A moving train crosses a man standing on a platform and a bridge 300 metres long in 10 seconds and 25 seconds respectively. What will be the time taken by the train to cross a platform 200 metres long?

Options

A. `16(2)/(3)` seconds
B. 18 seconds
C. 20 seconds
D. 22 seconds

Answer

To solve the problem, we need to determine the time taken by the train to cross a platform that is 200 meters long. We will first find the length of the train and its speed, then use that information to calculate the time taken to cross the 200-meter platform.

### Step 1: Find the speed of the train when it crosses the man
When the train crosses a man standing on the platform, it covers its own length (let's denote the length of the train as \( L \)) in 10 seconds.

- Speed of the train \( v = \frac{L}{10} \) meters per second.

### Step 2: Find the speed of the train when it crosses the bridge
When the train crosses a bridge that is 300 meters long, it covers the distance equal to its length plus the length of the bridge in 25 seconds.

- Distance covered = \( L + 300 \)
- Speed of the train \( v = \frac{L + 300}{25} \) meters per second.

### Step 3: Set the two expressions for speed equal to each other
Since both expressions represent the speed of the train, we can set them equal to each other:

\[ \frac{L}{10} = \frac{L + 300}{25} \]

### Step 4: Cross-multiply to solve for \( L \)
Cross-multiplying gives us:

\[ 25L = 10(L + 300) \]

Expanding the right side:

\[ 25L = 10L + 3000 \]

### Step 5: Rearrange the equation to isolate \( L \)
Subtract \( 10L \) from both sides:

\[ 25L - 10L = 3000 \]
\[ 15L = 3000 \]

### Step 6: Solve for \( L \)
Dividing both sides by 15 gives:

\[ L = \frac{3000}{15} = 200 \text{ meters} \]

### Step 7: Find the speed of the train
Now that we know the length of the train is 200 meters, we can find the speed of the train using the first equation:

\[ v = \frac{L}{10} = \frac{200}{10} = 20 \text{ meters per second} \]

### Step 8: Calculate the time taken to cross a 200-meter long platform
When the train crosses a platform that is 200 meters long, it has to cover its own length plus the length of the platform:

- Total distance to cover = \( L + 200 = 200 + 200 = 400 \) meters.

Using the speed we found:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{400}{20} = 20 \text{ seconds} \]

### Final Answer
The time taken by the train to cross a platform 200 meters long is **20 seconds**.

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