On a long journey if 10% of the distance is covered at an average speed of 40 km/h and 90% of the distance is covered at an average speed of 90 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 km/h and 90% of the duration is covered at average speed of 90 km/h.
First case:
Total distance = L (km)
Time at 40 km/h = 0.1 L/40
Time at 90 km/h = 0.9 L/90
Total time = (.0025+.01)L
Average speed = L/(.0125) = 80 km/h
Second case:
Total trip time = T
Distance covered at 40 km/h = 40*0.1T = 4.0 T km
Distance travelled at 90 km/h = 90*0.9T = 81 T km
Average speed = 85T/T = 85 km/h
To find the average speed for the entire journey, we need to calculate the total distance covered and the total time taken.
1. When the percentages relate to distance:
Let's assume the total distance of the journey is D km.
If 10% of the distance is covered at an average speed of 40 km/h, then the distance covered at this speed is 0.10D km.
Similarly, if 90% of the distance is covered at an average speed of 90 km/h, then the distance covered at this speed is 0.90D km.
To find the total distance covered, we need to add the distances covered at each speed:
Total distance = (0.10D km) + (0.90D km) = D km
Now, let's calculate the total time taken:
Time taken at a speed of 40 km/h = distance/speed
Time taken = (0.10D km) / (40 km/h) = (0.10D/40) hours
Time taken at a speed of 90 km/h = distance/speed
Time taken = (0.90D km) / (90 km/h) = (0.90D/90) hours
To find the average speed, we divide the total distance by the total time taken:
Average speed = (Total distance) / (Total time taken)
Average speed = D km / [(0.10D/40) hours + (0.90D/90) hours]
Simplifying this expression will give us the average speed for the entire journey.
2. When the percentages relate to time:
To find the average speed when percentages relate to time, we follow a similar approach. The only difference is that instead of calculating the total distance covered, we calculate the total time taken.
Let's assume the total duration of the journey is T hours.
If 10% of the duration is covered at a speed of 40 km/h, then the time taken at this speed is 0.10T hours.
Similarly, if 90% of the duration is covered at an average speed of 90 km/h, then the time taken at this speed is 0.90T hours.
To find the total time taken, we add the time taken at each speed:
Total time taken = (0.10T hours) + (0.90T hours) = T hours
Now, let's calculate the total distance covered:
Distance covered at a speed of 40 km/h = speed × time
Distance covered = 40 km/h × 0.10T hours = 4T km
Distance covered at a speed of 90 km/h = speed × time
Distance covered = 90 km/h × 0.90T hours = 81T km
To find the average speed, we divide the total distance by the total time taken:
Average speed = (Total distance) / (Total time taken)
Average speed = (4T km + 81T km) / T hours
Simplifying this expression will give us the average speed for the entire journey. Note that in this case, the average speed will be in km/h (distance per hour).