Mathematrix: Aufgabenbeispiele/ Grund und Bruchrechnungen Große Kombination

$({2{\frac {109}{120}}}+{\dfrac {\ {\frac {2}{3}}-{\frac {3}{4}}\ }{{\frac {3}{5}}-{\frac {3}{4}}}}-{3{\frac {5}{24}}}):{\frac {46}{45}}+({\frac {12}{7}}-{\frac {3}{8}})\cdot ({\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}})$

• Erste Klammer:
  • Mathematrix: Aufgabenbeispiele/ Doppelbrüche:
    
    ${\dfrac {\ {\frac {2}{3}}-{\frac {3}{4}}\ }{{\frac {3}{5}}-{\frac {3}{4}}}}={\dfrac {\ {\frac {2}{3}}{\underline {\xcancel {-}}}{\frac {3}{4}}\ }{{\frac {3}{5}}{\underline {\xcancel {-}}}{\frac {3}{4}}}}={\dfrac {\ \ {\frac {2\cdot 4-3\cdot 3\ \ }{3\cdot 4}}}{\frac {3\cdot 4-5\cdot 3}{5\cdot 4}}}={\dfrac {\ \ {\frac {-1}{12}}\ \ }{\frac {-3}{20}}}={\frac {-1}{12}}:{\frac {-3}{20}}={\frac {-1}{\cancelto {3}{12}}}\cdot {\frac {\cancelto {5}{20}}{-3}}={\frac {5}{9}}$
  • Gemischte Zahlen:
    
    $2{\frac {109}{120}}={\frac {2\cdot 120+109}{120}}={\frac {349}{120}}\qquad 3{\frac {5}{24}}={\frac {3\cdot 24+5}{24}}={\frac {77}{24}}$
  • kgV mit PFZ finden
    
    ${\begin{array}{ll}120&=2\cdot 2\cdot 2\cdot 3\cdot 5\\\hline 9&=3\cdot 3\\\hline 24&=2\cdot 2\cdot 2\cdot 3\\\hline \hline kgV&=2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5\end{array}}$
    
    Also $\textstyle {\frac {349}{120}}$ wird mit $\ 360:120=3$,$\textstyle {\frac {5}{9}}$ mit $\ 360:9=40$ und $\textstyle {\frac {77}{24}}$ mit $\ 360:24=15$ erweitert.
  • Brüche zusammenrechnen
    
    $({\cancelto {\frac {349}{120}}{2{\frac {109}{120}}}}+{\cancelto {\frac {5}{9}}{\dfrac {\ {\frac {2}{3}}-{\frac {3}{4}}\ }{{\frac {3}{5}}-{\frac {3}{4}}}}}-{\cancelto {\frac {77}{24}}{3{\frac {5}{24}}}})={\overset {3}{\frac {\breve {349}}{120}}}+{\overset {40}{\frac {\breve {5}}{9}}}-{\overset {15}{\frac {\breve {77}}{24}}}={\frac {1047+200-1155}{360}}={\frac {92}{360}}={\frac {23}{90}}$
• Zweite Klammer

${\frac {12}{7}}-{\frac {3}{8}}={\frac {12}{7}}{\underline {\xcancel {-}}}{\frac {3}{8}}={\frac {12\cdot 8-7\cdot 3\ \ }{7\cdot 8}}={\frac {75}{56}}$

• Dritte Klammer

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

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

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

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

• Jetzt Klammer ersetzten

${\cancelto {\frac {23}{90}}{({2{\frac {109}{120}}}+{\dfrac {\ {\frac {2}{3}}-{\frac {3}{4}}\ }{{\frac {3}{5}}-{\frac {3}{4}}}}-{3{\frac {5}{24}}})}}:{\frac {46}{45}}+{\cancelto {\frac {75}{56}}{({\frac {12}{7}}-{\frac {3}{8}})}}\cdot {\cancelto {-{\frac {14}{15}}}{({\frac {2}{5}}-{\frac {16}{21}}:{\frac {4}{7}})}}=$

$={\frac {23}{90}}:{\frac {46}{45}}+{\frac {75}{56}}\cdot (-{\frac {14}{15}})={\frac {\cancelto {1}{23}}{\cancelto {2}{90}}}\cdot {\frac {\cancelto {1}{45}}{\cancelto {2}{46}}}+{\frac {\cancelto {5}{75}}{\cancelto {4}{56}}}\cdot (-{\frac {\cancelto {1}{14}}{\cancelto {1}{15}}})={\frac {1}{4}}-{\frac {5}{4}}=-{\frac {4}{4}}=-1$