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 step by step, we will follow these calculations:

### Step 1: Define Variables
Let the length of the train be \( x \) meters.

### Step 2: Calculate Speed of the Train
1. When the train crosses a man standing on the platform in 10 seconds, the speed of the train can be calculated as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{x}{10} \quad \text{(1)} \]

2. When the train crosses a bridge that is 300 meters long in 25 seconds, the distance covered is the length of the train plus the length of the bridge:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{x + 300}{25} \quad \text{(2)} \]

### Step 3: Equate the Two Speeds
Since both expressions represent the speed of the train, we can set them equal to each other:
\[ \frac{x}{10} = \frac{x + 300}{25} \]

### Step 4: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 25x = 10(x + 300) \]
Expanding the right side:
\[ 25x = 10x + 3000 \]

### Step 5: Rearranging the Equation
Subtract \( 10x \) from both sides:
\[ 25x - 10x = 3000 \]
\[ 15x = 3000 \]

### Step 6: Solve for \( x \)
Dividing both sides by 15:
\[ x = \frac{3000}{15} = 200 \text{ meters} \]

### Step 7: Calculate Speed of the Train
Now that we know the length of the train \( x = 200 \) meters, we can calculate the speed of the train:
\[ \text{Speed} = \frac{x}{10} = \frac{200}{10} = 20 \text{ meters/second} \]

### Step 8: Calculate Time to Cross a 200 Meter Long Platform
To find the time taken to cross a platform that is 200 meters long, we need to consider the total distance the train will cover, which is the length of the train plus the length of the platform:
\[ \text{Total Distance} = 200 + 200 = 400 \text{ meters} \]

Now, using the speed we calculated:
\[ \text{Time} = \frac{\text{Total Distance}}{\text{Speed}} = \frac{400}{20} = 20 \text{ seconds} \]

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

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