math problem. need help. please!?

Question

Jennifer is considering a strategy for an upcoming 2-mile bicycle race. During practice she maintains a speed of 20 miles per hour for the first mile, but fatigue reduces her speed to 10 miles per hour for the second mile.

Explain why Jennifer's average speed over these 2 miles is not the same as tha arithmetic mean of 20 miles per hour and 10 miles per hour. Find Jennifer's average speed for these 2 miles.

Answer 1

Basic formula: d = rt.

For the first mile, she goes at 20mph. d = 1, r = 20, find t:

1 = 20t
t = 1/20 hours (= 3 minutes)

For the second mile, she goes at 10mph. d = 1, r = 10, find t:

1 = 10t
t = 1/10 hours (= 6 minutes)

Then her average speed is the total distance (2 miles) over the total time (1/10 + 1/20 = 3/20 hr):

r = d/t = 2/(3/20) = 2/1 x 20/3 = 40/3 = 13 1/3 mph

But the arithmetic mean would have been (10 mph + 20 mph)/2 = 30 mph/2 = 15 mph

The reason the numbers are different is the fact that there's an inverse relationship between rate and time. When you have an inverse relationship, you end up with a geometric mean instead of an arithmetic mean.