Chapter 4.6, Problem 43E

a.

To determine

To find: The equation whose solution is the harmonic mean of $30$ and $45$ .

Answer to Problem 43E

The equation is $\frac{1}{x}=\left(\frac{1}{30}+\frac{1}{45}\right)÷2$ .

Explanation of Solution

Given:A number $x$ is said to be harmonic mean of $y$ and $z$ if $\frac{1}{x}$ is the average of $\frac{1}{y}$ and $\frac{1}{z}$ .

Calculation:

The equation of harmonic mean is given by the formula,

  $\text{Average=}\frac{\text{sum}}{\text{total}}$ .

Hence,

  $\frac{1}{x}=\left(\frac{1}{y}+\frac{1}{z}\right)÷2$

Here,

  $y=30,z=45$ .

Substitute the value in the given formula,

  $\begin{array}{l}\frac{1}{x}=\left(\frac{1}{y}+\frac{1}{z}\right)÷2\\ \frac{1}{x}=\left(\frac{1}{30}+\frac{1}{45}\right)÷2\end{array}$

Therefore, the equation is $\frac{1}{x}=\left(\frac{1}{30}+\frac{1}{45}\right)÷2$ .

b.

To determine

To find: The harmonic mean of $30$ and $45$ .

Answer to Problem 43E

The harmonic mean is $36$ .

Explanation of Solution

Given:

A number $x$ is said to be harmonic mean of $y$ and $z$ if $\frac{1}{x}$ is the average of $\frac{1}{y}$ and $\frac{1}{z}$ .

Calculation:

The equation of harmonic mean is given by,

  $\frac{1}{x}=\left(\frac{1}{y}+\frac{1}{z}\right)÷2$

Solve further,

  $\begin{array}{c}\frac{1}{x}=\left(\frac{1}{y}+\frac{1}{z}\right)÷2\\ \frac{2}{x}=\frac{1}{30}+\frac{1}{45}\\ \frac{2}{x}=\frac{750}{1350}\\ x=36\end{array}$

Therefore, the harmonic mean is $36$ .