The time taken for two trains of lengths, a metres and b metres, running at x km/h and y km/h in the opposite directions to cross each other = Time taken to cover (a + b) metres at _______ km/h.

Answer

To solve the problem, we need to determine the time taken for two trains of lengths \( a \) meters and \( b \) meters, running at speeds \( x \) km/h and \( y \) km/h in opposite directions, to cross each other.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two trains: one of length \( a \) meters and the other of length \( b \) meters.
- They are moving towards each other with speeds \( x \) km/h and \( y \) km/h.

2. **Total Distance to be Covered**:
- When the two trains cross each other, they need to cover a distance equal to the sum of their lengths.
- Therefore, the total distance to be covered is \( a + b \) meters.

3. **Converting Speeds to the Same Units**:
- The speeds \( x \) and \( y \) are given in km/h. To work with the distance in meters, we need to convert the speeds into meters per second.
- We know that \( 1 \) km/h = \( \frac{1000}{3600} \) m/s = \( \frac{5}{18} \) m/s.
- Thus, \( x \) km/h = \( \frac{5}{18} x \) m/s and \( y \) km/h = \( \frac{5}{18} y \) m/s.

4. **Relative Speed**:
- Since the trains are moving towards each other, their relative speed is the sum of their speeds.
- Therefore, the relative speed in meters per second is:
\[ \text{Relative Speed} = \frac{5}{18} x + \frac{5}{18} y = \frac{5}{18} (x + y) \text{ m/s} \]

5. **Time Taken to Cross Each Other**:
- The time taken to cross each other can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Here, the distance is \( a + b \) meters and the speed is \( \frac{5}{18} (x + y) \) m/s.
- Therefore, the time taken to cross each other is:
\[ \text{Time} = \frac{a + b}{\frac{5}{18} (x + y)} = \frac{(a + b) \cdot 18}{5(x + y)} \text{ seconds} \]

6. **Final Answer**:
- The time taken for the two trains to cross each other is equivalent to the time taken to cover \( a + b \) meters at a speed of \( (x + y) \) km/h.

### Conclusion:
The time taken for two trains of lengths \( a \) meters and \( b \) meters, running at \( x \) km/h and \( y \) km/h in opposite directions to cross each other is equal to the time taken to cover \( (a + b) \) meters at \( (x + y) \) km/h.