Question

What is `0.49bar(9)` expressed as a simple fraction?

Answer 1

`49/100`

Explanation:

Given this decimal, 0.49 goes up to the hundredths place, we'll simplify it as over `100`. Since this fraction is 0.4999999, we'll assume it's just 0.49 (adding the nines makes the number more accurate when doing calculations).

So to do this, we simply take the 0.49, and multiply it by 100.

`0.49*100=49`

We have the first part. Great!

Since we are given that it goes to the hundredths place, we simply put it over `100` now.

Answer: `49/100`

Let's take another decimal though. `0.049`. As you can see, the `4` is in the thousandths place. Therefore, it's over `1000`.

Answer would be: `49/1000`

Answer 2

`0.499bar9=1/2`
(the bar over the `9` indicates that this repeats infinitely)

Explanation:

Let `x` = `0.49bar9`
Therefore `10x =4.99bar9`

#{: (10x,=,4.99bar9), (ul(-1x),=,ul(0.49bar9)), (color(white)"x"9x,=,4.5) :}#

`x=4.5/9 = 1/2`

~~~~~~~~~~~~~~~~~~~~~~~~~~

We normally think: `1/2=0.500bar0` (and this is true)

but if we try to subtract
`color(white)("XXXX")0.500bar0`
`color(white)("XXX")-ul(0.499bar9)`
we find we need to go an infinite distance (which we can't really do) to get anything other than `0`'s.

That is `0.499bar9` and `0.500bar0` actually represent the same value!