Question

What is the value of? 1/3÷4

Answer 1

`1/12` is the value.

Explanation:

What you do is the KCF method. Keep, Change, Flip. You would keep the `1/3`. Then you change the divide sign to a multiply sign. Then you flip the `4` to `1/4`. You do that since `1/4` is the reciprocal of `4`.

`1/3 div 4 = 1/3 xx 1/4`

Answer 2

`1/12`

Explanation:

You can work it out using the usual fraction division process, or just through what is happening...

If you take one third and cut it in half ( same as dividing by `2`), then each piece will be `1/6`. (More pieces, therefore they get smaller)

If you take `1/6` and cut it in half, the pieces get smaller again. Each piece will be `1/12`

`1/3 div 4 = 1/3 div 2 div 2 = 1/12`

A nifty short cut: To divide a fraction in half, either halve the top (if it is even) or double the bottom:

`2/3 div 2 = 1/3`

`4/11 div 2 = 2/11" "larr` pretty obvious if you think about it!!

`5/9 div 2 = 5/18`

`7/8 div 2 = 7/16`

In the same way: To divide a fraction by `3` in half, either divide the by `3` (if possible) or treble the bottom:

`6/11 div 3 = 2/11" "larr` share out `6` portions equally.

`5/8 div 3 = 5/24`

Answer 3

This is why the 'turn upside down and multiply' works.

Explanation:

`color(blue)("Answering the question using the shortcut method")`

Write as `1/3-: 4/1`

giving: `1/3xx1/4= (1xx1)/(3xx4)=1/12`
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`color(white)()`

`color(blue)("The teaching bit")`

A fraction structure is such that we have:

`("numerator")/("denominator") ->("count")/("size indicator of what you are counting")`

YOU CAN NOT `color(red)(ul("DIRECTLY"))` ADD, SUBTRACT OR DIVIDE ONLY THE COUNTS UNLESS THE SIZE INDICATORS ARE THE SAME.

You have been applying this rule for years without realising it!
Consider the numbers: 1,2,3,4,5 and so on. Did you know that it mathematically correct to write them as: `1/1,2/1,3/1,4/1,5/1` and so on. So their SIZE INDICATORS ARE THE SAME.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Explaining the principle using a different example")`

`color(brown)("I have chosen to use a different example as I wished")` `color(brown)("to avoid using 1's. In avoiding 1's the behaviour is more obvious.")`

Consider the example `color(green)(3/color(red)(4)-:2/color(red)(8)")`

Turn upside down and change the sign to multiply

`color(green)(3/color(red)(4)xxcolor(red)(8)/2 larr" as per the method"`

Note that: `4xx2=8 =2xx4.` This is commutative.

Using the principle of being commutative swap the 4 and 2 round the other way giving:

`color(green)(color(white)("ddd")ubrace(3/2)color(white)("ddd")xxcolor(white)("ddd")color(red)(ubrace(8/4))`

`color(green)("directly dividing ") color(red)("Converting the")`
`color(green)(color(white)("dd")"the counts")color(white)("ddddddd") color(red)("counts")`

Now split them up like this:

`( color(green)( 3)xxcolor(red)(8/4)) -:color(green)(2)`

`color(magenta)(color(white)("ddd") 6 color(white)("dddd")-:2)`

And compare to the original of `color(green)([3/color(red)(4)]-:2/color(red)(8)")`

`color(white)()`

`color(green)([3/color(red)(4)color(black)(xx2/2)] color(green)(-:)2/color(red)(8))color(white)("dddd")->color(white)("dddd")color(magenta)(6)/8-:color(magenta)(2)/8`
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So the `color(red)(8/4)` is the equivalent action of making the size indicators the same and adjusting the counts to suit.

`color(red)("IT IS A CONVERSION FACTOR")`
So by turning upside down' and multiplying you are applying a conversion and directly dividing the counts all at once.