A boat takes `4 hr` upstream and `2 hr` down the stream for covering the same distance.The ratio of velocity of boat to the water in river is.

Options

A. `1:3`
B. `3:1`
C. `1:sqrt3`
D. `sqrt3:1`

Explanation

To solve the problem, we need to find the ratio of the velocity of the boat (VB) to the velocity of the river (VR) based on the time taken to travel upstream and downstream.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( D \) be the distance covered by the boat.
- Let \( VB \) be the velocity of the boat in still water.
- Let \( VR \) be the velocity of the river.

2. **Write the Equations for Upstream and Downstream:**
- When the boat is going upstream (against the current), the effective velocity is \( VB - VR \). The time taken to cover the distance \( D \) upstream is 4 hours.
\[ D = (VB - VR) \times 4 \]
- When the boat is going downstream (with the current), the effective velocity is \( VB + VR \). The time taken to cover the same distance \( D \) downstream is 2 hours.
\[ D = (VB + VR) \times 2 \]

3. **Set the Two Equations Equal to Each Other:**
Since both expressions equal \( D \), we can set them equal to each other:
\[ (VB - VR) \times 4 = (VB + VR) \times 2 \]

4. **Expand and Rearrange the Equation:**
Expanding both sides:
\[ 4VB - 4VR = 2VB + 2VR \]
Rearranging gives:
\[ 4VB - 2VB = 2VR + 4VR \]
Simplifying further:
\[ 2VB = 6VR \]

5. **Solve for the Ratio \( \frac{VB}{VR} \):**
Dividing both sides by \( VR \):
\[ \frac{VB}{VR} = \frac{6}{2} = 3 \]

6. **Express the Ratio:**
Therefore, the ratio of the velocity of the boat to the velocity of the river is:
\[ \frac{VB}{VR} = 3:1 \]

### Final Answer:
The ratio of the velocity of the boat to the velocity of the river is \( 3:1 \).