Chapter 11.6, Problem 22WE

To determine

To calculate:

Given repeating decimal as a common fraction.

Answer to Problem 22WE

Fraction of $\mathrm{1.45045045...}$ is $\frac{145}{999}$ .

Explanation of Solution

Give information:

  $\mathrm{1.45045045...}$

Calculation:

Given is $\mathrm{1.45045045...}$ ,

Let $x=\mathrm{1.45045045...}$

  $1.45045045=1.450+1.450\left(0.001\right)+1.450{\left(0.001\right)}^2+\mathrm{...}$

Initial term is $a_1=1.450$

Common ratio is $r=0.001$

Sum of infinite series is

  $\begin{array}{l}S=\frac{a_1}{1-r}\\ S=\frac{1.450}{1-0.001}\\ S=\frac{1.450}{0.999}\\ S=\frac{145}{999}\end{array}$

Thus, fraction of $\mathrm{1.45045045...}$ is $\frac{145}{999}$ .