A train crosses a platform 122 metre long in 17 sec. and a bridge of 210 metre long in 25 sec. at the same speed. Speed of train is-

Options

A. 46.5 km./hr.
B. 37.5 km./hr.
C. 37.6 km./hr.
D. 39.6 km./hr.

Answer

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

### Step 1: Understand the problem
We know that the train crosses a platform and a bridge, and we need to find the speed of the train based on the distances and times provided.

### Step 2: Identify the lengths and times
- Length of the platform (LP) = 122 meters
- Length of the bridge (LB) = 210 meters
- Time to cross the platform = 17 seconds
- Time to cross the bridge = 25 seconds

### Step 3: Set up the equations
When the train crosses the platform, it covers its own length (LT) plus the length of the platform:
\[ \text{Distance covered while crossing platform} = LP + LT \]

When the train crosses the bridge, it covers its own length (LT) plus the length of the bridge:
\[ \text{Distance covered while crossing bridge} = LB + LT \]

### Step 4: Write the equations for speed
The speed of the train can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

For the platform:
\[ \text{Speed} = \frac{LP + LT}{17} \]

For the bridge:
\[ \text{Speed} = \frac{LB + LT}{25} \]

### Step 5: Set the two speed equations equal to each other
Since the speed is the same in both cases, we can set the equations equal to each other:
\[ \frac{122 + LT}{17} = \frac{210 + LT}{25} \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 25(122 + LT) = 17(210 + LT) \]

### Step 7: Expand both sides
Expanding both sides:
\[ 3050 + 25LT = 3570 + 17LT \]

### Step 8: Rearrange the equation to solve for LT
Rearranging gives:
\[ 25LT - 17LT = 3570 - 3050 \]
\[ 8LT = 520 \]
\[ LT = \frac{520}{8} = 65 \text{ meters} \]

### Step 9: Calculate the speed of the train
Now, we can find the speed using the length of the train:
Using the platform equation:
\[ \text{Speed} = \frac{122 + 65}{17} = \frac{187}{17} \]
Calculating this gives:
\[ \text{Speed} = 11 \text{ m/s} \]

### Step 10: Convert speed to km/h
To convert the speed from m/s to km/h:
\[ \text{Speed in km/h} = 11 \times \frac{18}{5} = \frac{198}{5} = 39.6 \text{ km/h} \]

### Final Answer
The speed of the train is **39.6 km/h**.

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