Udai travels half of his journey by train at the speed of 120 km/h and rest half by car at 80 km/h. What is the average speed ?

Answer

To find the average speed of Udai's journey, we can follow these steps:

### Step 1: Understand the Journey
Udai travels half of his journey by train at a speed of 120 km/h and the other half by car at a speed of 80 km/h.

### Step 2: Define the Total Distance
Let the total distance of the journey be \( D \). Therefore, half of the journey is \( \frac{D}{2} \).

### Step 3: Calculate Time Taken for Each Half
1. **Time taken by train**:
\[ \text{Time}_{\text{train}} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{2}}{120} = \frac{D}{240} \text{ hours} \]

2. **Time taken by car**:
\[ \text{Time}_{\text{car}} = \frac{\frac{D}{2}}{80} = \frac{D}{160} \text{ hours} \]

### Step 4: Calculate Total Time
Now, we can find the total time taken for the journey:
\[ \text{Total Time} = \text{Time}_{\text{train}} + \text{Time}_{\text{car}} = \frac{D}{240} + \frac{D}{160} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 240 and 160 is 480.

Rewriting the fractions:
\[ \frac{D}{240} = \frac{2D}{480} \]
\[ \frac{D}{160} = \frac{3D}{480} \]

Adding them together:
\[ \text{Total Time} = \frac{2D}{480} + \frac{3D}{480} = \frac{5D}{480} = \frac{D}{96} \text{ hours} \]

### Step 5: Calculate Average Speed
Average speed is defined as total distance divided by total time. Therefore:
\[ \text{Average Speed} = \frac{D}{\text{Total Time}} = \frac{D}{\frac{D}{96}} = 96 \text{ km/h} \]

### Final Answer
The average speed of Udai's journey is **96 km/h**.

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