How many Unique numbers?

Solution 1:

Presumably digits of $0$ are not allowed, and we don't include trivialities such as $xx/xx = x/x$. The fractions $<1$ with numerators and denominators of up to $3$ digits that work (cancelling all but one digit of numerator and denominator) are $$\eqalign{\frac{16}{64},& \frac{19}{95}, \frac{26}{65}, \frac{49}{98}, \frac{124}{217}, \frac{127}{762}, \frac{138}{184}, \frac{139}{973}, \frac{145}{435}, \frac{148}{185},\cr \frac{163}{326}, &\frac{166}{664}, \frac{182}{819}, \frac{187}{748}, \frac{199}{995}, \frac{218}{981}, \frac{244}{427}, \frac{266}{665}, \frac{273}{728}, \frac{316}{632},\cr \frac{327}{872}, &\frac{364}{637}, \frac{412}{721}, \frac{424}{742}, \frac{436}{763}, \frac{448}{784}, \frac{455}{546}, \frac{484}{847}, \frac{499}{998}, \frac{545}{654}} $$

Infinite families include $$\frac{16\ldots 6}{6\ldots 64} = \frac{1}{4}$$ $$ \frac{19\ldots 9}{9\ldots 95} = \frac{1}{5}$$ $$ \frac{26\ldots 6}{6 \ldots 65} = \frac{2}{5}$$ and $$ \frac{49\ldots9}{9\ldots98} = \frac{4}{8}$$