Question

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.

Answer 1

See the demonstration in the explanation

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

Explanation:

`color(blue)("Preamble")`

Consider the example `1/4 -:1/8 = 2`

This is the same as:`" "(1/4color(magenta)(xx1))-:1/8=2`

`(1/4color(magenta)(xx2/2))-:1/8=2`

`(1xx2)/(4xx2)-:1/8=2`

`2/8-:1/8=2`

`color(brown)("Notice that if we just have the numerators it gives the same answer")`

`2-:1=2`

`color(brown)("However, for direct division of numerators to work the denominators must be the same")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Using an example to demonstrate the principle")`

Selecting numbers that are obviously different.

Suppose we had: `color(brown)([2/4])color(green)(-: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.

`color(brown)([(2xx6/4)/(4xx6/4)]) -:8/6`

Now that both the denominators are in `6^("ths")` we can ignore them and just consider the numerators.

`2xx6/4 -:8" "->" "2-:8xx6/4`

`2/8xx6/4" "->" "(2xx6)/(8xx4)`

Swap the 8 and 4 round

`(2xx6)/(4xx8)" "=" "color(brown)(2/4)color(green)(xx6/8)`

`color(purple)("The "8/6" has now been inverted and the divide ")`
`color(purple)("has become multiply. The principle has been demonstrated.")`

Answer 2

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 `color(red)(a) div color(blue)(b) = color(magenta)(k)`
means `color(red)(a)` is equivalent to `color(blue)(b)` "pieces" each of size `color(magenta)(k)`
or`color(white)("XXX")color(red)(a)=color(magenta)k xx color(blue)(b)`

When dividing by a fraction, say `color(blue)(p/q)` instead of `color(blue)(b)`
we could write `color(white)("XX")color(red)(a)divcolor(blue)(p/q)=color(magenta)(k)`
meaning `color(white)("XX")color(red)(a)=color(magenta)(k)xxcolor(blue)(p/q)`

Provided `color(green)(q/p)!=0`
we also know that we can multiply both sides of an equation by `color(green)(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 `color(red)(a)divcolor(blue)(p/q)=color(magenta)(k)`
it follows that
`color(white)("XXX")color(red)(a) divcolor(blue)(p/q)=color(red)(a)xxcolor(green)(q/p)`

Answer 3

`color(magenta)("I added this for completeness of my solution")`

`color(magenta)("It is a general solution")`
Solution 2 of 2
Also see my equivalent using numbers ( 1 of 2)

Explanation:

Suppose we hade `color(blue)(a/b -: c/d)`
Making the denominators so that they are all `d`

`color(blue)([a/bxx1]-:c/d)`

`color(blue)([(axx d/b)/(bxxd/b)] -:c/d)`
'...................................................................................
Note that: `" "bxxd/b" " ->" " b/bxxd" " ->" " 1xxd" "=" "d`
'......................................................................................

This becomes

`color(blue)([(axxd/b)/d] -:c/d)`

This gives the same answer as:

`color(blue)(axxd/b -:c)rarr" compare to "[4/8-:2/8]=2=[4-:2]`

`color(blue)(axxd/bxx1/c)`

`color(blue)(a/bxxd/c)`

`color(green)("Notice that the "-:c/d " has now become "xxd/c)`

`color(magenta)("Thus the rule, invert and multiply is true")`

Answer 4

An alternate but similar approach added.

Explanation:

What does a fraction when represented as `a/b` mean?
By definition it means for any two numbers `a and b`, with the condition that `b!=0`

`a` divided by `b`
or symbolically

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

We also know that `a/b` can also be written as

`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"`

Answer 5

The explanation is really simple...

Consider first `24div3`

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 `1/2` in each?

`4 = 1/2+1/2+1/2+1/2+1/2+1/2+1/2+1/2" "larr` there are 8

This is because each `1` has two halves in it and `4 xx2 =8`

What about `6 div 3/4?`

We are asking " how many groups of `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 xx4 = 24` quarters

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

`6 = (1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)+(1/4+1/4+1/4)`

There are 8 groups with three quarters in each.

`6 div 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 `3/4`

`6 div 3/4 = 6 xx4 div 3`

`= 6 xx4/3`

`=8`

In the same way. `4 div 2/5`

Make everything into fifths by `xx5`, then `div2` to make groups of `2/5`

`4 div 2/5 = 4 xx5 div 2`

`=4 xx5/2`

`=20/2`

`=10`