The speed of a boat in still water is 9 km/hr. It can go 12 km upstream and 12 km downstream in 3 hours. Find the speed of the stream.

Answer

To solve the problem, we need to find the speed of the stream given the speed of the boat in still water and the total time taken for the journey upstream and downstream.

### Step-by-Step Solution:

1. **Identify Given Information:**
- Speed of the boat in still water (B) = 9 km/hr
- Distance upstream = 12 km
- Distance downstream = 12 km
- Total time for the journey = 3 hours

2. **Let the Speed of the Stream be \( y \) km/hr.**

3. **Calculate Speed of the Boat Upstream and Downstream:**
- Speed upstream = Speed of boat - Speed of stream = \( 9 - y \) km/hr
- Speed downstream = Speed of boat + Speed of stream = \( 9 + y \) km/hr

4. **Calculate Time Taken for Upstream and Downstream:**
- Time taken to go upstream = Distance / Speed = \( \frac{12}{9 - y} \) hours
- Time taken to go downstream = Distance / Speed = \( \frac{12}{9 + y} \) hours

5. **Set Up the Equation for Total Time:**
- According to the problem, the total time taken for both upstream and downstream is 3 hours:
\[ \frac{12}{9 - y} + \frac{12}{9 + y} = 3 \]

6. **Simplify the Equation:**
- Multiply through by \( (9 - y)(9 + y) \) to eliminate the denominators:
\[ 12(9 + y) + 12(9 - y) = 3(9 - y)(9 + y) \]
- This simplifies to:
\[ 12 \times 9 + 12y + 12 \times 9 - 12y = 3(81 - y^2) \]
- Combine like terms:
\[ 216 = 243 - 3y^2 \]

7. **Rearranging the Equation:**
- Move all terms to one side:
\[ 3y^2 = 243 - 216 \]
- Simplifying gives:
\[ 3y^2 = 27 \]
- Dividing by 3:
\[ y^2 = 9 \]

8. **Find the Value of \( y \):**
- Taking the square root:
\[ y = 3 \text{ km/hr} \]

9. **Conclusion:**
- The speed of the stream is \( 3 \) km/hr.