A train moves at the speed of 108 km/hr, passes a platform and a bridge in 15 second 18 sec respectively. If the length of platform is 50% of length of bridge, then find the length of train.

Options

A. 280 m
B. 360 m
C. 340 m
D. 320 m

Answer

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

### Step 1: Convert the speed of the train from km/hr to m/s
The speed of the train is given as 108 km/hr. To convert this to meters per second, we use the formula:

\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]

Calculating:

\[ \text{Speed} = 108 \times \frac{5}{18} = 30 \text{ m/s} \]

**Hint:** Remember that to convert km/hr to m/s, multiply by \(\frac{5}{18}\).

### Step 2: Set up the equations for the lengths of the platform and the bridge
Let the length of the bridge be \(2x\) meters. Since the length of the platform is 50% of the length of the bridge, the length of the platform will be \(x\) meters.

### Step 3: Calculate the distance covered when passing the platform
When the train passes the platform, it takes 15 seconds. The distance covered in this time is the length of the train plus the length of the platform:

\[ \text{Distance} = T + x \]

Using the speed we calculated, we can express the distance as:

\[ \text{Distance} = \text{Speed} \times \text{Time} = 30 \times 15 = 450 \text{ meters} \]

Thus, we have our first equation:

\[ T + x = 450 \quad \text{(1)} \]

### Step 4: Calculate the distance covered when passing the bridge
When the train passes the bridge, it takes 18 seconds. The distance covered in this time is the length of the train plus the length of the bridge:

\[ \text{Distance} = T + 2x \]

Again, using the speed:

\[ \text{Distance} = 30 \times 18 = 540 \text{ meters} \]

Thus, we have our second equation:

\[ T + 2x = 540 \quad \text{(2)} \]

### Step 5: Solve the equations simultaneously
We have two equations:

1. \(T + x = 450\)
2. \(T + 2x = 540\)

From equation (1), we can express \(T\) in terms of \(x\):

\[ T = 450 - x \quad \text{(3)} \]

Now, substitute equation (3) into equation (2):

\[ (450 - x) + 2x = 540 \]

Simplifying this:

\[ 450 + x = 540 \]

Subtracting 450 from both sides:

\[ x = 540 - 450 = 90 \text{ meters} \]

### Step 6: Calculate the length of the train
Now that we have \(x\), we can find \(T\) using equation (3):

\[ T = 450 - x = 450 - 90 = 360 \text{ meters} \]

### Final Answer
The length of the train is **360 meters**.

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