As far as I know, $a/b$ indicates the following things:
The fraction $a/b$, i.e. there are '$a$' of certain object(s) for '$b$' of certain other object(s).
The quotient obtained when $a$ is divided by $b$, i.e. the number of times $b$ can be subtracted from $a$, the number of times $b$ is contained in $a$, i.e. the number of times $a$ is greater than $b$ by stating what multiple $a$ is of $b$.
The ratio of $a$ to $b$, i.e. the number of times $a$ is greater than $b$ by stating what multiple is $a$ of $b$.
Point numbers 2 and 3 are basically the same.
Is there anything else that $a/b$ indicates?