**Aimed at: KS3, GCSE and A-level**

**First we will see why it seems to make sense that you can divide by a fraction by simply flipping it upside down and multiplying; then we will prove it works. **

**INTUITIVELY:
**Let’s see if we can find a rule for dividing by $\frac{1}{2}$.

The $\frac{1}{2}$ times table goes like this: $\frac{1}{2}, 1, 1\frac{1}{2}, 2, 2\frac{1}{2}, 3, …$. Therefore:

**$1\div \frac{1}{2} = 2$** because $\frac{1}{2}$ goes into one two times.

**$2\div \frac{1}{2} = 4$** because $\frac{1}{2}$ goes into two four times.

**$3\div \frac{1}{2} = 6$** because $\frac{1}{2}$ goes into three six times.

Hmm, it’s looking like dividing by $\frac{1}{2}$ is the same as multiplying by 2. This suggests the rule that $\div \frac{a}{b}$ is the same as $\times \frac{b}{a}$

Now let’s prove it!

**THE PROOF:**

As with most mathematical proofs, algebra is the language that lets us deal with all cases at once: in this case we will divide $\frac{a}{b} \div \frac{c}{d}$ rather than choosing specific numbers.

We start by recalling the following:

**If you multiply the top and bottom of a fraction by the same number, you get an equivalent fraction.**

**And if you divide the top and bottom of a fraction by the same number, you again get an equivalent fraction (this means we can “cancel” any multiplier on the numerator of a fraction with the same multiplier on the denominator).**

Now then:

$\frac{a}{b} \div \frac{c}{d}$

$ = \Large{\frac{\frac{a}{b}}{\frac{c}{d}}}$

Now we multiply top and bottom by $bd$ to clear the “fractions within fractions” – we can do this and still have an equivalent fraction:

$ = \Large{\frac{\frac{a}{b} \times bd}{\frac{c}{d} \times bd}}$

$ = \large{\frac{a\times d}{b \times c}}$ (because the b’s cancel on top, and the d’s cancel on the denominator)

$ = \large{\frac{a}{b} \times \frac{d}{c}}$

#### so it is as if we had simply **kept** the first fraction, **changed **$\div$ to $\times$, and **flipped **the second fraction: **KEEP, FLIP, CHANGE**.

Magic!