# How do you write 0.3666... as a fraction? The 6 repeating.

Jan 2, 2018

$\frac{18333}{50000}$

#### Explanation:

Put the number over 1( make denominator 1)
$\frac{0.36666}{1}$
and multiply by 10 for every number after the decimal point.
as there is 5 numbers you have to multiply by 100000
=$\frac{0.36666 \cdot 100000}{1 \cdot 100000}$

=$\frac{36666}{100000}$
and divide by the greatest factor, which is 2

=$\frac{\frac{36666}{2}}{\frac{100000}{2}}$

=$\frac{18333}{50000}$ final answer.

Jan 2, 2018

See a solution process below:

#### Explanation:

First, we can write:

$x = 0.3 \overline{6}$

Next, we can multiply each side by $10$ giving:

$10 x = 3. \overline{6}$

Then we can subtract each side of the first equation from each side of the second equation giving:

$10 x - x = 3. \overline{6} - 0.3 \overline{6}$

We can now solve for $x$ as follows:

$10 x - 1 x = \left(3.6 + 0.0 \overline{6}\right) - \left(0.3 + 0.0 \overline{6}\right)$

$\left(10 - 1\right) x = 3.6 + 0.0 \overline{6} - 0.3 - 0.0 \overline{6}$

$9 x = \left(3.6 - 0.3\right) + \left(0.0 \overline{6} - 0.0 \overline{6}\right)$

$9 x = 3.3 + 0$

$9 x = 3.3$

$\frac{9 x}{\textcolor{red}{9}} = \frac{3.3}{\textcolor{red}{9}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{9}}} x}{\cancel{\textcolor{red}{9}}} = \frac{3 \times 1.1}{\textcolor{red}{3 \times 3}}$

$x = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 1.1}{\textcolor{red}{\textcolor{b l a c k}{\cancel{\textcolor{red}{3}}} \times 3}}$

$x = \frac{1.1}{3}$

$x = \frac{10}{10} \times \frac{1.1}{3}$

$x = \frac{11}{30}$

Jan 2, 2018

$0.366 \ldots = \frac{11}{30}$

#### Explanation:

There's nice algebraic trick to get rid of repeating tail:

$x = 0.366 \ldots$

$10 x = 3.666 \ldots$

$10 x - x = 3.6 \cancel{66. . .} - 0.3 \cancel{66. . .}$

$9 x = 3.6 - 0.3 = 3.3$

$90 x = 33$

$x = \frac{33}{90} = \frac{11}{30}$

If there's $n$ digits in repeating element, then multiply by ${10}^{n}$

$x = 0.36565 \ldots$

$100 x = 36.56565 \ldots$

$100 x - x = 36.5 \cancel{6565. . .} - 0.3 \cancel{6565. . .}$

$99 x = 36.5 - 0.3 = 36.2$

$990 x = 362$

$x = \frac{362}{990} = \frac{181}{495}$