# How do you find the repeating decimal 0.82 with 82 repeated as a fraction?

Dec 8, 2016

$0. \overline{82} = \frac{82}{99}$

#### Explanation:

In case you have not encountered it, let me introduce you to some notation:

A repeating decimal can be represented using a bar placed over the repeating pattern of decimals.

So instead of writing $0.828282 \ldots$, you can write $0. \overline{82}$

If we multiply $0. \overline{82}$ by $\left(100 - 1\right)$ we get an integer:

$\left(100 - 1\right) 0. \overline{82} = 100 \cdot 0. \overline{82} - 1 \cdot 0. \overline{82}$

$\textcolor{w h i t e}{\left(100 - 1\right) 0. \overline{82}} = 82. \overline{82} - 0. \overline{82}$

$\textcolor{w h i t e}{\left(100 - 1\right) 0. \overline{82}} = 82$

Notice that the $100$ shifts the number two places to the left - the length of the repeating pattern. Then the $- 1$ cancels out the repeating tail.

Next divide both ends by $\left(100 - 1\right)$ and simplify:

$0. \overline{82} = \frac{82}{100 - 1} = \frac{82}{99}$

$82$ and $99$ have no common factor, so this is in simplest terms.

$\textcolor{w h i t e}{}$
Alternative method

An alternative method recognises that:

$0. \overline{82} = 0.82 + 0.0082 + 0.000082 + \ldots$

is a geometric series, with initial term $a = 0.82$ and common ratio $r = \frac{1}{100}$.

This has sum given by the formula:

${s}_{\infty} = \frac{a}{1 - r} = \frac{0.82}{1 - \frac{1}{100}} = \frac{0.82}{\frac{99}{100}} = \frac{0.82 \cdot 100}{99} = \frac{82}{99}$