Solving for Average Speed Over Equal Distances

To find the average speed when covering three equal distances at different speeds, we use the formula:

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

Setting Up the Problem

Let the distance for each part of the journey be '$d$'. The total distance is therefore '$3d$'.

The speeds for the three parts are:

• Speed 1 ($v_1$): $30\text{ km/h}$
• Speed 2 ($v_2$): $45\text{ km/h}$
• Speed 3 ($v_3$): $90\text{ km/h}$

Calculating Time for Each Segment

Time taken for each distance is calculated using $ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $.

• Time 1 ($t_1$) = $ \frac{d}{30} $ hours
• Time 2 ($t_2$) = $ \frac{d}{45} $ hours
• Time 3 ($t_3$) = $ \frac{d}{90} $ hours

Calculating Total Time

Total time ($T$) is the sum of the times for each segment:

$ T = t_1 + t_2 + t_3 = \frac{d}{30} + \frac{d}{45} + \frac{d}{90} $

To add these fractions, find a common denominator, which is 90:

$ T = \frac{3d}{90} + \frac{2d}{90} + \frac{d}{90} = \frac{3d + 2d + d}{90} = \frac{6d}{90} $

Simplifying the total time:

$ T = \frac{d}{15} \text{ hours} $

Calculating Average Speed

Now, apply the average speed formula using the total distance ($3d$) and total time ($ \frac{d}{15} $):

$ \text{Average Speed} = \frac{3d}{\frac{d}{15}} $

$ \text{Average Speed} = 3d \times \frac{15}{d} $

$ \text{Average Speed} = 3 \times 15 = 45 \text{ km/h} $

Conclusion

The average speed of the car for the entire journey is $45\text{ km/h}$.