Chapter 14.2, Problem 57PS

The mean defined in the text is sometimes called the arithmetic mean to distinguish it from other possible means. For example, a different mean, called the harmonic mean (H.M.), is used to find average speeds. This mean is defined to be the sum of the reciprocals of all scores divided into the number of scores. For example, the harmonic mean of the numbers $4,5,6,6,7,8$ is

$\begin{array}{l}\text{H}\text{.M}\text{.}=\frac{n}{{\displaystyle \sum \frac{1}{x}}}\\ =\frac{6}{\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}}\\ \simeq 5.7\end{array}$

a. Find the arithmetic and harmonic mean of the numbers $2,2,5,5,7,8,8,9,9,10.$

b. A trip from San Francisco to Disneyland is approximately $460$ miles. If the southbound trip averaged $52$ mph and the return trip averaged $61$ mph, what is the average speed for the round trip? Compare the arithmetic and harmonic means.