# 3.83 repeating as a fraction?

Jun 25, 2018

It depends upon how much of the given number you mean to be repeating.

#### Explanation:

$3.8 \overline{3} \textcolor{w h i t e}{\text{xxxxx}}$only the last digit $3$ repeats:

$10 x = \textcolor{w h i t e}{\text{x}} 38.33333 \ldots$
ul(-x)color(white)(=)ul(-color(white)(3)3.83333...
$99 x = \textcolor{w h i t e}{\text{x}} 34.5$
$\rightarrow x = \frac{34.5}{90} = \frac{23}{60}$

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$3. \overline{83} \textcolor{w h i t e}{\text{xxxxxx}}$ the last 2 digits are repeating:

$100 x = \textcolor{w h i t e}{\text{x..}} 383.838383 \ldots$
$\underline{- \textcolor{w h i t e}{\text{..")x)=ul(-color(white)("xx}} 3.838383 \ldots}$
$\textcolor{w h i t e}{\text{.")99x=color(white)("xx}} 380.0$
$\rightarrow x = \frac{380}{99}$

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$\overline{3.83} \textcolor{w h i t e}{\text{xxxxxx}}$all 3 digits repeat:

$1000 x = \textcolor{w h i t e}{\text{..}} 3833.83383383 \ldots$
$\underline{- \textcolor{w h i t e}{\text{xx")x)=ul(-color(white)("....}} 3.83383383383 \ldots}$
$\textcolor{w h i t e}{\text{.")999x=color(white)(".}} 3830.0$
$\rightarrow x = \frac{3830}{999}$