A man who swim 48m/minute in still water, swims 200 m against the current and 200m with the current. The difference between the time taken by him against the stream and with the stream is 10 minutes.
Quantity I: speed of current.
Quantity II: Speed of a man who completes 3 rounds of a circular path of radius 49 m in 14 minutes.

Options

A. Quantity I ` gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge `Quantity II
D. Quantity I `le` QuantityII

Answer

To solve the problem step by step, we will break it down into two parts: finding the speed of the current (Quantity I) and finding the speed of the man who completes three rounds of a circular path (Quantity II).

### Step 1: Finding the Speed of the Current (Quantity I)

1. **Define Variables**:
- Let \( W \) be the speed of the current in meters per minute.
- The speed of the man in still water is given as 48 m/min.

2. **Calculate Time Taken**:
- Time taken to swim 200 m against the current:
\[ \text{Time}_{\text{against}} = \frac{200}{48 - W} \]
- Time taken to swim 200 m with the current:
\[ \text{Time}_{\text{with}} = \frac{200}{48 + W} \]

3. **Set Up the Equation**:
- According to the problem, the difference in time taken is 10 minutes:
\[ \frac{200}{48 - W} - \frac{200}{48 + W} = 10 \]

4. **Cross Multiply**:
- Multiply both sides by \((48 - W)(48 + W)\):
\[ 200(48 + W) - 200(48 - W) = 10(48 - W)(48 + W) \]
- Simplifying the left side:
\[ 200W + 200W = 10(2304 - W^2) \]
\[ 400W = 23040 - 10W^2 \]

5. **Rearrange the Equation**:
- Rearranging gives:
\[ 10W^2 + 400W - 23040 = 0 \]
- Dividing the entire equation by 10:
\[ W^2 + 40W - 2304 = 0 \]

6. **Factor the Quadratic Equation**:
- Factoring gives:
\[ (W + 72)(W - 32) = 0 \]
- Thus, \( W = -72 \) or \( W = 32 \).

7. **Select the Positive Value**:
- Since speed cannot be negative, we take \( W = 32 \) m/min.

### Step 2: Finding the Speed of the Man (Quantity II)

1. **Calculate the Distance**:
- The radius of the circular path is given as 49 m. The circumference of one round is:
\[ C = 2\pi r = 2 \times \frac{22}{7} \times 49 = 2 \times 22 = 44 \text{ m} \]
- For three rounds:
\[ \text{Total Distance} = 3 \times 44 = 132 \text{ m} \]

2. **Calculate Speed**:
- The time taken to complete the three rounds is 14 minutes. Therefore, the speed is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{132}{14} = 9.42857 \text{ m/min} \]

3. **Convert to m/min**:
- To express this in a simpler form:
\[ \text{Speed} = \frac{66}{7} \approx 9.42857 \text{ m/min} \]

### Conclusion

- **Quantity I**: Speed of the current = 32 m/min
- **Quantity II**: Speed of the man = 9.42857 m/min

### Comparison

- Since \( 32 \text{ m/min} > 9.42857 \text{ m/min} \), we conclude that Quantity I is greater than Quantity II.

### Final Answer

- **Option B**: Quantity I is greater than Quantity II.