Quantity 1: 'x' : Train 'A' running at a speed of 25 m/sec crosses Train 'B' coming from opposite diection running at a speed of 15 m/sec in 12 seconds Length of train 'A' is twice of train 'B'. Length of train 'A' is 'x'
Quantity II: 160 meters.

Options

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

Explanation

To solve the problem step by step, we will analyze the information given about the two trains and derive the required quantities.

### Step 1: Understand the Problem
We have two trains, Train A and Train B, moving towards each other. We know:
- Speed of Train A = 25 m/s
- Speed of Train B = 15 m/s
- Time taken to cross each other = 12 seconds
- Length of Train A = x
- Length of Train B = x/2 (since Train A is twice the length of Train B)

### Step 2: Calculate the Total Distance Covered
When two objects move towards each other, the total distance covered when they cross each other is the sum of their lengths. Therefore, the total distance covered when Train A crosses Train B is:
\[ \text{Total Distance} = \text{Length of Train A} + \text{Length of Train B} = x + \frac{x}{2} \]
This can be simplified to:
\[ \text{Total Distance} = x + 0.5x = \frac{3x}{2} \]

### Step 3: Calculate the Relative Speed
Since the trains are moving towards each other, their relative speed is the sum of their speeds:
\[ \text{Relative Speed} = \text{Speed of Train A} + \text{Speed of Train B} = 25 \text{ m/s} + 15 \text{ m/s} = 40 \text{ m/s} \]

### Step 4: Use the Formula for Distance
We know that:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the known values:
\[ \frac{3x}{2} = 40 \text{ m/s} \times 12 \text{ s} \]

### Step 5: Calculate the Right Side
Calculating the right side:
\[ 40 \times 12 = 480 \text{ meters} \]

### Step 6: Set Up the Equation
Now we have:
\[ \frac{3x}{2} = 480 \]

### Step 7: Solve for x
To solve for x, multiply both sides by 2:
\[ 3x = 960 \]
Now, divide by 3:
\[ x = \frac{960}{3} = 320 \text{ meters} \]

### Step 8: Compare with Quantity II
Now we compare Quantity I (x = 320 meters) with Quantity II (160 meters):
- Quantity I = 320 meters
- Quantity II = 160 meters

### Conclusion
Since 320 meters is greater than 160 meters, we conclude:
\[ \text{Quantity I} > \text{Quantity II} \]

### Final Answer
The answer is that Quantity I is greater than Quantity II.

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