Decimal to fraction ?

Jan 20, 2018

$= \frac{33}{101}$.

Explanation:

In general, when you have a number of the form $0.$[repeating part] you can convert it to the fraction

$\frac{n}{{10}^{c} - 1}$

where $n$ is the repeating part (the integer created by the decimal digits of one repetition, in our case it is $3267$) and $c$ is the number of digits in the repeating part. It's equally easy with the form

$a .$[repeating part], you just get

$a + \frac{n}{{10}^{c} - 1}$

The reason this works is because if you multiply that repeating decimal by ${10}^{c}$, you'll get $n$ plus the repeating decimal. In our case, $c = 4$ and $n = 3267$:

$0.3267 \ldots \cdot {10}^{4} = 3267.3267 \ldots = 3267 + 0.3267 \ldots$

Therefore,

$0.3267 \ldots \cdot {10}^{4} - 0.3267 \ldots = 3267$

$0.3267 \ldots \cdot \left({10}^{4} - 1\right) = 3267$

$0.3267 \ldots = \frac{3267}{{10}^{4} - 1} = \frac{3267}{9999} = \frac{33}{101}$

It's also simple to find ${10}^{c} - 1$, just write $c$ nines.

Jan 20, 2018

$\frac{33}{101}$

Explanation:

$0. \textcolor{red}{3267} 32673267. \ldots = 0. \overline{3267}$

$\text{the bar represents the digits being repeated}$

$\text{these form the terms of a "color(blue)"geometric sequence}$

$\frac{3267}{10000} , \frac{3267}{100000000} , \ldots$

$\text{with } a = \frac{3267}{10000}$

$\text{and } r = {a}_{2} / {a}_{1}$

$\Rightarrow r = \frac{\cancel{3267}}{10000 \cancel{0000}} \times \frac{1 \cancel{0000}}{\cancel{3267}} = \frac{1}{10000}$

â€¢color(white)(x)S_oo=a/(1-r)

$\Rightarrow {S}_{\infty} = \frac{\frac{3267}{10000}}{1 - \frac{1}{10000}}$

$= \frac{\frac{3267}{10000}}{\frac{9999}{10000}}$

$= \frac{3267}{9999} = \frac{33}{101}$