A boat takes `4 hrs` to travel certain distance in a river in down stream and it takes `6 hrs` to travel the same distance in upstream. Then the time taken by the boat to travel the same distance in still water is

Options

A. `4.8 hrs`
B. `9.8 hrs`
C. `24 hrs`
D. `10 hrs`

Explanation

To solve the problem, we need to determine the time taken by the boat to travel the same distance in still water, given that it takes 4 hours downstream and 6 hours upstream.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the distance traveled by the boat be \( D \).
- Let the speed of the boat in still water be \( V_b \).
- Let the speed of the river current be \( V_w \).

2. **Set Up Equations for Downstream and Upstream:**
- When the boat is traveling downstream, the effective speed is \( V_b + V_w \).
- The time taken to travel downstream is given by:
\[ \text{Time}_{\text{downstream}} = \frac{D}{V_b + V_w} = 4 \text{ hours} \]
- When the boat is traveling upstream, the effective speed is \( V_b - V_w \).
- The time taken to travel upstream is given by:
\[ \text{Time}_{\text{upstream}} = \frac{D}{V_b - V_w} = 6 \text{ hours} \]

3. **Express Distances in Terms of Speed:**
- From the downstream equation:
\[ D = 4(V_b + V_w) \]
- From the upstream equation:
\[ D = 6(V_b - V_w) \]

4. **Set the Two Equations for Distance Equal:**
- Since both expressions equal \( D \), we can set them equal to each other:
\[ 4(V_b + V_w) = 6(V_b - V_w) \]

5. **Expand and Rearrange the Equation:**
- Expanding both sides gives:
\[ 4V_b + 4V_w = 6V_b - 6V_w \]
- Rearranging terms yields:
\[ 4V_w + 6V_w = 6V_b - 4V_b \]
\[ 10V_w = 2V_b \]
- Simplifying gives:
\[ V_b = 5V_w \]

6. **Substitute Back to Find Time in Still Water:**
- Now that we have \( V_b \) in terms of \( V_w \), we can find the time taken to travel the same distance \( D \) in still water:
\[ D = 4(V_b + V_w) = 4(5V_w + V_w) = 24V_w \]
- The time taken in still water is:
\[ \text{Time}_{\text{still water}} = \frac{D}{V_b} = \frac{24V_w}{5V_w} = \frac{24}{5} \text{ hours} = 4.8 \text{ hours} \]

### Final Answer:
The time taken by the boat to travel the same distance in still water is **4.8 hours**.