Ours is not to reason why; just invert and multiply.
- Anonymous

Rule number one of Fractions club is: do NOT let the Mathematical Ninja hear you talking like that, otherwise you’re not going to have ears to hear rule number two.

I mean - that is a way to divide fractions, and done correctly it works fine. However, it’s not something I often see done correctly.

(As an aside: the word ‘just’ is probably the most dangerous word in maths. “Do I just…?” will get my hackles up every time.)

Inspired by a recent post from @mathsjem, I thought it’d be a good idea to show some other explanations of why we divide fractions the way we do. In all of these, I’ll use the same example in Jo’s post: $\frac{2}{3} \div \frac{4}{5}$

The opposite of divide

My favourite way to answer the question is to think about what ÷ means: it’s “what do you multiply the second thing by to get the first thing?” For example, 18 ÷ 3 asks “What do you multiply 3 by to get 18?”

For this example, it’s “what do you multiply $\frac{4}{5}$ by to get $\frac{2}{3}$?”, and it’s easier if we call the answer $x$. Let’s go:

$\frac{4}{5}x = \frac{2}{3}$

We want to get $x$ on its own, so multiply both sides by 5:

$4x = \frac{10}{3}$

… and divide both sides by 4 (which is the same as multiplying the bottom by 4):

$x = \frac{10}{12} = \frac{5}{6}$. Easy.

Treat ÷ like -

A fraction is just a divide sum - and divide is the opposite of multiply. It’s VERY similar to how subtract is the opposite of add.

And you know how, when you work out something like $(3 + 5) - (4-3)$, it becomes $3 + 5 - 4 + 3$? It turns out that $(2 \div 3) \div (4 \div 5) = 2 \div 3 \div 4 \times 5$ - or, better, $(2 \times 5) \div (3 \times 4) = 10 \div 12 = \frac {5}{6}$

Common denominator method

You can also ask “how many $\frac{4}{5}$ths do you need to make up $\frac{2}{3}$?”

Let’s make the bottoms the same, even if that’s not what your teacher would tell you to do:

Four-fifths is the same as $\frac{12}{15}$; two-thirds is the same as $\frac{10}{15}$.

So how many twelve-fifteenths do you need to make ten-fifteenths? It’s the same as how many 12s you need to make 10. Or, if you prefer, $10 \div 12$, which is $\frac{5}{6}$.

Have you got any other favourite ways to divide fractions?

* Edited 2014-01-09 to fix a typo.