A river is running at 2 km/hr. It took a man twice as long to row up as to row down the river. The rate (in km/hr) of the man in still water is

Options

A. 8
B. 10
C. 4
D. 6

Answer

To solve the problem, we will use the information given about the river's speed and the time taken to row upstream and downstream.

### Step-by-step Solution:

1. **Define Variables:**
- Let the speed of the man in still water be \( x \) km/hr.
- The speed of the river is given as \( 2 \) km/hr.
- The distance traveled by the man is \( d \) km.

2. **Calculate Speeds:**
- The speed of the man rowing downstream (with the current) is:
\[ \text{Speed downstream} = x + 2 \text{ km/hr} \]
- The speed of the man rowing upstream (against the current) is:
\[ \text{Speed upstream} = x - 2 \text{ km/hr} \]

3. **Set Up Time Equations:**
- Time taken to row downstream is given by:
\[ \text{Time downstream} = \frac{d}{x + 2} \]
- Time taken to row upstream is given by:
\[ \text{Time upstream} = \frac{d}{x - 2} \]

4. **Use the Given Condition:**
- It is given that the time taken to row upstream is twice the time taken to row downstream:
\[ \frac{d}{x - 2} = 2 \cdot \frac{d}{x + 2} \]

5. **Eliminate \( d \):**
- Since \( d \) is common in both terms, we can cancel it out (assuming \( d \neq 0 \)):
\[ \frac{1}{x - 2} = \frac{2}{x + 2} \]

6. **Cross Multiply:**
- Cross multiplying gives:
\[ x + 2 = 2(x - 2) \]

7. **Simplify the Equation:**
- Expanding the right side:
\[ x + 2 = 2x - 4 \]
- Rearranging the equation:
\[ 2 + 4 = 2x - x \]
\[ 6 = x \]

8. **Conclusion:**
- The speed of the man in still water is \( x = 6 \) km/hr.

### Final Answer:
The rate of the man in still water is **6 km/hr**.