Question

Use equation (1.8) to find the fractions that are equivalent to the following repeating decimals:

$\mathbf{0}\mathbf{.}\mathbf{55555}\mathbf{.}\mathbf{.}\mathbf{.}$

Short answer

The equivalent fraction for$0.55555...$ is $\frac{5}{9}$.

Step-by-step solution

Explanation of solution

Given number is $0.55555...$.

The expression for sum of infinite GP series is $S=\frac{a}{1-r}$.

Where, S is the sum of GP up to infinite terms.

Equivalent fraction

Fractions having the same numerator and denominator but different numerators and denominators are called equivalent fractions.

Calculation

An example of how to write the number is as follows:

$0.55555...=0.5+0.05+0.005+0.0005+....+0.5{\left(\frac{1}{10}\right)}^n$

The words will create GP with $a=0.5$and $r=1/10$.

Substitute 0.5 for a and 1/10 for r in above equation.

$\begin{array}{rcl}S& =& \frac{0.5}{1-\frac{1}{10}}\\ & =& \frac{\frac{5}{10}}{\frac{9}{10}}\\ & =& \frac{5}{9}\\ & & \end{array}$

Hence, the fraction is $\frac{5}{9}$.