Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. Fast train completely passes a man sitting in the slower train in 5 seconds. What is the length of the fast train?

Options

A. 23 m
B. `23(2)/(9)m`
C. 27 m
D. `27(7)/(9)m`

Answer

To solve the problem, we need to find the length of the fast train that overtakes a man sitting in the slower train. Here’s a step-by-step solution:

### Step 1: Determine the speeds of the trains
- The speed of the fast train = 40 km/hr
- The speed of the slow train = 20 km/hr

### Step 2: Calculate the relative speed
Since both trains are moving in the same direction, the relative speed is calculated by subtracting the speed of the slower train from the speed of the faster train.
- Relative speed = Speed of fast train - Speed of slow train
- Relative speed = 40 km/hr - 20 km/hr = 20 km/hr

### Step 3: Convert the relative speed from km/hr to m/s
To convert km/hr to m/s, we use the conversion factor \( \frac{5}{18} \).
- Relative speed in m/s = \( 20 \times \frac{5}{18} = \frac{100}{18} = \frac{50}{9} \) m/s

### Step 4: Use the time taken to pass the man
The fast train completely passes the man in 5 seconds. We can use the formula for distance, which is:
- Distance = Speed × Time

### Step 5: Calculate the length of the fast train
Using the relative speed we calculated and the time taken:
- Length of the fast train = Relative speed × Time
- Length of the fast train = \( \frac{50}{9} \) m/s × 5 s = \( \frac{250}{9} \) meters

### Step 6: Simplify the length
To express \( \frac{250}{9} \) in a more understandable form:
- \( \frac{250}{9} \) = 27.78 meters (approximately)

Thus, the length of the fast train is \( \frac{250}{9} \) meters or approximately 27.78 meters.

### Summary of the Solution:
- Length of the fast train = \( \frac{250}{9} \) meters or approximately 27.78 meters.

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