Two trains running in opposite directions cross a man standing on the platfor min 54 s and 34 s respectively and they cross each other in 46 s. Find the ratio of their speeds.

Options

A. a. `3:2`
B. b. `2:3`
C. c. `5:3`
D. d. `3:5`

Answer

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

### Step 1: Define the Variables
Let the speed of Train A be \( x \) km/s and the speed of Train B be \( y \) km/s.

### Step 2: Calculate the Lengths of the Trains
- The time taken by Train A to cross a man standing on the platform is 54 seconds. Therefore, the length of Train A can be calculated as:
\[ \text{Length of Train A} = \text{Speed} \times \text{Time} = x \times 54 = 54x \text{ km} \]

- The time taken by Train B to cross the same man is 34 seconds. Therefore, the length of Train B can be calculated as:
\[ \text{Length of Train B} = \text{Speed} \times \text{Time} = y \times 34 = 34y \text{ km} \]

### Step 3: Set Up the Equation for Crossing Each Other
When the two trains cross each other, they take 46 seconds. The total length when they cross each other is the sum of their lengths:
\[ \text{Total Length} = \text{Length of Train A} + \text{Length of Train B} = 54x + 34y \]

Since they cross each other in 46 seconds, we can express this as:
\[ \text{Total Length} = (\text{Speed of Train A} + \text{Speed of Train B}) \times \text{Time} \]
This gives us:
\[ 54x + 34y = (x + y) \times 46 \]

### Step 4: Expand and Rearrange the Equation
Expanding the right side:
\[ 54x + 34y = 46x + 46y \]

Rearranging the equation to isolate terms involving \( x \) and \( y \):
\[ 54x + 34y - 46x - 46y = 0 \]
This simplifies to:
\[ 8x - 12y = 0 \]

### Step 5: Solve for the Ratio of Speeds
Rearranging the equation gives us:
\[ 8x = 12y \]
Dividing both sides by 4:
\[ 2x = 3y \]
Thus, the ratio of the speeds \( x:y \) is:
\[ \frac{x}{y} = \frac{3}{2} \]

### Conclusion
The ratio of the speeds of Train A and Train B is \( 3:2 \).

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