Question

Dada a dízima periódica, diga de qual é a fração geratriz: *

a) 0,44444... b) 0,12525... c) 0,54545... d) 0,04777...

Answer 1

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☞ a) 4/9; b) 124/900; c) 54/99; d) 43/900 ✅

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$\green{\rm\underline{EXPLICAC_{\!\!\!,}\tilde{A}O~PASSO{-}A{-}PASSO~~~}}$✍

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☔ Oi, Nycollas. Chamando nossa dízima de X podemos agora considerar o seguinte algoritmo para encontrar a sua fração geratriz:

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1. Identificar qual é o período;
2. Multiplicar o nosso número X por uma potência de $\sf 10^m$ de forma que 1 único período da dízima fique do lado esquerdo da vírgula;
3. Subtrair pelo nosso número X multiplicado por uma potência de $\sf 10^n$ de forma que a dízima esteja exatamente à direita da vírgula;
4. Igualar a subtração à $\sf (10^m - 10^n) \cdot x$;
5. Substituir os valores de X na esquerda da igualdade e encontrar o valor de X da direita da igualdade.

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Ⓐ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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$\LARGE\gray{\boxed{\sf\blue{~~0,\overline{4}~~}}}$

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$\LARGE\blue{\sf 1.~~~~~ P = 4 }$

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$\LARGE\blue{\sf 2.~~~~~ x \cdot 10^1 = 4,\overline{4} }$

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$\LARGE\blue{\sf 3.~~~~~ x \cdot 10^0 = 0,\overline{4} }$

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$\LARGE\blue{\sf 4.~~~~~ (10^1 - 10^0) \cdot x = 9x }$

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$\large\blue{\sf 5.~~~~~ x \cdot 10^1 - x \cdot 10^0 = (10^1 - 10^0) \cdot x }$

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⠀ ⠀  $\Large\blue{\sf 4,\overline{4} - 0,\overline{4} = 9x }$

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⠀ ⠀  $\Large\blue{\sf 4 + 0,\overline{4} - 0,\overline{4} = 9x }$

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⠀ ⠀  $\Large\blue{\sf 4 = 9x }$

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⠀ ⠀  $\Large\blue{\sf x = \dfrac{4}{9} }$

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$\LARGE\green{\boxed{\rm~~~\red{ A)}~\gray{0,\overline{4}}~\pink{=}~\blue{ \dfrac{4}{9} }~~~}}$ ✅

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Ⓑ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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$\LARGE\gray{\boxed{\sf\blue{~~0,1\overline{25}~~}}}$

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$\large\blue{\sf x \cdot 10^3 - x \cdot 10^1 = (10^3 - 10^1) \cdot x }$

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⠀ ⠀  $\Large\blue{\sf 125,\overline{25} - 1,\overline{25} = 990x }$

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⠀ ⠀  $\Large\blue{\sf 124 + 1,\overline{25} - 1,\overline{25} = 990x }$

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⠀ ⠀  $\Large\blue{\sf 124 = 990x }$

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⠀ ⠀  $\Large\blue{\sf x = \dfrac{124}{990} }$

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$\LARGE\green{\boxed{\rm~~~\red{ B)}~\gray{0,1\overline{25}}~\pink{=}~\blue{ \dfrac{124}{990} }~~~}}$ ✅

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Ⓒ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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$\LARGE\gray{\boxed{\sf\blue{~~0,\overline{54}~~}}}$

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$\large\blue{\sf x \cdot 10^2 - x \cdot 10^0 = (10^2 - 10^0) \cdot x }$

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⠀ ⠀  $\Large\blue{\sf 54,\overline{54} - 0,\overline{54} = 99x }$

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⠀ ⠀  $\Large\blue{\sf 54 + 0,\overline{54} - 0,\overline{54} = 99x }$

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⠀ ⠀  $\Large\blue{\sf 54 = 99x }$

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⠀ ⠀  $\Large\blue{\sf x = \dfrac{54}{99} }$

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$\LARGE\green{\boxed{\rm~~~\red{ C)}~\gray{0,\overline{54}}~\pink{=}~\blue{ \dfrac{54}{99} }~~~}}$ ✅

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Ⓓ$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad}}$✍

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$\LARGE\gray{\boxed{\sf\blue{~~0,04\overline{7}~~}}}$

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$\large\blue{\sf x \cdot 10^3 - x \cdot 10^2 = (10^3 - 10^2) \cdot x }$

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⠀ ⠀  $\Large\blue{\sf 47,\overline{7} - 4,\overline{7} = 900x }$

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⠀ ⠀  $\Large\blue{\sf 43 + 4,\overline{7} - 4,\overline{7} = 900x }$

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⠀ ⠀  $\Large\blue{\sf 43 = 900x }$

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⠀ ⠀  $\Large\blue{\sf x = \dfrac{43}{900} }$

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$\LARGE\green{\boxed{\rm~~~\red{ D)}~\gray{0,04\overline{7}}~\pink{=}~\blue{ \dfrac{43}{900} }~~~}}$ ✅

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$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}$☁

☕ $\bf\Large\blue{Bons\ estudos.}$

($\orange{D\acute{u}vidas\ nos\ coment\acute{a}rios}$) ☄

$\bf\large\red{\underline{\qquad \qquad \qquad \qquad \qquad \qquad \quad }}\LaTeX$✍

❄☃ $\sf(\gray{+}~\red{cores}~\blue{com}~\pink{o}~\orange{App}~\green{Brainly})$ ☘☀

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$\gray{"Absque~sudore~et~labore~nullum~opus~perfectum~est."}$