A train travelling at 72 kmph takes 45 seconds to cross a platform double its (the train's) length. How much time (in seconds) will the train take to cross (from the moment they meet) another train of length 400 m, travelling in opposite direction at a speed of 54 kmph?

Options

A. 10
B. 12
C. 20
D. 15

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Convert the speeds from km/h to m/s
The speed of the first train (Train A) is 72 km/h. To convert this to meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed of Train A} = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

The speed of the second train (Train B) is 54 km/h. We will convert this speed as well.

\[ \text{Speed of Train B} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 2: Determine the length of Train A
We know that Train A takes 45 seconds to cross a platform that is double its length. Let the length of Train A be \( x \). Therefore, the length of the platform is \( 2x \). The total distance covered while crossing the platform is:

\[ \text{Distance} = \text{Length of Train A} + \text{Length of Platform} = x + 2x = 3x \]

Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

\[ 45 = \frac{3x}{20} \]

### Step 3: Solve for \( x \)
Rearranging the equation gives:

\[ 3x = 45 \times 20 \]

Calculating the right side:

\[ 3x = 900 \]

Now, divide by 3:

\[ x = 300 \text{ m} \]

So, the length of Train A is 300 m.

### Step 4: Determine the total distance when the two trains meet
When Train A meets Train B, the total distance to be covered to completely cross each other is the sum of their lengths:

\[ \text{Total Distance} = \text{Length of Train A} + \text{Length of Train B} = 300 + 400 = 700 \text{ m} \]

### Step 5: Calculate the relative speed of the two trains
Since the trains are moving in opposite directions, we add their speeds:

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

### Step 6: Calculate the time taken to cross each other
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

\[ \text{Time} = \frac{700}{35} \]

Calculating this gives:

\[ \text{Time} = 20 \text{ seconds} \]

### Final Answer
The time taken for Train A to cross Train B is **20 seconds**.

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