A boat travels upstream from point B to point A and then downstream from point A back to point B in a total time 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 boats and streams problem gives you the speed of a boat in still water, the speed of the current, and the total time for an upstream and downstream journey between two points. You are asked to find the distance between the points. Because the distance each way is the same, the time relation can be used directly with upstream and downstream speeds.

Given Data / Assumptions:

• Speed of boat in still water = 9 km/h.
• Speed of current = 3 km/h.
• Let the distance between A and B be D km.
• Total time for upstream (B to A) and downstream (A to B) journeys = 3 hours.
• Upstream speed = 9 - 3 = 6 km/h; downstream speed = 9 + 3 = 12 km/h.

Concept / Approach:
When a boat goes a certain distance upstream and the same distance downstream, the total time is the sum of upstream time and downstream time. We use the formula time = distance / speed for each leg, express the total time in terms of D, and then solve for D. Since speeds are already known, the algebra is straightforward.

Step-by-Step Solution:
Step 1: Upstream speed = 9 - 3 = 6 km/h. Step 2: Downstream speed = 9 + 3 = 12 km/h. Step 3: Time taken to go upstream from B to A = D / 6 hours. Step 4: Time taken to go downstream from A to B = D / 12 hours. Step 5: Total time for both journeys = D / 6 + D / 12 = 3 hours. Step 6: Combine terms: D / 6 + D / 12 = (2D / 12) + (D / 12) = 3D / 12 = D / 4. Step 7: So D / 4 = 3 ⇒ D = 3 * 4 = 12 km.

Verification / Alternative check:
Check the times using D = 12 km. Upstream time = 12 / 6 = 2 hours. Downstream time = 12 / 12 = 1 hour. Total time = 2 + 1 = 3 hours, which exactly matches the given total time. Therefore, the distance of 12 km is consistent with all data.

Why Other Options Are Wrong:
If the distance were 9, 10 or 11 km, the combined upstream and downstream times would not total 3 hours when calculated with speeds 6 km/h and 12 km/h. For example, with 9 km, upstream time is 1.5 hours and downstream time is 0.75 hours, totaling 2.25 hours, not 3. Only 12 km gives a total of exactly 3 hours.

Common Pitfalls:
Many learners forget that the distance upstream and downstream is the same and mistakenly use different distances. Others neglect to simplify the fractions correctly when summing times. Remember that upstream speed is lower, so most of the total time is spent going against the current, and the final answer must reflect that balance.

Final Answer:
The distance between A and B is 12 km.