Question

Dada a expressão (0,080808…) / (0,11111…). Resolva e deixe em forma de fração a/b.

Answer 1

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$\boxed{\begin{array}{l}\sf a=0,080808...\cdot100\\\sf 100a=8,080808...\\-\underline{\begin{cases}\sf 100a=8,080808...\\\sf a=0,080808...\end{cases}}\\\sf 99a=8\\\sf a=\dfrac{8}{99}\\\sf b=0,111...\cdot10\\\sf 10b=1,111....\\-\underline{\begin{cases}\sf 10b=1,111...\\\sf b=0,111...\end{cases}}\\\sf 9b=1\\\sf b=\dfrac{1}{9}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf\dfrac{0,080808...}{0,111....}=\dfrac{\frac{8}{99}}{\frac{1}{9}}\\\\\sf\dfrac{0,080808...}{0,111...}=\dfrac{8}{\diagdown\!\!\!\!\!\!99_{11}}\cdot\dfrac{\diagup\!\!\!9}{1}\\\\\sf\dfrac{0,080808...}{0,111....}=\dfrac{8}{11}\end{array}}$