A car covers first half of total distance at speed of 50 km/h and other half at the rate of 80 km/h. The average speed of car would be

Options

A. 60.5 km/h
B. 61.5 km/h
C. 100 km/h
D. 58.5 km/h

Answer

To find the average speed of the car that covers the first half of the total distance at a speed of 50 km/h and the second half at a speed of 80 km/h, we can follow these steps:

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

### Step 2: Calculate the distance for each half
Since the car covers the first half of the distance and the second half, each half will be:
- First half distance = \( \frac{X}{2} \)
- Second half distance = \( \frac{X}{2} \)

### Step 3: Calculate the time taken for the first half
Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
- Speed for the first half, \( V_1 = 50 \) km/h
- Time taken for the first half, \( T_1 = \frac{\frac{X}{2}}{50} = \frac{X}{100} \) hours

### Step 4: Calculate the time taken for the second half
Using the same formula:
- Speed for the second half, \( V_2 = 80 \) km/h
- Time taken for the second half, \( T_2 = \frac{\frac{X}{2}}{80} = \frac{X}{160} \) hours

### Step 5: Calculate the total time taken
The total time taken for the journey is the sum of the times for both halves:
\[ T = T_1 + T_2 = \frac{X}{100} + \frac{X}{160} \]
To add these fractions, we need a common denominator. The least common multiple of 100 and 160 is 800.
\[ T = \frac{8X}{800} + \frac{5X}{800} = \frac{13X}{800} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{X}{T} \]
Substituting the total distance and total time:
\[ \text{Average Speed} = \frac{X}{\frac{13X}{800}} = \frac{800}{13} \text{ km/h} \]

### Step 7: Calculate the numerical value
Calculating \( \frac{800}{13} \):
\[ \text{Average Speed} \approx 61.54 \text{ km/h} \]

Thus, the average speed of the car is approximately **61.54 km/h**.