Rewrite as a simplified fraction.

Question

$$0.\bar{2}=?$$

Answer 1

To rewrite $0.\overline{2}$ as a simplified fraction, we can call it $x$ and set up an equation, as follows:
$$x=0.2222\dots$$
Since the digit $2$ repeats indefinitely, we can multiply both sides of the equation by $10$ to shift the decimal point to the right:
$$10x=2.2222\dots$$
Now, we can subtract the original equation from the second equation and eliminate the repeating portion of the decimal:
$$10x-x=2.2222\dots -0.2222\dots$$
This simplifies to:
$$9x=2$$
Finally, dividing both sides of the equation by $9$ gives us the value of $x$:
$$x={\displaystyle \frac{2}{9}}$$
Therefore, $0.\overline{2} = \boxed{\dfrac{2}{9}}$.