A moter car covers `(1)/(3)` rd part to total distance with the speed `30` km/h second `(1)/(3)`rd part with `60` km/hr and rest `(1)/(3)` rd part with `180` km/h . What is the average speed of tha car ?

Options

A. `54` km/hr
B. `45` km/hr
C. `6` km/hr
D. `22.5` km/hr

Explanation

To find the average speed of the motor car, we will follow these steps:

### Step 1: Define the total distance
Let's assume the total distance covered by the motor car is \( d \). Since the car covers the distance in three equal parts, each part will be \( \frac{d}{3} \).

### Step 2: Calculate the time taken for each part of the journey
1. **First part**:
- Distance = \( \frac{d}{3} \)
- Speed = \( 30 \) km/h
- Time taken \( t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{d}{3}}{30} = \frac{d}{90} \) hours

2. **Second part**:
- Distance = \( \frac{d}{3} \)
- Speed = \( 60 \) km/h
- Time taken \( t_2 = \frac{\frac{d}{3}}{60} = \frac{d}{180} \) hours

3. **Third part**:
- Distance = \( \frac{d}{3} \)
- Speed = \( 180 \) km/h
- Time taken \( t_3 = \frac{\frac{d}{3}}{180} = \frac{d}{540} \) hours

### Step 3: Calculate the total time taken
Now, we can find the total time taken \( T \) by summing up the times for each part:
\[ T = t_1 + t_2 + t_3 = \frac{d}{90} + \frac{d}{180} + \frac{d}{540} \]

### Step 4: Find a common denominator and simplify
The least common multiple (LCM) of \( 90, 180, \) and \( 540 \) is \( 540 \). We can rewrite each term with this common denominator:
\[ T = \frac{6d}{540} + \frac{3d}{540} + \frac{d}{540} = \frac{6d + 3d + d}{540} = \frac{10d}{540} = \frac{d}{54} \text{ hours} \]

### Step 5: Calculate the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{\frac{d}{54}} = 54 \text{ km/h} \]

### Final Answer
The average speed of the car is **54 km/h**.

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