A coolie standing on a railway platform observes that a train going in one direction takes 4 seconds to pass him. Another train of same length going in opposite direction takes 5 seconds to pass him. The time taken (in seconds) by the two trains to cross each other will be :

Options

A. 35
B. 36.5
C. `40/9`
D. none of these

Answer

To solve the problem, we will follow these steps:

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

### Step 2: Calculate the Speed of Each Train
1. **For the first train**: It takes 4 seconds to pass the coolie.
\[ \text{Speed of first train} (V_1) = \frac{L}{4} \text{ m/s} \]

2. **For the second train**: It takes 5 seconds to pass the coolie.
\[ \text{Speed of second train} (V_2) = \frac{L}{5} \text{ m/s} \]

### Step 3: Find the Relative Speed of the Two Trains
When the two trains are moving in opposite directions, their relative speed is the sum of their speeds:
\[ \text{Relative Speed} = V_1 + V_2 = \frac{L}{4} + \frac{L}{5} \]

To add these fractions, we need a common denominator:
\[ \text{Relative Speed} = \frac{5L}{20} + \frac{4L}{20} = \frac{9L}{20} \text{ m/s} \]

### Step 4: Calculate the Total Length to be Crossed
When the two trains cross each other, the total length to be crossed is the sum of the lengths of both trains:
\[ \text{Total Length} = L + L = 2L \]

### Step 5: Calculate the Time Taken to Cross Each Other
The time taken to cross each other can be calculated using the formula:
\[ \text{Time} = \frac{\text{Total Length}}{\text{Relative Speed}} = \frac{2L}{\frac{9L}{20}} \]

Simplifying this:
\[ \text{Time} = 2L \times \frac{20}{9L} = \frac{40}{9} \text{ seconds} \]

### Final Answer
The time taken by the two trains to cross each other is \( \frac{40}{9} \) seconds.

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