We can solve this problem by using the concepts of relative speed in still water and in the stream.

Let:

• $D$ be the distance from A to B (the unknown to be found).
• The speed of the boat in still water is $12 \, \text{km/h}$.
• The speed of the stream is $3 \, \text{km/h}$.

For downstream travel, the effective speed of the boat is the sum of the boat's speed and the stream's speed: $\text{Downstream speed} = 12 + 3 = 15 \, \text{km/h}.$ For upstream travel, the effective speed of the boat is the difference between the boat's speed and the stream's speed: $\text{Upstream speed} = 12 - 3 = 9 \, \text{km/h}.$

The total journey involves traveling downstream from A to B and then upstream from B to C, where point C is halfway between A and B. This means the upstream journey covers a distance of $D/2$.

Given:

The total time taken for the entire trip is 15 hours.

Time spent traveling downstream:

$\text{Downstream time} = \frac{D}{15} \, \text{hours}.$

Time spent traveling upstream (from B to C):

$\text{Upstream time} = \frac{D/2}{9} = \frac{D}{18} \, \text{hours}.$

The total time is the sum of downstream and upstream times: $\frac{D}{15} + \frac{D}{18} = 15.$

Solving this equation:

To simplify, find the least common denominator (LCD) of 15 and 18, which is 90: $\frac{6D}{90} + \frac{5D}{90} = 15.$ $\frac{11D}{90} = 15.$ Multiply both sides by 90: $11D = 1350.$ Now divide by 11: $D = \frac{1350}{11} \approx 122.7 \, \text{km}.$

Final Answer:

The distance from A to B is approximately $122.7 \, \text{km}$.

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Do you want further details on any step or have any questions?
Here are some related questions to expand on this:

1. How would the problem change if the speed of the stream increased?
2. What if the speed of the boat in still water was reduced by 2 km/h?
3. Can you calculate the time taken for the downstream journey alone?
4. How would this problem change if point C was not halfway but a quarter of the way between A and B?
5. What if the boat had no speed in still water? How would the journey time be affected?

Tip: When working with relative speeds, always carefully distinguish between downstream (helped by the current) and upstream (resisted by the current) speeds.