---
tags:
  - prealgebra
  - fractions
---

# Dividing fractions

Suppose you have the following shape:

![draw.io-Page-9.drawio 1.png](draw.io-Page-9.drawio.png)

One part is shaded. This represents one-eighth of the original shape.

![one-eighth-a.png](one-eighth-a.png)

Now imagine there are four instances of the shape and one-eighth remains shaded.
How man one-eighths are there in four?

![draw.io-Page-9.drawio 2.png](draw.io-Page-9.drawio.png)

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](Reciprocals.md) 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}


$$
