A train travels some distance with a speed of `30 km//h` and returns with a speed of `45 km//h` . Calculate the average speed of the train.

Explanation

To find the average speed of the train, we need to consider the total distance traveled and the total time taken for the journey. Let's break down the problem step by step.

### Step-by-Step Solution:

1. **Define the Distance:**
Let's assume the distance between points A and B is \( L \) kilometers.

2. **Calculate Time for the First Leg (A to B):**
The train travels from A to B with a speed of 30 km/h.
\[ \text{Time taken for A to B} = T_1 = \frac{L}{30} \text{ hours} \]

3. **Calculate Time for the Return Leg (B to A):**
The train returns from B to A with a speed of 45 km/h.
\[ \text{Time taken for B to A} = T_2 = \frac{L}{45} \text{ hours} \]

4. **Total Distance Traveled:**
The train travels from A to B and back from B to A, so the total distance is:
\[ \text{Total Distance} = L + L = 2L \text{ kilometers} \]

5. **Total Time Taken:**
The total time taken for the journey is the sum of the time taken for each leg:
\[ \text{Total Time} = T_1 + T_2 = \frac{L}{30} + \frac{L}{45} \]

6. **Simplify the Total Time Expression:**
To add the fractions, find a common denominator:
\[ \frac{L}{30} + \frac{L}{45} = \frac{3L}{90} + \frac{2L}{90} = \frac{5L}{90} = \frac{L}{18} \text{ hours} \]

7. **Calculate the Average Speed:**
The average speed is the total distance divided by the total time:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2L}{\frac{L}{18}} = 2L \times \frac{18}{L} = 36 \text{ km/h} \]

### Final Answer:
The average speed of the train is \( 36 \) km/h.