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# Mathematrix: Aufgabenbeispiele/ Bruchrechnungen und Vorrang

${\displaystyle {\frac {2}{3}}+\left({\frac {6}{7}}-{\frac {9}{7}}\right)\cdot \left({\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}}\right)}$

• Erste Klammer

${\displaystyle {\frac {6}{7}}-{\frac {9}{7}}={\frac {6-9}{7}}=-{\frac {3}{7}}}$

• Zweite Klammer

${\displaystyle {\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}}}$

${\displaystyle {\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}}={\frac {2}{5}}-{\frac {16}{21}}\cdot {\frac {7}{4}}={\frac {2}{5}}-{\frac {16\cdot 7}{21\cdot 4}}}$

${\displaystyle {\frac {2}{5}}-{\frac {{\cancelto {4}{16}}\cdot 7}{21\cdot {\cancelto {1}{4}}}}={\frac {2}{5}}-{\frac {4\cdot {\cancelto {1}{7}}}{{\cancelto {3}{21}}\cdot 1}}={\frac {2}{5}}-{\frac {4}{3}}}$

${\displaystyle \textstyle \left(^{\text{Vorgang: }}{\frac {2}{5}}{\underline {\xcancel {-}}}{\frac {4}{3}}\right)\quad }$${\displaystyle ={\frac {2\cdot 3-4\cdot 5}{5\cdot 3}}={\frac {6-20}{15}}=-{\frac {14}{15}}}$

• Jetzt Klammer ersetzten

${\displaystyle {\frac {2}{3}}+{\cancelto {-{\frac {3}{7}}}{\left({\frac {6}{7}}-{\frac {9}{7}}\right)}}\cdot {\cancelto {-{\frac {14}{15}}}{\left({\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}}\right)}}=}$

${\displaystyle {\frac {2}{3}}+\left(-{\frac {3}{7}}\right)\cdot \left(-{\frac {14}{15}}\right)}$

${\displaystyle {\frac {2}{3}}+{\frac {{\cancelto {1}{3}}\cdot {\cancelto {2}{14}}}{{\cancelto {1}{7}}\cdot {\cancelto {5}{15}}}}}$${\displaystyle ^{\text{(hier erst kürzen)}}={\frac {2}{3}}+{\frac {1\cdot 2}{1\cdot 5}}={\frac {2}{3}}+{\frac {2}{5}}=}$

${\displaystyle {\frac {2\cdot 5+2\cdot 3}{3\cdot 5}}={\frac {16}{15}}}$