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Dividing fractions
Suppose you have the following shape:
One part is shaded. This represents one-eighth of the original shape.
Now imagine there are four instances of the shape and one-eighth remains shaded. How man one-eighths are there in four?
The shaded proportion represents \frac{1}{8} of the shape. Imagine four of
these shapes, how many eighths are there?
This is a division statement: to find how many one-eighths there are we would calculate:
4 \div \frac{1}{8}
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}.
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:
4 \div \frac{1}{8} = 4 \cdot 8 = 32
Note that we omit the numerator but that technically the answer would be
\frac{1}{32}.
Formal specification of how to divide fractions
We combine the foregoing (that it is easier to divide by fractional amounts using multiplication) with the concept of a reciprocol to arrive at a definitive method for dividing two fractions. It boils down to: invert and multiply:
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}We invert the divisor (the second factor) and change the operator from division to multiplication.
Demonstration
Divide \frac{1}{2} by \frac{3}{5}
\begin{split}
\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \cdot \frac{5}{3} \
= \frac{5}{5}
\end{split}
Divide \frac{-6}{x} by \frac{-12}{x^2}
\begin{split}
\frac{-6}{x} \div \frac{12}{x^2} = \frac{-6}{x} \cdot \frac{x^2}{-12} \ =
\frac{(\cancel{3} \cdot \cancel{2} )}{\cancel{x}} \cdot \frac{(\cancel{x} \cdot \cancel{x} )}{\cancel{3} \cdot \cancel{2} \cdot 2} \ =
\frac{x}{2}
\end{split}