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}

$$