Base 12 is better

We will begin by stating that $b\neq0$ , or else the (duo)decimal form would have only one digit to the right.

Another important point is that multiplying by 12 effectively shifts the decimal place to the right one digit, and so multiplying by 144 should transform $0.ab$ to $ab.0$ . Multiplying each of the fractions by 144 allows us to cross out two answers immediately ( $\frac{1}{5}$ and $\frac{1}{7}$ ) since they do not yield whole numbers.

Now, multiplying each of $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{4}$ , and $\frac{1}{6}$ by 144 yields whole numbers, but each of these results is itself divisible by 12, implying that $b=0$ , a contradiction.

Therefore, $\frac{1}{8}$ is the only fraction left standing, and we can verify that it is equal to $0.16$ either by long division in base 12 or by noting that $\frac{1}{8}\cdot 144 = 16$ and shifting the decimal place.

Great question!

$\dfrac 18=0.16_{12}$
$\dfrac 12, \dfrac 13, \dfrac 14, \dfrac 16$ has only one digit.
$\dfrac 15, \dfrac 17$ has infinite.