Question

Convert 0.45 (5 is a repeating number) to a fraction

Answer 1

Let the given number be 'x',

`\begin{gathered} x=0.4\bar{5} \\ x=0.45555555\ldots\ldots \end{gathered}`

Multiply both sides by 10,

`\begin{gathered} 10x=4.\bar{5} \\ 10x=4.5555555\ldots\ldots \end{gathered}`

Subtract the equations,

`\begin{gathered} 10x-x=4.5555\ldots-0.45555\ldots \\ 9x=4.5+0.0\bar{5}-(0.4+0.0\bar{5}) \\ 9x=4.5+0.0\bar{5}-0.4-0.0\bar{5} \\ 9x=4.1 \\ 9x=(41)/(10) \\ x=(41)/(90) \end{gathered}`

Thus, the recurring decimal number is equivalent to the fraction,

`0.4\bar{5}=(41)/(90)`