Show that $0.1\overline{6} = \frac{1}{6}$.

(N/A) Let $x = 0.1\overline{6}$. This can be written as $x = 0.1666...$ (Equation $1$).
Multiply both sides by $10$ to shift the decimal point:
$10x = 1.6666...$ (Equation $2$).
Subtract Equation $1$ from Equation $2$:
$10x - x = (1.6666...) - (0.1666...)$
$9x = 1.5$
$x = \frac{1.5}{9}$
To remove the decimal,multiply the numerator and denominator by $10$:
$x = \frac{15}{90}$
Simplify the fraction by dividing both by $15$:
$x = \frac{1}{6}$.
Thus,$0.1\overline{6} = \frac{1}{6}$.