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$\color{#FF6800}{ 0.2 \dot{ 1 } }$

$ $ Set the repeating decimal number to x $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ 0.2 \dot{ 1 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ 0.2 \dot{ 1 } }$

$ $ Multiply both sides by an appropriate power of 10 to make two expressions with the same part of the prime number $ $

$\begin{cases} \color{#FF6800}{ 100 } \color{#FF6800}{ x } = \color{#FF6800}{ 21. \dot{ 1 } } \\ \color{#FF6800}{ 10 } \color{#FF6800}{ x } = \color{#FF6800}{ 2. \dot{ 1 } } \end{cases}$

$\begin{cases} \color{#FF6800}{ 100 } \color{#FF6800}{ x } = \color{#FF6800}{ 21. \dot{ 1 } } \\ \color{#FF6800}{ 10 } \color{#FF6800}{ x } = \color{#FF6800}{ 2. \dot{ 1 } } \end{cases}$

$ $ Since the prime number part of the right side of the two expressions is the same, only the integer part remains $ $

$\color{#FF6800}{ 90 } \color{#FF6800}{ x } = \color{#FF6800}{ 19 }$

$\color{#FF6800}{ 90 } \color{#FF6800}{ x } = \color{#FF6800}{ 19 }$

$ $ Divide both sides by the same number $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 19 } { 90 } }$