If the first one - third of q journey is travelled at 20 km `h^(-1)` , Next one - third at `40 km h^(-1)` And the last one third at `60 km h^(-1)` then the average speed for the whole journey will be

Options

A. `32.7 km h^(-1)`
B. `35 km h^(-1)`
C. `40 km h ^(-1)`
D. `45 km h ^(-1)`

Answer

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

### Step 1: Define the total distance
Let the total distance of the journey be \( x \) km. Since the journey is divided into three equal parts, each part will be \( \frac{x}{3} \) km.

### Step 2: Calculate the time for each segment
We will calculate the time taken for each segment of the journey using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

- **For the first segment** (speed = 20 km/h):
\[ t_1 = \frac{\frac{x}{3}}{20} = \frac{x}{60} \text{ hours} \]

- **For the second segment** (speed = 40 km/h):
\[ t_2 = \frac{\frac{x}{3}}{40} = \frac{x}{120} \text{ hours} \]

- **For the third segment** (speed = 60 km/h):
\[ t_3 = \frac{\frac{x}{3}}{60} = \frac{x}{180} \text{ hours} \]

### Step 3: Calculate the total time
Now, we will sum the times for all three segments:
\[ \text{Total time} = t_1 + t_2 + t_3 = \frac{x}{60} + \frac{x}{120} + \frac{x}{180} \]

To add these fractions, we need a common denominator. The least common multiple of 60, 120, and 180 is 360. We convert each term:
\[ t_1 = \frac{x}{60} = \frac{6x}{360}, \quad t_2 = \frac{x}{120} = \frac{3x}{360}, \quad t_3 = \frac{x}{180} = \frac{2x}{360} \]
Now, adding these:
\[ \text{Total time} = \frac{6x}{360} + \frac{3x}{360} + \frac{2x}{360} = \frac{11x}{360} \text{ hours} \]

### Step 4: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{x}{\frac{11x}{360}} = \frac{x \cdot 360}{11x} = \frac{360}{11} \text{ km/h} \]

### Step 5: Calculate the numerical value
Now, we calculate \( \frac{360}{11} \):
\[ \frac{360}{11} \approx 32.73 \text{ km/h} \]

Thus, the average speed for the whole journey is approximately **32.73 km/h**.

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