A train is moving with uniform speed crosses two tunnels of lengths 270 metres and 450 metres in 12 seconds and 15 seconds respectively. What is the speed (in km/hr) of the train?

Options

A. 144
B. 169
C. 196
D. 216

Answer

To find the speed of the train, we can follow these steps:

### Step 1: Understand the Problem
We have a train that crosses two tunnels of lengths 270 meters and 450 meters in 12 seconds and 15 seconds respectively. We need to find the speed of the train in km/hr.

### Step 2: Set Up the Equations
Let the length of the train be \( L \) meters. When the train crosses a tunnel, the distance covered by the train is the length of the train plus the length of the tunnel.

For the first tunnel:
\[ \text{Distance} = L + 270 \quad \text{and} \quad \text{Time} = 12 \text{ seconds} \]
Thus, the speed can be expressed as:
\[ \frac{L + 270}{12} \]

For the second tunnel:
\[ \text{Distance} = L + 450 \quad \text{and} \quad \text{Time} = 15 \text{ seconds} \]
Thus, the speed can be expressed as:
\[ \frac{L + 450}{15} \]

### Step 3: Equate the Two Speeds
Since the speed of the train is constant, we can set the two expressions for speed equal to each other:
\[ \frac{L + 270}{12} = \frac{L + 450}{15} \]

### Step 4: Cross-Multiply to Solve for \( L \)
Cross-multiplying gives us:
\[ 15(L + 270) = 12(L + 450) \]
Expanding both sides:
\[ 15L + 4050 = 12L + 5400 \]

### Step 5: Rearranging the Equation
Now, rearranging the equation to isolate \( L \):
\[ 15L - 12L = 5400 - 4050 \]
\[ 3L = 1350 \]
\[ L = 450 \text{ meters} \]

### Step 6: Calculate the Total Distance
Now that we have the length of the train, we can calculate the total distance covered when crossing both tunnels:
\[ \text{Total Distance} = L + 270 + L + 450 = 450 + 270 + 450 + 450 = 720 \text{ meters} \]

### Step 7: Calculate the Speed in Meters per Second
The total time taken to cross both tunnels is:
\[ \text{Total Time} = 12 + 15 = 27 \text{ seconds} \]
Now, we can calculate the speed in meters per second:
\[ \text{Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{720}{27} \approx 26.67 \text{ m/s} \]

### Step 8: Convert Speed to Kilometers per Hour
To convert the speed from meters per second to kilometers per hour, we use the conversion factor \( \frac{18}{5} \):
\[ \text{Speed in km/hr} = 26.67 \times \frac{18}{5} = 26.67 \times 3.6 \approx 96 \text{ km/hr} \]

### Final Answer
The speed of the train is approximately **96 km/hr**.