Quantity I : a train can cross a pole in 24 sec with a speed of 75 km / h. Length of train
Quantity II : A train can cross a pole in 12 sec and a tunnel in 55.2 sec. If length of tunnel is 1800 m. length of train

Options

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

Answer

To solve the question, we need to find the lengths of the train in both Quantity I and Quantity II and then compare them.

### Quantity I:
1. **Given Data**:
- Speed of the train = 75 km/h
- Time taken to cross a pole = 24 seconds

2. **Convert Speed to m/s**:
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]
\[ \text{Speed} = 75 \times \frac{5}{18} = \frac{375}{18} = 20.83 \text{ m/s} \]

3. **Calculate Length of the Train**:
- The length of the train can be calculated using the formula:
\[ \text{Length of train} = \text{Speed} \times \text{Time} \]
\[ \text{Length} = 20.83 \text{ m/s} \times 24 \text{ s} = 500 \text{ m} \]

### Quantity II:
1. **Given Data**:
- Time taken to cross a pole = 12 seconds
- Time taken to cross a tunnel = 55.2 seconds
- Length of the tunnel = 1800 m

2. **Let the speed of the train be \( x \) km/h and the length of the train be \( L \) m.**

3. **Set Up the Equations**:
- From crossing the pole:
\[ \frac{L}{x} = 12 \quad \text{(1)} \]
- From crossing the tunnel:
\[ \frac{L + 1800}{x} = 55.2 \quad \text{(2)} \]

4. **Express \( L \) from Equation (1)**:
\[ L = 12x \quad \text{(3)} \]

5. **Substitute \( L \) in Equation (2)**:
\[ \frac{12x + 1800}{x} = 55.2 \]
\[ 12 + \frac{1800}{x} = 55.2 \]
\[ \frac{1800}{x} = 55.2 - 12 \]
\[ \frac{1800}{x} = 43.2 \]
\[ x = \frac{1800}{43.2} = 41.67 \text{ km/h} \]

6. **Find Length of the Train Using \( x \)**:
- Substitute \( x \) back into Equation (3):
\[ L = 12 \times 41.67 = 500 \text{ m} \]

### Conclusion:
- **Quantity I**: Length of the train = 500 m
- **Quantity II**: Length of the train = 500 m

Thus, both quantities are equal:
\[ \text{Quantity I} = \text{Quantity II} = 500 \text{ m} \]

### Final Answer:
Both quantities are equal.

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