Divisionsregler
| Divisionsregler | |
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\[ \frac{a}{b} \frac{c}{d} =\frac{a}{b}\cdot\frac{c}{d} =\frac{ac}{bd},\] då \[ b,d \neq 0. \] |
\[ a\frac{b}{c} =a \cdot \frac{b}{c} = \frac{a}{1} \cdot \frac{b}{c} = \frac{ab}{c} \] då \[ \quad c \neq 0 .\] |
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\[ \frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}\] eller \[ \frac{-a}{-b} = \frac{a}{b}\] då \[ b \neq 0. \] |
\[ \frac{a}{b} \big/ \frac{c}{d} = \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} =\frac{ad}{bc} \] då \[ b,c \neq 0. \] |
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\[ \frac{a/b}{c} = \frac{\frac{a}{b}}{c} = \frac{a}{b} \cdot \frac{1}{c} = \frac{a}{bc}\] då \[ b,c \neq 0. \] |
\[ \frac{a}{b/c} = \frac{a}{\frac{b}{c}} = \frac{a}{1} \cdot \frac{c}{b} = \frac{ac}{b}\] då \[b,c \neq 0. \] |
Exempel:
\[ \frac{3}{2} \cdot \frac{4}{7} \]
\[ \begin{align}\frac{3}{2} \cdot \frac{4}{7} &= \frac{3\cdot4}{2\cdot 7}\\ &=\frac{3\cdot2 \cdot \cancel{2} }{\cancel{2}\cdot 7} \\& = \frac{6}{7} \end{align} \]
\[ 3\frac{2}{12} \]
\[ \begin{align} 3\frac{2}{12} &= 3\cdot\frac{2}{12} \\ &= \frac{3}{1}\cdot\frac{2}{12} \\ &= \frac{3\cdot 2}{1\cdot 12} \\ &= \frac{6}{12} \\ &= \frac{6}{6\cdot 2} \\ &= \frac{\cancel{6} }{\cancel{6}\cdot 2} \\ &= \frac{1}{2}. \end{align} \]
\[ \frac{3}{4} \big/ \frac{6}{8} \]
\[ \begin{align} \frac{3}{4} \big/ \frac{6}{8} &= \frac{\frac{3}{4}}{\frac{6}{8}} \\&= \frac{3}{4}\cdot \frac{8}{6} \\&= \frac{3 \cdot 8}{4 \cdot 6} \\&= \frac{\cancel{3} \cdot \cancel{2} \cdot \cancel{4} }{ \cancel{4} \cdot \cancel{3} \cdot \cancel{2}} \\&= 1. \end{align} \]