81 lines
2.2 KiB
Markdown
81 lines
2.2 KiB
Markdown
---
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tags:
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- prealgebra
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- fractions
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---
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# Dividing fractions
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Suppose you have the following shape:
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One part is shaded. This represents one-eighth of the original shape.
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Now imagine there are four instances of the shape and one-eighth remains shaded.
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How man one-eighths are there in four?
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The shaded proportion represents $\frac{1}{8}$ of the shape. Imagine four of
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these shapes, how many eighths are there?
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This is a division statement: to find how many one-eighths there are we would
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calculate:
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$$
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4 \div \frac{1}{8}
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$$
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But actually it makes more sense to think of this as a multiplication. There are
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four shapes of eight parts meaning there are $4 \cdot 8$ parts in total, 32. One
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of these parts is shaded making it equal to $\frac{1}{32}$.
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From this we realise that when we divide fractions by an amount, we can express
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the calculation in terms of multiplication and arrive at the correct answer:
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$$
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4 \div \frac{1}{8} = 4 \cdot 8 = 32
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$$
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Note that we omit the numerator but that technically the answer would be
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$\frac{1}{32}$.
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### Formal specification of how to divide fractions
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We combine the foregoing (that it is easier to divide by fractional amounts
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using multiplication) with the concept of a [reciprocol](Reciprocals.md) to
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arrive at a definitive method for dividing two fractions. It boils down to:
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_invert and multiply_:
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If $\frac{a}{b}$ and $\frac{c}{d}$ are fractions then:
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$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$
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We invert the divisor (the second factor) and change the operator from division
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to multiplication.
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#### Demonstration
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Divide $\frac{1}{2}$ by $\frac{3}{5}$
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$$
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\begin{split}
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\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \cdot \frac{5}{3} \\
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= \frac{5}{5}
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\end{split}
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$$
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Divide $\frac{-6}{x}$ by $\frac{-12}{x^2}$
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$$
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\begin{split}
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\frac{-6}{x} \div \frac{12}{x^2} = \frac{-6}{x} \cdot \frac{x^2}{-12} \\ =
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\frac{(\cancel{3} \cdot \cancel{2} )}{\cancel{x}} \cdot \frac{(\cancel{x} \cdot \cancel{x} )}{\cancel{3} \cdot \cancel{2} \cdot 2} \\ =
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\frac{x}{2}
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\end{split}
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$$
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