Calculating Average Speed for the Bus Journey

This problem requires us to calculate the average speed of a bus based on the distances covered and the time taken for two different segments of its journey. We need to find the overall average speed in kilometers per hour (km/h).

Understanding the Concept of Average Speed

Average speed is a fundamental concept in physics and is calculated by dividing the total distance traveled by the total time elapsed. The formula is:

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

To achieve the desired unit of km/h, the distance must be in kilometers (km) and the time must be in hours (h).

Analyzing the Bus Journey Segments

The bus undertakes its journey in two distinct parts:

• First Part: Covered a distance of 70 km in 40 minutes.
• Remaining Part: Covered a distance of 60 km in 30 minutes.

Calculating Total Distance

The total distance covered by the bus is the sum of the distances from both parts of the journey:

$$ \text{Total Distance} = 70 \text{ km} + 60 \text{ km} = 130 \text{ km} $$

Calculating Total Time and Unit Conversion

First, we sum the time taken for both parts to get the total time elapsed:

$$ \text{Total Time} = 40 \text{ minutes} + 30 \text{ minutes} = 70 \text{ minutes} $$

To use the average speed formula effectively for km/h, we must convert this total time from minutes to hours. Since there are 60 minutes in an hour, we perform the conversion:

$$ \text{Total Time in hours} = \frac{70 \text{ minutes}}{60 \text{ minutes/hour}} $$

Simplifying the fraction:

$$ \text{Total Time in hours} = \frac{7}{6} \text{ hours} $$

Calculating the Average Speed

Now, we apply the average speed formula using the total distance and the total time in hours:

$$ \text{Average Speed} = \frac{130 \text{ km}}{\frac{7}{6} \text{ hours}} $$

Dividing by a fraction is equivalent to multiplying by its reciprocal:

$$ \text{Average Speed} = 130 \times \frac{6}{7} \text{ km/h} $$

Performing the multiplication:

$$ \text{Average Speed} = \frac{130 \times 6}{7} \text{ km/h} = \frac{780}{7} \text{ km/h} $$

Expressing the Result as a Mixed Number

The average speed is $\frac{780}{7}$ km/h. To express this improper fraction as a mixed number, we divide 780 by 7:

$$ 780 \div 7 = 111 \text{ with a remainder of } 3 $$

This division result can be written as a mixed number:

$$ \frac{780}{7} = 111\frac{3}{7} $$

Thus, the average speed of the bus for the entire journey is $111\frac{3}{7}$ km/h.