A boat travels upstream from B to A and then downstream from A to B in a total of 3 hours. The speed of the boat in still water is 9 km/h and the speed of the current is 3 km/h. What is the distance (in km) between A and B?

Introduction / Context:
This problem describes a boat travelling from B to A upstream and from A to B downstream in a total of 3 hours. The boat's speed in still water and the speed of the current are known. We must determine the distance between A and B. This is a direct application of time = distance / speed for two legs of a journey with different effective speeds.

Given Data / Assumptions:

• Speed of the boat in still water b = 9 km/h.
• Speed of the current c = 3 km/h.
• Upstream speed = b - c = 9 - 3 = 6 km/h.
• Downstream speed = b + c = 9 + 3 = 12 km/h.
• Total time for upstream plus downstream journey = 3 hours.
• Let the distance between A and B be d km.
• We must find d.

Concept / Approach:
For each leg of the journey, we write time as distance / speed. The upstream leg takes d / 6 hours, and the downstream leg takes d / 12 hours. Since the total time is 3 hours, we set the sum of these times equal to 3 and solve for d. The arithmetic is simple if we combine the fractions correctly.

Step-by-Step Solution:
Step 1: Express upstream and downstream times. Upstream time = d / 6 hours. Downstream time = d / 12 hours. Step 2: Use the total time condition. d / 6 + d / 12 = 3. Step 3: Combine the fractions on the left. d / 6 + d / 12 = (2d + d) / 12 = 3d / 12 = d / 4. So d / 4 = 3. Step 4: Solve for d. d = 3 * 4 = 12 km.
Verification / Alternative check:
With d = 12 km, upstream time = 12 / 6 = 2 hours. Downstream time = 12 / 12 = 1 hour. Total time = 2 + 1 = 3 hours, which matches the given total.
Why Other Options Are Wrong:
If d were 9, 10 or 11 km, the sum of upstream and downstream times would be less than 3 hours. If d were 8 km, total time would be 8 / 6 + 8 / 12, which also does not equal 3 hours.
Common Pitfalls:
A common error is to average the upstream and downstream speeds to get 9 km/h and then divide the total distance by this average, which does not correctly represent the time for two different speeds over equal distances. Another pitfall is misadding the fractions d / 6 and d / 12 without finding a common denominator.
Final Answer:
The distance between A and B is 12 km.