A train X running at 72 km/h crosses another train Y running at 54 km/h in opposite direction in 15 seconds. If the length of Y is two - fifth that of X, then what is the length (in meter) of X ?

Options

A. 375
B. 150
C. 300
D. 225

Answer

To solve the problem, we need to find the length of train X given the information about both trains and their speeds. Let's break it down step by step.

### Step 1: Convert the speeds from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

- Speed of train X:
\[ 72 \text{ km/h} = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

- Speed of train Y:
\[ 54 \text{ km/h} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 2: Calculate the relative speed of the two trains
Since the trains are moving in opposite directions, we add their speeds to get the relative speed.

\[ \text{Relative Speed} = 20 \text{ m/s} + 15 \text{ m/s} = 35 \text{ m/s} \]

### Step 3: Use the time taken to cross each other to find the combined length of the trains
We know that the trains cross each other in 15 seconds. The distance covered while crossing each other is given by:

\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]

Substituting the values:

\[ \text{Distance} = 35 \text{ m/s} \times 15 \text{ s} = 525 \text{ meters} \]

### Step 4: Set up the equation for the lengths of the trains
Let the length of train X be \( L_X \) and the length of train Y be \( L_Y \). According to the problem, we have:

\[ L_Y = \frac{2}{5} L_X \]

The total length when both trains cross each other is:

\[ L_X + L_Y = 525 \]

Substituting \( L_Y \) in the equation:

\[ L_X + \frac{2}{5} L_X = 525 \]

### Step 5: Solve for \( L_X \)
Combine the terms:

\[ L_X + \frac{2}{5} L_X = \frac{5}{5} L_X + \frac{2}{5} L_X = \frac{7}{5} L_X \]

Now, set the equation:

\[ \frac{7}{5} L_X = 525 \]

To find \( L_X \), multiply both sides by \( \frac{5}{7} \):

\[ L_X = 525 \times \frac{5}{7} = 375 \text{ meters} \]

### Conclusion
The length of train X is \( 375 \) meters.

---