A river is flowing from A to B. The speed of the boat is 6 km/hr. The distance between A to B is 20 km. The boat goes from A to B then B to A and then again A to B. The total time taken by boat in this journey is 10 hours. Then find the speed of the stream.

Let's solve the problem step-by-step.

### Given:
1. Speed of the boat in still water: \( v_b = 6 \) km/hr.
2. Distance between A and B: \( D = 20 \) km.
3. Total time taken for the journey \( A \to B \to A \to B \): \( T = 10 \) hours.

### To Find:
- Speed of the stream \( v_s \).

### Steps to Solve:

1. **Define the Speeds in Each Direction:**

   - **Downstream speed (from A to B):**  
   The speed of the boat downstream (with the current) is the sum of the boat's speed and the stream's speed:
   \[
   v_{AB} = v_b + v_s = 6 + v_s \, \text{km/hr}
   \]

   - **Upstream speed (from B to A):**  
   The speed of the boat upstream (against the current) is the difference between the boat's speed and the stream's speed:
   \[
   v_{BA} = v_b - v_s = 6 - v_s \, \text{km/hr}
   \]

2. **Calculate Time for Each Leg of the Journey:**

   - **Time from A to B (downstream):**  
   \[
   t_{AB} = \frac{D}{v_{AB}} = \frac{20}{6 + v_s}
   \]

   - **Time from B to A (upstream):**  
   \[
   t_{BA} = \frac{D}{v_{BA}} = \frac{20}{6 - v_s}
   \]

   - **Time for the second trip from A to B (downstream):**  
   This is the same as the first trip downstream:
   \[
   t_{AB} = \frac{20}{6 + v_s}
   \]

3. **Set Up the Total Time Equation:**

   The total time for the entire journey is the sum of all three times:
   \[
   t_{AB} + t_{BA} + t_{AB} = 10
   \]

   Substitute the expressions for \( t_{AB} \) and \( t_{BA} \):
   \[
   \frac{20}{6 + v_s} + \frac{20}{6 - v_s} + \frac{20}{6 + v_s} = 10
   \]

   Combine like terms:
   \[
   2 \cdot \frac{20}{6 + v_s} + \frac{20}{6 - v_s} = 10
   \]

   Simplify further:
   \[
   \frac{40}{6 + v_s} + \frac{20}{6 - v_s} = 10
   \]

4. **Solve for \( v_s \):**

   To solve for \( v_s \), clear the denominators by multiplying through by \( (6 + v_s)(6 - v_s) \):
   \[
   40(6 - v_s) + 20(6 + v_s) = 10(6 + v_s)(6 - v_s)
   \]

   Expand both sides:
   \[
   240 - 40v_s + 120 + 20v_s = 10(36 - v_s^2)
   \]

   Combine like terms:
   \[
   360 - 20v_s = 360 - 10v_s^2
   \]

   Rearrange to form a quadratic equation:
   \[
   10v_s^2 - 20v_s = 0
   \]

   Factor out \( 10v_s \):
   \[
   10v_s(v_s - 2) = 0
   \]

   Set each factor to zero:
   \[
   v_s = 0 \quad \text{or} \quad v_s = 2
   \]

   Since the stream cannot have a speed of 0 (because it would mean no stream), we have:
   \[
   v_s = 2 \, \text{km/hr}
   \]

### Answer:
The speed of the stream is **2 km/hr**.

Would you like more details or have any questions about this solution? 

Here are some related questions for further exploration:

1. How would the total time change if the speed of the stream increased to 3 km/hr?
2. What would the total distance covered by the boat be in this journey?
3. If the speed of the boat in still water was 8 km/hr, how would the speed of the stream affect the total time?
4. Can you find the speed of the stream if the total journey time was 9 hours instead of 10?
5. What happens to the travel time if the stream is faster than the boat's speed in still water?

**Tip:** Always remember to check if your answers are physically meaningful, especially when solving quadratic equations!

Learn how to calculate the speed of a stream affecting a boat's journey with a detailed step-by-step solution. This problem involves applying speed, distance, and time formulas along with solving a quadratic equation. Suitable for high school students.

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