A car covers `(1)/(3)` part of total distance with a speed of `20 km hr^-1` and second `(1)/(3)` part with a speed of `30 km hr^-1` and the last `(1)/(3)` part with a speed of `60 km hr^-1`. The average speed of the car is.

Options

A. `55 kmhr^-1`
B. `30 kmhr^-1`
C. `45 kmhr^-1`
D. `37.3 kmhr^-1`

Explanation

To find the average speed of the car that covers different parts of a distance at different speeds, we can follow these steps:

### Step 1: Define the total distance
Let the total distance be \( x \) km. Since the car covers \( \frac{1}{3} \) of the total distance at different speeds, we can express each segment of the journey.

### Step 2: Calculate the distance for each segment
Each segment of the journey is:
- First segment: \( \frac{x}{3} \) km at a speed of \( 20 \) km/hr
- Second segment: \( \frac{x}{3} \) km at a speed of \( 30 \) km/hr
- Third segment: \( \frac{x}{3} \) km at a speed of \( 60 \) km/hr

### Step 3: Calculate the time taken for each segment
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

1. **Time for the first segment** (\( t_1 \)):
\[ t_1 = \frac{\frac{x}{3}}{20} = \frac{x}{60} \text{ hours} \]

2. **Time for the second segment** (\( t_2 \)):
\[ t_2 = \frac{\frac{x}{3}}{30} = \frac{x}{90} \text{ hours} \]

3. **Time for the third segment** (\( t_3 \)):
\[ t_3 = \frac{\frac{x}{3}}{60} = \frac{x}{180} \text{ hours} \]

### Step 4: Calculate the total time taken
Now, we sum the times for all segments:
\[ t = t_1 + t_2 + t_3 = \frac{x}{60} + \frac{x}{90} + \frac{x}{180} \]

To add these fractions, we find a common denominator. The least common multiple of \( 60, 90, \) and \( 180 \) is \( 180 \):
- Convert \( \frac{x}{60} \) to \( \frac{3x}{180} \)
- Convert \( \frac{x}{90} \) to \( \frac{2x}{180} \)
- \( \frac{x}{180} \) remains \( \frac{x}{180} \)

Now, summing them:
\[ t = \frac{3x}{180} + \frac{2x}{180} + \frac{x}{180} = \frac{6x}{180} = \frac{x}{30} \text{ hours} \]

### Step 5: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x}{\frac{x}{30}} = 30 \text{ km/hr} \]

### Final Answer
The average speed of the car is \( 30 \) km/hr.

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