A train covers the first half of the distance between two stations with a speed of 40 km/h and the other half with 60 km/h. Then its average speed is :

Options

A. 50 km/h
B. 48 km/h
C. 52 km/h
D. 100 km/h

Answer

To find the average speed of the train covering two halves of a distance at different speeds, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance between the two stations (A and B) be \(2s\). Therefore, the first half of the distance is \(s\) and the second half is also \(s\).

2. **Determine Speeds for Each Half**:
The speed for the first half (from A to C) is \(40 \text{ km/h}\) and for the second half (from C to B) is \(60 \text{ km/h}\).

3. **Calculate Time Taken for Each Half**:
- Time taken to cover the first half (A to C) is given by:
\[ t_1 = \frac{s}{40} \]
- Time taken to cover the second half (C to B) is given by:
\[ t_2 = \frac{s}{60} \]

4. **Find Total Time Taken**:
The total time taken \(T\) is the sum of the times for both halves:
\[ T = t_1 + t_2 = \frac{s}{40} + \frac{s}{60} \]
To add these fractions, we need a common denominator. The least common multiple of \(40\) and \(60\) is \(120\):
\[ T = \frac{3s}{120} + \frac{2s}{120} = \frac{5s}{120} = \frac{s}{24} \]

5. **Calculate Average Speed**:
The average speed \(V_{avg}\) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2s}{\frac{s}{24}} \]
Simplifying this expression:
\[ V_{avg} = 2s \times \frac{24}{s} = 48 \text{ km/h} \]

### Final Answer:
The average speed of the train is \(48 \text{ km/h}\).

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