Convert Repeating Decimals to Fraction

Question

If there is a number $x = 0.333333333$ where 3 is repeated infinitely how to convert it to fraction ? It is said that $-> 0.(3) = \frac{3}{10} - 1$ and $0.33 = \frac{33}{100} - 1$, But I can't understand Why ? I searched for similar Questions like this and i found answers but i didn't understand the prove.

Answer 1

Suppose $$x=0.\overline3\tag1$$ Then, $$10x=3.\overline3\tag2$$

Performing the subtraction $(2)-(1)$ yields \begin{align} 10x-x&=3.\overline3-0.\overline3\\ 9x&=3\\ x&=\frac13 \end{align}

Answer 2

If $$x = 0.33333333....$$ , then $$10x = 3.33333333....$$
Subtracting : $10x - x = 3$ , The reason being that since the decimal is repeated till infinity , removing one term would not change it. [Think about it!]

Evaluating : $$9x = 3$$ $$x = \frac{1}{3}$$