# 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.

##### 5 Answers

See the demonstration in the explanation

Solution 1 of 2

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

#### Explanation:

Consider the example

This is the same as:

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Selecting numbers that are obviously different.

Suppose we had:

Change the 4 in

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

Now that both the denominators are in

Swap the 8 and 4 round

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

means

or

When dividing by a fraction, say

we could write

meaning

Provided

we also know that we can multiply both sides of an equation by

So

and since from our original specification that

it follows that

Solution 2 of 2

Also see my equivalent using numbers ( 1 of 2)

#### Explanation:

Suppose we hade

Making the denominators so that they are all

'...................................................................................

Note that:

'......................................................................................

This becomes

This gives the same answer as:

An alternate but similar approach added.

#### Explanation:

What does a fraction when represented as

By definition it means for any two numbers

#a# divided by#b#

or symbolically

#=>a -: b# ........(1)

We also know that

#a" multipled by "1/ b#

or symbolically

#=>axx1/ b# .....(2)

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

#a -: b-=axx1/ b#

This is same as saying that

#"division is multiplication with the reciprocal"#

The explanation is really simple...

Consider first

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.

What about

We are asking how many groups can be made with

This is because each

What about

We are asking " how many groups of

First we need to change

Each 1 has four quarters in it.

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

There are 8 groups with three quarters in each.

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

In the same way.

Make everything into fifths by