Quantity I : A boat can cover distance of 480 km each in downstream and in upstream in total 11 hours. If ratio of speed of boat in still water to that of stream is 11 : 1, then find speed of boat in still water (in km /hr)
Quantity II: A boat can cover a distance of 300 km in downstream in 3 . 5 hours and can cover a distance of 380 km in upstream in 5 hours. Find speed of boat in still water (in km / hr )

Options

A. Quantity I ` lt ` Quantity II
B. Quantity I ` le ` Quantity II
C. Quantity I ` gt ` Quantity II
D. Quantity I = Quantity II or no relation

Answer

To solve the problem, we will break it down into two parts: Quantity I and Quantity II.

### Quantity I

1. **Define Variables**:
Let the speed of the boat in still water be \( 11x \) km/hr and the speed of the stream be \( x \) km/hr.

2. **Set Up the Equations**:
The speed of the boat downstream is \( 11x + x = 12x \) km/hr and the speed upstream is \( 11x - x = 10x \) km/hr.

3. **Calculate Time for Downstream and Upstream**:
The time taken to cover 480 km downstream is:
\[ \text{Time}_{\text{downstream}} = \frac{480}{12x} = \frac{40}{x} \text{ hours} \]
The time taken to cover 480 km upstream is:
\[ \text{Time}_{\text{upstream}} = \frac{480}{10x} = \frac{48}{x} \text{ hours} \]

4. **Total Time**:
According to the problem, the total time for both downstream and upstream is 11 hours:
\[ \frac{40}{x} + \frac{48}{x} = 11 \]
Simplifying this gives:
\[ \frac{88}{x} = 11 \]

5. **Solve for \( x \)**:
Multiplying both sides by \( x \):
\[ 88 = 11x \implies x = 8 \]

6. **Find Speed of Boat in Still Water**:
Now substituting \( x \) back to find the speed of the boat in still water:
\[ \text{Speed of boat in still water} = 11x = 11 \times 8 = 88 \text{ km/hr} \]

### Quantity II

1. **Define Variables**:
Let the speed of the boat in still water be \( a \) km/hr and the speed of the stream be \( b \) km/hr.

2. **Set Up the Equations**:
From the downstream distance of 300 km in 3.5 hours:
\[ \frac{300}{a + b} = 3.5 \implies a + b = \frac{300}{3.5} = 85.71 \text{ km/hr} \]

3. **Set Up the Upstream Equation**:
From the upstream distance of 380 km in 5 hours:
\[ \frac{380}{a - b} = 5 \implies a - b = \frac{380}{5} = 76 \text{ km/hr} \]

4. **Solve the System of Equations**:
Now we have two equations:
\[ a + b = 85.71 \quad (1) \]
\[ a - b = 76 \quad (2) \]
Adding these two equations:
\[ 2a = 161.71 \implies a = 80.855 \text{ km/hr} \]

5. **Find Speed of Boat in Still Water**:
Thus, the speed of the boat in still water is approximately \( 80.86 \) km/hr.

### Conclusion
- **Quantity I**: Speed of boat in still water = 88 km/hr
- **Quantity II**: Speed of boat in still water = 80.86 km/hr

### Comparison
Since \( 88 \) km/hr (Quantity I) is greater than \( 80.86 \) km/hr (Quantity II), we conclude that Quantity I is greater than Quantity II.