Average Speed Calculation for a Journey

This solution explains how to find the average speed when a journey involves traveling different parts of the distance at different speeds.

Understanding Average Speed

Average speed is defined as the total distance traveled divided by the total time taken for the journey. The formula is:

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

Analyzing the Journey Details

Let's break down the information given in the question:

• The journey is divided into two parts based on distance.
• Part 1: Two-fifth ($ \frac{2}{5} $) of the total distance is traveled.
• The speed during Part 1 ($ S_1 $) is 40 km/h.
• Part 2: The remaining distance is traveled.
• The speed during Part 2 ($ S_2 $) is 60 km/h.

Step-by-Step Calculation

To calculate the average speed, we need the total distance and the total time.

1. Define Total Distance

Let the total distance of the journey be represented by $ D $ kilometers.

2. Calculate Distances for Each Part

• Distance for Part 1 ($ D_1 $): $ D_1 = \frac{2}{5} D $ km.
• Distance for Part 2 ($ D_2 $): This is the remaining distance. $ D_2 = D - D_1 = D - \frac{2}{5} D = \frac{3}{5} D $ km.

3. Calculate Time Taken for Each Part

We use the formula: Time = Distance / Speed.

• Time for Part 1 ($ T_1 $): $$ T_1 = \frac{D_1}{S_1} = \frac{\frac{2}{5} D}{40 \text{ km/h}} = \frac{2D}{5 \times 40} \text{ h} = \frac{2D}{200} \text{ h} = \frac{D}{100} \text{ h} $$
• Time for Part 2 ($ T_2 $): $$ T_2 = \frac{D_2}{S_2} = \frac{\frac{3}{5} D}{60 \text{ km/h}} = \frac{3D}{5 \times 60} \text{ h} = \frac{3D}{300} \text{ h} = \frac{D}{100} \text{ h} $$

4. Calculate Total Time

The total time ($ T $) is the sum of the times for Part 1 and Part 2.

$$ T = T_1 + T_2 = \frac{D}{100} \text{ h} + \frac{D}{100} \text{ h} = \frac{2D}{100} \text{ h} = \frac{D}{50} \text{ h} $$

5. Calculate Average Speed

Now, we use the average speed formula with the Total Distance ($ D $) and Total Time ($ T $).

$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{50} \text{ h}} $$

$$ \text{Average Speed} = D \times \frac{50}{D} \text{ km/h} = 50 \text{ km/h} $$

Summary of Calculation

The average speed for the entire journey is 50 km/h.

Average Speed = Total Distance / Total Time = $ D / (\frac{D}{50}) = 50 $ km/h.