Step 3: Calculating the average rate Now it’s time to tackle the original question.
Going from point A to point B, the cheetah traveled at an average rate of 70 mph. Returning to point A, the cheetah traveled at an average rate of 40 mph.
Can we say that this cheetah’s average rate was 55 mph?
We cannot say truthfully that the average rate is 55 mi/h.
a) The average rate for the trip is equal to the total distance traveled divided by the total time traveled. The following equations represent the distance traveled on each leg of the trip.

Write an equation for the average rate for the trip. Remember, the cheetah runs from point A to point B and back to point A.
First leg of trip: AB = 70·t1 ⇒ t1 = AB/70
Second leg of trip: AB = 40·t2 ⇒ t2 = AB/40
The average rate is the ratio of total distance to total time:
average rate = (AB +AB)/(t1 +t2)
average rate = 2AB/(AB/70 +AB/40) = 2/(1/70 +1/40) = 2(40)(70)/(70+40)
average rate = 560/11 = 50 10/11 mph

For two rates over the same distance (as above), the average is their harmonic mean
average rate = 2r1·r2/(r1+r2)

c) The following equation represents the average rate for the trip. Is this equation equivalent to the one you wrote above? Explain why or why not.

The cheetah's average rate was 50 10/11 mph ≈ 50.91 mph.

d) The following steps show how the equation for the average rate can be transformed so that it is written in terms of only the rates for each leg of the trip. Write an algebraic justification for each step. Think about the number operations and properties that you know.

e) Using the equation for average rate above, determine the cheetah’s average rate for the entire trip.