The ratio of the distance carried away by the water current, downstream, in crossing a river, by a person, making same angle with downstream and upstream is `2 : 1`. The ratio of the speed of person to the water current cannot be less than.

Options

A. `1//3`
B. `4//5`
C. `2//5`
D. `4//3`

Explanation

To solve the problem, we need to analyze the motion of a person crossing a river while being affected by the water current. The problem states that the ratio of the distance carried away by the water current downstream to the distance carried upstream is 2:1. We need to find the minimum ratio of the speed of the person (P) to the speed of the water current (W).

### Step-by-Step Solution:

1. **Define the Scenario**:
- Let the width of the river be \( D \).
- The person crosses the river at an angle \( \theta \) with respect to the direction of the current.

2. **Resolve the Velocities**:
- The speed of the person is \( P \).
- The speed of the water current is \( W \).
- The component of the person's speed across the river (perpendicular to the current) is \( P \sin \theta \).
- The component of the person's speed along the river (parallel to the current) is \( P \cos \theta \).

3. **Time Taken to Cross the River**:
- The time taken to cross the river downstream is given by:
\[ t_{\text{down}} = \frac{D}{P \sin \theta} \]
- The time taken to cross the river upstream is the same:
\[ t_{\text{up}} = \frac{D}{P \sin \theta} \]

4. **Distance Carried Away by the Current**:
- The distance carried downstream while crossing is:
\[ \text{Distance}_{\text{down}} = (W + P \cos \theta) \cdot t_{\text{down}} = (W + P \cos \theta) \cdot \frac{D}{P \sin \theta} \]
- The distance carried upstream while crossing is:
\[ \text{Distance}_{\text{up}} = (W - P \cos \theta) \cdot t_{\text{up}} = (W - P \cos \theta) \cdot \frac{D}{P \sin \theta} \]

5. **Set Up the Ratio**:
- According to the problem, the ratio of the distances is given as:
\[ \frac{\text{Distance}_{\text{down}}}{\text{Distance}_{\text{up}}} = \frac{2}{1} \]
- Substituting the expressions:
\[ \frac{(W + P \cos \theta)}{(W - P \cos \theta)} = 2 \]

6. **Cross-Multiply and Solve**:
- Cross-multiplying gives:
\[ W + P \cos \theta = 2(W - P \cos \theta) \]
- Expanding and simplifying:
\[ W + P \cos \theta = 2W - 2P \cos \theta \]
\[ 3P \cos \theta = W \]

7. **Find the Ratio \( \frac{P}{W} \)**:
- Rearranging gives:
\[ \frac{P}{W} = \frac{1}{3 \cos \theta} \]

8. **Determine the Minimum Ratio**:
- Since \( \cos \theta \) can take values between 0 and 1, the minimum value of \( \cos \theta \) is 1 (when \( \theta = 0 \)).
- Therefore, the minimum ratio of \( \frac{P}{W} \) cannot be less than:
\[ \frac{P}{W} \geq \frac{1}{3} \]

### Final Answer:
The ratio of the speed of the person to the speed of the water current cannot be less than \( \frac{1}{3} \).