Converting a decimal number with non-repeating and repeating digits to an improper fraction

Question

What are the steps to convert $3.2\overline{901234567}$ to an improper fraction?

Answer 1

$$x = 3.2\overline{901234567}$$

$$10^9 x = 3290123456.7\overline{901234567}$$

$$(10^9 - 1)x = 329123453.5$$

so

$$x = \frac{3290123453.5}{999999999}$$

Answer 2

$3.2\overline{901234567}=\dfrac{32}{10}+\dfrac{901234567}{9999999990}=\dfrac{32}{10}+\dfrac{73}{810}.$ Can you take it from here?

Answer 3

You can re-write this repeating decimal using a infinite geometric series: \begin{align*} 3.2\overline{901234567} &= 3.2+901234567(10^{-10}+10^{-19}+10^{-28}+\cdots)\\ &=3.2 + 901234567\cdot 10^{-10}\sum_{k=0}^\infty (10^{-9})^k\\ &=3.2 + 901234567\cdot \frac{1}{10^{10}}\frac{1}{1-10^{-9}}\\ &=\frac{32}{10}+\frac{901234567}{10^{10}-10} \end{align*} and now you can find a common denominator and add these fractions.