Math Help - How to convert decimals to fractions

1. How to convert decimals to fractions

How would we solve problems such as 16 and 17? These problems do not repeat in numbers such as 0.235235235.

Like for 16, how would we put that in geometric form first?
I tried 0.122 + 0.0012 + 0.000012 ect but cannot find out how to get a fraction that will give that answer.

2. Solution13: Let x=0.232323............. (1)
multiply (1) with 100,we get
100x=23.232323............ (2)
subtract (1) from (2),we get
99x=23
x=23/99

3. 16.

$0.1222222\dots = \frac{1}{10} + \frac{2}{100} + \frac{2}{1000} + \frac{2}{10\,000} + \dots$

$= \frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k}$.

In this case, the geometric series has $a = \frac{1}{100}$ and $r = \frac{1}{10}$.

Since $S_{\infty} = \frac{a}{1 - r}$

$S_{\infty} = \frac{\frac{1}{100}}{1 - \frac{1}{10}}$

$= \frac{\frac{1}{100}}{\frac{9}{10}}$

$= \frac{10}{900}$

$= \frac{1}{90}$.

So $\frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k} = \frac{1}{10} + 2\left(\frac{1}{90}\right)$

$= \frac{9}{90} + \frac{2}{90}$

$= \frac{11}{90}$.

Therefore $0.122222\dots = \frac{11}{90}$.

4. 17.

$0.478888\dots = \frac{47}{100} + \frac{8}{1000} + \frac{8}{10\,000} + \frac{8}{100\,000} + \dots$

$= \frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k}$

In this case, the geometric series has $a = \frac{1}{1000}$ and $r = \frac{1}{10}$.

Therefore $S_{\infty} = \frac{\frac{1}{1000}}{1 - \frac{1}{10}}$

$= \frac{\frac{1}{1000}}{\frac{9}{10}}$

$= \frac{10}{9000}$

$= \frac{1}{900}$.

Thus $\frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k} = \frac{47}{100} + 8\left(\frac{1}{900}\right)$

$= \frac{423}{900} + \frac{8}{900}$

$= \frac{431}{900}$.

Therefore $0.478888\dots = \frac{431}{900}$.