Two trains cross each other in 12 seconds , when they travel in opposite directions and they take 60 seconds when they travel in the same direction. The possible speeds of the trains, (inm/sec), can be:

Options

A. 15, 45
B. 18, 40
C. 15, 30
D. 20, 30

Answer

To solve the problem of two trains crossing each other, we need to analyze the information given about their crossing times in both opposite and same directions.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the speed of the first train be \( S_1 \) m/s.
- Let the speed of the second train be \( S_2 \) m/s.
- The time taken to cross each other when traveling in opposite directions is 12 seconds.
- The time taken to cross each other when traveling in the same direction is 60 seconds.

2. **Setting Up the Equations**:
- When the trains cross each other in opposite directions, the distance covered is the sum of their lengths, which can be expressed as:
\[ \text{Distance} = (S_1 + S_2) \times 12 \]
- When the trains cross each other in the same direction, the distance covered is the difference of their lengths:
\[ \text{Distance} = (S_1 - S_2) \times 60 \]

3. **Equating the Distances**:
- Since the distance is the same in both cases, we can set the two equations equal to each other:
\[ (S_1 + S_2) \times 12 = (S_1 - S_2) \times 60 \]

4. **Expanding and Rearranging**:
- Expanding both sides gives:
\[ 12S_1 + 12S_2 = 60S_1 - 60S_2 \]
- Rearranging the equation:
\[ 12S_1 + 12S_2 + 60S_2 = 60S_1 \]
\[ 12S_1 + 72S_2 = 60S_1 \]
- Bringing all terms involving \( S_1 \) to one side:
\[ 72S_2 = 60S_1 - 12S_1 \]
\[ 72S_2 = 48S_1 \]

5. **Finding the Ratio of Speeds**:
- Dividing both sides by 12 gives:
\[ 6S_2 = 4S_1 \]
- Rearranging gives the ratio of the speeds:
\[ \frac{S_1}{S_2} = \frac{6}{4} = \frac{3}{2} \]

6. **Identifying Possible Speeds**:
- This ratio indicates that for every 3 units of speed of the first train, the second train has 2 units of speed.
- If we assume \( S_2 = 2x \) and \( S_1 = 3x \), we can substitute different values for \( x \) to find possible speeds.

7. **Conclusion**:
- The possible speeds of the trains can be expressed in the ratio \( 3:2 \).
- Therefore, if we check the options provided in the question, we can find which pair of speeds maintains this ratio.