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:sqrt(3)`
D. `sqrt(3):1`

Answer

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. **Understanding Upstream and Downstream:**
- When the boat is going upstream, it is moving against the current of the river. Therefore, the effective velocity of the boat upstream (V_upstream) is given by:
\[ V_{upstream} = V_B - V_R \]
- When the boat is going downstream, it is moving with the current of the river. Thus, the effective velocity of the boat downstream (V_downstream) is:
\[ V_{downstream} = V_B + V_R \]

2. **Using Time and Distance:**
- Let the distance covered by the boat in both cases be \(D\).
- The time taken to travel upstream is given as 4 hours, and the time taken to travel downstream is given as 2 hours.
- Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \), we can write:
\[ \text{Upstream: } 4 = \frac{D}{V_B - V_R} \]
\[ \text{Downstream: } 2 = \frac{D}{V_B + V_R} \]

3. **Setting Up Equations:**
- From the upstream equation, we can express \(D\):
\[ D = 4(V_B - V_R) \]
- From the downstream equation, we can express \(D\) as well:
\[ D = 2(V_B + V_R) \]

4. **Equating the Two Expressions for Distance:**
- Since both expressions equal \(D\), we can set them equal to each other:
\[ 4(V_B - V_R) = 2(V_B + V_R) \]

5. **Expanding and Rearranging:**
- Expanding both sides gives:
\[ 4V_B - 4V_R = 2V_B + 2V_R \]
- Rearranging the equation:
\[ 4V_B - 2V_B = 2V_R + 4V_R \]
\[ 2V_B = 6V_R \]

6. **Finding the Ratio:**
- Dividing both sides by \(2V_R\) gives:
\[ \frac{V_B}{V_R} = \frac{6}{2} = 3 \]

### Final Answer:
The ratio of the velocity of the boat to the velocity of the water in the river is:
\[ \frac{V_B}{V_R} = 3 \]