Dezimalentwicklung/Periodisch/Summe/Aufgabe/Lösung

Es ist

${}{\begin{aligned}0{,}2{\overline {13}}&={\frac {2}{10}}+0{,}0{\overline {13}}\\&={\frac {2}{10}}+{\frac {1}{10}}\cdot 0{,}{\overline {13}}\\&={\frac {2}{10}}+{\frac {1}{10}}\cdot 0{,}{\overline {131313}}\\&={\frac {2}{10}}+{\frac {1}{10}}\cdot {\frac {131313}{999999}}\end{aligned}}$

und

${}{\begin{aligned}0{,}{\overline {127}}&=0{,}{\overline {127127}}\\&={\frac {127127}{999999}}.\end{aligned}}$

Somit ist

${}{\begin{aligned}0{,}2{\overline {13}}+0{,}{\overline {127}}&={\frac {2}{10}}+{\frac {1}{10}}\cdot {\frac {131313}{999999}}+{\frac {127127}{999999}}\\&={\frac {2}{10}}+{\frac {1}{10}}\cdot {\left({\frac {131313}{999999}}+{\frac {1271270}{999999}}\right)}\\&={\frac {2}{10}}+{\frac {1}{10}}\cdot {\frac {1402583}{999999}}\\&={\frac {2}{10}}+{\frac {1}{10}}+{\frac {1}{10}}\cdot {\frac {402584}{999999}}\\&={\frac {3}{10}}+{\frac {1}{10}}\cdot 0{,}{\overline {402584}}\\&=0{,}3{\overline {402584}}.\end{aligned}}$

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