Chapter 3.1, Problem 36EC

Harmonic Mean The harmonic mean (HM) is defined as the number of values divided by the sum of the reciprocals of each value. The formula is

$\text{HM}=\frac{n}{{\displaystyle \sum (\frac{1}{x})}}$

For example, the harmonic mean of 1, 4, 5, and 2 is

$\text{HM}=\frac{4}{\frac{1}{1}+\frac{1}{4}+\frac{1}{5}+\frac{1}{2}}\approx 2.051$

This mean is useful for finding the average speed. Suppose a person drove 100 miles at 40 miles per hour and returned driving 50 miles per hour. The average miles per hour is not 45 miles per hour, which is found by adding 40 and 50 and dividing by 2. The average is found as shown. Since

Time = distance ÷ rate

Then

$\begin{array}{l}\text{Time}\text{1=}\frac{\text{100}}{\text{40}}\text{=2}\text{.5}\text{hours}\text{to}\text{make}\text{the}\text{trip}\\ \text{Time}\text{2=}\frac{\text{100}}{\text{50}}\text{=2}\text{hours}\text{to}\text{return}\end{array}$

Hence, the total time is 4.5 hours, and the total miles driven are 200. Now, the average speed is

$\text{Rate=}\frac{\text{distance}}{\text{time}}\text{=}\frac{\text{200}}{\text{4}\text{.5}}\approx \text{44}\text{.444}\text{miles}\text{per}\text{hour}$

This value can also be found by using the harmonic mean formula

$\text{HM=}\frac{2}{1/40+1/50}\approx 44.444$

Using the harmonic mean, find each of these.

a. A salesperson drives 300 miles round trip at 30 miles per hour going to Chicago and 45 miles per hour returning home. Find the average miles per hour.

b. A bus driver drives the 50 miles to West Chester at 40 miles per hour and returns driving 25 miles per hour. Find the average miles per hour.

c. A carpenter buys $500 worth of nails at $50 per pound and $500 worth of nails at $10 per pound. Find the average cost of 1 pound of nails.