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Formula
Convert decimals to fractions
$7. \dot{ 1 } =$
$\dfrac { 64 } { 9 }$
Convert the repeating decimal number to a fraction
$\color{#FF6800}{ 7. \dot{ 1 } }$
 Set the repeating decimal number to x 
$\color{#FF6800}{ x } = \color{#FF6800}{ 7. \dot{ 1 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ 7. \dot{ 1 } }$
 Multiply both sides by an appropriate power of 10 to make two expressions with the same part of the prime number 
$\begin{cases} \color{#FF6800}{ 10 } \color{#FF6800}{ x } = \color{#FF6800}{ 71. \dot{ 1 } } \\ \color{#FF6800}{ 1 } \color{#FF6800}{ x } = \color{#FF6800}{ 7. \dot{ 1 } } \end{cases}$
$\begin{cases} \color{#FF6800}{ 10 } \color{#FF6800}{ x } = \color{#FF6800}{ 71. \dot{ 1 } } \\ \color{#FF6800}{ 1 } \color{#FF6800}{ x } = \color{#FF6800}{ 7. \dot{ 1 } } \end{cases}$
 Since the prime number part of the right side of the two expressions is the same, only the integer part remains 
$\color{#FF6800}{ 9 } \color{#FF6800}{ x } = \color{#FF6800}{ 64 }$
$\color{#FF6800}{ 9 } \color{#FF6800}{ x } = \color{#FF6800}{ 64 }$
 Divide both sides by the same number 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 64 } { 9 } }$
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