream speed from this ratio.

Given Data / Assumptions:
- Speed of the motorboat in still water b = 8 km/h. - Time taken upstream is three times the time taken downstream for the same distance. - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c. - Let the distance each way be L km.

Concept / Approach:
Time equals distance divided by speed. For distance L, upstream time is L / (b - c), downstream time is L / (b + c). The statement that upstream time is three times downstream time gives L / (b - c) = 3L / (b + c). Cancelling L and solving for c in terms of b leads to a simple linear equation. Substituting b = 8 gives the numerical value of c, which is the current speed.

Step-by-Step Solution:
Step 1: Upstream time T up = L / (8 - c). Step 2: Downstream time T down = L / (8 + c). Step 3: Given T up = 3 * T down. Step 4: So L / (8 - c) = 3L / (8 + c). Step 5: Cancel L: 1 / (8 - c) = 3 / (8 + c). Step 6: Cross multiply: 8 + c = 3(8 - c). Step 7: Expand right side: 8 + c = 24 - 3c. Step 8: Bring terms together: c + 3c = 24 - 8. Step 9: 4c = 16, so c = 16 / 4 = 4 km/h.

Verification / Alternative check:
Take b = 8 km/h and c = 4 km/h. Then upstream speed = 8 - 4 = 4 km/h, downstream speed = 8 + 4 = 12 km/h. For some distance such as 12 km, downstream time = 12 / 12 = 1 hour. Upstream time for the same distance = 12 / 4 = 3 hours, which is exactly three times the downstream time, confirming the correctness of c = 4 km/h.

Why Other Options Are Wrong:
- 6 km/h or 5 km/h would make upstream speed too low, leading to a time ratio more than three. - 3 km/h or 2 km/h would not produce a time ratio of exactly three when used with 8 km/h as the still water speed.

Common Pitfalls:
Learners often invert the ratio, using downstream time equal to three times upstream time, or they average speeds instead of setting up a time equation. Another mistake is to forget the minus sign for upstream speed. Rewriting the relationship clearly as T up = 3 T down before substituting formulas for time helps avoid these issues.

Final Answer:
The speed of the current is 4 km/h.