Two trains of same length running in opposite site direction crosses a student who is standing by side of railway track in 18 sec. and 12 sec. respectively. How much time will they take to cross each other ?

Options

A. 7.2 second
B. 9.6 second
C. 10.8 second
D. 14.4 second

Answer

To solve the problem step by step, we will follow these calculations:

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

### Step 2: Calculate the speed of each train
- For the first train, which crosses the student in 18 seconds:
\[ \text{Speed of Train 1} = \frac{\text{Distance}}{\text{Time}} = \frac{x}{18} \text{ m/s} \]

- For the second train, which crosses the student in 12 seconds:
\[ \text{Speed of Train 2} = \frac{x}{12} \text{ m/s} \]

### Step 3: Calculate the relative speed of the two trains
Since the trains are moving in opposite directions, their speeds add up:
\[ \text{Relative Speed} = \text{Speed of Train 1} + \text{Speed of Train 2} = \frac{x}{18} + \frac{x}{12} \]

To add these fractions, we need a common denominator. The least common multiple of 18 and 12 is 36:
\[ \frac{x}{18} = \frac{2x}{36} \quad \text{and} \quad \frac{x}{12} = \frac{3x}{36} \]
Now, adding these:
\[ \text{Relative Speed} = \frac{2x}{36} + \frac{3x}{36} = \frac{5x}{36} \text{ m/s} \]

### Step 4: Calculate the total distance when the trains cross each other
When the two trains cross each other, the total distance they need to cover is the sum of their lengths:
\[ \text{Total Distance} = x + x = 2x \text{ meters} \]

### Step 5: Calculate the time taken to cross each other
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
\[ \text{Time} = \frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{2x}{\frac{5x}{36}} = 2x \times \frac{36}{5x} \]
The \( x \) cancels out:
\[ \text{Time} = \frac{72}{5} = 14.4 \text{ seconds} \]

### Final Answer
The time taken for the two trains to cross each other is **14.4 seconds**.

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