A motorboat takes exactly twice as long to cover a certain distance upstream as it takes to cover the same distance downstream. What is the ratio of the speed of the current to the speed of the boat in still water?

Introduction / Context:
Here the key information is about the relative times to travel the same distance upstream and downstream. The motorboat takes twice the time going upstream, which tells us how its speed in still water compares to the speed of the current. The question asks specifically for the ratio of the current speed to the boat speed in still water.

Given Data / Assumptions:
- Time taken upstream is twice the time taken downstream for the same distance. - Let the distance be D km. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c.

Concept / Approach:
Time equals distance divided by speed. For the same distance D, upstream time is D / (b - c) and downstream time is D / (b + c). Given that upstream time is twice downstream time, we have D / (b - c) = 2D / (b + c). Cancelling D gives a relation between b and c. From this relation we can form the ratio c : b that is required.

Step-by-Step Solution:
Step 1: Upstream time T up = D / (b - c). Step 2: Downstream time T down = D / (b + c). Step 3: Given T up = 2 * T down. Step 4: So D / (b - c) = 2D / (b + c). Step 5: Cancel D 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, which gives -b + 3c = 0. Step 9: So 3c = b, or c = b / 3. Step 10: Therefore the ratio c : b = 1 : 3.

Verification / Alternative check:
Assume a convenient value such as b = 3 units and c = 1 unit. Then downstream speed = 3 + 1 = 4 units, upstream speed = 3 - 1 = 2 units. For distance D, downstream time is D / 4 and upstream time is D / 2. The ratio of times is (D / 2) : (D / 4) = 2 : 1, so upstream time is indeed twice downstream time, confirming that c : b = 1 : 3 is correct.

Why Other Options Are Wrong:
- Ratios 3 : 2, 2 : 3, 3 : 1, and 1 : 2 all lead to different relationships between upstream and downstream speeds and do not produce a strict two to one ratio between upstream and downstream times.

Common Pitfalls:
Some learners accidentally reverse the ratio and give b : c instead of c : b, or they misread the condition as downstream taking twice as long. Others try to use numerical values for b and c without setting up the equation properly. Always translate the time condition precisely into a fraction equation before solving for the ratio.

Final Answer:
The ratio of the speed of the current to the speed of the boat in still water is 1 : 3.