A car traveled 75% of the way from town A to town B at an average speed of 50 miles per hour. The car travels at an average speed of S miles per hour for the remaining part of the trip. The average speed for the entire trip was 40 miles per hour. What is S ?
Say the entire distance is 200 miles.
75% of the distance = 150 miles.
25% of the distance = 50 miles.
Total time = 200/40 = 5 hours;
Time spent to cover 150 miles = 150/50 = 3 hours.
Thus 50 miles was covered in 5-3=2 hours --> S = (speed) = (distance)/(time) = 50/2 = 25 miles per hour.
I tried to solve it a different way... using weighted averages but hit a wall...
Can you see where my logic fails?
Since the first part of the trip was 3/4 and the last was 1/4, this is what I got in my diagram:
When doing the cross multiplication, I get (50-40)/(40-S) = 3/1 -> I get S=110/3.
Why is this failing?
The weight when calculating average speed is time, not distance. This means that when you write (50-40)/(40-S) = 3/1, you are assuming that the car traveled 75% of the TIME at speed S and 25% of the time at speed 50 mph.
Ratio of 'Distance traveled' cannot act as the weight. See this post for a discussion of this concept: