# In division by a fraction why is it that we invert and then multiply? I posted this question so that I could explain why this works.

Sep 26, 2016

See the demonstration in the explanation

Solution 1 of 2
Also see my equivalent using algebra. (2 of 2)

#### Explanation:

$\textcolor{b l u e}{\text{Preamble}}$

Consider the example $\frac{1}{4} \div \frac{1}{8} = 2$

This is the same as:$\text{ } \left(\frac{1}{4} \textcolor{m a \ge n t a}{\times 1}\right) \div \frac{1}{8} = 2$

$\left(\frac{1}{4} \textcolor{m a \ge n t a}{\times \frac{2}{2}}\right) \div \frac{1}{8} = 2$

$\frac{1 \times 2}{4 \times 2} \div \frac{1}{8} = 2$

$\frac{2}{8} \div \frac{1}{8} = 2$

$\textcolor{b r o w n}{\text{Notice that if we just have the numerators it gives the same answer}}$

$2 \div 1 = 2$

$\textcolor{b r o w n}{\text{However, for direct division of numerators to work the denominators must be the same}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Using an example to demonstrate the principle}}$

Selecting numbers that are obviously different.

Suppose we had: $\textcolor{b r o w n}{\left[\frac{2}{4}\right]} \textcolor{g r e e n}{\div \frac{8}{6}}$

Change the 4 in color(brown)([2/4] such that it is in 6^("ths")" instead " -> color(red)(4)xx6/4

For multiply or divide, what we do to the bottom we do to the top.

$\textcolor{b r o w n}{\left[\frac{2 \times \frac{6}{4}}{4 \times \frac{6}{4}}\right]} \div \frac{8}{6}$

Now that both the denominators are in ${6}^{\text{ths}}$ we can ignore them and just consider the numerators.

$2 \times \frac{6}{4} \div 8 \text{ "->" } 2 \div 8 \times \frac{6}{4}$

$\frac{2}{8} \times \frac{6}{4} \text{ "->" } \frac{2 \times 6}{8 \times 4}$

Swap the 8 and 4 round

$\frac{2 \times 6}{4 \times 8} \text{ "=" } \textcolor{b r o w n}{\frac{2}{4}} \textcolor{g r e e n}{\times \frac{6}{8}}$

$\textcolor{p u r p \le}{\text{The "8/6" has now been inverted and the divide }}$
$\textcolor{p u r p \le}{\text{has become multiply. The principle has been demonstrated.}}$

Sep 26, 2016

See below for an alternate (perhaps more abstract) explanation than the one provided by Tony.

#### Explanation:

In part this question deals with what it means to divide.

In general $\textcolor{red}{a} \div \textcolor{b l u e}{b} = \textcolor{m a \ge n t a}{k}$
means $\textcolor{red}{a}$ is equivalent to $\textcolor{b l u e}{b}$ "pieces" each of size $\textcolor{m a \ge n t a}{k}$
or$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} = \textcolor{m a \ge n t a}{k} \times \textcolor{b l u e}{b}$

When dividing by a fraction, say $\textcolor{b l u e}{\frac{p}{q}}$ instead of $\textcolor{b l u e}{b}$
we could write $\textcolor{w h i t e}{\text{XX}} \textcolor{red}{a} \div \textcolor{b l u e}{\frac{p}{q}} = \textcolor{m a \ge n t a}{k}$
meaning $\textcolor{w h i t e}{\text{XX}} \textcolor{red}{a} = \textcolor{m a \ge n t a}{k} \times \textcolor{b l u e}{\frac{p}{q}}$

Provided $\textcolor{g r e e n}{\frac{q}{p}} \ne 0$
we also know that we can multiply both sides of an equation by $\textcolor{g r e e n}{\frac{q}{p}}$ and the equation will remain valid.

So
color(white)("XXX")color(red)(a)xxcolor(green)(q/p) = color(magenta)(k)xxcolor(blue)(cancel(p)/cancel(q)) xx color(green)(cancel(q)/cancel(p)

and since from our original specification that $\textcolor{red}{a} \div \textcolor{b l u e}{\frac{p}{q}} = \textcolor{m a \ge n t a}{k}$
it follows that
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} \div \textcolor{b l u e}{\frac{p}{q}} = \textcolor{red}{a} \times \textcolor{g r e e n}{\frac{q}{p}}$

Sep 28, 2016

$\textcolor{m a \ge n t a}{\text{I added this for completeness of my solution}}$

$\textcolor{m a \ge n t a}{\text{It is a general solution}}$
Solution 2 of 2
Also see my equivalent using numbers ( 1 of 2)

#### Explanation:

Suppose we hade $\textcolor{b l u e}{\frac{a}{b} \div \frac{c}{d}}$
Making the denominators so that they are all $d$

$\textcolor{b l u e}{\left[\frac{a}{b} \times 1\right] \div \frac{c}{d}}$

$\textcolor{b l u e}{\left[\frac{a \times \frac{d}{b}}{b \times \frac{d}{b}}\right] \div \frac{c}{d}}$
'...................................................................................
Note that: $\text{ "bxxd/b" " ->" " b/bxxd" " ->" " 1xxd" "=" } d$
'......................................................................................

This becomes

$\textcolor{b l u e}{\left[\frac{a \times \frac{d}{b}}{d}\right] \div \frac{c}{d}}$

This gives the same answer as:

$\textcolor{b l u e}{a \times \frac{d}{b} \div c} \rightarrow \text{ compare to } \left[\frac{4}{8} \div \frac{2}{8}\right] = 2 = \left[4 \div 2\right]$

$\textcolor{b l u e}{a \times \frac{d}{b} \times \frac{1}{c}}$

$\textcolor{b l u e}{\frac{a}{b} \times \frac{d}{c}}$

$\textcolor{g r e e n}{\text{Notice that the "-:c/d " has now become } \times \frac{d}{c}}$

$\textcolor{m a \ge n t a}{\text{Thus the rule, invert and multiply is true}}$

Jan 2, 2017

An alternate but similar approach added.

#### Explanation:

What does a fraction when represented as $\frac{a}{b}$ mean?
By definition it means for any two numbers $a \mathmr{and} b$, with the condition that $b \ne 0$

$a$ divided by $b$
or symbolically

$\implies a \div b$ ........(1)

We also know that $\frac{a}{b}$ can also be written as

$a \text{ multipled by } \frac{1}{b}$
or symbolically

$\implies a \times \frac{1}{b}$ .....(2)

For expressions in lines (1) and (2) to be equal

$a \div b \equiv a \times \frac{1}{b}$

This is same as saying that

$\text{division is multiplication with the reciprocal}$

Jan 2, 2017

The explanation is really simple...

Consider first $24 \div 3$

What we are actually asking is "

If I have 24 of anything, how many groups can I make with 3 in each group?"

This could be shown like this:
24 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
= (1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1 +1)+(1+1+1)

We can see that 8 possible groups can be made.

$24 \div 3 = 8$

What about 4 div 1/2?

We are asking how many groups can be made with $\frac{1}{2}$ in each?

$4 = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} \text{ } \leftarrow$ there are 8

This is because each $1$ has two halves in it and $4 \times 2 = 8$

What about 6 div 3/4?

We are asking " how many groups of $\frac{3}{4}$ can be made from 6"?
First we need to change $6$ into quarters, and then group them into threes.

Each 1 has four quarters in it.

$6 = 6 \times 4 = 24$ quarters

"How many groups of 3 quarters can be made from 24 quarters?"

$6 = \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right)$

There are 8 groups with three quarters in each.

$6 \div \frac{3}{4} = 8$

What did we do? We changed everything into quarters by multiplying by 4 and then divided by 3 to make groups of $\frac{3}{4}$

$6 \div \frac{3}{4} = 6 \times 4 \div 3$

$= 6 \times \frac{4}{3}$

$= 8$

In the same way. $4 \div \frac{2}{5}$

Make everything into fifths by $\times 5$, then $\div 2$ to make groups of $\frac{2}{5}$

$4 \div \frac{2}{5} = 4 \times 5 \div 2$

$= 4 \times \frac{5}{2}$

$= \frac{20}{2}$

$= 10$