Question

How do you write 27/99 as a decimal?

Answer 1

`0.27272727...->0.27bar(27)`

Explanation:

A fraction written as a decimal is either terminating (has a fixed number of decimal places) or has a cycle of digits that repeat for ever. As we have 99 in the denominator I suspect that the decimal has an infinitely repeating set of digits.

`color(blue)("Using a calculator I get "0.272727....)`

The repeating part can be indicated by putting a bar over the appropriate digits. So I would chose to write this as:

`0.27bar(27)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("What if you do not have a calculator?")`

We are dividing 99 into a number that is less (do NOT use the word 'smaller')

However I can do a sort of cheat. 27 is the same as `270xx1/10`

This idea can be repeated as many times as you wish as long as you apply the `xx1/10xx1/10xx` however many `1/10` you end up with. This will be clearer when I use it.

`color(white)()`
`color(white)()`

`27 ->color(white)("ddd")270 color(magenta)(xx1/10)`

`color(red)(2)xx99-> ul(198larr" Subtract")`
`color(white)("ddddddddd")72`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
But `72<99" so write it as "720xx1/10`

`color(white)("ddddddddd")720color(magenta)(xx1/10)`

`color(red)(7)xx99->color(white)("d")ul( 693 larr" Subtract")`
`color(white)("dddddddddd")27`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(purple)("And so the cycle goes on and on for ever.")`

Thus so far we have:

`color(red)(27)color(magenta)(xx1/10xx1/10) = 0.27`

But the repeats give: `0.27272727...... ->0.27bar27`