A man takes `3 h` to cover a certain distance along the flow and takes `6h` to cover the same distance opposite to flow. In how much time, he will cross this distance in still water.

Options

A. `3.5h`
B. `4h`
C. `4.5h`
D. `5h`

Explanation

To solve the problem, we need to find the time it takes for a man to cross a certain distance in still water, given the time it takes him to cover the same distance along and against the flow of water.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( D \) be the distance covered.
- Let \( V \) be the speed of the man in still water (in km/h).
- Let \( U \) be the speed of the water flow (in km/h).

2. **Set Up Equations**:
- When moving along the flow, the effective speed is \( V + U \). The time taken to cover distance \( D \) is 3 hours.
\[ D = (V + U) \times 3 \quad \text{(1)} \]
- When moving against the flow, the effective speed is \( V - U \). The time taken to cover the same distance \( D \) is 6 hours.
\[ D = (V - U) \times 6 \quad \text{(2)} \]

3. **Equate the Two Expressions for Distance**:
From equations (1) and (2), we can set the two expressions for \( D \) equal to each other:
\[ (V + U) \times 3 = (V - U) \times 6 \]

4. **Expand and Rearrange**:
Expanding both sides gives:
\[ 3V + 3U = 6V - 6U \]
Rearranging the equation:
\[ 3U + 6U = 6V - 3V \]
\[ 9U = 3V \]
Dividing both sides by 3:
\[ V = 3U \quad \text{(3)} \]

5. **Substitute Back to Find Time in Still Water**:
Now, we need to find the time taken to cover distance \( D \) in still water. The speed in still water is \( V = 3U \). We can substitute this back into either of the original distance equations. Using equation (1):
\[ D = (3U + U) \times 3 \]
\[ D = 4U \times 3 = 12U \]

6. **Calculate Time in Still Water**:
The time taken to cover distance \( D \) in still water is:
\[ \text{Time} = \frac{D}{V} = \frac{12U}{3U} = 4 \text{ hours} \]

### Final Answer:
The man will take **4 hours** to cross the distance in still water.

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