Exercise 37 Page 687

We want to write the repeating decimal $2.\overline{18}$ as a fraction. To do so, we will start by writing this number as a sum of decimals and as a sum of fractions.

Number	$2.\overline{18}$
Sum of Decimals	$2+0.18+0.0018+0.000018+\dots$
Sum of Fractions	$\frac{2}{1}+\frac{18}{100}+\frac{18}{10000}+\frac{18}{1000000}+\dots$

Consider the sum of fractions above. Note that from the $2^{\text{nd}}$ term on, we can think of it as a geometric series that has a first term of $a_1=\frac{18}{100}.$ Let's find its common ratio $r.$

The common ratio is $r=\frac{1}{100}.$ Let's substitute this value — together with $a_1=\frac{18}{100}$ — in the formula for the sum of an infinite geometric series.

$S=\frac{a_1}{1-r}$

SubstituteII

$r=\frac{1}{100}$, $a_1=\frac{18}{100}$

$S=\frac{\frac{18}{100}}{1-\frac{1}{100}}$

▼

Simplify right-hand side

OneToFrac

Rewrite $1$ as $\frac{100}{100}$

$S=\frac{\frac{18}{100}}{\frac{100}{100}-\frac{1}{100}}$

SubFrac

Subtract fractions

$S=\frac{\frac{18}{100}}{\frac{99}{100}}$

FracToDiv

$\frac{a}{b}=a÷b$

$S=\frac{18}{100}÷\frac{99}{100}$

DivFracByFracDiv

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

$S=\frac{18}{100}\left(\frac{100}{99}\right)$

MultFrac

Multiply fractions

$S=\frac{1800}{9900}$

ReduceFrac

$\frac{a}{b}=\frac{a/900}{b/900}$

$S=\frac{2}{11}$

We found that the series formed by the sum of the terms, starting from the $2^{\text{nd}},$ is $\frac{2}{11}.$ With this information we can express the given number as a sum of two fractions.

$$\begin{array}{c}2.\overline{18}=\frac{2}{1}+\frac{2}{11}\end{array}$$

Finally, we will add the fractions to obtain the value of $2.\overline{18}$ expressed as a single fraction.

$2.\overline{18}=\frac{2}{1}+\frac{2}{11}$

ExpandFrac

$\frac{a}{b}=\frac{a\cdot 11}{b\cdot 11}$

$2.\overline{18}=\frac{22}{11}+\frac{2}{11}$

AddFrac

Add fractions

$2.\overline{18}=\frac{24}{11}$