# How do you convert 0.83 (3 repeating) to a fraction?

Mar 16, 2016

$\text{ so } x = 0.8 \overline{3} = \frac{5}{6}$

#### Explanation:

To format this question type you write: 0.8bar3

But you use the hash key just before 0.8bar3
and also at the end. So you end up with

$\text{ } 0.8 \overline{3}$

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

Let $x = 0.8 \overline{3}$

Then $10 x = 8. \overline{3}$

So $\left(10 x - x\right) = \text{ } 8.3333 \overline{3}$
color(white)("bbnnn.nnnnnnnbb")underline(0.8333bar3- Subtracting
$\textcolor{w h i t e}{\text{bbbbb.bbbbbbbbb}} 7.5$

$\text{ } 9 x = 7.5$

Multiply both sides by 10

$\text{ } 90 x = 75$

Divide both sides by 90

$\text{ } x = \frac{75}{90} = \frac{5}{6}$

$\text{ so } x = 0.8 \overline{3} = \frac{5}{6}$

Aug 23, 2017

Here's a method using a calculator to help...

#### Explanation:

Here's another way you can convert decimals to fractions if you have a calculator to hand.

We use the calculator to find the terminating continued fraction expansion for the given number, then unwrap it to a regular fraction.

For our example, type $0.83333333$ into your calculator.

Note that the portion before the decimal point is $0$, so write that down:

$\textcolor{b l u e}{0} +$

Take the reciprocal of the given number to get a result something like: $1.2000000048$. We can ignore the trailing digits $48$ as they are just a rounding error. So with our new result $1.2$ note that the number before the decimal point is $1$. Write that down as the next coefficient in the continued fraction:

$\textcolor{b l u e}{0} + \frac{1}{\textcolor{b l u e}{1}}$

then subtract it to get $0.2$. Take the reciprocal, getting the result $5.0$. This has the number $5$ before the decimal point and no remainder. So add that to our continued fraction as the next reciprocal to get:

$\textcolor{b l u e}{0} + \frac{1}{\textcolor{b l u e}{1} + \frac{1}{\textcolor{b l u e}{5}}} = 0 + \frac{1}{\frac{6}{5}} = \frac{5}{6}$

$\textcolor{w h i t e}{}$
Another example

Just to make the method a little clearer, let us consider a more complex example:

Given:

$3.82857142857$

Note the $\textcolor{b l u e}{3}$, subtract it and take the reciprocal to get:

$1.20689655173$

Note the $\textcolor{b l u e}{1}$, subtract it and take the reciprocal to get:

$4.83333333320 \text{ "color(lightgrey)"Note the rounding error}$

Note the $\textcolor{b l u e}{4}$, subtract it and take the reciprocal to get:

$1.20000000019$

Note the $\textcolor{b l u e}{1}$, subtract it and take the reciprocal to get:

$4.99999999525$

Let's call that $\textcolor{b l u e}{5}$ and stop.

Taking the numbers we have found, we have:

$3.82857142857 = \textcolor{b l u e}{3} + \frac{1}{\textcolor{b l u e}{1} + \frac{1}{\textcolor{b l u e}{4} + \frac{1}{\textcolor{b l u e}{1} + \frac{1}{\textcolor{b l u e}{5}}}}}$

$\textcolor{w h i t e}{3.82857142857} = \textcolor{b l u e}{3} + \frac{1}{\textcolor{b l u e}{1} + \frac{1}{\textcolor{b l u e}{4} + \frac{5}{6}}}$

$\textcolor{w h i t e}{3.82857142857} = \textcolor{b l u e}{3} + \frac{1}{\textcolor{b l u e}{1} + \frac{6}{29}}$

$\textcolor{w h i t e}{3.82857142857} = \textcolor{b l u e}{3} + \frac{29}{35}$

$\textcolor{w h i t e}{3.82857142857} = \frac{134}{35}$