$\require{cancel}$
I saw a meme that fraction $$\frac{163}{326}=\frac{1\cancel6\cancel3}{\cancel{3}\cancel{6}2}=\frac{1}{2}$$
And It means that
$$1\leq a_i,b_i \leq 9, a_i,b_i \in \mathbb{N},\\\sum_{i=1}^n 10^{i-1}a_i=\overline{a_n...a_2a_1}\\\frac{\overline{a_n...a_2a_1}} {\overline{b_n...b_2b_1}}=\frac{\prod_{i=1}^n{a_i}}{\prod_{i=1}^n{b_i}}$$
So,I have question that how many this kind of things. han I found some examples $$\frac{49}{98},\frac{19}{95},\frac{16}{64}, \frac{26}{65},\frac{14}{63},\frac{18}{45}, \frac{15}{24},\frac{22}{165},\frac{44}{198},\frac{62513}{312565}$$ They have something special? And it is infinite that$ (\{a_n,...,a_2,a_1\},\{b_n,...b_2,b_1\})$??
I got the answer.
If$$\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$$ Than$$ \frac{\overline{a_n...a_2a_1}} {\overline{b_n...b_2b_1}}=\frac{\sum_{i=1}^n{a_i}}{\sum_{i=1}^n{b_i}}$$
And this can solve when $a_n,b_n>9$ for example the case $$\frac{540540540}{864864864}=\frac{5+40+5+40+5+40}{8+64+8+64+8+64}=\frac{5}{8}$$or$$\frac{5707070}{9131312}=\frac{5+70+70+70}{8+112+112+112}=\frac{5}{8}$$ Therefore we could crate like $$\frac{21818...18}{98181...81}=\frac{2}{9}$$
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 $$ \frac{49\ldots9}{9\ldots98} = \frac{4}{8}$$