A train X travelling at 60 km/h overtakes another train Y, 225 m long, and completely passes it in 72 seconds. If the trains had been going in opposite directions, they would have passed each other in 18 seconds. The length (in m) of X and the speed (in km/h) of Y are, respectively:

Options

A. 255 and 36
B. 255 and 40
C. 245 and 54
D. 245 and 45

Answer

To solve the problem step by step, we will use the information provided about the two trains, X and Y.

### Step 1: Define Variables
Let:
- Length of train X = L (in meters)
- Speed of train Y = y (in km/h)

### Step 2: Convert Speed of Train X
The speed of train X is given as 60 km/h. To work with consistent units, we can convert this speed to meters per second:
\[ \text{Speed of X in m/s} = \frac{60 \times 1000}{3600} = 16.67 \, \text{m/s} \]

### Step 3: Use the Overtaking Information
When train X overtakes train Y, the total distance covered is the sum of the lengths of both trains, which is \( L + 225 \) meters. The time taken to overtake is 72 seconds.

Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
We can write:
\[ L + 225 = (16.67 - \frac{y \times 1000}{3600}) \times 72 \]

### Step 4: Use the Opposite Direction Information
When the trains are moving in opposite directions, they would pass each other in 18 seconds. The total distance is still \( L + 225 \) meters, but the speeds add up:
\[ L + 225 = (16.67 + \frac{y \times 1000}{3600}) \times 18 \]

### Step 5: Set Up the Equations
From the overtaking scenario:
\[ L + 225 = (16.67 - \frac{y \times 1000}{3600}) \times 72 \quad \text{(1)} \]
From the opposite direction scenario:
\[ L + 225 = (16.67 + \frac{y \times 1000}{3600}) \times 18 \quad \text{(2)} \]

### Step 6: Solve the First Equation
Expanding equation (1):
\[ L + 225 = 1200 - 72 \times \frac{y \times 1000}{3600} \]
Simplifying:
\[ L + 225 = 1200 - 20y \quad \text{(3)} \]

### Step 7: Solve the Second Equation
Expanding equation (2):
\[ L + 225 = 300 + 18 \times \frac{y \times 1000}{3600} \]
Simplifying:
\[ L + 225 = 300 + 5y \quad \text{(4)} \]

### Step 8: Equate and Solve for y
From equations (3) and (4):
\[ 1200 - 20y = 300 + 5y \]
Rearranging gives:
\[ 1200 - 300 = 20y + 5y \]
\[ 900 = 25y \]
\[ y = \frac{900}{25} = 36 \, \text{km/h} \]

### Step 9: Substitute y Back to Find L
Substituting \( y = 36 \) into equation (3):
\[ L + 225 = 1200 - 20 \times 36 \]
\[ L + 225 = 1200 - 720 \]
\[ L + 225 = 480 \]
\[ L = 480 - 225 = 255 \, \text{m} \]

### Final Answer
The length of train X is 255 meters and the speed of train Y is 36 km/h.