Question

0,121212... • 0,333...???

Answer 1

x = 0,121212
100x = 12,121212
100x - x = 12 - 0
99x = 12
x = 12/99

x = 0,333
10x = 3,333
10x - x = 3 - 0
9x = 3
x = 3/9


12/99 . 3/9 = 36/891

Answer 2

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Primeiro vamos encontrar as frações geratrizes das dízimas periódicas envolvidas:
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$\mathsf{\bullet~~x=0,121212\ldots~~~~~~\mathbf{(i)}}$


Multiplicando os dois lados por 100, temos

$\mathsf{100x=12,121212\ldots~~~~~~\mathbf{(ii)}}$


Subtraindo $\mathbf{(ii)}-\mathbf{(i)},$ temos

$\mathsf{100x-x=12,121212\ldots-0,121212\ldots}\\\\ \mathsf{100x-x=12}\\\\ \mathsf{99x=12}\\\\ \mathsf{x=\dfrac{12}{99}\begin{array}{c}^{\mathsf{\div 3}}\\ ^{\mathsf{\div 3}} \end{array}}\\\\\\ \mathsf{x=\dfrac{4}{33}}\\\\\\ \therefore~~\boxed{\begin{array}{c}\mathsf{0,121212\ldots=\dfrac{4}{33}} \end{array}}$

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$\mathsf{\bullet~~y=0,333\ldots~~~~~~\mathbf{(iii)}}$


Multiplicando os dois lados por 10, temos

$\mathsf{10y=3,333\ldots~~~~~~\mathbf{(iv)}}$


Subtraindo $\mathbf{(iv)}-\mathbf{(iii)},$ temos

$\mathsf{10y-y=3,333\ldots-0,333\ldots}\\\\ \mathsf{10y-y=3}\\\\ \mathsf{9y=3}\\\\ \mathsf{y=\dfrac{3}{9}\begin{array}{c}^{\mathsf{\div 3}}\\^{\mathsf{\div 3}} \end{array}}\\\\\\ \mathsf{y=\dfrac{1}{3}}\\\\\\ \therefore~~\boxed{\begin{array}{c}\mathsf{0,333\ldots=\dfrac{1}{3}} \end{array}}$

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Dessa forma, temos que

$\mathsf{0,121212\ldots~\cdot~ 0,333\ldots}\\\\ \mathsf{=\dfrac{4}{33}\cdot \dfrac{1}{3}}\\\\\\ \mathsf{=\dfrac{4}{99}}\\\\\\ \mathsf{=0,040404\ldots}$