Two trains running in opposite directions cross a man standing on the platform in 25 seconds and 32 seconds respectively and they cross each other in 30 seconds. The ratio of their speed is:

Options

A. `4:3`
B. `2:5`
C. `5:6`
D. `1:3`

Answer

To solve the problem, we need to find the ratio of the speeds of two trains that cross a man standing on a platform in different times and also cross each other in a specific time. Here’s the step-by-step solution:

### Step 1: Define Variables
Let the speed of Train 1 be \( x \) meters per second and the speed of Train 2 be \( y \) meters per second.

### Step 2: Calculate Lengths of the Trains
- The length of Train 1 can be calculated using the time it takes to cross the man:
\[ \text{Length of Train 1} = \text{Speed} \times \text{Time} = x \times 25 = 25x \text{ meters} \]
- The length of Train 2 can be calculated similarly:
\[ \text{Length of Train 2} = y \times 32 = 32y \text{ meters} \]

### Step 3: Total Length When Trains Cross Each Other
When the two trains cross each other, the total length is the sum of the lengths of both trains:
\[ \text{Total Length} = 25x + 32y \]

### Step 4: Calculate Speed When Trains Cross Each Other
When the trains cross each other, their relative speed is the sum of their individual speeds since they are moving in opposite directions:
\[ \text{Relative Speed} = x + y \]
They cross each other in 30 seconds, so we can set up the equation:
\[ \text{Total Length} = \text{Relative Speed} \times \text{Time} \]
Thus,
\[ 25x + 32y = (x + y) \times 30 \]

### Step 5: Expand and Rearrange the Equation
Expanding the right side gives:
\[ 25x + 32y = 30x + 30y \]
Rearranging the equation:
\[ 25x + 32y - 30x - 30y = 0 \]
This simplifies to:
\[ -5x + 2y = 0 \]
or
\[ 2y = 5x \]

### Step 6: Find the Ratio of Speeds
From the equation \( 2y = 5x \), we can express the ratio of \( x \) to \( y \):
\[ \frac{x}{y} = \frac{2}{5} \]
Thus, the ratio of the speeds of Train 1 to Train 2 is:
\[ x : y = 2 : 5 \]

### Final Answer
The ratio of their speeds is \( 2 : 5 \).

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