A vehicle covers first half of its total distance at a speed 20 kmph. and the other half of the distance at 40 kmph. The mean speed over the entire distance.

Options

A. 30 kmph
B. 50 kmph
C. `40/3` kmph
D. `80/3` kmph

Answer

To find the mean speed of the vehicle over the entire distance, we can follow these steps:

### Step 1: Define the total distance
Let the total distance covered by the vehicle be \( x \) kilometers.

### Step 2: Calculate the distance for each half
The vehicle covers the first half of the distance, which is \( \frac{x}{2} \) kilometers, at a speed of 20 km/h.

### Step 3: Calculate the time taken for the first half
The time taken to cover the first half of the distance can be calculated using the formula:
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{x}{2}}{20} = \frac{x}{40} \text{ hours} \]

### Step 4: Calculate the time taken for the second half
The vehicle covers the second half of the distance, which is also \( \frac{x}{2} \) kilometers, at a speed of 40 km/h. The time taken for this half can be calculated as:
\[ t_2 = \frac{\frac{x}{2}}{40} = \frac{x}{80} \text{ hours} \]

### Step 5: Calculate the total time taken
The total time taken for the entire journey is the sum of the time taken for both halves:
\[ t_{\text{total}} = t_1 + t_2 = \frac{x}{40} + \frac{x}{80} \]
To add these fractions, we need a common denominator:
\[ t_{\text{total}} = \frac{2x}{80} + \frac{x}{80} = \frac{3x}{80} \text{ hours} \]

### Step 6: Calculate the mean speed
The mean speed (average speed) is defined as the total distance divided by the total time:
\[ \text{Mean Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x}{\frac{3x}{80}} = \frac{x \cdot 80}{3x} = \frac{80}{3} \text{ km/h} \]

### Conclusion
Thus, the mean speed of the vehicle over the entire distance is:
\[ \frac{80}{3} \text{ km/h} \approx 26.67 \text{ km/h} \]