A boat's speed in still water is 10 km/h, while river is flowing with a speed of 2 km/h and time taken to cover a certain distance upstream is 4 h more than time taken to rover the same distance downstream. Find the distance.

Answer

To solve the problem step by step, we will use the information provided about the boat's speed, the river's speed, and the relationship between the time taken to travel upstream and downstream.

### Step-by-Step Solution:

1. **Identify the given data:**
- Speed of the boat in still water (b) = 10 km/h
- Speed of the river (s) = 2 km/h
- Let the distance to be covered be \( d \) km.
- Time taken to row upstream is 4 hours more than the time taken to row downstream.

2. **Calculate effective speeds:**
- Upstream speed = \( b - s = 10 - 2 = 8 \) km/h
- Downstream speed = \( b + s = 10 + 2 = 12 \) km/h

3. **Set up the time equations:**
- Time taken to row upstream \( (t_u) = \frac{d}{8} \) hours
- Time taken to row downstream \( (t_d) = \frac{d}{12} \) hours

4. **Write the relationship between the times:**
- According to the problem, \( t_u = t_d + 4 \)
- Therefore, we can write the equation:
\[ \frac{d}{8} = \frac{d}{12} + 4 \]

5. **Clear the fractions by finding a common denominator:**
- The least common multiple of 8 and 12 is 24.
- Multiply the entire equation by 24 to eliminate the denominators:
\[ 24 \cdot \frac{d}{8} = 24 \cdot \frac{d}{12} + 24 \cdot 4 \]
- This simplifies to:
\[ 3d = 2d + 96 \]

6. **Solve for \( d \):**
- Rearranging the equation gives:
\[ 3d - 2d = 96 \]
- Thus, we have:
\[ d = 96 \text{ km} \]

7. **Conclusion:**
- The distance to be covered is \( 96 \) km.