College Algebra, Exercise P, Exercise P.2, Section Exercise P.2, Problem 42

Express each repeating decimal as a fraction.
a.) $5.\overline{23}$
A repeating decimal $x = 5.232323$ is a rational number.
To convert it to a ratio of two integers, we have

$\begin{array}{l} & & 100x & = & 523.232323 ....... \\ \\ - & & x& = & 5.232323 .......\\ \hline\\ & & 99x & = & 518.0 \end{array}$

Thus, $\displaystyle x = \frac{518}{99}$

b.) $1.3\overline{7}$
A repeating decimal $x = 1.377777$ is a rational number.
To convert it to a ratio of two integers, we have

$\begin{array}{l} & & 100x & = & 137.77777 ....... \\ \\ - & & 10x& = & 13.7777 .......\\ \hline\\ & & 90x & = & 124.0 \end{array}$

Thus, $\displaystyle x = \frac{124}{90} \text{ or } x = \frac{62}{45}$

c.) $2.1\overline{35}$
A repeating decimal $x = 2.135353535$ is a rational number.
To convert it to a ratio of two integers, we have

$\begin{array}{l} & & 1000x & = & 2135.353535 ....... \\ \\ - & & 10x& = & 21.353535 .......\\ \hline\\ & & 990x & = & 2114.0 \end{array}$

Thus, $\displaystyle x = \frac{2114}{990} \text{ or } x = \frac{1057}{495}$