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# MathemaTriX ⋅ Aufgabenheft Antworten

 ${\displaystyle {\color {white}\mathbf {MATHE} \mu \alpha T\mathbb {R} ix}}$ DEINE FESTE BEGLEITERIN FÜR DIE SCHULMATHEMATIK
EINFACH UND
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## Grundniveau 1

### G1.1 Grundrechenartenvorrang

#### Typ 1

 ${\displaystyle 24}$${\displaystyle 9}$ ${\displaystyle -9}$${\displaystyle 0}$ ${\displaystyle -27}$${\displaystyle 7}$ ${\displaystyle 49}$${\displaystyle 8}$ ${\displaystyle 10}$${\displaystyle 21}$ ${\displaystyle 98}$${\displaystyle 15}$ ${\displaystyle -27}$${\displaystyle 13}$ ${\displaystyle -12}$${\displaystyle -1}$

### G1.2 Strich und Punkt Bruchrechnungen

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

### G1.3 Direkte Proportionalität

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

### G1.4 Grundaufgaben der Prozentrechnung

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

### G1.5 Ausmultiplizieren mit einer oder zwei Klammer

 ${\displaystyle \ 6x^{7}-14x^{2}+10x^{3}}$${\displaystyle \ 6m^{4}-8m^{2}-15m^{2}+20=6m^{4}-23m^{2}+20}$ ${\displaystyle \ 14b^{9}+8b^{8}-14b^{7}}$${\displaystyle \ 10w^{7}-8w^{5}-25w^{5}+20w^{3}=10w^{7}-33w^{5}+20w^{3}}$ ${\displaystyle \ 8s^{8}+20s^{9}-28s^{5}}$${\displaystyle \ 15w^{8}-12w^{6}+5w^{6}-4w^{4}=15w^{8}-7w^{6}-4w^{4}}$ ${\displaystyle \ 28v^{11}+12v^{12}-8v^{8}}$${\displaystyle \ 10g^{7}+8g^{6}-15g^{5}-12g^{4}}$ ${\displaystyle 14n^{1}4-14n^{9}+35n^{7}}$${\displaystyle 15c^{2}-20c^{4}-18+24c^{2}=39c^{2}-20c^{4}-18}$ ${\displaystyle 6z^{6}+12z^{7}-21z^{5}}$${\displaystyle 8p^{5}-10p^{7}-12p^{3}+15p^{5}=23p^{5}-10p^{7}-12p^{3}}$ ${\displaystyle 8s^{8}+20s^{5}-28s^{7}}$${\displaystyle 10w^{7}-8w^{6}+5w^{5}-4w^{4}}$ ${\displaystyle 4v^{5}+12v^{6}-8v^{2}}$${\displaystyle 10a^{7}-8a^{5}+15a^{5}-12a^{4}}$

### G1.6 Textaufgaben zu den Grundrechenarten

#### Typ 1

 34241−4 52119−42 42812−46 3310-1826 34241−4 52119−42 42812−46 3310-1826

#### Typ 2

 1C, 2A, 3B 1C, 2B, 3A 1A, 2B, 3C 1B, 2A, 3C 1C, 2A, 3B 1C, 2B, 3A 1A, 2B, 3C 1B, 2A, 3C

### G1.1 Grundrechenartenvorrang mit Plus-Minus Regel

#### Typ 2

 ${\displaystyle -17}$ ${\displaystyle 73}$ ${\displaystyle 39}$ ${\displaystyle -5}$ ${\displaystyle 29}$ ${\displaystyle -5}$ ${\displaystyle -13}$ ${\displaystyle 0}$

#### Typ 3

 ${\displaystyle 9}$ ${\displaystyle 0}$ ${\displaystyle -15}$ ${\displaystyle 16}$ ${\displaystyle 21}$ ${\displaystyle 0}$ ${\displaystyle -13}$ ${\displaystyle -1}$

#### Typ 4

 ${\displaystyle -7}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle -1}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$

#### Typ 5

 ${\displaystyle {\color {red}21+56}:7-(79-2\cdot 5):3=}$${\displaystyle {\color {red}77}:7-(79-10):3=}$...→ Punkt vor Strich6 ${\displaystyle 43+56:8-({\color {red}30-2}\cdot 5):4=}$${\displaystyle 43+7-({\color {red}28}\cdot 5):4=}$...→ Punkt vor Strich45 ${\displaystyle 43+72:8-(56-2\cdot {\color {red}20}):{\color {red}4}=}$${\displaystyle 43+9-(56-2\cdot {\color {red}5})=}$... → Klammer vor Punkt48 ${\displaystyle 63-(56:7-2)\cdot {\color {red}9+72}:3=}$${\displaystyle 63-(8-2)\cdot {\color {red}81}:3=}$...→ Punkt vor Strich33 ${\displaystyle 21+56:7-({\color {red}77-2\cdot 4}):3=}$${\displaystyle 21+8-({\color {red}75}\cdot 4):3=}$...→ Punkt vor Strich6 ${\displaystyle {\color {red}40+56:8}(30-2\cdot 5):4=}$${\displaystyle {\color {red}96:8}(30-10):4=}$...→ Punkt vor Strich45 ${\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 Punkt38 ${\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 Strich73

## Grundniveau 2

### Gemischte Zahlen

#### Gemischte Zahl in unechten Bruch

 ${\displaystyle \ {\tfrac {40}{9}}\qquad }$${\displaystyle \ {\tfrac {120}{13}}\qquad }$${\displaystyle \ {\tfrac {12}{7}}}$ ${\displaystyle \ {\tfrac {53}{7}}\qquad }$${\displaystyle \ {\tfrac {9}{5}}\qquad }$${\displaystyle \ {\tfrac {35}{3}}}$ ${\displaystyle \ {\tfrac {11}{3}}\qquad }$${\displaystyle \ {\tfrac {76}{11}}\qquad }$${\displaystyle \ {\tfrac {17}{9}}}$ ${\displaystyle \ {\tfrac {37}{7}}\qquad }$${\displaystyle \ {\tfrac {7}{6}}\qquad }$${\displaystyle \ {\tfrac {89}{9}}}$ ${\displaystyle \ {\tfrac {37}{9}}\qquad }$${\displaystyle \ {\tfrac {111}{13}}\qquad }$${\displaystyle \ {\tfrac {36}{7}}}$ ${\displaystyle \ {\tfrac {60}{7}}\qquad }$${\displaystyle \ {\tfrac {19}{5}}\qquad }$${\displaystyle \ {\tfrac {32}{3}}}$ ${\displaystyle \ {\tfrac {5}{3}}\qquad }$${\displaystyle \ {\tfrac {72}{11}}\qquad }$${\displaystyle \ {\tfrac {25}{13}}}$ ${\displaystyle \ {\tfrac {47}{7}}\qquad }$${\displaystyle \ {\tfrac {13}{12}}\qquad }$${\displaystyle \ {\tfrac {82}{9}}}$

#### Unechten Bruch in gemischte Zahl

 ${\displaystyle \ 35{\tfrac {7}{11}}\qquad }$${\displaystyle \ 7\qquad }$${\displaystyle \ 11{\tfrac {1}{12}}}$ ${\displaystyle \ 37{\tfrac {3}{12}}\qquad }$${\displaystyle \ 6{\tfrac {3}{5}}\qquad }$${\displaystyle \ 11}$ ${\displaystyle \ 34\qquad }$${\displaystyle \ 1{\tfrac {4}{5}}\qquad }$${\displaystyle \ 12{\tfrac {1}{8}}}$ ${\displaystyle \ 40{\tfrac {7}{11}}\qquad }$${\displaystyle \ 1{\tfrac {2}{7}}\qquad }$${\displaystyle \ 7}$ ${\displaystyle \ 35{\tfrac {4}{11}}\qquad }$${\displaystyle \ 9\qquad }$${\displaystyle \ 11{\tfrac {3}{12}}}$ ${\displaystyle \ 37{\tfrac {7}{12}}\qquad }$${\displaystyle \ 7{\tfrac {4}{5}}\qquad }$${\displaystyle \ 11}$ ${\displaystyle \ 35\qquad }$${\displaystyle \ 2{\tfrac {1}{5}}\qquad }$${\displaystyle \ 11{\tfrac {7}{8}}}$ ${\displaystyle \ 37\qquad }$${\displaystyle \ 1{\tfrac {1}{7}}\qquad }$${\displaystyle \ 7{\tfrac {3}{5}}}$

#### Subtraktion

 ${\displaystyle \ {\tfrac {11}{9}}\qquad }$${\displaystyle \ 8{\tfrac {10}{13}}\qquad }$${\displaystyle \ -1{\tfrac {1}{7}}}$ ${\displaystyle \ 3{\tfrac {4}{7}}\qquad }$${\displaystyle \ {\tfrac {1}{5}}\qquad }$${\displaystyle \ {\tfrac {11}{3}}}$ ${\displaystyle \ -5{\tfrac {1}{3}}\qquad }$${\displaystyle \ {\tfrac {56}{11}}\qquad }$${\displaystyle \ {\tfrac {1}{9}}}$ ${\displaystyle \ -{\tfrac {17}{7}}\qquad }$${\displaystyle \ {\tfrac {5}{6}}\qquad }$${\displaystyle \ {\tfrac {53}{9}}}$ ${\displaystyle \ -{\tfrac {2}{9}}\qquad }$${\displaystyle \ -{\tfrac {1}{8}}\qquad }$${\displaystyle \ {\tfrac {34}{7}}}$ ${\displaystyle \ {\tfrac {52}{7}}\qquad }$${\displaystyle \ -1{\tfrac {4}{5}}\qquad }$${\displaystyle \ {\tfrac {9}{11}}}$ ${\displaystyle \ {\tfrac {5}{7}}\qquad }$${\displaystyle \ -{\tfrac {1}{11}}\qquad }$${\displaystyle \ 4{\tfrac {1}{13}}}$ ${\displaystyle \ -2{\tfrac {3}{7}}\qquad }$${\displaystyle \ {\tfrac {11}{12}}\qquad }$${\displaystyle \ 8{\tfrac {1}{9}}}$

### Bruchkürzen

 ${\displaystyle \ {\frac {2}{5}}\qquad }$${\displaystyle \ {\frac {3}{7}}\qquad }$${\displaystyle \ {\frac {3}{2}}\qquad }$${\displaystyle \ {\frac {5}{2}}\qquad }$${\displaystyle \ {\frac {7}{4}}\qquad }$ ${\displaystyle \ {\frac {2}{3}}\qquad }$${\displaystyle \ {\frac {3}{5}}\qquad }$${\displaystyle \ {\frac {3}{4}}\qquad }$${\displaystyle \ {\frac {5}{6}}\qquad }$${\displaystyle \ {\frac {2}{3}}\qquad }$ ${\displaystyle \ {\frac {2}{3}}\qquad }$${\displaystyle \ {\frac {10}{11}}\qquad }$${\displaystyle \ {\frac {9}{11}}\qquad }$${\displaystyle \ {\frac {5}{6}}\qquad }$${\displaystyle \ {\frac {3}{2}}\qquad }$ ${\displaystyle \ {\frac {3}{2}}\qquad }$${\displaystyle \ {\frac {7}{4}}\qquad }$${\displaystyle \ {\frac {4}{13}}\qquad }$${\displaystyle \ {\frac {3}{4}}\qquad }$${\displaystyle \ {\frac {1}{3}}\qquad }$

### Umformen Grundwissen Gegenrechnungen

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

### Einheiten und physikalische Größen

#### Typ 1

1.  Ordnen Sie richtig zu: Länge einer Zunge cm ${\displaystyle \qquad }$ ${\displaystyle \ }$cm³ Dauer eines Filmes h ${\displaystyle \ }$km Dauer eines Herzschlags s ${\displaystyle \ }$m Länge eines Zuges m ${\displaystyle \ }$h Abstand zwischen Paris und Rom${\displaystyle \quad }$ km${\displaystyle \ \ \ }$ ${\displaystyle \ }$s Volumen einer Spritze cm³ ${\displaystyle \ }$cm
1.  Ordnen Sie richtig zu: Höhe eines Fernsehturms m ${\displaystyle \qquad }$ ${\displaystyle \ }$cm³ Volumen eines Ölkanisters ${\displaystyle {\boldsymbol {\ell }}}$ ${\displaystyle \ }$km Dauer einer Unterrichtspause min ${\displaystyle \ }$m Volumen eines LKWs m3 ${\displaystyle \ }$m3 Abstand Mogadischu-Kambala${\displaystyle \quad }$ km${\displaystyle \quad }$ ${\displaystyle \ }$min Volumen einer Spritze cm3 ${\displaystyle \ \ell }$
1.  Ordnen Sie richtig zu: Fläche eines Fingernagels mm2 ${\displaystyle \qquad }$ ${\displaystyle \ }$m2 Dauer einer Flugreise h ${\displaystyle \ }$km2 Höhe eines Hauses m ${\displaystyle \ }$h Fläche eines Zimmers m2 ${\displaystyle \ }$m Abstand zwischen den Augen${\displaystyle \quad }$ cm${\displaystyle \quad }$ ${\displaystyle \ }$mm2 Fläche eines Staates km2 ${\displaystyle \ }$cm
1.  Ordnen Sie richtig zu: Fläche eines Staates km² ${\displaystyle \qquad }$ ${\displaystyle \ }$m2 Dauer einer Flugreise h ${\displaystyle \ }$km2 Dauer eine Schulpause min ${\displaystyle \ }$h Fläche eines Zimmers m² ${\displaystyle \ }$min Abstand zwischen den Augen${\displaystyle \quad }$ cm${\displaystyle \quad }$ s${\displaystyle \ }$ Dauer eines Atemzugs s cm${\displaystyle \ }$

#### Typ 2

 ${\displaystyle \ m^{2}\qquad }$${\displaystyle \ m\qquad }$${\displaystyle \ h\qquad }$${\displaystyle \ km\qquad }$${\displaystyle \ g\qquad }$${\displaystyle \ m\qquad }$ ${\displaystyle \ s\qquad }$${\displaystyle \ dm^{2}\qquad }$${\displaystyle \ kg\qquad }$${\displaystyle \ km^{2}\qquad }$${\displaystyle \ mm\qquad }$${\displaystyle \ t\qquad }$ ${\displaystyle \ m^{2}\qquad }$${\displaystyle \ min\qquad }$${\displaystyle \ m^{3}\qquad }$${\displaystyle \ mm\qquad }$${\displaystyle \ g\qquad }$${\displaystyle \ cm^{3}\qquad }$ ${\displaystyle \ dm^{2}\qquad }$${\displaystyle \ cm^{3}\qquad }$${\displaystyle \ t\qquad }$${\displaystyle \ mm\qquad }$${\displaystyle \ m^{3}\qquad }$${\displaystyle \ min\qquad }$ ${\displaystyle \ cm^{2}\qquad }$${\displaystyle \ km\qquad }$${\displaystyle \ min\qquad }$${\displaystyle \ dm\qquad }$${\displaystyle \ t\qquad }$${\displaystyle \ cm\qquad }$ ${\displaystyle \ min\qquad }$${\displaystyle \ dm^{2}\qquad }$${\displaystyle \ cm^{2}\qquad }$${\displaystyle \ m^{2}\qquad }$${\displaystyle \ cm\qquad }$${\displaystyle \ kg\qquad }$ ${\displaystyle m^{2}}$${\displaystyle s}$${\displaystyle m^{3}}$${\displaystyle mm}$${\displaystyle kg}$${\displaystyle m\ell \ (cm^{3})}$ ${\displaystyle cm^{2}}$${\displaystyle cm^{3}}$${\displaystyle g}$${\displaystyle mm}$${\displaystyle mm^{3}}$${\displaystyle s}$

### Einheiten ohne Hochzahl

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

### Lageparameter

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

### Säulendiagramm

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

### Kürzen mit Primfaktorzerlegung

 ${\displaystyle \ {\frac {34}{45}}\qquad }$${\displaystyle \ {\frac {91}{99}}\qquad }$${\displaystyle \ {\frac {77}{68}}}$ ${\displaystyle \ {\frac {10}{9}}\qquad }$${\displaystyle \ {\frac {121}{225}}\qquad }$${\displaystyle \ {\frac {56}{55}}}$ ${\displaystyle \ {\frac {21}{22}}\qquad }$${\displaystyle \ {\frac {65}{77}}\qquad }$${\displaystyle \ {\frac {136}{273}}}$ ${\displaystyle \ {\frac {52}{35}}\qquad }$${\displaystyle \ {\frac {13}{33}}\qquad }$${\displaystyle \ {\frac {77}{45}}}$ ${\displaystyle \ {\frac {17}{9}}\qquad }$${\displaystyle \ {\frac {455}{297}}\qquad }$${\displaystyle \ {\frac {385}{306}}}$ ${\displaystyle \ {\frac {4}{9}}\qquad }$${\displaystyle \ {\frac {121}{25}}\qquad }$${\displaystyle \ {\frac {168}{275}}}$ ${\displaystyle \ {\frac {27}{22}}\qquad }$${\displaystyle \ {\frac {65}{77}}\qquad }$${\displaystyle \ {\frac {4}{3}}}$ ${\displaystyle \ {\frac {136}{245}}\qquad }$${\displaystyle \ {\frac {13}{33}}\qquad }$${\displaystyle \ {\frac {11}{5}}}$

### Prozentrechnung bei Wachstum und Abnahme

 ${\displaystyle 1755\ {\text{€}}\qquad 45\ {\text{€}}}$ ${\displaystyle 24\ {\text{cm}}\qquad 36\ {\text{cm Unterschied (das ist 60}}\ \%{\text{ von 60 cm)}}}$ ${\displaystyle 60\ {\text{cm}}\qquad 36\ {\text{cm Unterschied (das ist 150}}\ \%{\text{ von 24 cm)}}}$ ${\displaystyle 685100\ {\text{€}}\qquad 35100\ {\text{€ mehr}}}$ ${\displaystyle 71{,}4\ {\text{kg}}\qquad 3{,}4\ {\text{kg}}}$ ${\displaystyle 68{,}4\ {\text{kg}}\qquad 3{,}6\ {\text{kg}}}$ ${\displaystyle 3{,}91\ {\text{min}}\qquad 0{,}51\ {\text{min}}}$ ${\displaystyle 50\ {\text{Jahre}}\qquad 30\ {\text{Jahre mehr}}}$

### Einheiten mit Hochzahl

 ${\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 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 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 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 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 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 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 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 }$

### Formel Einsetzen in der ebenen Geometrie

 ${\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 \ 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 \ 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 \ 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 \ 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 \ 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 \ 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 \ 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 }$

### Liniendiagramm

 ${\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 }$${\displaystyle {\text{ca. }}10^{40}\quad 16^{15}\quad 19\quad {\text{und }}21^{50}\ {\text{Uhr}}\qquad }$ ${\displaystyle \ {\text{ca. }}4\quad 1\quad 1\quad {\text{bzw. }}6\ ^{\circ }C\qquad }$${\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}$${\displaystyle \ {\text{ca. }}-1\quad {\text{bzw. }}6{,}6\ m}$ ${\displaystyle \ {\text{ca. }}3\quad 2\quad 5{,}5\quad {\text{bzw. }}8\quad {\text{F/min}}\qquad }$${\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 }$${\displaystyle \ {\text{ca. um }}\ 5\ {\text{und um}}\ 12\qquad }$ ${\displaystyle \ {\text{ca. }}200\quad 250\quad {\text{bzw. }}280\quad {\text{ppmv}}\qquad }$${\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 \ 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 {\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 {\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 }$${\displaystyle {\text{ca. }}10^{40}\quad 16^{15}\quad 19\quad {\text{und }}21^{50}\ {\text{Uhr}}\qquad }$ ${\displaystyle {\text{ca. }}\ 36{,}1^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}5^{\circ }C\quad 36{,}4^{\circ }C\quad \qquad }$${\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 }$

### Indirekte Proportionalität

#### Typ 1

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

#### Typ 2

 1,5 Tage später 34 Tage 20 Jahre 7 Kinder je 3 Stücke und 14 Kinder je 6 Stücke 1,5 Tage später 34 Tage 15,5 Tage 100000 €

### Textaufgaben zu den Bruchrechnungen

 ${\displaystyle \ 3{,}76\ {\text{Mill. Liter}}\qquad \ 30080{\text{€}}\qquad }$${\displaystyle A\ 42000\ {\text{€}}\qquad B\ 35000\ {\text{€}}\qquad C\ und\ D\ je\ 14000\ {\text{€}}\qquad }$ 360 t Kart., 280 t Tom., 189 t Gur., 11 t Karot., 420 t Getr.${\displaystyle \textstyle {\frac {1}{3}}\ {\text{der Ernte}}}$ 462 Öst., 168 Serb., 132 Türk., 162 Rest${\displaystyle \textstyle {\frac {27}{154}}\ {\text{der SchülerInnen}}}$ ${\displaystyle \textstyle {\frac {1}{98000}}\ {\text{der Menschen}}\qquad \ 352\ {\text{Millionen €}}\qquad }$${\displaystyle \textstyle {\frac {1}{3675}}\ {\text{der Menschen}}\qquad \ 400\ {\text{Millionen €}}\qquad }$${\displaystyle \textstyle {\frac {4}{11}}\ {\text{der Menschen}}\qquad \ 480\ {\text{Millionen €}}\qquad }$${\displaystyle \textstyle {\text{fast}}\ {\frac {7}{11}}\ {\text{der Menschen}}\qquad \ 88\ {\text{Millionen €}}\qquad }$ die Orangen${\displaystyle 128\ kg}$ Saskia${\displaystyle 5\ Punkte}$ die Hosen${\displaystyle 28}$ Elektrolytkondensatoren${\displaystyle 313}$g

### Sachaufgaben zu den Grundrechenarten

 50,4 €das 4-Fache 93 126 93

## Vertiefendes Niveau 1

### Umkehraufgaben der Prozentrechnung

 ${\displaystyle 4{,}5\ {\text{kW}}}$ ${\displaystyle 3\ {\text{m}}}$ ${\displaystyle 35\ {\text{cm}}}$ ${\displaystyle 0{,}24\ {\text{kW}}}$

### Bruchstrichrechnungen mit Primfaktorzerlegung

 ${\displaystyle -{\frac {313}{1800}}}$ ${\displaystyle {\frac {296}{60}}\ also\ {\frac {74}{15}}}$ ${\displaystyle -{\frac {232}{252}}\ also\ -{\frac {58}{63}}}$ ${\displaystyle {\frac {649}{504}}}$ ${\displaystyle -{\frac {85}{100}}\ also\ -{\frac {17}{20}}}$ ${\displaystyle -{\frac {32}{36}}\ also\ -{\frac {8}{9}}}$ ${\displaystyle -{\frac {37}{300}}}$ ${\displaystyle {\frac {1082}{180}}\ also\ {\frac {541}{90}}}$

### Umformen einfache Kombinationen

 ${\displaystyle x=5}$8 ${\displaystyle b=-2}$−3 ${\displaystyle m=0{,}4}$−1 ${\displaystyle z=-2}$−1 ${\displaystyle z=2{,}5}$2 ${\displaystyle z=-{\frac {1}{3}}}$−1 ${\displaystyle z=5}$−2 ${\displaystyle z=-1}$1

### Vergleich direkter und indirekter Proportionalität

 1,65 kWh${\displaystyle 3{,}{\dot {8}}\ {\text{kWh pro h}}}$1,25 Milliarden Menschen${\displaystyle 36{,}9{\dot {4}}\ {\text{kWh}}}$ 525 €87,5 €3937,5 € 9,375 mal (durchschnittlich)7 malca. 7,2 mal (durchschnittlich) 3,15 h14 h19,6875 h ${\displaystyle 325\ t}$11,7 Tage6 Tage 40,5 Tageca. 11,6 Tageca. 3.8 Tage 14 Kinder9 Kinder20 Tage 4,2 Tage25 Tage27 Arbeiter

### Punktrechnungen von zwei Potenzen mit der gleichen Basis

 ${\displaystyle \ a^{-2}\qquad }$${\displaystyle \ 4^{3+b}\qquad }$${\displaystyle \ c^{-4}\qquad }$${\displaystyle \ w^{-3}}$ ${\displaystyle \ a^{b-x}\qquad }$${\displaystyle \ 4^{4}\qquad }$${\displaystyle \ c^{-14}\qquad }$${\displaystyle \ w^{12}}$ ${\displaystyle \ 1\qquad }$${\displaystyle \ t^{-3b}\qquad }$${\displaystyle \ c^{3-b}\qquad }$${\displaystyle \ w^{-b-12}}$ ${\displaystyle \ 3\qquad }$${\displaystyle \ b^{-7t}\qquad }$${\displaystyle \ b^{2}\qquad }$${\displaystyle \ w^{12-b}}$ ${\displaystyle \ c^{-7}\qquad }$${\displaystyle \ 3^{2t-7}\qquad }$${\displaystyle \ 5^{6}\qquad }$${\displaystyle \ w^{-5b}}$ ${\displaystyle \ 3^{0}\ also\ 1\qquad }$${\displaystyle \ a^{z-2t}\qquad }$${\displaystyle \ b^{-2}\qquad }$${\displaystyle \ 3^{-4w}}$ ${\displaystyle \ 7^{b}\qquad }$${\displaystyle \ t^{t-3}\qquad }$${\displaystyle \ 20^{2}\qquad }$${\displaystyle \ u^{b+12}}$ ${\displaystyle \ 7\qquad }$${\displaystyle \ j^{2w}\qquad }$${\displaystyle \ 3^{4b}\qquad }$${\displaystyle \ b^{-w-5}}$

### Textaufgaben linearer Gleichungssysteme mit 2 Variablen

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

### Kombinationsaufgaben der Prozentrechnung

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

### Vorrang und Bruchrechnungen

#### Vorrang mit Klammern in Klammern

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

#### Bruchrechnungen und Vorrang

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

### Umformen in der ebenen Geometrie konkret

 ${\displaystyle A=9\ cm^{2}}$ ${\displaystyle u\approx 12{,}23\ cm\qquad }$ ${\displaystyle A=90\ cm^{2}\qquad }$ ${\displaystyle A\approx 11{,}46\ cm^{2}\qquad }$ ${\displaystyle u=96\ cm\qquad }$ ${\displaystyle u=18\ dm\qquad }$ ${\displaystyle u=7{,}95\ dm\qquad }$ ${\displaystyle A\approx 6{,}93\ cm^{2}\qquad }$

### Mittelwerte bei einem Säulendiagramm

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

## Vertiefendes Niveau 2

### Prozentrechnung und Brüche

 ✘, ✔, ✔ ✘, ✘, ✘ ✘, ✔, ✔ ✔, ✘, ✔ ✘, ✔, ✔ ✘, ✘, ✘ ✘, ✔, ✔ ✔, ✘, ✔

### Umformen in der ebenen Geometrie abstrakt

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

### Lineare Funktion Diagramm

 ca. 2500 €, 3700 €, −600 € bzw. −1200 €ca. 1 t, 1200 €ca. 2,6, 3,8 bzw. 5,8 t ca.60, 50, 35 bzw. 10 Hzca. 70 Hz, 140 cmca. 100, 60, 40 bzw. 120 cm ca. 85, 70, bzw. 52 g/Lca. 100 g/L, 250 °Cca. 0, 75, 100 bzw. 150 °C ca. 71 Jahreca. 77 Jahreca. 30 Zig./Tagca. 34 Zig./Tagca. 85 Jahre ${\displaystyle 600\ m}$${\displaystyle 346\ m}$${\displaystyle 2{,}2\ km}$${\displaystyle 1{,}5\ km}$ ${\displaystyle 3{,}5\ t\ CO_{2}}$${\displaystyle 4\ t\ CO_{2}}$${\displaystyle ca.\ 260\ g}$${\displaystyle ca.\ 620\ g}$ ${\displaystyle ca.\ 3{,}8\ m}$${\displaystyle ca.\ 5{,}3\ m}$${\displaystyle 1{,}1\ h}$${\displaystyle 1{,}75\ h}$ ca. 71 Jahreca. 77 Jahreca. 30 Zig./Tagca. 34 Zig./Tagca. 85 Jahre

### Kreisdiagramm

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

### Vergleichen von Mittelwerten

 Verteilung möglicherweise gleichmäßig Verteilung möglicherweise gleichmäßig Verteilung eher ungleichmäßig (?) DE: Verteilung ungleichmäßigGR: Verteilung möglicherweise gleichmäßig Verteilung ungleichmäßig Verteilung eher ungleichmäßig (?) AT: Verteilung ungleichmäßigPO: Verteilung möglicherweise gleichmäßig Verteilung ungleichmäßig

### Wachstum und Abnahme

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

### Satz von Pythagoras

#### Typ 1

 ${\displaystyle 145\ {\text{mm}}}$ ${\displaystyle 112\ \ {\text{dm}}}$ ${\displaystyle 18541\ \ {\text{cm}}}$ ${\displaystyle 119\ \ {\text{mm}}}$

#### Typ 2

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

### Umsatzsteuer und Rabatt

#### Umsatzsteuer (USt.)

 BVP: 56 €, USt.: 6 € NVP: 60 € 12,5% NVP: 75 €

#### Rabatt

 BVP: 66 € PnR: 572 € 12% ${\displaystyle 685100\ {\text{€}}\qquad 35100\ {\text{€ mehr}}}$
{{clear}

#### USt. und Rabatt Gegebener Endwert

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

#### USt. und Rabatt Kombinationsaufgaben

1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€
1.  ${\displaystyle \ }$Berechnen Sie die fehlenden Werte in der Tabelle:${\displaystyle \ }$ ${\displaystyle \ }$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€

### Eine lineare Funktion mit Hilfe von zwei Punkten ermitteln

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

### Geometrie Beweise

1. hier
1. hier
1. hier
1. hier
1. hier
1. hier
1.  + =
1.  ${\displaystyle -}$ =

### Zinsrechnung

 ${\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 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 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\%}$