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.