# MathemaTriX ⋅ Aufgabenheft Antworten

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

#### 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

### 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 }$

### Direkte Proportionalität

 1980 €132000 Flaschen ${\displaystyle \ x=0{,}0224\ \ l\quad }$ ${\displaystyle \ x\approx 14344\ {\text{ min}}}$ ${\displaystyle \ x=0{,}0735\ \ kg\quad }$ ${\displaystyle \ x\approx 916{,}7\ l}$ ${\displaystyle \ x\approx 0{,}125\ \ t\quad }$ ${\displaystyle \ x=7,3\ Tage}$ ${\displaystyle \ x=519{,}5\ \ Liter\quad }$ ${\displaystyle \ x=92{,}25\ \ t\quad }$ ${\displaystyle \ x=20\ {\text{Kühe}}}$ ${\displaystyle \ x=115\ {\text{Menschen}}}$${\displaystyle \ x=17{,}5\ \ km^{2}\quad }$ ${\displaystyle \ x=320\ m}$${\displaystyle \ x=28\ \ g\quad }$

### 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 }$

### 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}}$

### 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

## 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 }$${\displaystyle \ 7{\tfrac {3}{5}}\qquad }$${\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=-5514\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 }$${\displaystyle \ {\text{ca. um }}\ 5\ {\text{und um}}\ 12\qquad }$ ${\displaystyle \ {\text{ca. }}200\quad 230\quad {\text{bzw. }}270\quad {\text{ppmv}}\qquad }$${\displaystyle \ {\text{ca. }}410\ \ 320\ \ 260\ \ 130\ \ {\text{und }}\ 0\ }$Tausende Jahre her.${\displaystyle \ {\text{ca. }}350\ \ 260\ \ 160\ \ 50\ \ {\text{und }}\ 20\ }$Tausende Jahre her.Temperatur und CO2 Konzentration ändern sich fast genau in der gleichen Weise. ${\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 14{,}{\dot {4}}\ t}$11,7 Tage6 Tage 40,5 Tageca. 11,6 Tageca. 3.8 Tage 14 Kinder9 Kinder63 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. Höhe auf einer Seite ziehen
1. Formel für Diagonale in Bezug auf die Seite
beweisen und in die Fläche einsetzen
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 6153{,}17\ {\text{€}}\quad }$${\displaystyle eZs=1{,}35\%}$${\displaystyle G_{158}\approx 52989{,}79\ {\text{€}}}$ ${\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 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 }$