A boat takes 4 hours for travelling downstream from point A to point B and coming back to point A upstream. If the velocity of the stream is 2 km/hr and the speed of the boat in still water is 4 km/hr, what is the distance between A and B?

Answer

To solve the problem, we need to find the distance between points A and B given the time taken for the round trip and the speeds involved.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Velocity of the stream (current) = 2 km/hr
- Speed of the boat in still water = 4 km/hr
- Total time for the round trip = 4 hours

2. **Calculate the Effective Speeds**:
- **Downstream Speed**: When the boat is going downstream, it moves with the current. Therefore, the effective speed downstream is:
\[ \text{Downstream Speed} = \text{Speed of the boat} + \text{Speed of the stream} = 4 \text{ km/hr} + 2 \text{ km/hr} = 6 \text{ km/hr} \]
- **Upstream Speed**: When the boat is going upstream, it moves against the current. Therefore, the effective speed upstream is:
\[ \text{Upstream Speed} = \text{Speed of the boat} - \text{Speed of the stream} = 4 \text{ km/hr} - 2 \text{ km/hr} = 2 \text{ km/hr} \]

3. **Set Up the Distance Equation**:
- Let the distance between points A and B be \( x \) km.
- The time taken to travel downstream from A to B is:
\[ \text{Time Downstream} = \frac{x}{6} \text{ hours} \]
- The time taken to travel upstream from B to A is:
\[ \text{Time Upstream} = \frac{x}{2} \text{ hours} \]

4. **Write the Total Time Equation**:
- The total time for the round trip is given as 4 hours. Therefore, we can write:
\[ \frac{x}{6} + \frac{x}{2} = 4 \]

5. **Solve the Equation**:
- To solve for \( x \), first find a common denominator for the fractions on the left side. The common denominator for 6 and 2 is 6:
\[ \frac{x}{6} + \frac{3x}{6} = 4 \]
- Combine the fractions:
\[ \frac{4x}{6} = 4 \]
- Simplify the fraction:
\[ \frac{2x}{3} = 4 \]
- Multiply both sides by 3:
\[ 2x = 12 \]
- Divide both sides by 2:
\[ x = 6 \]

6. **Conclusion**:
- The distance between points A and B is \( 6 \) km.