Ratio of speed of two train are 7 : 9 and `1^(st)` train crosses a telephone pole in 4 sec and `2^(nd)` train cross the same pole in 6 sec. Find the how much both train will crosses each other. If both are running in opposite direction.

Options

A. `3 (1)/(8)` sec.
B. `5 (1)/(8)` sec.
C. `7 (1)/(8)`
D. `7 (1)/(3)`

Answer

To solve the problem step by step, we will follow the given information and calculate the required time for the two trains to cross each other when they are running in opposite directions.

### Step 1: Understand the given data
- Ratio of speeds of Train 1 and Train 2 = 7 : 9
- Time taken by Train 1 to cross a pole = 4 seconds
- Time taken by Train 2 to cross the same pole = 6 seconds

### Step 2: Assign variables for speed
Let the speed of Train 1 be \( 7x \) and the speed of Train 2 be \( 9x \), where \( x \) is a common multiplier.

### Step 3: Calculate the lengths of the trains
- Length of Train 1 (\( T1 \)) = Speed × Time = \( 7x \times 4 = 28x \)
- Length of Train 2 (\( T2 \)) = Speed × Time = \( 9x \times 6 = 54x \)

### Step 4: Calculate the total distance when both trains cross each other
When the two trains cross each other, the total distance is the sum of their lengths:
\[ \text{Total Distance} = T1 + T2 = 28x + 54x = 82x \]

### Step 5: Calculate the relative speed
Since the trains are moving in opposite directions, their speeds add up:
\[ \text{Relative Speed} = 7x + 9x = 16x \]

### Step 6: Calculate the time taken to cross each other
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{82x}{16x} \]
The \( x \) cancels out:
\[ \text{Time} = \frac{82}{16} = \frac{41}{8} \text{ seconds} \]
This can be converted to a mixed fraction:
\[ \frac{41}{8} = 5 \frac{1}{8} \text{ seconds} \]

### Final Answer:
The time taken for both trains to cross each other when running in opposite directions is **5 \(\frac{1}{8}\) seconds**.

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