What is the length of a running train ?
(i) The train crosses a man in 9 seconds
(ii) The train crosses a 240 metre long platform in 24 seconds.

Answer

To find the length of the running train, we can use the information provided in the two statements. Let's break down the steps:

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

### Step 2: Use the First Statement
From the first statement, the train crosses a man in 9 seconds. The speed of the train can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
In this case, the distance is the length of the train \( x \) and the time is 9 seconds. Therefore, the speed \( s \) of the train can be expressed as:
\[ s = \frac{x}{9} \]

### Step 3: Use the Second Statement
From the second statement, the train crosses a 240-meter long platform in 24 seconds. When the train crosses the platform, it covers its own length plus the length of the platform. Therefore, the distance is \( x + 240 \) meters, and the time is 24 seconds. The speed can also be expressed as:
\[ s = \frac{x + 240}{24} \]

### Step 4: Set the Two Speed Equations Equal
Since the speed of the train is the same in both scenarios, we can set the two equations for speed equal to each other:
\[ \frac{x}{9} = \frac{x + 240}{24} \]

### Step 5: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives us:
\[ 24x = 9(x + 240) \]

### Step 6: Expand and Simplify
Expanding the right side:
\[ 24x = 9x + 2160 \]

### Step 7: Rearrange the Equation
Now, rearranging the equation to isolate \( x \):
\[ 24x - 9x = 2160 \]
\[ 15x = 2160 \]

### Step 8: Solve for \( x \)
Dividing both sides by 15:
\[ x = \frac{2160}{15} = 144 \]

### Conclusion
The length of the running train is \( 144 \) meters.

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