What do I study to find another description of this subset of the rationals?

Highway signs in the US generally depict the distance to several of the nearest exits, generally in whole numbers, or in quarter- or half-miles:

Inspired by this observation, I started thinking about fractions that could be represented using only the (base-ten) digits $1$ through $4$. In particular, I thought: Suppose $H$ is the set of positive integers with this type of expansion, and $Q_H$ is the set of rationals equivalent to $a / b$ where $a$ and $b$ are in $H$.

After playing around with the set, I've discovered that (for example) $\frac{1}{5}$ is not in $Q_H$, since for all $k$, $5k$ is congruent to either $5$ or $0$ mod $10$. Similarly $\frac{1}{9}$ is not, since when $k$ is in $H$ (and therefore congruent to $1$, $2$, $3$, or $4$ mod $10$), $9k$ is congruent to $9$, $8$, $7$, or $6$. ($\frac{2}{9} = \frac{32}{144}$, however, is in $Q_H$).

But I'd like to find another way to characterize the (positive) rationals in, or not in, $Q_H$.

I did get a BA in theoretical mathematics 25 years ago, but I don't remember taking any course where we discussed questions like this; so I don't even know where to start—or for that matter what I need to know in order to get to where I need to start. Is this, for example, a number theory question? Where should I start looking to get the skills I need to try and answer this?

Some ideas for where to start...

As you've started to see, the set of positive integers $I$ whose last digit is 1, 2, 3, or 4 is relatively easy to reason about. So try characterizing $Q_I$ instead. If you can solve that problem, then it may help to tackle $Q_H$.

Instead of base ten, try using a prime base like three or five. That should also simplify matters.

If you're stuck, throw a computer at the problem. Try to compute some fragment of $Q_I$ by brute force; find a way to visualize the result; and look for patterns.