A train covers the first half of the distance between two stations at a speed of `40 km//h` and the other half at 60km/h. Thenits average speed :-

Options

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

Answer

To find the average speed of the train that covers the first half of the distance at 40 km/h and the second half at 60 km/h, we can follow these steps:

### Step 1: Define the distances
Let the total distance between the two stations be \( D \). Since the train covers the first half and the second half, each half will be:
- Distance for the first half, \( S = \frac{D}{2} \)
- Distance for the second half, \( S = \frac{D}{2} \)

### Step 2: Calculate the time taken for each half
The time taken to cover the first half of the distance at 40 km/h is:
\[ T_1 = \frac{S}{40} = \frac{\frac{D}{2}}{40} = \frac{D}{80} \text{ hours} \]

The time taken to cover the second half of the distance at 60 km/h is:
\[ T_2 = \frac{S}{60} = \frac{\frac{D}{2}}{60} = \frac{D}{120} \text{ hours} \]

### Step 3: Calculate the total time taken
The total time \( T \) taken to travel the entire distance \( D \) is the sum of the two times:
\[ T = T_1 + T_2 = \frac{D}{80} + \frac{D}{120} \]

To add these fractions, we need a common denominator. The least common multiple of 80 and 120 is 240. Therefore, we convert the fractions:
\[ T = \frac{D \cdot 3}{240} + \frac{D \cdot 2}{240} = \frac{3D + 2D}{240} = \frac{5D}{240} = \frac{D}{48} \text{ hours} \]

### Step 4: Calculate the average speed
The average speed \( V_{avg} \) is given by the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{D}{\frac{D}{48}} = 48 \text{ km/h} \]

### Final Answer
The average speed of the train is **48 km/h**.

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