Recurring Decimals

$x = 0.\overline{7}$

One digit repeats, so multiply both sides by 10 -- that shifts the repeating pattern exactly one place, lining it back up.

$10x = 7.777\ldots$

Subtract the original x -- the repeating parts cancel exactly.

$10x - x = 7 \Rightarrow 9x = 7 \Rightarrow x = \dfrac{7}{9}$