MathemaTriX ⋅ Aufgabenheft Antworten

${\displaystyle \ }$

${\displaystyle 24}$ ${\displaystyle 9}$

${\displaystyle \ }$

${\displaystyle -9}$ ${\displaystyle 0}$

${\displaystyle \ }$

${\displaystyle -27}$ ${\displaystyle 7}$

${\displaystyle \ }$

${\displaystyle 49}$ ${\displaystyle 8}$

${\displaystyle \ }$

${\displaystyle 10}$ ${\displaystyle 21}$

${\displaystyle \ }$

${\displaystyle 98}$ ${\displaystyle 15}$

${\displaystyle \ }$

${\displaystyle -27}$ ${\displaystyle 13}$

${\displaystyle \ }$

${\displaystyle -14}$ ${\displaystyle -1}$

${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {3}{13}}\qquad }$ ${\displaystyle \textstyle \ {\frac {23}{12}}\qquad }$ ${\displaystyle \textstyle \ {\frac {15}{28}}\qquad }$ ${\displaystyle \textstyle \ {\frac {35}{12}}\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ -{\frac {57}{44}}\qquad }$ ${\displaystyle \textstyle \ {\frac {35}{44}}\qquad }$ ${\displaystyle \textstyle \ -{\frac {12}{11}}\qquad }$ ${\displaystyle \textstyle \ {\frac {20}{77}}\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {63}{55}}\qquad }$ ${\displaystyle \textstyle \ -{\frac {8}{45}}\qquad }$ ${\displaystyle \textstyle \ {\frac {77}{45}}\qquad }$ ${\displaystyle \textstyle \ {\frac {18}{9}}(=2)\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {143}{24}}\qquad }$ ${\displaystyle \textstyle \ {\frac {119}{39}}\qquad }$ ${\displaystyle \textstyle \ -{\frac {13}{13}}(=-1)\qquad }$ ${\displaystyle \textstyle \ {\frac {88}{39}}\qquad }$
${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {4}{7}}\qquad }$ ${\displaystyle \textstyle \ {\frac {60}{77}}\qquad }$ ${\displaystyle \textstyle \ {\frac {137}{28}}\qquad }$ ${\displaystyle \textstyle \ {\frac {165}{28}}\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {26}{13}}(=2)\qquad }$ ${\displaystyle \textstyle \ {\frac {165}{104}}\qquad }$ ${\displaystyle \textstyle \ {\frac {143}{120}}\qquad }$ ${\displaystyle \textstyle \ {\frac {23}{104}}\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ {\frac {3}{13}}\qquad }$ ${\displaystyle \textstyle \ {\frac {23}{20}}\qquad }$ ${\displaystyle \textstyle \ {\frac {28}{15}}\qquad }$ ${\displaystyle \textstyle \ {\frac {12}{35}}\qquad }$	${\displaystyle \ }$	${\displaystyle \textstyle \ -{\frac {57}{55}}\qquad }$ ${\displaystyle \textstyle \ {\frac {44}{35}}\qquad }$ ${\displaystyle \textstyle \ -{\frac {12}{11}}\qquad }$ ${\displaystyle \textstyle \ {\frac {77}{20}}\qquad }$

${\displaystyle \ }$	${\displaystyle \ x\approx 23170\%\quad }$ ${\displaystyle \ x=1225{,}67\ kg\quad }$ ${\displaystyle \ x\approx 0{,}432\ {\text{kg}}\quad }$	${\displaystyle \ }$	${\displaystyle \ x\approx 61786\ {\text{V}}\quad }$ ${\displaystyle \ x\approx 0{,}162\%\quad }$ ${\displaystyle \ x\approx 1{,}94\ {\text{V}}\quad }$	${\displaystyle \ }$	${\displaystyle \ x\approx 58{,}82\%\quad }$ ${\displaystyle \ x=170\ {\text{h}}\quad }$ ${\displaystyle \ x=1{,}7\ {\text{h}}\quad }$	${\displaystyle \ }$	${\displaystyle \ x\approx 0{,}0026\ {\text{h}}\quad }$ ${\displaystyle \ x\approx 3830000\%\quad \ }$ ${\displaystyle x\approx 11{,}1\ {\text{h}}\quad }$
${\displaystyle \ }$	${\displaystyle \ x=40\ {\text{h}}\quad }$ ${\displaystyle \ x=625000\%\quad \ }$ ${\displaystyle x=0{,}016\ {\text{h}}\quad }$	${\displaystyle \ }$	${\displaystyle \ x=180000\ {\text{h}}\quad }$ ${\displaystyle \ x=50\ {\text{h}}\quad \ }$ ${\displaystyle x=200\%\quad }$	${\displaystyle \ }$	${\displaystyle \ x=125000\ {\text{Volt}}\quad }$ ${\displaystyle \ x=0{,}08\%\quad \ }$ ${\displaystyle x=0{,}98\ {\text{Volt}}\quad }$	${\displaystyle \ }$	${\displaystyle \ x=250\%\quad }$ ${\displaystyle \ x=40\ {\text{h}}\quad \ }$ ${\displaystyle x=7{,}225\ {\text{h}}\quad }$

${\displaystyle \ }$	${\displaystyle \ x=0{,}0224\ \ l\quad }$ ${\displaystyle \ x\approx 14344\ {\text{ min}}}$	${\displaystyle \ }$	${\displaystyle \ x=0{,}0735\ \ kg\quad }$ ${\displaystyle \ x\approx 916{,}7\ l}$	${\displaystyle \ }$	1980 € 132000 Flaschen	${\displaystyle \ }$	${\displaystyle \ x\approx 0{,}125\ \ t\quad }$ ${\displaystyle \ x=7,3\ Tage}$
${\displaystyle \ }$	${\displaystyle \ x\approx 26717\ \ Liter\quad }$	${\displaystyle \ }$	${\displaystyle \ x=92{,}25\ \ t\quad }$ ${\displaystyle \ x=20\ {\text{Kühe}}}$	${\displaystyle \ }$	${\displaystyle \ x=115\ {\text{Menschen}}}$ ${\displaystyle \ x=17{,}5\ \ km^{2}\quad }$	${\displaystyle \ }$	${\displaystyle \ x=320\ m}$ ${\displaystyle \ x=28\ \ g\quad }$

${\displaystyle \ }$	${\displaystyle \ 6x^{7}-14x^{2}+10x^{3}}$ ${\displaystyle \ 6m^{5}+8m^{3}+15m^{7}+20m^{5}=26m^{5}+8m^{3}+15m^{7}}$	${\displaystyle \ }$	${\displaystyle \ 14b^{9}+8b^{8}-14b^{7}}$ ${\displaystyle \ 10w^{7}+8w^{5}+25w^{5}+20w^{3}=10w^{7}+33w^{5}+20w^{3}}$
${\displaystyle \ }$	${\displaystyle \ 8s^{8}+4s^{9}-28s^{7}}$ ${\displaystyle \ 15w^{8}+12w^{6}+35w^{6}+28w^{4}=15w^{8}+47w^{6}+28w^{4}}$	${\displaystyle \ }$	${\displaystyle \ 28v^{11}+12v^{12}-8v^{8}}$ ${\displaystyle \ 10g^{7}+8g^{6}+15g^{5}+12g^{4}}$
${\displaystyle \ }$	${\displaystyle 14n^{1}4-14n^{9}+35n^{7}}$ ${\displaystyle 15c^{5}+20c^{4}+18c^{4}+24c^{3}=15c^{5}+38c^{4}+24c^{3}}$	${\displaystyle \ }$	${\displaystyle 6z^{6}+12z^{7}-21z^{5}}$ ${\displaystyle 8p^{5}+10p^{7}+12p^{3}+15p^{5}=23p^{5}+10p^{7}+12p^{3}}$
${\displaystyle \ }$	${\displaystyle 8s^{8}+4s^{5}-28s^{7}}$ ${\displaystyle 10w^{7}+8w^{6}+5w^{5}+4w^{4}}$	${\displaystyle \ }$	${\displaystyle 4v^{5}+12v^{6}-8v^{2}}$ ${\displaystyle 10a^{7}+8a^{5}+15a^{5}+12a^{4}}$

Typ 1

${\displaystyle \ }$

3 42 41 −4

${\displaystyle \ }$

5 21 19 −42

${\displaystyle \ }$

4 28 12 −46

${\displaystyle \ }$

33 10 -18 26

${\displaystyle \ }$

3 42 41 −4

${\displaystyle \ }$

5 21 19 −42

${\displaystyle \ }$

4 28 12 −46

${\displaystyle \ }$

33 10 -18 26

Typ 2

${\displaystyle \ }$

1C, 2A, 3B

${\displaystyle \ }$

1C, 2B, 3A

${\displaystyle \ }$

1A, 2B, 3C

${\displaystyle \ }$

1B, 2A, 3C

${\displaystyle \ }$

1C, 2A, 3B

${\displaystyle \ }$

1C, 2B, 3A

${\displaystyle \ }$

1A, 2B, 3C

${\displaystyle \ }$

1B, 2A, 3C

${\displaystyle \ }$	${\displaystyle \ 130\qquad }$ ${\displaystyle \ G=5B+8A\qquad }$ ${\displaystyle \ 790g\ \ ..\ 1048g\qquad }$	${\displaystyle \ }$	${\displaystyle \ -24\qquad }$ ${\displaystyle A=4k+7b\qquad }$ ${\displaystyle 86\ \ ..\ 92Bet.\qquad }$	${\displaystyle \ }$	${\displaystyle \ 18\qquad }$ ${\displaystyle z=6k+9g\qquad }$ ${\displaystyle 86,1s\ \ ..\ 109,8s\qquad }$	${\displaystyle \ }$	${\displaystyle \ -3,5\qquad }$ ${\displaystyle \ T=6x+9y\qquad }$ ${\displaystyle 108s\ \ ..\ 153s\qquad }$
${\displaystyle \ }$	${\displaystyle -1\qquad }$ ${\displaystyle p=3k+4m\qquad }$ ${\displaystyle 317g\ \ ..\ 397g\qquad }$	${\displaystyle \ }$	${\displaystyle 46\qquad }$ ${\displaystyle \ t=15a+4b\qquad }$ ${\displaystyle 70,5s\ \ ..\ 68s\qquad }$	${\displaystyle \ }$	${\displaystyle \ 1\qquad }$ ${\displaystyle \ s=A+B\qquad }$ ${\displaystyle 594\ \ ..\ 612\ Sitzpl.\qquad }$	${\displaystyle \ }$	${\displaystyle \ 23\qquad }$ ${\displaystyle \ z=10k+6h\qquad }$ ${\displaystyle 100s\ \ ..\ 98s\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {40}{9}}\qquad }$${\displaystyle \ {\tfrac {120}{13}}\qquad }$${\displaystyle \ {\tfrac {12}{7}}}$	${\displaystyle \ 35{\tfrac {7}{11}}\qquad }$${\displaystyle \ 7\qquad }$${\displaystyle \ 11{\tfrac {1}{12}}}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ {\tfrac {11}{9}}\qquad }$${\displaystyle \ 8{\tfrac {10}{13}}\qquad }$${\displaystyle \ -1{\tfrac {1}{7}}\qquad }$${\displaystyle \ 3{\tfrac {3}{5}}}$		${\displaystyle \textstyle \ {\frac {30}{3}}=10\qquad }$ ${\displaystyle \textstyle \ {\frac {1275}{12}}=106,25\qquad }$ ${\displaystyle \textstyle \ {\frac {63}{7}}=9\qquad }$ ${\displaystyle \textstyle \ {\frac {6}{13}}\approx 0,46\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {53}{7}}\qquad }$${\displaystyle \ {\tfrac {9}{5}}\qquad }$${\displaystyle \ {\tfrac {35}{3}}}$	${\displaystyle \ 37{\tfrac {3}{12}}\qquad }$${\displaystyle \ 6{\tfrac {3}{5}}\qquad }$${\displaystyle \ 11}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ 3{\tfrac {4}{7}}\qquad }$${\displaystyle \ {\tfrac {1}{5}}\qquad }$${\displaystyle \ {\tfrac {11}{3}}\qquad }$${\displaystyle \ -2{\tfrac {1}{4}}}$		${\displaystyle \textstyle \ {\frac {14}{20}}=0,7\qquad }$ ${\displaystyle \textstyle \ {\frac {1515}{24}}=63,125\qquad }$ ${\displaystyle \textstyle \ {\frac {231}{11}}=21\qquad }$ ${\displaystyle \textstyle \ {\frac {156}{13}}=12\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {11}{3}}\qquad }$${\displaystyle \ {\tfrac {76}{11}}\qquad }$${\displaystyle \ {\tfrac {17}{9}}}$	${\displaystyle \ 34\qquad }$${\displaystyle \ 1{\tfrac {4}{5}}\qquad }$${\displaystyle \ 12{\tfrac {1}{8}}}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ -5{\tfrac {1}{3}}\qquad }$${\displaystyle \ {\tfrac {56}{11}}\qquad }$${\displaystyle \ {\tfrac {1}{9}}\qquad }$${\displaystyle \ -4{\tfrac {8}{13}}}$		${\displaystyle \textstyle \ {\frac {3}{4}}=0,75\qquad }$ ${\displaystyle \textstyle \ {\frac {924}{8}}=115,5\qquad }$ ${\displaystyle \textstyle \ {\frac {56}{7}}=8\qquad }$ ${\displaystyle \textstyle \ {\frac {156}{4}}=39\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {37}{7}}\qquad }$${\displaystyle \ {\tfrac {7}{6}}\qquad }$${\displaystyle \ {\tfrac {89}{9}}}$	${\displaystyle \ 40{\tfrac {7}{11}}\qquad }$${\displaystyle \ 1{\tfrac {2}{7}}\qquad }$${\displaystyle \ 7}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ -{\tfrac {17}{7}}\qquad }$${\displaystyle \ {\tfrac {5}{6}}\qquad }$${\displaystyle \ {\tfrac {53}{9}}\qquad }$${\displaystyle \ -{\tfrac {1}{17}}}$		${\displaystyle \textstyle \ {\frac {60}{5}}=12\qquad }$ ${\displaystyle \textstyle \ {\frac {12}{27}}=0,{\dot {4}}\qquad }$ ${\displaystyle \textstyle \ {\frac {105}{6}}=17,5\qquad }$ ${\displaystyle \textstyle \ {\frac {627}{11}}=57\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {37}{9}}\qquad }$${\displaystyle \ {\tfrac {111}{13}}\qquad }$${\displaystyle \ {\tfrac {36}{7}}}$	${\displaystyle \ 35{\tfrac {4}{11}}\qquad }$${\displaystyle \ 9\qquad }$${\displaystyle \ 11{\tfrac {3}{12}}}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ -{\tfrac {2}{9}}\qquad }$${\displaystyle \ -{\tfrac {1}{8}}\qquad }$${\displaystyle \ {\tfrac {34}{7}}\qquad }$${\displaystyle \ -{\tfrac {6}{7}}}$		${\displaystyle \textstyle \ {\frac {510}{5}}=102\qquad }$ ${\displaystyle \textstyle \ {\frac {3}{5}}=0,6\qquad }$ ${\displaystyle \textstyle \ {\frac {66}{11}}=6\qquad }$ ${\displaystyle \textstyle \ {\frac {35}{12}}=2,91{\dot {6}}\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {60}{7}}\qquad }$${\displaystyle \ {\tfrac {19}{5}}\qquad }$${\displaystyle \ {\tfrac {32}{3}}}$	${\displaystyle \ 37{\tfrac {7}{12}}\qquad }$${\displaystyle \ 7{\tfrac {4}{5}}\qquad }$${\displaystyle \ 11}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ {\tfrac {52}{7}}\qquad }$${\displaystyle \ -1{\tfrac {4}{5}}\qquad }$${\displaystyle \ {\tfrac {9}{11}}\qquad }$${\displaystyle \ -3{\tfrac {5}{14}}}$		${\displaystyle \textstyle \ {\frac {4}{5}}=0,8\qquad }$ ${\displaystyle \textstyle \ {\frac {1001}{13}}=77\qquad }$ ${\displaystyle \textstyle \ {\frac {210}{7}}=30\qquad }$ ${\displaystyle \textstyle \ {\frac {21}{10}}=2,1\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {5}{3}}\qquad }$${\displaystyle \ {\tfrac {72}{11}}\qquad }$${\displaystyle \ {\tfrac {25}{13}}}$	${\displaystyle \ 35\qquad }$${\displaystyle \ 2{\tfrac {1}{5}}\qquad }$${\displaystyle \ 11{\tfrac {7}{8}}}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ {\tfrac {5}{7}}\qquad }$${\displaystyle \ -{\tfrac {1}{11}}\qquad }$${\displaystyle \ 4{\tfrac {1}{13}}\qquad }$${\displaystyle \ -1{\tfrac {1}{9}}}$		${\displaystyle \textstyle \ {\frac {1422}{3}}=474\qquad }$ ${\displaystyle \textstyle \ {\frac {28}{20}}=1,4\qquad }$ ${\displaystyle \textstyle \ {\frac {6}{11}}=0,{\overline {54}}\qquad }$ ${\displaystyle \textstyle \ {\frac {12}{5}}=2,4\qquad }$

${\displaystyle \ }$	${\displaystyle \qquad }$a und b: Addition:${\displaystyle \qquad \qquad }$	${\displaystyle \qquad }$c: Unechten Bruch in${\displaystyle \qquad }$ ${\displaystyle \qquad \qquad }$gemischte Zahl
	${\displaystyle \ {\tfrac {47}{7}}\qquad }$${\displaystyle \ {\tfrac {13}{12}}\qquad }$${\displaystyle \ {\tfrac {82}{9}}}$	${\displaystyle \ 37\qquad }$${\displaystyle \ 1{\tfrac {1}{7}}\qquad }$${\displaystyle \ 7{\tfrac {3}{5}}}$
${\displaystyle \qquad }$d: Subtraktion		${\displaystyle \qquad }$e: Multiplikation
${\displaystyle \ -2{\tfrac {3}{7}}\qquad }$${\displaystyle \ {\tfrac {11}{12}}\qquad }$${\displaystyle \ 8{\tfrac {1}{9}}\qquad }$${\displaystyle \ -{\tfrac {1}{3}}}$		${\displaystyle \textstyle \ {\frac {10}{16}}=0,625\qquad }$ ${\displaystyle \textstyle \ {\frac {50}{5}}=10\qquad }$ ${\displaystyle \textstyle \ {\frac {651}{7}}=93\qquad }$ ${\displaystyle \textstyle \ {\frac {78}{13}}=6\qquad }$

${\displaystyle \ }$	${\displaystyle c=-4111\qquad }$ ${\displaystyle k=55\qquad }$ ${\displaystyle f=38\qquad }$ ${\displaystyle x=1208\ }$ ${\displaystyle \textstyle m={\tfrac {214}{23}}\approx 9{,}3\qquad }$ ${\displaystyle w=19{,}{\overline {45}}}$	${\displaystyle \ }$	${\displaystyle c=1992\qquad }$ ${\displaystyle k=52\qquad }$ ${\displaystyle f=45\qquad }$ ${\displaystyle x=3983\ }$ ${\displaystyle \textstyle m=22\qquad }$ ${\displaystyle w={\tfrac {214}{13}}=16{,}{\overline {461538}}\approx 16{,}46}$	${\displaystyle \ }$	${\displaystyle c=-4011\qquad }$ ${\displaystyle k=60\qquad }$ ${\displaystyle f=56\qquad }$ ${\displaystyle x=-3654\ }$ ${\displaystyle \textstyle m={\tfrac {72}{11}}=6{,}{\overline {54}}\qquad }$ ${\displaystyle w=12{\tfrac {10}{17}}\approx 12{,}59}$	${\displaystyle \ }$	${\displaystyle c=-2603\qquad }$ ${\displaystyle k=91\qquad }$ ${\displaystyle f=28\qquad }$ ${\displaystyle x=-4834\ }$ ${\displaystyle \textstyle m={\tfrac {493}{29}}=17\qquad }$ ${\displaystyle w=17{\tfrac {11}{17}}\approx 17{,}65}$
${\displaystyle \ }$	${\displaystyle c=-4111\qquad }$ ${\displaystyle k=55\qquad }$ ${\displaystyle f=38\qquad }$ ${\displaystyle x=1208\ }$ ${\displaystyle \textstyle m={\tfrac {214}{23}}\approx 9{,}3\qquad }$ ${\displaystyle w=19{,}{\overline {45}}}$	${\displaystyle \ }$	${\displaystyle c=1992\qquad }$ ${\displaystyle k=52\qquad }$ ${\displaystyle f=45\qquad }$ ${\displaystyle x=3983\ }$ ${\displaystyle \textstyle m=22\qquad }$ ${\displaystyle w={\tfrac {214}{13}}=16{,}{\overline {461538}}\approx 16{,}46}$	${\displaystyle \ }$	${\displaystyle c=-4011\qquad }$ ${\displaystyle k=60\qquad }$ ${\displaystyle f=56\qquad }$ ${\displaystyle x=-3654\ }$ ${\displaystyle \textstyle m={\tfrac {72}{11}}=6{,}{\overline {54}}\qquad }$ ${\displaystyle w=12{\tfrac {10}{17}}\approx 12{,}59}$	${\displaystyle \ }$	${\displaystyle c=-2603\qquad }$ ${\displaystyle k=91\qquad }$ ${\displaystyle f=28\qquad }$ ${\displaystyle x=-5514\ }$ ${\displaystyle \textstyle m={\tfrac {493}{29}}=17\qquad }$ ${\displaystyle w=17{\tfrac {11}{17}}\approx 17{,}65}$

${\displaystyle \ }$	...0,04... m ...110... min ...0,8... s ...12000... cm ...950... km ...21... ${\displaystyle {\mathbf {m\ell }}}$	${\displaystyle \ }$	...0,0055... t ...150... dm ...5... dm³ ...700... s ...14,5... m² ...730000... g	${\displaystyle \ }$	...0,48... cm² ...320... min ...290... cm³ ...0,002... km ...24500... g ...4,9... t	${\displaystyle \ }$	...560... km² ...0,15... dm ...0,55... ${\displaystyle {\mathbf {\ell }}}$ ...0,002... h ...0,021... kg ...6100... s
${\displaystyle \ }$	...5... cm ...5000... s ...0,01... min ...0,15... km ...950000... m ...21... cm³	${\displaystyle \ }$	...550000... mg ...0,015... m ...5... ${\displaystyle {\mathbf {\ell }}}$ ...0,25... h ...1450... dm² ...190000... km	${\displaystyle \ }$	...48... mm² ...0,15... Tage ...320... ${\displaystyle {\mathbf {m\ell }}}$ ...2000... mm ...0,0245... t ...5430... kg	${\displaystyle \ }$	...80000... km² ...15... mm ...0,55... dm³ ...0,25... min ...24500... mg ...0,08... Tage

${\displaystyle \ }$	${\displaystyle 53700000\ cm\qquad }$ ${\displaystyle 0{,}537\ m\qquad }$ ${\displaystyle 13{,}95\ h\qquad }$ ${\displaystyle 470\ g\qquad }$ ${\displaystyle 2764{,}8\ s}$	${\displaystyle \ }$	${\displaystyle 44500\ cm\qquad }$ ${\displaystyle 4{,}45\ dm\qquad }$ ${\displaystyle 3{,}15\ Tage\qquad }$ ${\displaystyle 445\ g\qquad }$ ${\displaystyle 178{,}2\ s}$	${\displaystyle \ }$	${\displaystyle 0{,}000793\ kg\qquad }$ ${\displaystyle 79300\ cm\qquad }$ ${\displaystyle 0{,}0000793\ km\qquad }$ ${\displaystyle 0{,}793\ g\qquad }$ ${\displaystyle 1{,}8792\ h}$	${\displaystyle \ }$	${\displaystyle 0{,}000577\ m\qquad }$ ${\displaystyle 5770000\ km\qquad }$ ${\displaystyle 793000\ mg\qquad }$ ${\displaystyle 0{,}00001305\ min\qquad }$ ${\displaystyle 111{,}312\ min}$
${\displaystyle \ }$	${\displaystyle 53700000\ mm\qquad }$ ${\displaystyle 0{,}537\ km\qquad }$ ${\displaystyle 13{,}95\ min\qquad }$ ${\displaystyle 470\ mg\qquad }$ ${\displaystyle 2764{,}8\ s}$	${\displaystyle \ }$	${\displaystyle 44500\ m\qquad }$ ${\displaystyle 4{,}45\ m\qquad }$ ${\displaystyle 3{,}15\ Tage\qquad }$ ${\displaystyle 445\ mg\qquad }$ ${\displaystyle 178{,}2\ s}$	${\displaystyle \ }$	${\displaystyle 0{,}000793\ t\qquad }$ ${\displaystyle 79300\ mm\qquad }$ ${\displaystyle 0{,}0000793\ m\qquad }$ ${\displaystyle 0{,}793\ mg\qquad }$ ${\displaystyle 1{,}8792\ h}$	${\displaystyle \ }$	${\displaystyle 0{,}000577\ km\qquad }$ ${\displaystyle 5770000\ mm\qquad }$ ${\displaystyle 793000\ g\qquad }$ ${\displaystyle 0{,}00001305\ h\qquad }$ ${\displaystyle 111{,}312\ min}$

${\displaystyle \ }$	${\displaystyle \ 5,\ \ 0,\ \ 4,\ \ 9,\ \ 6,\ \ {\text{und }}8\ {\text{Pack.}}}$	${\displaystyle \ }$	${\displaystyle \ 0,\ \ 6,\ \ 2,\ \ 10,\ \ 14,\ \ {\text{und }}11\ {\text{Tische}}}$	${\displaystyle \ }$	${\displaystyle \ 2,\ \ 0,\ \ 5,\ \ 12,\ \ 13,\ \ {\text{und }}10\ {\text{T.}}}$
${\displaystyle \ }$	${\displaystyle \ 2,\ \ 6,\ \ 1,\ \ 5,\ \ 18,\ \ {\text{und }}7\ {\text{Punkte}}}$	${\displaystyle \ }$	${\displaystyle \ 0,\ \ 1,\ \ 3,\ \ 15,\ \ 9,\ \ {\text{und }}6\ {\text{Sch.}}}$	${\displaystyle \ }$	${\displaystyle \ 4,\ \ 3,\ \ 3,\ \ 8,\ \ 12,\ \ {\text{und }}10\ {\text{Autob.}}}$
${\displaystyle \ }$	${\displaystyle \ 5,\ \ 6,\ \ 2,\ \ 14,\ \ 17,\ \ {\text{und }}8\ {\text{Packungen}}}$	${\displaystyle \ }$	${\displaystyle \ 6,\ \ 6,\ \ 5,\ \ 14,\ \ 19,\ \ {\text{und }}10\ {\text{Packungen}}}$

${\displaystyle \ }$	${\displaystyle D\approx 57{,}86\ kg\qquad Med=54\ kg\qquad Mod=65\ kg}$	${\displaystyle \ }$	${\displaystyle D=58{,}75\ kg\qquad Med=58{,}5\ kg\qquad Mod=45\ und\ 65\ kg}$
${\displaystyle \ }$	${\displaystyle D=11{,}{\dot {2}}\qquad Med=5{,}5\qquad Mod=2\ und\ 7}$	${\displaystyle \ }$	DE: ${\displaystyle D=36\qquad Med=10\qquad Mod=1\ \&\ 10\quad }$ GR: ${\displaystyle D=16\qquad Med=10\qquad Mod=1}$
${\displaystyle \ }$	${\displaystyle D=6\ {\text{bzw.}}\ 21\qquad Med=1\qquad Mod=1}$	${\displaystyle \ }$	${\displaystyle D=2{,}875\qquad Med=3{,}5\qquad Mod=2\ und\ 5}$
${\displaystyle \ }$	AT: ${\displaystyle D=35{,}6\qquad Med=9\qquad Mod=2\ \&\ 10\quad }$ PO: ${\displaystyle D=16\qquad Med=11\qquad Mod=1\&\ 11}$	${\displaystyle \ }$	${\displaystyle D=12\ {\text{bzw.}}\ 42\qquad Med=2\qquad Mod=2}$

Strich- und Punktrechnungen mit ± Regel Strich ${\displaystyle \ -3}$ ${\displaystyle \ 9}$ ${\displaystyle \ 7}$ ${\displaystyle \ -3}$ ${\displaystyle -3}$ ${\displaystyle -7}$ ${\displaystyle 9}$ ${\displaystyle -7}$ Punkt ${\displaystyle \ -71}$ ${\displaystyle \ 71}$ ${\displaystyle \ 71}$ ${\displaystyle \ -71}$ ${\displaystyle -61}$ ${\displaystyle -61}$ ${\displaystyle -71}$ ${\displaystyle 61}$

Kombinationen Typ A ${\displaystyle -17}$ ${\displaystyle 73}$ ${\displaystyle 39}$ ${\displaystyle -5}$ ${\displaystyle 29}$ ${\displaystyle -5}$ ${\displaystyle -13}$ ${\displaystyle 0}$ Typ B ${\displaystyle 9}$ ${\displaystyle 0}$ ${\displaystyle -15}$ ${\displaystyle 16}$ ${\displaystyle 21}$ ${\displaystyle 0}$ ${\displaystyle -13}$ ${\displaystyle -1}$ Typ C ${\displaystyle -7}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle -1}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$

Klammer auflösen
${\displaystyle \ 6m^{4}-8m^{2}-15m^{2}+20=6m^{4}-23m^{2}+20}$ ${\displaystyle \ 10w^{7}-8w^{5}-25w^{5}+20w^{3}=10w^{7}-33w^{5}+20w^{3}}$ ${\displaystyle \ 15w^{8}-12w^{6}+5w^{6}-4w^{4}=15w^{8}-7w^{6}-4w^{4}}$ ${\displaystyle \ 10g^{7}+8g^{6}-15g^{5}-12g^{4}}$	${\displaystyle 15c^{2}-20c^{4}-18+24c^{2}=39c^{2}-20c^{4}-18}$ ${\displaystyle 8p^{5}-10p^{7}-12p^{3}+15p^{5}=23p^{5}-10p^{7}-12p^{3}}$ ${\displaystyle 10w^{7}-8w^{6}+5w^{5}-4w^{4}}$ ${\displaystyle 10a^{7}-8a^{5}+15a^{5}-12a^{4}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {34}{45}}\qquad }$ ${\displaystyle \ {\frac {91}{99}}\qquad }$ ${\displaystyle \ {\frac {77}{68}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {10}{9}}\qquad }$ ${\displaystyle \ {\frac {121}{225}}\qquad }$ ${\displaystyle \ {\frac {56}{55}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {21}{22}}\qquad }$ ${\displaystyle \ {\frac {65}{77}}\qquad }$ ${\displaystyle \ {\frac {136}{273}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {52}{35}}\qquad }$ ${\displaystyle \ {\frac {13}{33}}\qquad }$ ${\displaystyle \ {\frac {77}{45}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {17}{9}}\qquad }$ ${\displaystyle \ {\frac {455}{297}}\qquad }$ ${\displaystyle \ {\frac {385}{306}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {4}{9}}\qquad }$ ${\displaystyle \ {\frac {121}{25}}\qquad }$ ${\displaystyle \ {\frac {168}{275}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {27}{22}}\qquad }$ ${\displaystyle \ {\frac {65}{77}}\qquad }$ ${\displaystyle \ {\frac {4}{3}}}$

${\displaystyle \ }$

${\displaystyle \ {\frac {136}{245}}\qquad }$ ${\displaystyle \ {\frac {13}{33}}\qquad }$ ${\displaystyle \ {\frac {11}{5}}}$

${\displaystyle \ }$

${\displaystyle 1755\ {\text{€}}}$ ${\displaystyle 45\ {\text{€}}}$

${\displaystyle \ }$

${\displaystyle 24\ {\text{cm}}}$ ${\displaystyle 36\ {\text{cm}}}$

${\displaystyle \ }$

${\displaystyle 60\ {\text{cm}}}$ ${\displaystyle 36\ {\text{cm}}}$

${\displaystyle \ }$

${\displaystyle 685100\ {\text{€}}}$ ${\displaystyle 35100\ {\text{€}}}$

${\displaystyle \ }$

${\displaystyle 71{,}4\ {\text{kg}}}$ ${\displaystyle 3{,}4\ {\text{kg}}}$

${\displaystyle \ }$

${\displaystyle 68{,}4\ {\text{kg}}}$ ${\displaystyle 3{,}6\ {\text{kg}}}$

${\displaystyle \ }$

${\displaystyle 3{,}91\ {\text{min}}}$ ${\displaystyle 0{,}51\ {\text{min}}}$

${\displaystyle \ }$

${\displaystyle 50\ {\text{Jahre}}}$ ${\displaystyle 30\ {\text{Jahre}}}$

${\displaystyle \ }$	${\displaystyle \ u\approx 175{,}9\ cm\quad A\approx 2463\ cm^{2}\qquad }$ ${\displaystyle \ u=164\ cm\quad A=16\ dm^{2}\qquad }$ ${\displaystyle \ u=9{,}6cm\quad A\approx 4{,}43\ cm^{2}\qquad }$	${\displaystyle \ }$	${\displaystyle \ u=112\ cm\quad A=784\ cm^{2}\qquad }$ ${\displaystyle \ u=164\ cm\quad A=16\ dm^{2}\qquad }$ ${\displaystyle \ u=10{,}1cm\quad A\approx 8{,}04\ cm^{2}\qquad }$	${\displaystyle \ }$	${\displaystyle \ u\approx 25{,}1\ cm\quad A\approx 50{,}3\ cm^{2}\qquad }$ ${\displaystyle \ u=10\ dm\quad A=6\ dm^{2}\qquad }$ ${\displaystyle \ u=14{,}8\ dm\quad A=13{,}44\ dm^{2}\qquad }$
${\displaystyle \ }$	${\displaystyle \ u=58{,}4\ dm\quad A=165{,}55\ dm^{2}\qquad }$ ${\displaystyle \ u=8{,}80\ dm\quad A=6{,}16\ dm^{2}\qquad }$ ${\displaystyle \ u=114\ cm\quad A=306\ cm^{2}\qquad }$	${\displaystyle \ }$	${\displaystyle \ u\approx 88\ cm\quad A\approx 616\ cm^{2}\qquad }$ ${\displaystyle \ u=8{,}2\ dm\quad A=4\ dm^{2}\qquad }$ ${\displaystyle \ u=4{,}8\ cm\quad A\approx 1{,}11\ cm^{2}\qquad }$	${\displaystyle \ }$	${\displaystyle \ u=56\ cm\quad A=196\ cm^{2}\qquad }$ ${\displaystyle \ u=82\ cm\quad A=4\ dm^{2}\qquad }$ ${\displaystyle \ u\approx 5{,}1\ cm\quad A\approx 2\ cm^{2}\qquad }$
${\displaystyle \ }$	${\displaystyle \ u\approx 12{,}55\ cm\quad A\approx 12{,}58\ cm^{2}\qquad }$ ${\displaystyle \ u=5\ dm\quad A=1{,}5\ dm^{2}\qquad }$ ${\displaystyle \ u=7{,}4\ dm\quad A=3{,}36\ dm^{2}\qquad }$	${\displaystyle \ }$	${\displaystyle \ u=29{,}2\ dm\quad A\approx 41{,}4\ dm^{2}\qquad }$ ${\displaystyle \ u=4{,}4\ dm\quad A=1{,}54\ dm^{2}\qquad }$ ${\displaystyle \ u=57\ cm\quad A=76.5\ cm^{2}\qquad }$

${\displaystyle \ }$	${\displaystyle 53700000000\ dm^{2}\qquad }$ ${\displaystyle 0{,}000537\ dm^{3}\qquad }$ ${\displaystyle 0{,}00000537\ km^{2}\qquad }$ ${\displaystyle 537000000\ dm^{3}\qquad }$ ${\displaystyle 0{,}00032\ m^{2}}$	${\displaystyle \ }$	${\displaystyle 0{,}374\ m^{3}\qquad }$ ${\displaystyle 374000000\ mm^{3}\qquad }$ ${\displaystyle 0{,}000374\ m^{2}\qquad }$ ${\displaystyle 374\ cm^{3}\qquad }$ ${\displaystyle 0{,}00000374\ m^{2}}$	${\displaystyle \ }$	${\displaystyle 2570000\ mm^{2}\qquad }$ ${\displaystyle 0{,}000000257\ km^{3}\qquad }$ ${\displaystyle 0{,}00000257\ km^{2}\qquad }$ ${\displaystyle 0{,}257\ mm^{3}\qquad }$ ${\displaystyle 0{,}000257\ dm^{2}}$	${\displaystyle \ }$	${\displaystyle 447000\ cm^{3}\qquad }$ ${\displaystyle 0{,}00000257\ km^{2}\qquad }$ ${\displaystyle 0{,}000311\ m^{2}\qquad }$ ${\displaystyle 3{,}35\ mm^{3}}$ ${\displaystyle 25700\ mm^{3}\qquad }$
${\displaystyle \ }$	${\displaystyle 53700000000\ mm^{2}\qquad }$ ${\displaystyle 0{,}000537\ m^{3}\qquad }$ ${\displaystyle 0{,}00000537\ m^{2}\qquad }$ ${\displaystyle 537000000\ mm^{3}\qquad }$ ${\displaystyle 0{,}00032\ cm^{2}}$	${\displaystyle \ }$	${\displaystyle 0{,}374\ cm^{3}\qquad }$ ${\displaystyle 374000000\ cm^{3}\qquad }$ ${\displaystyle 0{,}000374\ km^{2}\qquad }$ ${\displaystyle 374\ mm^{3}\qquad }$ ${\displaystyle 0{,}00000374\ dm^{2}}$	${\displaystyle \ }$	${\displaystyle 2570000\ cm^{2}\qquad }$ ${\displaystyle 0{,}000000257\ m^{3}\qquad }$ ${\displaystyle 0{,}00000257\ m^{2}\qquad }$ ${\displaystyle 0{,}257\ dm^{3}\qquad }$ ${\displaystyle 0{,}000257\ cm^{2}}$	${\displaystyle \ }$	${\displaystyle 447000\ mm^{3}\qquad }$ ${\displaystyle 0{,}00000257\ m^{2}\qquad }$ ${\displaystyle 0{,}000311\ km^{2}\qquad }$ ${\displaystyle 3{,}35\ cm^{3}}$ ${\displaystyle 25700\ cm^{3}\qquad }$

${\displaystyle \ }$	360 t Kart., 280 t Tom., 189 t Gur., 11 t Karot., 420 t Getr. ${\displaystyle \textstyle {\frac {1}{3}}\ {\text{der Ernte}}}$	${\displaystyle \ }$	462 Öst., 168 Serb., 132 Türk., 162 Rest ${\displaystyle \textstyle {\frac {27}{154}}\ {\text{der SchülerInnen}}}$	${\displaystyle \ }$	${\displaystyle \textstyle {\frac {1}{98000}}\ {\text{der Menschen}}\qquad \ 3,52\ {\text{Milliarden €}}\ }$ ${\displaystyle \textstyle {\frac {1}{3675}}\ {\text{der Menschen}}\qquad \ 4\ {\text{Milliarden €}}\ }$ ${\displaystyle \textstyle {\frac {4}{11}}\ {\text{der Menschen}}\qquad \ 4,8\ {\text{Milliarden €}}\ }$ ${\displaystyle \textstyle {\text{fast}}\ {\frac {7}{11}}\ {\text{der Menschen}}\qquad \ 880\ {\text{Millionen €}}\ }$
${\displaystyle \ }$	${\displaystyle \ 3{,}76\ {\text{Mill. Liter}}\qquad \ 30080{\text{€}}\ }$ ${\displaystyle A\ 42000\ {\text{€}}\qquad B\ 35000\ {\text{€}}\ }$${\displaystyle C\ und\ D\ je\ 14000\ {\text{€}}\ }$	${\displaystyle \ }$	die Orangen ${\displaystyle 128\ kg}$
${\displaystyle \ }$	Saskia ${\displaystyle 5\ Punkte}$	${\displaystyle \ }$	die Hosen ${\displaystyle 28}$	${\displaystyle \ }$	Elektrolytkondensatoren ${\displaystyle 313\ g}$

${\displaystyle \ }$	${\displaystyle {\text{ca. }}\ 36{,}1^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}4^{\circ }C\quad \qquad }$ ${\displaystyle {\text{ca. }}1^{30}\quad 6\quad {\text{und }}22\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}0\quad 8\quad 17^{40}\quad 18\quad {\text{und }}21^{55}\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}10^{40}\quad 16^{15}\quad 19\quad {\text{und }}21^{50}\ {\text{Uhr}}\qquad }$	${\displaystyle \ }$	${\displaystyle \ {\text{ca. }}4\quad 1\quad 1\quad {\text{bzw. }}6\ ^{\circ }C\qquad }$ ${\displaystyle \ {\text{ca. }}-0{,}8\quad 4\quad 5\quad 6\quad 6{,}8\ m}$ ${\displaystyle \ {\text{ca. }}-0{,}6\quad 0{,}8\quad 1\quad {\text{bzw. }}3{,}4\ m}$ ${\displaystyle \ {\text{ca. }}-0{,}4\quad 0\quad 1{,}8\quad {\text{bzw. }}2{,}6\ m}$ ${\displaystyle \ {\text{ca. }}-1\quad {\text{bzw. }}6{,}6\ m}$	${\displaystyle \ }$	${\displaystyle \ {\text{ca. }}3\quad 2\quad 5{,}5\quad {\text{bzw. }}8\quad {\text{F/min}}\qquad }$ ${\displaystyle \ {\text{ca. um }}4^{30}\ \ 6^{00}\ \ {\text{und }}\ 11^{00}\qquad }$ ${\displaystyle \ {\text{ca. um }}10^{00}\qquad }$ ${\displaystyle \ {\text{ca. um }}\ 1^{00}\ \ 3^{30}\ \ 6^{30}\ \ {\text{und }}\ 10^{30}\qquad }$ ${\displaystyle \ {\text{ca. um }}\ 5\ {\text{und um}}\ 12\qquad }$
${\displaystyle \ }$	${\displaystyle \ {\text{ca. }}200\quad 250\quad {\text{bzw. }}280\quad {\text{ppmv}}\qquad }$ ${\displaystyle \ {\text{ca. }}400\ \ 320\ \ 240\ \ 130\ \ {\text{und }}\ 0\ }$Tausende Jahre her. ${\displaystyle \ {\text{ca. }}360\ \ 345\ \ 270\ \ 260\ \ 165\ \ 150\ \ 40\ \ {\text{und }}\ 30\ }$Tausende Jahre her.	${\displaystyle \ }$	${\displaystyle \ 2\quad 3\quad {\text{ca. }}\ 6{,}2{\text{bzw. }}{\text{ca. }}\ 3{,}4\quad m^{3}\qquad }$ ${\displaystyle \ 0\ {\text{ca. }}1{,}3\ \ 3\ {\text{und }}\ {\text{ca. }}\ 1{,}8\ s}$ ${\displaystyle \ {\text{ca. }}\ 3{,}4\ {\text{und }}\ {\text{ca. }}\ 5{,}4\ s}$ War nicht ${\displaystyle {\text{ca. }}2{,}3\ ({\text{bzw. ca. }}-0{,}3)\ s\qquad }$	${\displaystyle \ }$	${\displaystyle {\text{ca. }}\ 3^{\circ }C\quad 5^{\circ }C\quad 1^{\circ }C\quad 1^{\circ }C\quad \qquad }$ ${\displaystyle {\text{ca. }}0\quad 1{,}2\quad 5{,}6\quad {\text{und }}6\ m\ {\text{(und -0,4 m)}}\qquad }$ ${\displaystyle {\text{ca. }}1{,}6\quad 3\quad 5{,}2\quad {\text{und }}6{,}2\ m\qquad }$ ${\displaystyle {\text{ca. }}4{,}3\quad 4{,}8\quad {\text{und }}6{,}3\ m\qquad }$ ${\displaystyle {\text{ca. }}0{,}2\quad 1{\text{und }}\quad 5{,}8\ m\ {\text{(und -0,4 m)}}\qquad }$
${\displaystyle \ }$	${\displaystyle {\text{ca. }}\ 36{,}1^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}4^{\circ }C\quad \qquad }$ ${\displaystyle {\text{ca. }}1^{30}\quad 6\quad {\text{und }}22\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}0\quad 8\quad 17^{40}\quad 18\quad {\text{und }}21^{55}\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}10^{40}\quad 16^{15}\quad 19\quad {\text{und }}21^{50}\ {\text{Uhr}}\qquad }$	${\displaystyle \ }$	${\displaystyle {\text{ca. }}\ 36{,}1^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}4^{\circ }C\quad \qquad }$ ${\displaystyle {\text{ca. }}1^{30}\quad 6\quad {\text{und }}22\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}0\quad 8\quad 17^{40}\quad 18\quad {\text{und }}21^{55}\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}10^{40}\quad 16^{15}\quad 19\quad {\text{und }}21^{50}\ {\text{Uhr}}\qquad }$

50,4 € das 4-Fache

9 3

12 6

9 3

Vertiefendes Niveau 1

V1.1 Bruchstrichrechnungen mit Primfaktorzerlegung

${\displaystyle \ }$

${\displaystyle -{\frac {313}{1800}}}$

${\displaystyle \ }$

${\displaystyle {\frac {296}{60}}\left(={\frac {74}{15}}\right)}$

${\displaystyle \ }$

${\displaystyle -{\frac {232}{252}}\left(=-{\frac {58}{63}}\right)}$

${\displaystyle \ }$

${\displaystyle {\frac {649}{504}}}$

${\displaystyle \ }$

${\displaystyle -{\frac {85}{100}}\left(=-{\frac {17}{20}}\right)}$

${\displaystyle \ }$

${\displaystyle -{\frac {32}{36}}\left(=-{\frac {8}{9}}\right)}$

${\displaystyle \ }$

${\displaystyle -{\frac {37}{300}}}$

${\displaystyle \ }$

${\displaystyle {\frac {1082}{180}}\left(={\frac {541}{90}}\right)}$

V1.2 Umformen einfache Kombinationen

${\displaystyle \ }$

${\displaystyle x=5}$ 8

${\displaystyle \ }$

${\displaystyle b=-2}$ −3

${\displaystyle \ }$

${\displaystyle m=0{,}4}$ −1

${\displaystyle \ }$

${\displaystyle z=-2}$ −1

${\displaystyle \ }$

${\displaystyle z=2{,}5}$ 2

${\displaystyle \ }$

${\displaystyle z=-{\tfrac {1}{3}}}$ −1

${\displaystyle \ }$

${\displaystyle z=5}$ −2

${\displaystyle \ }$

${\displaystyle z=-1}$ 1

V1.3 Indirekte Proportionalität

${\displaystyle \ }$ ${\displaystyle 9\ {\text{h}}\ }$ ${\displaystyle \ }$ ${\displaystyle 65\ {\text{g}}\ }$ ${\displaystyle \ }$ ${\displaystyle 371\ {\text{€}}\ }$ ${\displaystyle \ }$ ${\displaystyle 1{,}4\ {\text{kWh}}\ }$ ${\displaystyle \ }$ ${\displaystyle 7{,}8\ {\text{Tage}}\ }$ ${\displaystyle \ }$ ${\displaystyle 21\ {\text{Tage}}\ }$ ${\displaystyle \ }$ ${\displaystyle 3\ {\text{Stücke}}\ }$ ${\displaystyle \ }$ ${\displaystyle 100000\ {\text{€}}\ }$

${\displaystyle \ }$

${\displaystyle \ a^{-2}\qquad }$ ${\displaystyle \ 4^{3+b}\qquad }$ ${\displaystyle \ c^{-4}\qquad }$ ${\displaystyle \ w^{-3}}$

${\displaystyle \ }$

${\displaystyle \ a^{b-x}\qquad }$ ${\displaystyle \ 4^{4}\qquad }$ ${\displaystyle \ c^{-14}\qquad }$ ${\displaystyle \ w^{12}}$

${\displaystyle \ }$

${\displaystyle \ 1\qquad }$ ${\displaystyle \ t^{-3b}\qquad }$ ${\displaystyle \ c^{3-b}\qquad }$ ${\displaystyle \ w^{-b-12}}$

${\displaystyle \ }$

${\displaystyle \ 3\qquad }$ ${\displaystyle \ b^{-7t}\qquad }$ ${\displaystyle \ b^{2}\qquad }$ ${\displaystyle \ w^{12-b}}$

${\displaystyle \ }$

${\displaystyle \ c^{-7}\qquad }$ ${\displaystyle \ 3^{2t-7}\qquad }$ ${\displaystyle \ 5^{6}\qquad }$ ${\displaystyle \ w^{-5b}}$

${\displaystyle \ }$

${\displaystyle \ 3^{0}\ also\ 1\qquad }$ ${\displaystyle \ a^{z-2t}\qquad }$ ${\displaystyle \ b^{-2}\qquad }$ ${\displaystyle \ 3^{-4w}}$

${\displaystyle \ }$

${\displaystyle \ 7^{b}\qquad }$ ${\displaystyle \ t^{t-3}\qquad }$ ${\displaystyle \ 20^{2}\qquad }$ ${\displaystyle \ u^{b+12}}$

${\displaystyle \ }$

${\displaystyle \ 7\qquad }$ ${\displaystyle \ j^{2w}\qquad }$ ${\displaystyle \ 3^{4b}\qquad }$ ${\displaystyle \ b^{-w-5}}$

Einfache Bruchteile

${\displaystyle \ }$	${\displaystyle W-h,\ X-f,\ Y-g,\ Z-e}$	${\displaystyle \ }$	${\displaystyle P-b,\ Q-a,\ R-keiner,\ S-e}$	${\displaystyle \ }$	${\displaystyle K-c,\ L-b,\ M-g,\ N-e}$	${\displaystyle \ }$	${\displaystyle W-g,\ X-d,\ Y-keiner,\ Z-e}$
${\displaystyle \ }$	${\displaystyle P-c,\ Q-d,\ R-a,\ S-b}$	${\displaystyle \ }$	${\displaystyle N-h,\ M-c,\ L-g,\ K-a}$	${\displaystyle \ }$	${\displaystyle X-keiner,\ R-keiner,\ L-h,\ O-keiner}$	${\displaystyle \ }$	${\displaystyle Q-e,\ M-d,\ L-g,\ V-c}$

Ergänzung: Prozentsätze und Winkle

${\displaystyle \ }$	40%=144°,20%=72°, 10%=36°,5%=18°	${\displaystyle \ }$	25%=90°,20%=72°,16,5%≈60°, 12,5%=45°,8,5%≈30°,5%=18°	${\displaystyle \ }$	25%=90°,16,7%≈60°, 12,5%=45°,4,2%≈15°	${\displaystyle \ }$	37,5%=135°,16,7%≈60°, 12,5%=45°,8,3%≈30°
${\displaystyle \ }$	20%=72°,16,7%≈60°,10%=36°	${\displaystyle \ }$	25%=90°,17%≈60°	${\displaystyle \ }$	25%=90°,20%=72°, 15%=54°,10%=36°,5%=18°	${\displaystyle \ }$	25%=90°,17%≈60°, 15%=54°,10%=36°,8%≈30°

${\displaystyle \ }$	${\displaystyle 7\ {\text{Stunden}}}$ 77\% reduziert	${\displaystyle \ }$	${\displaystyle 2{,}4\ cm\qquad }$ 2,5\% zurückgegangen	${\displaystyle \ }$	${\displaystyle 1750\ {\text{€}}\qquad }$ 0,8\% erhöht	${\displaystyle \ }$	${\displaystyle 15\ {\text{t}}\qquad }$ 9,2\% erhöht
${\displaystyle \ }$	${\displaystyle 155\ {\text{cm}}}$ ca. 12,28\% reduziert	${\displaystyle \ }$	${\displaystyle 2{,}5\ cm\qquad }$ keine Änderung	${\displaystyle \ }$	${\displaystyle 350\ {\text{ml}}\qquad }$ ca. 43\% mehr	${\displaystyle \ }$	${\displaystyle 440\ {\text{ml}}\qquad }$ keine Änderung

${\displaystyle \ }$	${\displaystyle 11\ {\text{T. mit 3 B.}}\qquad \ 22\ {\text{T. mit 8 B.}}}$	${\displaystyle \ }$	${\displaystyle 5\ {\text{W. mit 40 S.}}\qquad \ 8\ {\text{W. mit 65 S.}}\qquad }$	${\displaystyle \ }$	${\displaystyle 37\ {\text{M. mit 2 K.}}\qquad \ 14\ {\text{M. mit 3 K.}}\qquad }$
${\displaystyle \ }$	${\displaystyle 9\ {\text{W mit 15 S.}}\qquad \ 9\ {\text{W.mit 20 S.}}\qquad }$	${\displaystyle \ }$	${\displaystyle 2\ {\text{T. mit 3 P.}}\qquad \ 6\ {\text{T. mit 5 P.}}}$	${\displaystyle \ }$	${\displaystyle 5\ {\text{T. mit 4 B.}}\qquad \ 18\ {\text{T. mit 7 B.}}\qquad }$
${\displaystyle \ }$	${\displaystyle 11\ {\text{F. mit 130 P.}}\qquad \ 15\ {\text{F. mit 150 P.}}\qquad }$	${\displaystyle \ }$	${\displaystyle 13\ {\text{Mädchen}}\qquad \ 11\ {\text{Jungen}}\qquad }$

Vorrang mit Klammern in Klammern

${\displaystyle \ }$ ${\displaystyle \ 22}$ ${\displaystyle \ }$ ${\displaystyle \ -10}$ ${\displaystyle \ }$ ${\displaystyle \ -25}$ ${\displaystyle \ }$ ${\displaystyle \ 51}$ ${\displaystyle \ }$ ${\displaystyle \ -37}$ ${\displaystyle \ }$ ${\displaystyle \ 55}$ ${\displaystyle \ }$ ${\displaystyle \ 53}$ ${\displaystyle \ }$ ${\displaystyle \ 49}$

Vorrangregeln Fehlerfindung

${\displaystyle \ }$	${\displaystyle {\color {red}21+56}:7-(79-2\cdot 5):3=}$ ${\displaystyle {\color {red}77}:7-(79-10):3=}$ ...→ Punkt vor Strich 6	${\displaystyle \ }$	${\displaystyle 43+56:8-({\color {red}30-2}\cdot 5):4=}$ ${\displaystyle 43+7-({\color {red}28}\cdot 5):4=}$ ...→ Punkt vor Strich 45	${\displaystyle \ }$	${\displaystyle 43+72:8-(56-2\cdot {\color {red}20}):{\color {red}4}=}$ ${\displaystyle 43+9-(56-2\cdot {\color {red}5})=}$ ... → Klammer vor Punkt 48	${\displaystyle \ }$	${\displaystyle 63-(56:7-2)\cdot {\color {red}9+72}:3=}$ ${\displaystyle 63-(8-2)\cdot {\color {red}81}:3=}$ ...→ Punkt vor Strich 33
${\displaystyle \ }$	${\displaystyle 21+56:7-({\color {red}77-2\cdot 4}):3=}$ ${\displaystyle 21+8-({\color {red}75}\cdot 4):3=}$ ...→ Punkt vor Strich 6	${\displaystyle \ }$	${\displaystyle {\color {red}40+56:8}(30-2\cdot 5):4=}$ ${\displaystyle {\color {red}96:8}(30-10):4=}$ ...→ Punkt vor Strich 45	${\displaystyle \ }$	${\displaystyle 33+72:8-(56-2\cdot 20):4=}$ ${\displaystyle 33+9-(56-2\cdot {\color {red}40}):{\color {red}4}=}$ ${\displaystyle 33+9-(56-2\cdot {\color {red}10})=}$ ... → Klammer vor Punkt 38	${\displaystyle \ }$	${\displaystyle 63-(56:7-2):3+72:6=}$ ${\displaystyle 63-(8-2):3+72:6=}$ ${\displaystyle {\color {red}63-6}:3+72:6=}$ ${\displaystyle {\color {red}57}:3-12=}$ ...→ Punkt vor Strich 73

Doppelbrüche ${\displaystyle \ }$ ${\displaystyle \textstyle {\frac {5}{9}}}$ ${\displaystyle \ }$ ${\displaystyle \textstyle 1{,}5}$ ${\displaystyle \ }$ ${\displaystyle \textstyle -{\frac {1}{8}}}$ ${\displaystyle \ }$ ${\displaystyle \textstyle 1{\frac {1}{3}}}$ ${\displaystyle \ }$ ${\displaystyle \textstyle {\frac {5}{6}}}$ ${\displaystyle \ }$ ${\displaystyle \textstyle -{\frac {3}{2}}}$ ${\displaystyle \ }$ ${\displaystyle \textstyle -2}$ ${\displaystyle \ }$ ${\displaystyle \textstyle -{\frac {3}{4}}}$

Bruchrechnungen und Vorrang ${\displaystyle \ }$ ${\displaystyle {\frac {16}{15}}}$ ${\displaystyle \ }$ ${\displaystyle -{\frac {1}{4}}}$ ${\displaystyle \ }$ ${\displaystyle -{\frac {383}{180}}}$ ${\displaystyle \ }$ ${\displaystyle -{\frac {35}{11}}}$ ${\displaystyle \ }$ ${\displaystyle -{\frac {1}{6}}}$ ${\displaystyle \ }$ ${\displaystyle -{\frac {1}{15}}}$ ${\displaystyle \ }$ ${\displaystyle {\frac {4}{15}}}$ ${\displaystyle \ }$ ${\displaystyle 0}$

${\displaystyle \ }$

${\displaystyle A=9\ cm^{2}}$

${\displaystyle \ }$

${\displaystyle u\approx 12{,}23\ cm\qquad }$

${\displaystyle \ }$

${\displaystyle A=90\ cm^{2}\qquad }$

${\displaystyle \ }$

${\displaystyle A\approx 11{,}46\ cm^{2}\qquad }$

${\displaystyle \ }$

${\displaystyle u=96\ cm\qquad }$

${\displaystyle \ }$

${\displaystyle u=18\ dm\qquad }$

${\displaystyle \ }$

${\displaystyle u=7{,}95\ dm\qquad }$

${\displaystyle \ }$

${\displaystyle A\approx 6{,}93\ cm^{2}\qquad }$

${\displaystyle \ }$	${\displaystyle D=2{,}2\ {\text{Ban./Pack.}},\ Mod=4}$	${\displaystyle \ }$	${\displaystyle D\approx 3{,}42\ {\text{St./Tisch}},\ Mod=2\ und\ 5}$	${\displaystyle \ }$	${\displaystyle D\approx 3{,}04\ {\text{Bl/T}},\ Mod=6}$	${\displaystyle \ }$	${\displaystyle D\approx 4{,}19\ {\text{Pun./Sch.}},\ Mod=5}$
${\displaystyle \ }$	${\displaystyle D\approx 2{,}78\ {\text{B/S}},\ Mod=1}$	${\displaystyle \ }$	${\displaystyle D\approx 3{,}06\ {\text{Aut/H}},\ Mod=4}$	${\displaystyle \ }$	${\displaystyle D=3{,}5\ {\text{Sch/Kl}},\ Mod=5\ und\ 6}$	${\displaystyle \ }$	${\displaystyle D=3{,}{\dot {2}}\ {\text{Schl./Haus}},\ Mod=3\ und\ 5}$

${\displaystyle \ }$	${\displaystyle \approx 1{,}73\ 10^{14}\ {\text{Personen!}}}$ ${\displaystyle \approx 40522\ {\text{Atome}}}$	${\displaystyle \ }$	${\displaystyle \approx 9{,}22\ 10^{18}\ {\text{Körner!}}}$ ${\displaystyle \approx 4488601\ {\text{Personen}}}$	${\displaystyle \ }$	${\displaystyle \approx 75{,}7\approx 75\ {\text{Bakterien}}}$ ${\displaystyle \approx 3{,}147\ {\text{Ampere}}}$	${\displaystyle \ }$	${\displaystyle \approx 340\ {\text{Millionen bzw. }}2{,}00\ {\text{Billionen Personen!}}}$ ${\displaystyle \approx 47847\ {\text{Atome}}}$
${\displaystyle \ }$	${\displaystyle \approx 328\ {\text{Bakterien}}}$ ${\displaystyle \approx 117686\ {\text{Bakterien}}}$	${\displaystyle \ }$	${\displaystyle \approx 318\ {\text{Menschen}}}$ ${\displaystyle \approx 34{,}2{\text{°C}}}$	${\displaystyle \ }$	${\displaystyle \approx 20636\ {\text{Bakterien}}}$ ${\displaystyle \approx 21700000\ {\text{Personen}}}$	${\displaystyle \ }$	${\displaystyle \approx 9{,}8\ {\text{bzw. }}59\ {\text{Millionen Menschen}}}$ ${\displaystyle \approx 136{,}12\ {\text{V}}}$

Rechenaufgaben

${\displaystyle \ }$	${\displaystyle {\frac {2}{5}}=0,4=40\%\qquad {\frac {5}{2}}=2,5=250\%\qquad {\frac {7}{9}}=0,{\dot {7}}=77,{\dot {7}}\%}$ ${\displaystyle {\frac {50}{4}}=12,5=1250\%\qquad {\frac {1}{2000}}=0,0005=0,05\%}$ ${\displaystyle 350\%=3,5={\frac {350}{100}}={\frac {7}{2}}\qquad 90000\%=900={\frac {90000}{100}}={\frac {900}{1}}}$ ${\displaystyle 0,06\%=0,0006={\frac {0,06}{100}}={\frac {3}{500}}\qquad 75\%=0,75={\frac {75}{100}}={\frac {1}{4}}}$ ${\displaystyle 7,5\%=0,075={\frac {7,5}{100}}={\frac {1}{40}}}$	${\displaystyle \ }$	${\displaystyle {\frac {3}{10}}=0,3=30\%\qquad {\frac {10}{3}}=3,{\dot {3}}=333,{\dot {3}}\%\qquad {\frac {8}{9}}=0,{\dot {8}}=88,{\dot {8}}\%}$ ${\displaystyle {\frac {48}{4}}=12=1200\%\qquad {\frac {1}{400}}=0,0025=0,25\%}$ ${\displaystyle 500\%=5={\frac {500}{100}}={\frac {5}{1}}\qquad 45000\%=450={\frac {45000}{100}}={\frac {450}{1}}}$ ${\displaystyle 0,08\%=0,0008={\frac {0,08}{100}}={\frac {1}{125}}\qquad 66,{\dot {6}}\%=0,{\dot {6}}={\frac {66,{\dot {6}}}{100}}={\frac {2}{3}}}$ ${\displaystyle 3\%=0,03={\frac {3}{100}}={\frac {3}{100}}}$
${\displaystyle \ }$	${\displaystyle {\frac {1}{5}}=0,2=20\%\qquad {\frac {5}{1}}=5=500\%\qquad {\frac {4}{9}}=0,{\dot {4}}=44,{\dot {4}}\%}$ ${\displaystyle {\frac {400}{4}}=100=10000\%\qquad {\frac {1}{80}}=0,0125=1,25\%}$ ${\displaystyle 200\%=2={\frac {200}{100}}={\frac {2}{1}}\qquad 2050\%=20,5={\frac {2050}{100}}={\frac {410}{20}}}$ ${\displaystyle 0,01\%=0,0001={\frac {0,01}{100}}={\frac {1}{10000}}\qquad 40\%=0,4={\frac {40}{100}}={\frac {2}{5}}}$ ${\displaystyle 7,{\dot {7}}\%=0,0{\dot {7}}={\frac {7,{\dot {7}}}{100}}={\frac {7}{90}}}$	${\displaystyle \ }$	${\displaystyle {\frac {4}{5}}=0,8=80\%\qquad {\frac {5}{4}}=1,25=125\%\qquad {\frac {6}{9}}=0,{\dot {6}}=66,{\dot {6}}\%}$ ${\displaystyle {\frac {102}{4}}=25,5=2550\%\qquad {\frac {1}{200}}=0,005=0,5\%}$ ${\displaystyle 750\%=7,5={\frac {750}{100}}={\frac {15}{2}}\qquad 20000\%=200={\frac {20000}{100}}={\frac {200}{1}}}$ ${\displaystyle 0,03\%=0,0003={\frac {0,03}{100}}={\frac {3}{10000}}\qquad 80\%=0,8={\frac {80}{100}}={\frac {4}{5}}}$ ${\displaystyle 2,5\%=0,025={\frac {2,5}{100}}={\frac {1}{40}}}$
${\displaystyle \ }$	${\displaystyle {\frac {6}{10}}=0,6=60\%\qquad {\frac {10}{6}}=1,{\dot {6}}=166,{\dot {6}}\%\qquad {\frac {2}{3}}=0,{\dot {6}}=66,{\dot {6}}\%}$ ${\displaystyle {\frac {600}{4}}=150=15000\%\qquad {\frac {1}{400}}=0,0025=0,25\%}$ ${\displaystyle 800\%=8={\frac {800}{100}}={\frac {8}{1}}\qquad 40000\%=400={\frac {40000}{100}}={\frac {400}{1}}}$ ${\displaystyle 0,05\%=0,0005={\frac {0,05}{100}}={\frac {1}{2000}}\qquad 50\%=0,5={\frac {50}{100}}={\frac {1}{2}}}$ ${\displaystyle 4\%=0,04={\frac {4}{100}}={\frac {1}{25}}}$	${\displaystyle \ }$	${\displaystyle {\frac {3}{5}}=0,6=60\%\qquad {\frac {5}{3}}=1,{\dot {6}}=166,{\dot {6}}\%\qquad {\frac {3}{9}}=0,{\dot {3}}=33,{\dot {3}}\%}$ ${\displaystyle {\frac {82}{4}}=20,5=2050\%\qquad {\frac {1}{100}}=0,01=1\%}$ ${\displaystyle 250\%=2,5={\frac {250}{100}}={\frac {5}{2}}\qquad 50050\%=500,5={\frac {50050}{100}}={\frac {1001}{2}}}$ ${\displaystyle 0,04\%=0,0004={\frac {0,04}{100}}={\frac {1}{2500}}\qquad 25\%=0,25={\frac {25}{100}}={\frac {1}{4}}}$ ${\displaystyle 8\%=0,08={\frac {8}{100}}={\frac {2}{25}}}$
${\displaystyle \ }$	${\displaystyle {\frac {9}{10}}=0,9=90\%\qquad {\frac {10}{9}}=1,{\dot {1}}=111,{\dot {1}}\%\qquad {\frac {1}{3}}=0,{\dot {3}}=33,{\dot {3}}\%}$ ${\displaystyle {\frac {40}{4}}=10=1000\%\qquad {\frac {1}{800}}=0,00125=0,125\%}$ ${\displaystyle 100\%=1={\frac {100}{100}}={\frac {1}{1}}\qquad 70000\%=700={\frac {70000}{100}}={\frac {700}{1}}}$ ${\displaystyle 0,02\%=0,0002={\frac {0,02}{100}}={\frac {1}{5000}}\qquad 60\%=0,6={\frac {60}{100}}={\frac {3}{5}}}$ ${\displaystyle 5\%=0,05={\frac {5}{100}}={\frac {1}{20}}}$	${\displaystyle \ }$	${\displaystyle {\frac {1}{10}}=0,1=10\%\qquad {\frac {10}{1}}=10=1000\%\qquad {\frac {8}{9}}=0,{\dot {8}}=88,{\dot {8}}\%}$ ${\displaystyle {\frac {90}{4}}=22,5=2250\%\qquad {\frac {1}{8000}}=0,000125=0,0125\%}$ ${\displaystyle 600\%=6={\frac {600}{100}}={\frac {6}{1}}\qquad 4500\%=45={\frac {4500}{100}}={\frac {45}{1}}}$ ${\displaystyle 0,07\%=0,0007={\frac {0,07}{100}}={\frac {7}{10000}}\qquad 33,{\dot {3}}\%=0,{\dot {3}}={\frac {33,{\dot {3}}}{100}}={\frac {1}{3}}}$ ${\displaystyle 9\%=0,09={\frac {9}{100}}={\frac {9}{100}}}$

Richtig/Falsch

${\displaystyle \ }$ ✘, ✔, ✔ ${\displaystyle \ }$ ✘, ✘, ✘ ${\displaystyle \ }$ ✘, ✔, ✔ ${\displaystyle \ }$ ✔, ✘, ✔ ${\displaystyle \ }$ ✘, ✔, ✔ ${\displaystyle \ }$ ✘, ✘, ✘ ${\displaystyle \ }$ ✘, ✔, ✔ ${\displaystyle \ }$ ✔, ✘, ✔

Aufgaben mit Prozentänderung

${\displaystyle \ }$	${\displaystyle 6340\%}$ ${\displaystyle ca.\ 1,58\%}$ ${\displaystyle 6240\%}$ mehr bzw. ca. ${\displaystyle 98,42\%}$ weniger	${\displaystyle \ }$	${\displaystyle 13066,{\dot {6}}\%}$ ${\displaystyle ca.\ 7,69\%}$ ${\displaystyle 12966,{\dot {6}}\%}$ mehr bzw. ca. ${\displaystyle 92,31\%}$ weniger	${\displaystyle \ }$	${\displaystyle 22825\%}$ ${\displaystyle ca.\ 4,38\%}$ ${\displaystyle 22725\%}$ mehr bzw. ca. ${\displaystyle 95,62\%}$ weniger	${\displaystyle \ }$	${\displaystyle 5280\%}$ ${\displaystyle 1,8{\overline {93}}\%}$ ${\displaystyle 5180\%}$ mehr bzw. ${\displaystyle 98,1{\overline {07}}\%}$ weniger
${\displaystyle \ }$	${\displaystyle 5575\%}$ ${\displaystyle ca.\ 1,79\%}$ ${\displaystyle 5475\%}$ mehr bzw. ca. ${\displaystyle 98,21\%}$ weniger	${\displaystyle \ }$	${\displaystyle 16060\%}$ ${\displaystyle ca.\ 0,623\%}$ ${\displaystyle 15960\%}$ mehr bzw. ca. ${\displaystyle 99,377\%}$ weniger	${\displaystyle \ }$	${\displaystyle 6033,{\dot {3}}\%}$ ${\displaystyle ca.\ 1,66\%}$ ${\displaystyle 5933,{\dot {3}}\%}$ mehr bzw. ca. ${\displaystyle 98,34\%}$ weniger	${\displaystyle \ }$	${\displaystyle 1820\%}$ ${\displaystyle ca.\ 5,49\%}$ ${\displaystyle 1720\%}$ mehr bzw. ca. ${\displaystyle 94,51\%}$ weniger

Direkte Anwendung der Formel

${\displaystyle \ }$ ${\displaystyle \approx 225,9\ {\text{mm}}}$ ${\displaystyle \ }$ ${\displaystyle \approx 12,37\ \ {\text{dm}}}$ ${\displaystyle \ }$ ${\displaystyle \approx 66,0\ \ {\text{cm}}}$ ${\displaystyle \ }$ ${\displaystyle 119\ \ {\text{mm}}}$

Beispiele mit mehreren Schritten

${\displaystyle \ }$ ${\displaystyle 105\ {\text{cm}}^{2}}$ ${\displaystyle \ }$ ${\displaystyle {\sqrt {3362}}\ \ {\text{cm}}\approx 58{,}0\ {\text{cm}}}$ ${\displaystyle \ }$ ${\displaystyle 90{\sqrt {2}}\ \ {\text{mm}}\approx 127{,}3\ {\text{mm}}}$ ${\displaystyle \ }$ ${\displaystyle 221\ \ {\text{cm}}}$ ${\displaystyle \ }$ ${\displaystyle 90{\sqrt {2}}\ \ {\text{mm}}\approx 127{,}3\ {\text{mm}}}$ ${\displaystyle \ }$ ${\displaystyle {\sqrt {3362}}\ \ {\text{cm}}\approx 58{,}0\ {\text{cm}}}$ ${\displaystyle \ }$ ${\displaystyle 105\ {\text{cm}}^{2}}$ ${\displaystyle \ }$ ${\displaystyle 221\ \ {\text{cm}}}$

Tabelle für eine lineare Funktion erstellen

${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -2&5&(-2|5)\\-1&3&(-1|3)\\0&1&(0|1)\\1&-1&(1|-1)\\2&-3&(2|-3)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline 2,5\\1,5\\0,5\\-0,5\\-1,5\\\end{array}}}$	${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -4&9&(-4|9)\\-2&5&(-2|5)\\0&1&(0|1)\\2&-3&(2|-3)\\4&-7&(4|-7)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline 1,5\\1\\0,5\\0\\-0,5\\\end{array}}}$	${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -2&-7&(-2|-7)\\-1&-4&(-1|-4)\\0&1&(0|-1)\\1&2&(1|2)\\2&5&(2|5)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline -1\\-0,{\dot {3}}\\0,{\dot {3}}\\1\\1,{\dot {6}}\\\end{array}}}$
${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -4&-13&(-4|-13)\\-2&-7&(-2|-7)\\0&-1&(0|-1)\\2&5&(2|5)\\4&11&(4|11)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline -0,{\dot {3}}\\0\\0,{\dot {3}}\\0,{\dot {6}}\\1\\\end{array}}}$	${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -2&3&(-2|3)\\-1&2,5&(-1|2,5)\\0&2&(0|2)\\1&1,5&(1|1,5)\\2&1&(2|1)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline 12\\8\\4\\0\\4\\\end{array}}}$	${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -4&4&(-4|4)\\-2&3&(-2|3)\\0&2&(0|2)\\2&1&(2|1)\\4&0&(4|0)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline 8\\6\\4\\2\\0\\\end{array}}}$
${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -2&-6&(-2|-6)\\-1&-4,5&(-1|-4,5)\\0&-3&(0|-3)\\1&-1,5&(1|-1,5)\\2&0&(2|0)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline -0,{\dot {6}}\\0,{\dot {6}}\\2\\3,{\dot {3}}\\4,{\dot {6}}\\\end{array}}}$	${\displaystyle \ }$	${\displaystyle {\begin{array}{|c|c||c|}x&y&(x|y)\\\hline -4&-9&(-4|-9)\\-2&-6&(-2|-6)\\0&-3&(0|-3)\\2&0&(2|0)\\4&3&(4|3)\\\end{array}}\qquad }$ ${\displaystyle {\begin{array}{r}x\ \ \\\hline 0,{\dot {6}}\\1,{\dot {3}}\\2\\2,{\dot {6}}\\3,{\dot {3}}\\\end{array}}}$

lineare Funktion Diagramm

${\displaystyle \ }$	ca. 2500 €, 3700 €, −600 € bzw. −1200 € ca. 1 t, 1200 € ca. 2,6, 3,8 bzw. 5,8 t	${\displaystyle \ }$	ca.24, 32, 36 bzw. fast 39 Liter ca. 3, 1,5, 1 bzw. 0,5 kg ca. 16 Liter ca. 3,5 kg	${\displaystyle \ }$	ca. 6, 5, 4,5 bzw. 3,5 Liter ca. 8 Liter, ca. 120 °C ca. 80, 60, 40 bzw. 30 °C	${\displaystyle \ }$	ca. 71 Jahre ca. 77 Jahre ca. 30 Zig./Tag ca. 34 Zig./Tag ca. 85 Jahre
${\displaystyle \ }$	ca.60, 50, 35 bzw. 10 Hz ca. 70 Hz, 140 cm ca. 100, 60, 40 bzw. 120 cm	${\displaystyle \ }$	ca. 3, 4, 6 bzw. 7,3 Pa ca. 1 Pa, ca. 115 °C ca. 45, 57, 70 bzw. 90 °C	${\displaystyle \ }$	ca.20, 28, 32 bzw. fast 34,2 Liter ca. 3, 2, 1,5, bzw. 1 kg ca. 12 Liter ca. 3,5 kg	${\displaystyle \ }$	ca. 79 Jahre ca. 74 Jahre ca. 26 Zig./Tag ca. 13 Zig./Tag ca. 85 Jahre

${\displaystyle \ }$	1,5 Tage später	${\displaystyle \ }$	34 Tage	${\displaystyle \ }$	20 Jahre	${\displaystyle \ }$	7 Kinder je 3 Stücke und 14 Kinder je 6 Stücke
${\displaystyle \ }$	1,5 Tage später	${\displaystyle \ }$	34 Tage	${\displaystyle \ }$	15,5 Tage	${\displaystyle \ }$	100000 €

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${\displaystyle \ }$

Erklärungen der Potenzzahleneigenschaften

${\displaystyle \ }$	${\displaystyle {\frac {z^{n}}{z^{m}}}=z^{n-m}\ {\text{daher bspw.}}\ {\frac {z^{3}}{z^{3}}}={\frac {z\cdot z\cdot z}{z\cdot z\cdot z}}={\frac {{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}}{{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}}}\ _{({\text{kürzen}})}=}$ ${\displaystyle {\text{...}}=1\quad {\text{aber auch ...}}=z^{3-3}=z^{0}\quad {\text{also}}\quad z^{0}=1}$	${\displaystyle \ }$	${\displaystyle {\frac {z^{7}}{z^{4}}}={\frac {z\cdot z\cdot z\cdot z\cdot z\cdot z\cdot z}{z\cdot z\cdot z\cdot z}}=}$ ${\displaystyle {\frac {{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}\cdot z\cdot z}{{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}}}\ _{({\text{kürzen}})}={\frac {z\cdot z\cdot z}{1}}=z^{3}\quad _{=(z^{7-4})}}$
${\displaystyle \ }$	${\displaystyle w^{7}\cdot w^{4}=\underbrace {w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w} _{7\ mal}\cdot \underbrace {w\cdot w\cdot w\cdot w} _{4\ mal}=}$ ${\displaystyle \underbrace {w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w\cdot w} _{11\ mal}=w^{1}1\quad _{(=w^{7+4})}}$	${\displaystyle \ }$	${\displaystyle {\frac {z^{n}}{z^{m}}}=z^{n-m}\ {\text{daher bspw.}}\ {\frac {z^{3}}{z^{4}}}={\frac {z\cdot z\cdot z}{z\cdot z\cdot z\cdot z}}=}$ ${\displaystyle ={\frac {{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}}{{\cancel {z}}\cdot {\cancel {z}}\cdot {\cancel {z}}\cdot z}}\ _{({\text{kürzen}})}={\frac {1}{z}}=z^{-1}\quad _{=(z^{3-4})}}$

Geometrische Erklärungen und Beweise

${\displaystyle \ }$	Das Ganze ist die Summe seiner Teilen ${\displaystyle \textstyle A_{gQ}=A_{kQ}+4\cdot A_{D}}$ ${\displaystyle \textstyle (a+b)^{2}=c^{2}+4\cdot {\frac {a\cdot b}{2}}}$ ${\displaystyle \textstyle a^{2}+2ab+b^{2}=c^{2}+2ab}$ ${\displaystyle \textstyle c^{2}=a^{2}+b^{2}\ }$	${\displaystyle \ }$	Das Ganze ist die Summe seiner Teilen ${\displaystyle \ \ \textstyle A_{gQ}=A_{1}+A_{2}+A_{3}+A_{4}}$ ${\displaystyle \ \ \textstyle (a+b)^{2}=a^{2}+a\ b+b\ a+b^{2}}$ ${\displaystyle \ \ \textstyle (a+b)^{2}=a^{2}+2\ a\ b+b^{2}}$	${\displaystyle \ }$	Das Ganze ist die Summe seiner Teilen ${\displaystyle \ \ \textstyle A_{gQ}=A_{1}+A_{2}+A_{3}+A_{4}}$ ${\displaystyle \ \ \textstyle a^{2}=(a-b)^{2}+(a-b)\cdot b+(a-b)\cdot b+b^{2}}$ ${\displaystyle \ \ \textstyle a^{2}=(a-b)^{2}+a\cdot b-b^{2}+a\cdot b-b^{2}+b^{2}}$ ${\displaystyle \ \ \textstyle (a-b)^{2}=a^{2}-2\cdot a\cdot b+b^{2}}$
${\displaystyle \ }$	Das Ganze ist die Summe seiner Teilen ${\displaystyle \textstyle A_{gQ}=A_{kQ}+4\cdot A_{D}}$ ${\displaystyle \textstyle c^{2}=(a-b)^{2}+4\cdot {\frac {a\cdot b}{2}}}$ ${\displaystyle \textstyle c^{2}=a^{2}-2ab+b^{2}+2ab}$ ${\displaystyle \textstyle c^{2}=a^{2}+b^{2}\ }$	${\displaystyle \ }$		${\displaystyle \ }$
${\displaystyle \ }$	+ =	${\displaystyle \ }$	${\displaystyle -}$ =

${\displaystyle \ }$	${\displaystyle G\approx 6397{,}19\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{38,21 € }}\quad }$ ${\displaystyle eZ\approx {\text{28,66 € }}\quad }$ ${\displaystyle KESt.\approx {\text{9,55 € }}}$ ${\displaystyle G=6340\ {\text{€}}\quad }$ ${\displaystyle eZs=0{,}45\%}$ ${\displaystyle G_{45}\approx 7794{,}47\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G\approx 7660{,}89\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{268,54 € }}\quad }$ ${\displaystyle eZ\approx {\text{201,41 € }}\quad }$ ${\displaystyle KESt.\approx {\text{67,14 € }}}$ ${\displaystyle G\approx 7263{,}37\ {\text{€}}\quad }$ ${\displaystyle eZs=2{,}7\%}$ ${\displaystyle G_{15}\approx 11124{,}11\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G\approx 6454{,}51\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{114,63 € }}\quad }$ ${\displaystyle eZ\approx {\text{85,98 € }}\quad }$ ${\displaystyle KESt.\approx {\text{28,66 € }}}$ ${\displaystyle G\approx 6283{,}70\ {\text{€}}\quad }$ ${\displaystyle eZs=1{,}35\%}$ ${\displaystyle G_{158}\approx 52989{,}79\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G=80840\ {\text{€}}\quad }$ ${\displaystyle Z={\text{1120 € }}\quad }$ ${\displaystyle eZ={\text{840 € }}\quad }$ ${\displaystyle KESt.={\text{280 € }}}$ ${\displaystyle G\approx 79168{,}73\ {\text{€}}\quad }$ ${\displaystyle eZs=1{,}05\%}$ ${\displaystyle G_{106}\approx 242069{,}28\ {\text{€}}}$
${\displaystyle \ }$	${\displaystyle G\approx 6397{,}19\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{38,21 € }}\quad }$ ${\displaystyle eZ\approx {\text{28,66 € }}\quad }$ ${\displaystyle KESt.\approx {\text{9,55 € }}}$ ${\displaystyle G=6340\ {\text{€}}\quad }$ ${\displaystyle eZs=0{,}45\%}$ ${\displaystyle G_{45}\approx 7794{,}47\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G\approx 7660{,}89\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{268,54 € }}\quad }$ ${\displaystyle eZ\approx {\text{201,41 € }}\quad }$ ${\displaystyle KESt.\approx {\text{67,14 € }}}$ ${\displaystyle G\approx 7263{,}37\ {\text{€}}\quad }$ ${\displaystyle eZs=2{,}7\%}$ ${\displaystyle G_{15}\approx 11124{,}11\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G\approx 6454{,}51\ {\text{€}}\quad }$ ${\displaystyle Z\approx {\text{114,63 € }}\quad }$ ${\displaystyle eZ\approx {\text{85,98 € }}\quad }$ ${\displaystyle KESt.\approx {\text{28,66 € }}}$ ${\displaystyle G\approx 6283{,}70\ {\text{€}}\quad }$ ${\displaystyle eZs=1{,}35\%}$ ${\displaystyle G_{158}\approx 52989{,}79\ {\text{€}}}$	${\displaystyle \ }$	${\displaystyle G=80840\ {\text{€}}\quad }$ ${\displaystyle Z={\text{1120 € }}\quad }$ ${\displaystyle eZ={\text{840 € }}\quad }$ ${\displaystyle KESt.={\text{280 € }}}$ ${\displaystyle G\approx 79168{,}73\ {\text{€}}\quad }$ ${\displaystyle eZs=1{,}05\%}$ ${\displaystyle G_{106}\approx 242069{,}28\ {\text{€}}}$

${\displaystyle \ }$	D Ö R F E R 3 1 5 5 4 3 4 6 7 8 Toten pro Dorf ${\displaystyle D=6{,}{\dot {1}}\ {\text{T./D.}},\ Med=6{,}5,\ }$ ${\displaystyle Mod=6\ und\ 7,\ S=5}$	${\displaystyle \ }$	R Ä Ü M E 1 2 5 5 3 0 1 2 4 8 Betten pro Raum ${\displaystyle D=3{,}5\ {\text{B./R.}},\ Med=3,\ }$ ${\displaystyle Mod=2\ und\ 4,\ S=8}$	${\displaystyle \ }$	T I S C H E 4 5 1 4 1 0 1 2 3 9 Personen/Tisch ${\displaystyle D=1{,}8{\dot {6}}\ {\text{P./T.}},\ Med=1,\ }$ ${\displaystyle Mod=1,\ S=9}$	${\displaystyle \ }$	P E R S O N 3 5 1 5 1 0 1 2 3 14 Part./Person ${\displaystyle D=2{,}4\ {\text{Par./Per.}},\ Med=1,\ }$ ${\displaystyle Mod=3,\ S=14}$
${\displaystyle \ }$	K I N D E R 5 2 3 5 2 4 5 6 7 8 Tonnen/Kind ${\displaystyle D\approx 5{,}82\ {\text{t/K}},\ Med=6,\ }$ ${\displaystyle Mod=4\ und\ 7,\ S=4}$	${\displaystyle \ }$	T Ö P F E 3 2 2 5 4 0 3 5 7 8 Blumen pro Topf ${\displaystyle D\approx 5{,}19\ {\text{Bl./T.}},\ Med=7,\ }$ ${\displaystyle Mod=7,\ S=8}$	${\displaystyle \ }$	M Ü T T E R 6 6 5 1 1 1 2 3 6 7 Kinder/Mutter ${\displaystyle D\approx 2{,}42\ {\text{K./M.}},\ Med=2,\ }$ ${\displaystyle Mod=1\ und\ 2,\ S=6}$	${\displaystyle \ }$	K I N D E R 3 5 1 5 1 0 1 2 3 14 Bücher pro Kind ${\displaystyle D=2{,}4\ {\text{B./K.}},\ Med=1,\ }$ ${\displaystyle Mod=1\ und\ 3,\ S=14}$

${\displaystyle \ }$	${\displaystyle \ y={\frac {5x-5}{4}}}$ x: Tonnen, y: 1000 €, S: 1000 €/t	${\displaystyle \ }$	${\displaystyle \ y=-0{,}5x+70}$ x: cm, y: Hz, S: Hz/cm	${\displaystyle \ }$	${\displaystyle \ y=-0{,}4x+100}$ x: °C, y: g/L, S: g/(L mal °C).	${\displaystyle \ }$	${\displaystyle \ y=-{\frac {17x}{30}}+85}$ x: Zig./Tag, y: Jahre, S: Jahre mal Tag/Zig.
${\displaystyle \ }$	${\displaystyle \ y=205{,}63x+346}$ x: km, y: m, S: m/km	${\displaystyle \ }$	${\displaystyle \ y=-0{,}004x+4}$ x: g Obst, y: t CO2, S: g/t	${\displaystyle \ }$	${\displaystyle \ y=-3x+5{,}25}$ x: h, y: m, S: m/h	${\displaystyle \ }$	${\displaystyle \ y=-{\frac {17x}{30}}+85}$ x: Zig./Tag, y: Jahre, S: Jahre mal Tag/Zig.

Umsatzsteuer

${\displaystyle \ }$ BVP: 56 €, USt.: 6 € ${\displaystyle \ }$ NVP: 60 € ${\displaystyle \ }$ 12,5% ${\displaystyle \ }$ NVP: 75 €

Rabatt

${\displaystyle \ }$ BVP: 66 € ${\displaystyle \ }$ PnR: 572 € ${\displaystyle \ }$ 12% ${\displaystyle \ }$ 6%

USt. und Rabatt Gegebener Endwert

${\displaystyle \ }$	NVP: 60 €, Rabatt: 9,9 €, USt.: 6 €	${\displaystyle \ }$	NVP: 85 €, Rabatt: 20%
${\displaystyle \ }$	NVP: 455 €, BVP: 527,8 €, Rabatt: 131,95 €, USt.: 72,8 €	${\displaystyle \ }$	NVP: 88 €, BVP: 110 €, Rabatt: 22 €, USt.: 22 €

USt. und Rabatt Kombinationsaufgaben

${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$55${\displaystyle \ }$ ${\displaystyle \ }$780€${\displaystyle \ }$ Umsatzsteuer % 60% 25% Umsatzsteuer € 33 195€ Bruttoverkaufspreis € 88€ 975 Rabatt % 37,5% 20% Rabatt € 33€ 195€ Preis nach dem Rabatt € 55€ 780€	${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$336000€${\displaystyle \ }$ ${\displaystyle \ }$420€${\displaystyle \ }$ Umsatzsteuer % 10% 20% Umsatzsteuer € 33600 84€ Bruttoverkaufspreis € 369600€ 504€ Rabatt % 3% 5% Rabatt € 11088€ 25,2€ Preis nach dem Rabatt € 358512€ 478,8€
${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$914,94${\displaystyle \ }$ ${\displaystyle \ }$780€${\displaystyle \ }$ Umsatzsteuer % 14,1% 25% Umsatzsteuer € 129€ 195€ Bruttoverkaufspreis € 1043,94€ 975 Rabatt % 10% 20%1 Rabatt € 104,39€ 95€ Preis nach dem Rabatt € 939,55€ 780€	${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$3500€${\displaystyle \ }$ ${\displaystyle \ }$84${\displaystyle \ }$ Umsatzsteuer % 14% 10% Umsatzsteuer € 490€ 8,4€ Bruttoverkaufspreis € 3990€ 92,4€ Rabatt % 10% 10% Rabatt € 399€ 9,24€ Preis nach dem Rabatt € 3591€ 83,16€
${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$55€${\displaystyle \ }$ ${\displaystyle \ }$3500€${\displaystyle \ }$ Umsatzsteuer % 60% 14% Umsatzsteuer € 33€ 490€ Bruttoverkaufspreis € 88€ 3990€ Rabatt % 37,5% 10% Rabatt € 33€ 399€ Preis nach dem Rabatt € 55€ 3591€	${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$420€${\displaystyle \ }$ ${\displaystyle \ }$55€${\displaystyle \ }$ Umsatzsteuer % 20% 60% Umsatzsteuer € 84€ 33€ Bruttoverkaufspreis € 504€ 88€ Rabatt % 5% 37,5% Rabatt € 25,2€ 33€ Preis nach dem Rabatt € 478,8€ 55€
${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$60€${\displaystyle \ }$ ${\displaystyle \ }$85€${\displaystyle \ }$ Umsatzsteuer % 10% 14% Umsatzsteuer € 6€ 11,9€ Bruttoverkaufspreis € 66€ 96,9€ Rabatt % 15% 20% Rabatt € 9,9€ 19,38€ Preis nach dem Rabatt € 56,1€ 77,52€	${\displaystyle \ }$	${\displaystyle \ }$Bsp. 1${\displaystyle \ }$ ${\displaystyle \ }$Bsp. 2${\displaystyle \ }$ Nettoverkaufspreis € ${\displaystyle \ }$455€${\displaystyle \ }$ ${\displaystyle \ }$40€${\displaystyle \ }$ Umsatzsteuer % 16% 10,5% Umsatzsteuer € 72,8€ 4,2€ Bruttoverkaufspreis € 527,8€ 44,2€ Rabatt % 25% 5% Rabatt € 131,95 2,21€ Preis nach dem Rabatt € 395,85€ 41,99€

${\displaystyle \ }$	${\displaystyle r={\sqrt {\tfrac {A}{\pi }}}}$	${\displaystyle \ }$	${\displaystyle a={\tfrac {u-2\cdot b}{2}}={\tfrac {u}{2}}-b}$	${\displaystyle \ }$	${\displaystyle b={\sqrt {d^{2}-a^{2}}}}$	${\displaystyle \ }$	${\displaystyle a={\sqrt {\tfrac {4A}{\sqrt {3}}}}\left(=2{\sqrt {\tfrac {A}{\sqrt {3}}}}={\sqrt {A\cdot {\tfrac {4}{\sqrt {3}}}\ }}\right)}$
${\displaystyle \ }$	${\displaystyle b={\tfrac {u-2\cdot a}{2}}={\tfrac {u}{2}}-a}$	${\displaystyle \ }$	${\displaystyle d={\frac {u}{\pi }}}$	${\displaystyle \ }$	${\displaystyle b={\sqrt {c^{2}-a^{2}}}}$	${\displaystyle \ }$	${\displaystyle d={\sqrt {\tfrac {4\,A}{\pi }}}}$

${\displaystyle \ }$	1,65 kWh ${\displaystyle 3{,}{\dot {8}}\ {\text{kWh pro h}}}$ 1,25 Milliarden Menschen ${\displaystyle 36{,}9{\dot {4}}\ {\text{kWh}}}$	${\displaystyle \ }$	525 € 87,5 € 3937,5 €	${\displaystyle \ }$	9,375 mal (durchschnittlich) 7 mal ca. 7,2 mal (durchschnittlich)	${\displaystyle \ }$	3,15 h 14 h 19,6875 h
${\displaystyle \ }$	${\displaystyle 325\ t}$ 11,7 Tage 6 Tage	${\displaystyle \ }$	40,5 Tage ca. 11,6 Tage ca. 3.8 Tage	${\displaystyle \ }$	14 Kinder 9 Kinder 20 Tage	${\displaystyle \ }$	4,2 Tage 25 Tage 27 Arbeiter