Answer
$=\dfrac {50771}{9900}$
Work Step by Step
$5.12\overline {83}=5.128383\ldots =\dfrac {512}{100}+\dfrac {1}{100}\left( \dfrac {83}{100}+\dfrac {83}{1002}+\dfrac {83}{1003}...\right) =\dfrac {512}{100}+\dfrac {1}{100}\left( \dfrac {a_{1}}{1-r}\right) =\dfrac {512}{100}+\dfrac {1}{100}\dfrac {\dfrac {83}{100}}{1-\dfrac {1}{100}}=\dfrac {1}{100}\left( 512+\dfrac {83}{99}\right) =\dfrac {50771}{9900}$
