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

Options

A. 88 km//hr
B. 92 km//hr
C. 96 km//hr
D. 100 km//hr

Answer

To find Arjun's average speed for his journey, we can follow these steps:

### Step 1: Define the total distance
Let's assume the total distance of Arjun's journey is \( D \) km. Since he travels half of his journey by train and half by car, each part of the journey will cover \( \frac{D}{2} \) km.

### Step 2: Calculate the time taken for each part of the journey
1. **Time taken by train**:
- Speed of the train = 120 km/hr
- Distance covered by train = \( \frac{D}{2} \) km
- Time taken by train = \( \text{Distance} / \text{Speed} = \frac{D/2}{120} = \frac{D}{240} \) hours

2. **Time taken by car**:
- Speed of the car = 80 km/hr
- Distance covered by car = \( \frac{D}{2} \) km
- Time taken by car = \( \text{Distance} / \text{Speed} = \frac{D/2}{80} = \frac{D}{160} \) hours

### Step 3: Calculate the total time taken for the journey
Total time taken = Time taken by train + Time taken by car
\[ \text{Total time} = \frac{D}{240} + \frac{D}{160} \]

To add these fractions, we need a common denominator. The least common multiple of 240 and 160 is 480.
- Convert \( \frac{D}{240} \) to a fraction with a denominator of 480:
\[ \frac{D}{240} = \frac{2D}{480} \]
- Convert \( \frac{D}{160} \) to a fraction with a denominator of 480:
\[ \frac{D}{160} = \frac{3D}{480} \]
Now, add them:
\[ \text{Total time} = \frac{2D}{480} + \frac{3D}{480} = \frac{5D}{480} = \frac{D}{96} \text{ hours} \]

### Step 4: Calculate the average speed
Average speed is defined as total distance divided by total time.
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{D}{\frac{D}{96}} = 96 \text{ km/hr} \]

### Final Answer
Arjun's average speed for the entire journey is **96 km/hr**.

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