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

Explanation

To solve the problem, we need to determine the time it takes for the man to cross the distance in still water. We can break down the problem step by step.

### Step 1: Define the Variables
Let:
- \( D \) = distance to be covered
- \( v \) = speed of the man in still water
- \( u \) = speed of the river

### Step 2: Set Up the Equations
When the man is moving along the flow of the river, his effective speed is \( v + u \). The time taken to cover the distance \( D \) along the flow is given as 3 hours. Thus, we can write the first equation as:
\[ D = (v + u) \times 3 \quad \text{(Equation 1)} \]

When the man is moving against the flow of the river, his effective speed is \( v - u \). The time taken to cover the same distance \( D \) against the flow is given as 6 hours. Thus, we can write the second equation as:
\[ D = (v - u) \times 6 \quad \text{(Equation 2)} \]

### Step 3: Equate the Distances
Since both equations represent the same distance \( D \), we can set them equal to each other:
\[ (v + u) \times 3 = (v - u) \times 6 \]

### Step 4: Expand and Rearrange the Equation
Expanding both sides gives:
\[ 3v + 3u = 6v - 6u \]
Now, rearranging the equation to isolate terms involving \( v \) and \( u \):
\[ 3u + 6u = 6v - 3v \]
\[ 9u = 3v \]
Dividing both sides by 3:
\[ v = 3u \quad \text{(Equation 3)} \]

### Step 5: Substitute Back to Find Time in Still Water
Now, we can substitute \( v \) back into either Equation 1 or Equation 2 to find \( D \). We will use Equation 1:
\[ D = (3u + u) \times 3 \]
\[ D = 4u \times 3 = 12u \]

### Step 6: Calculate Time in Still Water
Now, we want to find the time taken to cover distance \( D \) in still water:
\[ \text{Time} = \frac{D}{v} = \frac{12u}{3u} = 4 \text{ hours} \]

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

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