A man standing on a platform finds that a train takes 3 seconds to pass him and another train of the same length moving in the opposite direction, takes 4 seconds. The time taken by the trains to pass each other will be

Options

A. `2(3)/(7)` seconds
B. `3(3)/(7)` seconds
C. `4(3)/(7)` seconds
D. `5(3)/(7)` seconds

Answer

To solve the problem step by step, we will first determine the lengths and speeds of the trains, and then calculate the time taken for the two trains to pass each other.

### Step 1: Determine the length of the trains
Let the length of each train be \( L \).

### Step 2: Calculate the speed of the first train
The first train takes 3 seconds to pass the man standing on the platform. The speed of the first train \( V_1 \) can be calculated using the formula:
\[ V_1 = \frac{\text{Distance}}{\text{Time}} = \frac{L}{3} \]

### Step 3: Calculate the speed of the second train
The second train takes 4 seconds to pass the man. The speed of the second train \( V_2 \) is:
\[ V_2 = \frac{L}{4} \]

### Step 4: Calculate the effective speed when the two trains pass each other
When the two trains are moving towards each other, their speeds add up. Therefore, the effective speed \( V_{eff} \) is:
\[ V_{eff} = V_1 + V_2 = \frac{L}{3} + \frac{L}{4} \]
To add these fractions, we need a common denominator, which is 12:
\[ V_{eff} = \frac{4L}{12} + \frac{3L}{12} = \frac{7L}{12} \]

### Step 5: Calculate the total distance when the trains pass each other
When the two trains pass each other, the total distance they need to cover is the sum of their lengths:
\[ \text{Total Distance} = L + L = 2L \]

### Step 6: Calculate the time taken to pass each other
Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
\[ \text{Time} = \frac{2L}{V_{eff}} = \frac{2L}{\frac{7L}{12}} = 2L \cdot \frac{12}{7L} = \frac{24}{7} \text{ seconds} \]

### Step 7: Convert the time into a mixed number
\[ \frac{24}{7} = 3 \frac{3}{7} \text{ seconds} \]

### Final Answer
The time taken by the trains to pass each other is \( 3 \frac{3}{7} \) seconds.

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