Question

What is 36 divided by 396?

Answer 1

`0.9090909...` going on for ever

Written mathematically as `0.90bar(90)`

Explanation:

`color(blue)("Introduction to a very different approach")`

`color(purple)("They are expecting you to do long division")`

In this question we are dividing a lesser number by a greater number. Let me show you a trick.

Consider the example: `3-:6 ->3/6` This is smaller divided by larger
We know that this is `(3-:3)/(6-:3)=1/2`
..........................................................................
Suppose I turn this upside down then I have `6/3=2` I now have larger divide by smaller. It is perfectly correct to write `2" as "2/1`
and if I turn `2/1` upside down I get `1/2` which is the correct answer for `3/6`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Given:`" "36-:396 ->36/396`

`color(red)("Invert this (turn upside down) and then do the division")`

`color(green)("Write as "396/36) larr" deliberate act"`

`"Starting point "->396`
`color(magenta)(11)xx36 ->" "ul(396) larr" subtract"`
`" "00`

So `396-:36=11/1`

`color(red)("Turn this answer upside down to answer the question")`

This means that `36-:396 ->36/396 = 1/11`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Footnote")`

By calculator:

`1/11= 0.9090909...` going on for ever

Written mathematically as `0.90bar(90)`

If no one demonstrates long division I will return to this question.

Answer 2

`color(red)("Note that the question does not specify the method of division")`

Long division method.

`0.09bar(09)`

`0.09` to 2 decimal places

Explanation:

write as:

`396color(white)(.)bar(|color(red)(36))`

.............................................
`36" less than "396` so write as:

`" "0.`
`396color(white)(.)bar(|360)`
........................................................

`360" less than "396` so write as:

`" "0.0`
`396color(white)(.)bar(|3600)`

`3600" greater then "396` so we can do the division
.....................................................
`" "color(purple)(0.09)`
`" " 396color(white)(.)bar(|3600)`
`9xx396 ->ul( 3564) larr" subtract"`
`" "color(red)(36)`

The `color(red)(36)` is a repeat of the starting point so this means that we will have a never ending repeat of `09` giving:

`" "color(green)(ul(bar(|color(white)(2/2)color(purple)(0.09)color(white)(.)09090909..... color(white)(2/2)|)))`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(white)(.)`

`color(blue)("Footnote")`

This is how the actual process should look:

`" "color(purple)(0.09)`
`" "396color(white)(.)bar(|color(red)(36. color(white)() 00))`
`" "ul(35color(white)(,)64) -`
`" "36`
............................................................................................................
Note that some people much prefer the subtract sign to be to the left of 3564. So in that case it would look like.

`" "color(purple)(0.09)`
`" "396color(white)(.)bar(|color(red)(36. color(white)() 00))`
`" "-ul(35color(white)(,)64) larr" alternative format"`
`" "36`

`color(brown)("Its position is a matter of current preferences. This changes over time")`

Go back far enough and you end in the time that I was in school and then the subtract was put on the right.

In those days it was definitely:

`" "color(purple)(0.09)`
`" "396color(white)(.)bar(|color(red)(36. color(white)() 00))`
`" "ul(35color(white)(,)64) -`
`" "36`