((8 1/7+6 1/8)+7 1/9)+6 1/2+7/8+(3 5/8-5/9)

Und so dürfte es am einfachsten gehen:

$${\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{7}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{9}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{7}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{5}}}{{\mathtt{8}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$  =

= $${\mathtt{30}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{7}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{13}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{4}}}{{\mathtt{9}}}}$$           (  $${\frac{{\mathtt{1}}}{{\mathtt{2}}}} = {\frac{{\mathtt{4}}}{{\mathtt{8}}}}$$     ;   $${\frac{{\mathtt{13}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\mathtt{8}}}} = {\frac{{\mathtt{17}}}{{\mathtt{8}}}}$$ )

=  $${\mathtt{30}}{\mathtt{\,\small\textbf+\,}}{\frac{\left({\mathtt{72}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1\,071}}{\mathtt{\,-\,}}{\mathtt{224}}\right)}{{\mathtt{504}}}}$$    =  

$${\mathtt{30}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{919}}}{{\mathtt{504}}}} = {\mathtt{31.823\: \!412\: \!698\: \!412\: \!698\: \!4}}$$

=  $$31\frac{415}{504}$$

Gruß radix !

$${\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{7}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{9}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{7}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{5}}}{{\mathtt{8}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{9}}}} = {\mathtt{31.823\: \!412\: \!698\: \!412\: \!698\: \!4}}$$

$${\frac{{\mathtt{1}}}{{\mathtt{7}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{9}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{7}}}{{\mathtt{8}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{5}}}{{\mathtt{8}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{9}}}}$$          Hauptnenner = 504  

  =>   $${\frac{{\mathtt{919}}}{{\mathtt{504}}}} = {\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{415}}}{{\mathtt{504}}}}$$

$${\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{415}}}{{\mathtt{504}}}} = {\frac{{\mathtt{919}}}{{\mathtt{504}}}} = {\mathtt{1.823\: \!412\: \!698\: \!412\: \!698\: \!4}}$$

$${\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{30}}$$              ; 

  $${\mathtt{30}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{415}}}{{\mathtt{504}}}} = {\mathtt{31.823\: \!412\: \!698\: \!412\: \!698\: \!4}}$$   

   Ergebnis:     $$31\frac{415}{504}$$

Gruß radix !

Das geht mit dem Hauptnenner  (7  * 8 * 9 = 504 ) relativ einfach :

Siehe unten! 

Gruß radix !

Berechnen der Brüche:

Gruß radix !