Question

10. Determine a fração geratriz das dizimas periódicas simples, simplificando-a, se possivel.

a) 0,8888...

b) 0,141414...

c) 2,2222...

d) 1.125125125.​
por favor vale 10 pontos

Answer 1

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$\tt a)~\sf x=0,888...\cdot10\\\sf10x=8,888...\\-\underline{\begin{cases}\sf10x=8,\diagup\!\!\!8\diagup\!\!\!8\diagup\!\!\!8...\\\sf x=0,\diagup\!\!\!8\diagup\!\!\!8\diagup\!\!\!8...\end{cases}}\\\sf9x=8\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=\dfrac{8}{9}}}}}\blue{\checkmark}$

$\tt b)~\sf y=0,141414...\cdot100\\\sf 100y=14,141414...\\-\underline{\begin{cases}\sf 100y=14,141414...\\\sf y=0,141414...\end{cases}}\\\sf 99y=14\\\huge\boxed{\boxed{\boxed{\boxed{\sf y=\dfrac{14}{99}}}}}\blue{\checkmark}$

$\tt c)~\sf k=2,222...\cdot10\\\sf10k=22,222...\\-\underline{\begin{cases}\sf10k=22,\diagup\!\!\!2\diagup\!\!\!2\diagup\!\!\!2...\\\sf k=2,\diagup\!\!\!2\diagup\!\!\!2\diagup\!\!\!2... \end{cases}}\\\sf 9k=20\\\huge\boxed{\boxed{\boxed{\boxed{\sf k=\dfrac{20}{9}}}}}\blue{\checkmark}$

$\tt d)~\sf p=1,125125...\cdot1000\\\sf 1000p=1125,125125...\\-\underline{\begin{cases}\sf1000p=1125,\diagup\!\!\!1~\diagup\!\!\!2~\diagup\!\!\!5~\diagup\!\!\!1~\diagup\!\!\!2~\diagup\!\!\!5...\\\sf p=1,\diagup\!\!\!1~\diagup\!\!\!2~\diagup\!\!\!5~\diagup\!\!\!1~\diagup\!\!\!2~\diagup\!\!\!5... \end{cases}}\\\sf999p=1124\\\huge\boxed{\boxed{\boxed{\boxed{\sf p=\dfrac{1124}{999}}}}}\blue{\checkmark}$