Calculating Average Speed for a Multi-Part Journey

To find the average speed for the entire journey, we need to calculate the total distance covered and the total time taken for the entire trip. The formula for average speed is:

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

Step 1: Identify the given information

• Distance for the first part (d₁) = 100 km
• Speed for the first part (v₁) = 60 km/h
• Distance for the second part (d₂) = 200 km
• Speed for the second part (v₂) = 75 km/h

Step 2: Calculate the Total Distance

The total distance is the sum of the distances of the two parts of the journey.

$\text{Total Distance} = d_1 + d_2$

$\text{Total Distance} = 100 \text{ km} + 200 \text{ km} = 300 \text{ km}$

Step 3: Calculate the time taken for each part of the journey

The time taken for each part can be calculated using the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.

Time for the first part (t₁):

$t_1 = \frac{d_1}{v_1} = \frac{100 \text{ km}}{60 \text{ km/h}} = \frac{10}{6} \text{ hours} = \frac{5}{3} \text{ hours}$

Time for the second part (t₂):

$t_2 = \frac{d_2}{v_2} = \frac{200 \text{ km}}{75 \text{ km/h}}$

We can simplify $\frac{200}{75}$ by dividing both numbers by 25:

$\frac{200 \div 25}{75 \div 25} = \frac{8}{3}$

So, $t_2 = \frac{8}{3} \text{ hours}$.

Step 4: Calculate the Total Time taken

The total time is the sum of the time taken for the two parts.

$\text{Total Time} = t_1 + t_2$

$\text{Total Time} = \frac{5}{3} \text{ hours} + \frac{8}{3} \text{ hours} = \frac{5 + 8}{3} \text{ hours} = \frac{13}{3} \text{ hours}$

Step 5: Calculate the Average Speed

Now we use the total distance and total time to find the average speed.

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

$\text{Average Speed} = \frac{300 \text{ km}}{\frac{13}{3} \text{ hours}}$

$\text{Average Speed} = 300 \times \frac{3}{13} \text{ km/h} = \frac{900}{13} \text{ km/h}$

Step 6: Convert the fraction to a decimal and round

Now, we calculate the decimal value of $\frac{900}{13}$ and round it to one decimal place.

$\frac{900}{13} \approx 69.2307...$

Rounding to one decimal place, we get 69.2 km/h.

Summary of Calculations:

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{300}{\frac{13}{3}} = \frac{300 \times 3}{13} = \frac{900}{13} \approx 69.23$ km/h

Rounding to one decimal place gives 69.2 km/h.

Therefore, the average speed of the car for the whole journey is approximately 69.2 km/h.