Chapter 11.6, Problem 21WE

To determine

To calculate:

Given repeating decimal as a common fraction.

Answer to Problem 21WE

Fraction of $0.49999$ is $\frac{499}{999}$ .

Explanation of Solution

Give information:

  $0.49999$

Calculation:

Given is $0.49999$ ,

Let $x=0.49999$

  $0.49999=0.499+0.499\left(0.001\right)+0.499{\left(0.001\right)}^2+\mathrm{...}$

Initial term is $a_1=0.499$

Common ratio is $r=0.001$

Sum of infinite series is

  $\begin{array}{l}S=\frac{a_1}{1-r}\\ S=\frac{0.499}{1-0.001}\\ S=\frac{0.499}{0.999}\\ S=\frac{499}{999}\end{array}$

Thus, fraction of $0.49999$ is $\frac{499}{999}$ .