Convert the following into a vulgar fraction :

${0.1\overline{6}}$

Let $x = 0.1\overline{6} = 0.1666…$

Multiplying by 10 to move the non-repeating digit:

10x = 1.6666… $\qquad$….. (i)

Multiplying by 100 to move the decimal past the first repeating digit:

100x = 16.6666… $\qquad$….. (ii)

On subtracting (i) from (ii), we get:

$90x = 15 \\[1em] \Rightarrow x = \dfrac{15}{90} = \dfrac{1}{6}$

Hence, $0.1\overline{6} = \dfrac{1}{6}$.