Question

transforme em fração irredutível a dízimas periódicas 0.3444?​

Answer 1

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$\boxed{\begin{array}{l}\underline{\rm por~equac_{\!\!,}\tilde ao\!:}\\\sf x=0,3444...\cdot10\\\sf 10x=3,444....\cdot10\\\sf 100x=34,444...\\-\underline{\begin{cases}\sf 100x=34,444...\\\sf 10x=3,444...\end{cases}}\\\sf 90x=31\\\sf x=\dfrac{31}{90}\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm pela~soma~dos~termos~da~PG~infinita\!:}\\\sf 0,3444...=0,3+\underbrace{0,044...+0,00044...}_{\sf soma~dos~termos~da~PG~infinita}\\\sf a_1=0,044=\dfrac{44}{1000}~~a_2=0,00044=\dfrac{44}{100000}\\\sf q=\dfrac{a_2}{a_1}\\\sf q=\dfrac{\frac{44}{100000}}{\frac{44}{1000}}\\\sf q=\dfrac{\diagdown\!\!\!\!\!44}{100\backslash\!\!\!0\backslash\!\!\!0\backslash\!\!\!0}\cdot\dfrac{1\backslash\!\!\!0\backslash\!\!\!0\backslash\!\!\!0}{\diagdown\!\!\!\!44}=\dfrac{1}{100}\end{array}}$

$\boxed{\begin{array}{l}\sf S_n=\dfrac{a_1}{1-q}\\\sf S_n=\dfrac{\frac{44}{100}}{1-\frac{1}{100}}\\\sf S_n=\dfrac{\frac{44}{1000}}{\frac{99}{100}}\\\sf S_n=\dfrac{\diagdown\!\!\!\!\!\!44^4}{10\diagdown\!\!\!\!\!\!00}\cdot\dfrac{1\diagdown\!\!\!\!\!\!00}{\diagdown\!\!\!\!\!\!99_9}\\\sf S_n=\dfrac{4\div2}{90\div2}=\dfrac{2}{45}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf0,3444...=0,3+\dfrac{2}{45}\\\sf 0,3444...=\dfrac{3}{10}+\dfrac{2}{45}=\dfrac{27+4}{90}\\\\\sf 0,3444...=\dfrac{31}{90}\end{array}}$