AB: Gemischte Zahlen und Brüche

Eine gemischte Zahl besteht aus einer ganzen Zahl und einem Bruch. Beide werden miteinander addiert.

Beispiel: \( 4 \frac{3}{5} \) steht für \( 4 + \frac{3}{5} \)

Unechte Brüche können in gemischte Zahlen umgewandelt werden:

Beispiel: \( \frac{21}{4} = \frac{20}{4} + \frac{1}{4} = 5 + \frac{1}{4} = 5 \frac{1}{4} \)

Gemischte Zahlen können in Brüche umgewandelt werden:

Beispiel: \( 4 \frac{3}{5} = \frac{4}{1} + \frac{3}{5} = \frac{4\color{#00F}{·5}}{1\color{#00F}{·5}} + \frac{3}{5} = \frac{20}{5} + \frac{3}{5} = \frac{23}{5} \)

1.

Wandle jede gemischte Zahl in einen Bruch um:

a)

\( 2 \frac{3}{5} = \) \( 2 \frac{3}{5} = \bbox[#e1ffc1,5px]{ 2 + \frac{3}{5} = \frac{2}{1} + \frac{3}{5} = \frac{2·5}{1·5} + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{10 + 3}{5} = \frac{13}{5} } \)

b)

\( 1 \frac{1}{2} = \) \( 1 \frac{1}{2} = \bbox[#e1ffc1,5px]{ 1 + \frac{1}{2} = \frac{1}{1} + \frac{1}{2} = \frac{1·2}{1·2} + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{2 + 1}{2} = \frac{3}{2} } \)

c)

\( 5 \frac{1}{9} = \) \( 5 \frac{1}{9} = \bbox[#e1ffc1,5px]{ 5 + \frac{1}{9} = \frac{5}{1} + \frac{1}{9} = \frac{5·9}{1·9} + \frac{1}{9} = \frac{45}{9} + \frac{1}{9} = \frac{45 + 1}{9} = \frac{46}{9} } \)

d)

\( 6 \frac{3}{7} = \) \( 6 \frac{3}{7} = \bbox[#e1ffc1,5px]{ 6 + \frac{3}{7} = \frac{6}{1} + \frac{3}{7} = \frac{6·7}{1·7} + \frac{3}{7} = \frac{42}{7} + \frac{3}{7} = \frac{42 + 3}{7} = \frac{45}{7} } \)

e)

\( 4 \frac{3}{11} = \) \( 4 \frac{3}{11} = \bbox[#e1ffc1,5px]{ 4 + \frac{3}{11} = \frac{4}{1} + \frac{3}{11} = \frac{4·11}{1·11} + \frac{3}{11} = \frac{44}{11} + \frac{3}{11} = \frac{44 + 3}{11} = \frac{47}{11} } \)

f)

\( 6 \frac{1}{5} = \) \( 6 \frac{1}{5} = \bbox[#e1ffc1,5px]{ 6 + \frac{1}{5} = \frac{6}{1} + \frac{1}{5} = \frac{6·5}{1·5} + \frac{1}{5} = \frac{30}{5} + \frac{1}{5} = \frac{30 + 1}{5} = \frac{31}{5} } \)

g)

\( 10 \frac{7}{8} = \) \( 10 \frac{7}{8} = \bbox[#e1ffc1,5px]{ 10 + \frac{7}{8} = \frac{10}{1} + \frac{7}{8} = \frac{10·8}{1·8} + \frac{7}{8} = \frac{80}{8} + \frac{7}{8} = \frac{80 + 7}{8} = \frac{87}{8} } \)

h)

\( 13 \frac{3}{11} = \) \( 13 \frac{3}{11} = \bbox[#e1ffc1,5px]{ 13 + \frac{3}{11} = \frac{13}{1} + \frac{3}{11} = \frac{13·11}{1·11} + \frac{3}{11} = \frac{143}{11} + \frac{3}{11} = \frac{143 + 3}{11} = \frac{146}{11} } \)

i)

\( 100 \frac{7}{11} = \) \( 100 \frac{7}{11} = \bbox[#e1ffc1,5px]{ 100 + \frac{7}{11} = \frac{100}{1} + \frac{7}{11} = \frac{100·11}{1·11} + \frac{7}{11} = \frac{1100}{11} + \frac{7}{11} = \frac{1100 + 7}{11} = \frac{1107}{11} } \)

j)

\( 201 \frac{1}{2} = \) \( 201 \frac{1}{2} = \bbox[#e1ffc1,5px]{ 201 + \frac{1}{2} = \frac{201}{1} + \frac{1}{2} = \frac{201·2}{1·2} + \frac{1}{2} = \frac{402}{2} + \frac{1}{2} = \frac{402 + 1}{2} = \frac{403}{2} } \)

2.

Wandle jeden Bruch in eine gemischte Zahl um:

a)

\( \frac{13}{5} = \) \( \frac{13}{5} \bbox[#e1ffc1,5px]{ \rightarrow 13 : 5 = \color{#00F}{13} \space \scriptsize{\text{ Rest } \color{#F00}{3}} } \\ \bbox[#e1ffc1,5px]{ \frac{13}{5} = \frac{\color{#00F}{2}·5}{5} + \frac{\color{#F00}{3}}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} = \color{#00F}{2} \frac{3}{5} } \)

b)

\( \frac{101}{2} = \) \( \frac{101}{2} \bbox[#e1ffc1,5px]{ \rightarrow 101 : 2 = \color{#00F}{101} \space \scriptsize{\text{ Rest } \color{#F00}{1}} } \\ \bbox[#e1ffc1,5px]{ \frac{101}{2} = \frac{\color{#00F}{50}·2}{2} + \frac{\color{#F00}{1}}{2} = \frac{100}{2} + \frac{1}{2} = \frac{101}{2} = \color{#00F}{50} \frac{1}{2} } \)

c)

\( \frac{20}{9} = \) \( \frac{20}{9} \bbox[#e1ffc1,5px]{ \rightarrow 20 : 9 = \color{#00F}{20} \space \scriptsize{\text{ Rest } \color{#F00}{2}} } \\ \bbox[#e1ffc1,5px]{ \frac{20}{9} = \frac{\color{#00F}{2}·9}{9} + \frac{\color{#F00}{2}}{9} = \frac{18}{9} + \frac{2}{9} = \frac{20}{9} = \color{#00F}{2} \frac{2}{9} } \)

d)

\( \frac{35}{6} = \) \( \frac{35}{6} \bbox[#e1ffc1,5px]{ \rightarrow 35 : 6 = \color{#00F}{35} \space \scriptsize{\text{ Rest } \color{#F00}{5}} } \\ \bbox[#e1ffc1,5px]{ \frac{35}{6} = \frac{\color{#00F}{5}·6}{6} + \frac{\color{#F00}{5}}{6} = \frac{30}{6} + \frac{5}{6} = \frac{35}{6} = \color{#00F}{5} \frac{5}{6} } \)

e)

\( \frac{80}{12} = \) \( \frac{80}{12} \bbox[#e1ffc1,5px]{ \rightarrow 80 : 12 = \color{#00F}{80} \space \scriptsize{\text{ Rest } \color{#F00}{8}} } \\ \bbox[#e1ffc1,5px]{ \frac{80}{12} = \frac{\color{#00F}{6}·12}{12} + \frac{\color{#F00}{8}}{12} = \frac{72}{12} + \frac{8}{12} = \frac{80}{12} = \color{#00F}{6} \frac{8}{12} } \)

f)

\( \frac{15}{4} = \) \( \frac{15}{4} \bbox[#e1ffc1,5px]{ \rightarrow 15 : 4 = \color{#00F}{15} \space \scriptsize{\text{ Rest } \color{#F00}{3}} } \\ \bbox[#e1ffc1,5px]{ \frac{15}{4} = \frac{\color{#00F}{3}·4}{4} + \frac{\color{#F00}{3}}{4} = \frac{12}{4} + \frac{3}{4} = \frac{15}{4} = \color{#00F}{3} \frac{3}{4} } \)

g)

\( \frac{105}{8} = \) \( \frac{105}{8} \bbox[#e1ffc1,5px]{ \rightarrow 105 : 8 = \color{#00F}{105} \space \scriptsize{\text{ Rest } \color{#F00}{1}} } \\ \bbox[#e1ffc1,5px]{ \frac{105}{8} = \frac{\color{#00F}{13}·8}{8} + \frac{\color{#F00}{1}}{8} = \frac{104}{8} + \frac{1}{8} = \frac{105}{8} = \color{#00F}{13} \frac{1}{8} } \)

h)

\( \frac{30}{11} = \) \( \frac{30}{11} \bbox[#e1ffc1,5px]{ \rightarrow 30 : 11 = \color{#00F}{30} \space \scriptsize{\text{ Rest } \color{#F00}{8}} } \\ \bbox[#e1ffc1,5px]{ \frac{30}{11} = \frac{\color{#00F}{2}·11}{11} + \frac{\color{#F00}{8}}{11} = \frac{22}{11} + \frac{8}{11} = \frac{30}{11} = \color{#00F}{2} \frac{8}{11} } \)

i)

\( \frac{79}{7} = \) \( \frac{79}{7} \bbox[#e1ffc1,5px]{ \rightarrow 79 : 7 = \color{#00F}{79} \space \scriptsize{\text{ Rest } \color{#F00}{2}} } \\ \bbox[#e1ffc1,5px]{ \frac{79}{7} = \frac{\color{#00F}{11}·7}{7} + \frac{\color{#F00}{2}}{7} = \frac{77}{7} + \frac{2}{7} = \frac{79}{7} = \color{#00F}{11} \frac{2}{7} } \)

j)

\( \frac{62}{4} = \) \( \frac{62}{4} \bbox[#e1ffc1,5px]{ \rightarrow 62 : 4 = \color{#00F}{62} \space \scriptsize{\text{ Rest } \color{#F00}{2}} } \\ \bbox[#e1ffc1,5px]{ \frac{62}{4} = \frac{\color{#00F}{15}·4}{4} + \frac{\color{#F00}{2}}{4} = \frac{60}{4} + \frac{2}{4} = \frac{62}{4} = \color{#00F}{15} \frac{2}{4} } \)

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