Question

How do you write $0.3666...$ as a fraction? The $6$ repeating.

Answer 1

Step-1: Create two equations where the only numbers to the right of the decimal place are the repeating part:

Given the expression, $0.3666...$

The integer $6$ is repeating. so,

$\Rightarrow x=0.3\overline{6}$

Multiply both sides of $x=0.3\overline{6}$ by ${\displaystyle 10}$ to create the first equation:

$\begin{array}{rcl}& \Rightarrow & 10x=0.3\overline{6}\times 10\\ 10x& =& 3.\overline{6}\to \left(1\right)\end{array}$

Multiply both sides of $x=0.3\overline{6}$ by $100$ to create the second equation:

$\begin{array}{rcl}& \Rightarrow & 100x=0.3\overline{6}\times 100\\ 100x& =& 36.\overline{6}\to \left(2\right)\end{array}$

Step-2: Solve for $x$:

Subtract the equation (1) and (2):

$\begin{array}{c}\Rightarrow 100x=36.\overline{6}\\ 10x=3.\overline{6}\\ 90x=33\end{array}$

$\begin{array}{rcl}& \Rightarrow & 90x=33\\ x& =& \frac{33}{90}\\ x& =& \frac{3\left(11\right)}{3\left(30\right)}[\mathrm{Factor}3\mathrm{out}\mathrm{of}\mathrm{the}\mathrm{fraction}]\\ x& =& \frac{11}{30}\end{array}$

The fraction form of the given term $\mathbf{0}\mathbf{.}\mathbf{3666}\mathbf{.}\mathbf{.}\mathbf{.}$ is $\mathit{x}\mathbf{=}\frac{\mathbf{11}}{\mathbf{30}}$.