A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.

Explanation

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

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Speed of the motorboat in still water (B) = 9 km/hr
- Distance downstream (D) = 15 km
- Total time for the round trip = 3 hours 45 minutes = 3.75 hours

2. **Let the Speed of the Stream be x km/hr.**
- Therefore, the speed of the motorboat downstream (B + x) = 9 + x km/hr
- The speed of the motorboat upstream (B - x) = 9 - x km/hr

3. **Calculate the Time Taken for Each Part of the Journey:**
- Time taken to go downstream = Distance / Speed = \( \frac{15}{9 + x} \) hours
- Time taken to come back upstream = Distance / Speed = \( \frac{15}{9 - x} \) hours

4. **Set Up the Equation for Total Time:**
- Total time for the round trip = Time downstream + Time upstream
- Therefore, we have:
\[ \frac{15}{9 + x} + \frac{15}{9 - x} = 3.75 \]

5. **Multiply the Entire Equation by (9 + x)(9 - x) to Eliminate the Denominators:**
- This gives:
\[ 15(9 - x) + 15(9 + x) = 3.75(9 + x)(9 - x) \]
- Simplifying the left side:
\[ 15(9 - x) + 15(9 + x) = 135 \]
- The right side becomes:
\[ 3.75(81 - x^2) \]

6. **Combine and Simplify:**
- So we have:
\[ 135 = 3.75(81 - x^2) \]
- Divide both sides by 3.75:
\[ 36 = 81 - x^2 \]
- Rearranging gives:
\[ x^2 = 81 - 36 = 45 \]

7. **Solve for x:**
- Taking the square root:
\[ x = \sqrt{45} = 3\sqrt{5} \approx 6.71 \text{ km/hr} \]

8. **Conclusion:**
- The speed of the stream is approximately 6.71 km/hr.