Question

How do you convert 0.789 (789 repeating) to a fraction?

Answer 1

`0.789bar789 = 789/999`

Explanation:

This is written as `0.789bar789`

Let `x=0.789bar789` ...............................Equation (1)

Then `1000x = 789.789bar789` ............Equation (2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So `1000x-x=789`

`=>999x=789`

Thus `x= 789/999`

Answer 2

Do some algebra and reasoning to find `.bar(789)=263/333`.

Explanation:

The process for converting repeating decimals to fractions is confusing at first, but with practice it's pretty easy.

You begin by setting `x` equal to `.789789...`:
`x=.bar(789)`

Then, multiply the equation by `1000`:
`1000x=789.bar(789)`

We do this so we can move one chunk of the repeating part to the left of the decimal point. This sets us up for the next, most important step: subtracting `x` from both sides.
`1000x-x=789.bar(789)-x`

On the left side of the equation, this is simply `999x`. On the right side, change `x` back to `.bar(789)`:
`789.bar(789)-.bar(789)`

And take a good look at this subtraction problem:
`789.bar(789)`
`ul(-color(white)(L).bar(789))`
`?`

The `.bar(789)` cancels!
`789cancel(.bar(789))`
`ul(-color(white)(L)cancel(.bar(789)))`
`789`

The right side of the equation becomes `789`, so we have:
`999x=789`

To solve for `x`, we divide `789` by `999` and simplify:
`x=789/999=263/333`

Therefore, `263/333=.bar(789)`.