A car travels the first half of a distance between two places at a speed of `30 km// hr` and the second half of the distance at `50 km//hr` . The average speed of the car for the whole journey is

Options

A. `42.5 km//hr`
B. `40.0 km//hr`
C. `36.5 km//hr`
D. `35.0 km//hr`

Answer

To find the average speed of the car for the whole journey, we can follow these steps:

### Step 1: Define the total distance
Let the total distance between the two places be \( 2x \). Therefore, the first half of the distance is \( x \) and the second half is also \( x \).

### Step 2: Calculate the time taken for each half of the journey
- For the first half of the journey at a speed of \( 30 \, \text{km/hr} \):
\[ t_1 = \frac{x}{v_1} = \frac{x}{30} \]

- For the second half of the journey at a speed of \( 50 \, \text{km/hr} \):
\[ t_2 = \frac{x}{v_2} = \frac{x}{50} \]

### Step 3: Calculate the total time taken for the journey
The total time \( t \) taken for the entire journey is the sum of \( t_1 \) and \( t_2 \):
\[ t = t_1 + t_2 = \frac{x}{30} + \frac{x}{50} \]

### Step 4: Find a common denominator to simplify the total time
The least common multiple of \( 30 \) and \( 50 \) is \( 150 \). Therefore, we can rewrite the times:
\[ t = \frac{5x}{150} + \frac{3x}{150} = \frac{8x}{150} = \frac{4x}{75} \]

### Step 5: Calculate the average speed
The average speed \( v_{\text{avg}} \) is defined as the total distance divided by the total time:
\[ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{t} \]
Substituting \( t \):
\[ v_{\text{avg}} = \frac{2x}{\frac{4x}{75}} = 2x \times \frac{75}{4x} = \frac{150}{4} = 37.5 \, \text{km/hr} \]

### Final Answer
The average speed of the car for the whole journey is \( 37.5 \, \text{km/hr} \).

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