On a long journey if 10% of the distance is covered at an average speed of $$40\ \text{km/h}$$ and 90% of the distance is covered at an average speed of $$90\ \text{km/h}$$. What is the average speed for the entire journey? How does this answer change if the percentages relate to time rather than distance, that is, if 10% of the duration is covered at $$40\ \text{km/h}$$ and 90% of the duration is covered at average speed of $$90\ \text{km/h}$$.

Let the total distance be $$D$$. The total time is $$\frac{0.1D}{40} + \frac{0.9D}{90} = 0.0025D + 0.01D = 0.0125D$$, so the average speed is $$\frac{D}{0.0125D} = 80\ \text{km/h}$$. If instead the percentages refer to time, let the total time be $$t$$; the distance covered is $$0.1t \times 40 + 0.9t \times 90 = 4t + 81t = 85t$$. So the average speed becomes $$85\ \text{km/h}$$, i.e. it increases by $$5\ \text{km/h}$$, because more time is now spent at the higher speed.