Chapter 8.5, Problem 45PPSE

Solution

a.)

To calculate:The harmonic mean of two numbers $1$ and $\frac{1}{2}$ using the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

The value of harmonic mean of two numbers $1$ and $\frac{1}{2}$ of is $\frac{2}{3}$ .

Given information:

The two numbers $1$ and $\frac{1}{2}$ .

Formula Used:

The formula for harmonic mean is $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

Calculation:

Consider the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ,

The LCM of the factors $\frac{1}{a}+\frac{1}{b}$ is $ab$ .

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2}{\frac{a+b}{ab}}$

Divide numerator by denominator.

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2ab}{a+b}$

Substitute $1$ for $a$ and $\frac{1}{2}$ for $b$ .

  $\begin{array}{c}1+\frac{1}{2}=\frac{2\left(1\right)\left(\frac{1}{2}\right)}{1+\frac{1}{2}}\\ =\frac{1}{\frac{2+1}{2}}\\ =\frac{1}{\frac{3}{2}}\\ =\frac{2}{3}\end{array}$

Thus, the value ofharmonic mean of two numbers $1$ and $\frac{1}{2}$ of is $\frac{2}{3}$ .

b.)

To calculate: The harmonic mean of two numbers $\frac{3}{4}$ and $\frac{1}{2}$ using the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

The value of harmonic mean of two numbers $\frac{3}{4}$ and $\frac{1}{2}$ is $\frac{3}{5}$ .

Given information:

The two numbers $\frac{3}{4}$ and $\frac{1}{2}$ .

Formula Used:

The formula for harmonic mean is $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

Calculation:

Consider the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ,

The LCM of the factors $\frac{1}{a}+\frac{1}{b}$ is $ab$ .

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2}{\frac{a+b}{ab}}$

Divide numerator by denominator.

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2ab}{a+b}$

Substitute $\frac{3}{4}$ for $a$ and $\frac{1}{2}$ for $b$ .

  $\begin{array}{c}\frac{3}{4}+\frac{1}{2}=\frac{2\left(\frac{3}{4}\right)\left(\frac{1}{2}\right)}{\frac{3}{4}+\frac{1}{2}}\\ =\frac{\frac{3}{4}}{\frac{3+2}{4}}\\ =\frac{\frac{3}{4}}{\frac{5}{4}}\\ =\frac{3}{5}\end{array}$

Thus, the value of harmonic mean of two numbers $\frac{3}{4}$ and $\frac{1}{2}$ is $\frac{3}{5}$ .

c.)

To calculate: The harmonic mean of two numbers $\frac{3}{4}$ and $\frac{3}{5}$ using the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

The value of harmonic mean of two numbers $\frac{3}{4}$ and $\frac{3}{5}$ is $\frac{2}{3}$ .

Given information:

The two numbers $\frac{3}{4}$ and $\frac{3}{5}$ .

Formula Used:

The formula for harmonic mean is $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

Calculation:

Consider the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ,

The LCM of the factors $\frac{1}{a}+\frac{1}{b}$ is $ab$ .

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2}{\frac{a+b}{ab}}$

Divide numerator by denominator.

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2ab}{a+b}$

Substitute $\frac{3}{4}$ for $a$ and $\frac{3}{5}$ for $b$ .

  $\begin{array}{c}\frac{3}{4}+\frac{3}{5}=\frac{2\left(\frac{3}{4}\right)\left(\frac{3}{5}\right)}{\frac{3}{4}+\frac{3}{5}}\\ =\frac{\frac{18}{20}}{\frac{27}{20}}\\ =\frac{2}{3}\end{array}$

Thus, the value of harmonic mean of two numbers $\frac{3}{4}$ and $\frac{3}{5}$ is $\frac{2}{3}$ .

d.)

To calculate: The harmonic mean of two numbers $\frac{1}{2}$ and $\frac{1}{4}$ using the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

The value of harmonic mean of two numbers $\frac{1}{2}$ and $\frac{1}{4}$ is $\frac{1}{3}$ .

Given information:

The two numbers $\frac{1}{2}$ and $\frac{1}{4}$ .

Formula Used:

The formula for harmonic mean is $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ .

Calculation:

Consider the formula $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ,

The LCM of the factors $\frac{1}{a}+\frac{1}{b}$ is $ab$ .

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2}{\frac{a+b}{ab}}$

Divide numerator by denominator.

  $\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2ab}{a+b}$

Substitute $\frac{1}{2}$ for $a$ and $\frac{1}{4}$ for $b$ .

  $\begin{array}{c}\frac{1}{2}+\frac{1}{4}=\frac{2\left(\frac{1}{2}\right)\left(\frac{1}{4}\right)}{\frac{1}{2}+\frac{1}{4}}\\ =\frac{\frac{1}{4}}{\frac{2+1}{4}}\\ =\frac{\frac{1}{4}}{\frac{3}{4}}\\ =\frac{1}{3}\end{array}$

Thus, the value of harmonic mean of two numbers $\frac{1}{2}$ and $\frac{1}{4}$ is $\frac{1}{3}$ .