One third of a certain journey is covered at the speed of 40km/hr, one fourth at the speed of 25 km/hr and the rest at the speed of 50 km/hr. What will be the average speed (in km/hr) for the whole journey?

Options

A. `37.5`
B. `33.5`
C. `33.33`
D. `38.33`

Answer

To find the average speed for the entire journey, we will follow these steps:

### Step 1: Define the total distance
Let the total distance of the journey be \( D \). For simplicity, we can assume \( D = 120 \) km (this is a multiple of the denominators in the fractions of the journey).

### Step 2: Calculate the distance covered in each segment
1. **First segment**: One third of the journey is covered at 40 km/hr.
\[ \text{Distance}_1 = \frac{1}{3}D = \frac{1}{3} \times 120 = 40 \text{ km} \]

2. **Second segment**: One fourth of the journey is covered at 25 km/hr.
\[ \text{Distance}_2 = \frac{1}{4}D = \frac{1}{4} \times 120 = 30 \text{ km} \]

3. **Third segment**: The rest of the journey is covered at 50 km/hr.
\[ \text{Distance}_3 = D - \left(\text{Distance}_1 + \text{Distance}_2\right) = 120 - (40 + 30) = 50 \text{ km} \]

### Step 3: Calculate the time taken for each segment
1. **Time for the first segment**:
\[ \text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{40}{40} = 1 \text{ hour} \]

2. **Time for the second segment**:
\[ \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{30}{25} = 1.2 \text{ hours} \]

3. **Time for the third segment**:
\[ \text{Time}_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{50}{50} = 1 \text{ hour} \]

### Step 4: Calculate the total time taken
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3 = 1 + 1.2 + 1 = 3.2 \text{ hours} \]

### Step 5: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{120}{3.2} = 37.5 \text{ km/hr} \]

### Final Answer
The average speed for the whole journey is **37.5 km/hr**.

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