70 lines
2.2 KiB
Markdown
70 lines
2.2 KiB
Markdown
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---
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tags:
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- Mathematics
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- Prealgebra
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- fractions
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- division
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---
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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. 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 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 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 four shapes of eight parts meaning there are $4 \cdot 8$ parts in total, 32. One 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 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 $\frac{1}{32}$.
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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 using multiplication) with the concept of a [reciprocol](Reciprocals.md) to arrive at a definitive method for dividing two fractions.
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It boils down to: *invert and multiply*:
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>
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> If $\frac{a}{b}$ and $\frac{c}{d}$ are fractions then: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$
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We invert the divisor (the second factor) and change the operator from division 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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