Length and speed of train A is 'L' meters and 108 km/hr. It crosses a platform, whose length is 60% less than the length of train A in 28 sec. If train B crosses the same platform in 24 sec running at the speed of 90 km/hr., then find the time taken by train A to cross train B running in same direction?

Options

A. A)172 sec
B. B)198 sec
C. C)196 sec
D. D)192 sec

Answer

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

### Step 1: Determine the length of Train A
Given:
- Speed of Train A = 108 km/hr
- Time taken to cross the platform = 28 seconds
- Length of the platform = 60% less than the length of Train A

Let the length of Train A be \( L \) meters. The length of the platform will then be:
\[ \text{Length of platform} = L - 0.6L = 0.4L \]

The total distance covered by Train A while crossing the platform is:
\[ \text{Distance} = \text{Length of Train A} + \text{Length of platform} = L + 0.4L = 1.4L \]

Using the formula for distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

We convert the speed of Train A from km/hr to m/s:
\[ 108 \text{ km/hr} = 108 \times \frac{5}{18} = 30 \text{ m/s} \]

Now, substituting the values into the distance formula:
\[ 1.4L = 30 \times 28 \]
\[ 1.4L = 840 \]
\[ L = \frac{840}{1.4} = 600 \text{ meters} \]

### Step 2: Determine the length of the platform
Now that we have the length of Train A:
\[ \text{Length of platform} = 0.4L = 0.4 \times 600 = 240 \text{ meters} \]

### Step 3: Determine the length of Train B
Given:
- Speed of Train B = 90 km/hr
- Time taken to cross the platform = 24 seconds

Convert the speed of Train B from km/hr to m/s:
\[ 90 \text{ km/hr} = 90 \times \frac{5}{18} = 25 \text{ m/s} \]

Using the distance formula again:
\[ \text{Distance} = \text{Length of Train B} + \text{Length of platform} \]
Let the length of Train B be \( L_B \):
\[ L_B + 240 = 25 \times 24 \]
\[ L_B + 240 = 600 \]
\[ L_B = 600 - 240 = 360 \text{ meters} \]

### Step 4: Find the time taken by Train A to cross Train B
When two trains are moving in the same direction, the relative speed is the difference of their speeds.

Relative speed of Train A and Train B:
\[ \text{Relative speed} = 108 \text{ km/hr} - 90 \text{ km/hr} = 18 \text{ km/hr} \]
Convert to m/s:
\[ 18 \text{ km/hr} = 18 \times \frac{5}{18} = 5 \text{ m/s} \]

The total distance to be covered when Train A crosses Train B is the sum of their lengths:
\[ \text{Total distance} = L + L_B = 600 + 360 = 960 \text{ meters} \]

Now, using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{960}{5} = 192 \text{ seconds} \]

### Final Answer
The time taken by Train A to cross Train B is **192 seconds**.

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