a Boat takes 15 hours for travelling downstream from point a to point B incoming back to point C made away between A and B if the speed of stream is 3 km per hour and speed of water still water in 2 km per hour 12 km then what will be the approximately distance from A to B is 

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 \( 122.7 \, \text{km} \).

---

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.

A Boat takes 15 hours for travelling downstream from point A to point B and coming back to point C midway between A and B. If the speed of the stream is 3 km per hour and the speed of the boat in still water is 12 km per hour, what will be the approximate distance from A to B?

Discover how to calculate the distance from point A to point B for a boat traveling downstream and upstream in a stream. The boat's speed in still water is 12 km/h, and the stream's speed is 3 km/h. This detailed solution explores relative speed concepts and time-distance relationships. Suitable for Grades 9-10.

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