---
tags:
  - Mathematics
  - Prealgebra
  - fractions
  - division
---

Suppose you have the following shape:

![draw.io-Page-9.drawio 1.png](../../img/draw.io-Page-9.drawio%201.png)

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

![one-eighth-a.png](../../img/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](../../img/draw.io-Page-9.drawio%202.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}

$$
