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

Jun 24, 2016

$0.789 \overline{789} = \frac{789}{999}$

Explanation:

This is written as $0.789 \overline{789}$

Let $x = 0.789 \overline{789}$ ...............................Equation (1)

Then $1000 x = 789.789 \overline{789}$ ............Equation (2)

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

So $1000 x - x = 789$

$\implies 999 x = 789$

Thus $x = \frac{789}{999}$

Jun 24, 2016

Do some algebra and reasoning to find $. \overline{789} = \frac{263}{333}$.

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 $.789789 \ldots$:
$x = . \overline{789}$

Then, multiply the equation by $1000$:
$1000 x = 789. \overline{789}$

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.
$1000 x - x = 789. \overline{789} - x$

On the left side of the equation, this is simply $999 x$. On the right side, change $x$ back to $. \overline{789}$:
$789. \overline{789} - . \overline{789}$

And take a good look at this subtraction problem:
$789. \overline{789}$
$\underline{- \textcolor{w h i t e}{L} . \overline{789}}$
?

The $. \overline{789}$ cancels!
$789 \cancel{. \overline{789}}$
$\underline{- \textcolor{w h i t e}{L} \cancel{. \overline{789}}}$
$789$

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

To solve for $x$, we divide $789$ by $999$ and simplify:
$x = \frac{789}{999} = \frac{263}{333}$

Therefore, $\frac{263}{333} = . \overline{789}$.