Rational numbers are usually defined in terms of the integers.
I would define a rational number
as an ordered pair of integers
$(a, b)$ such that $b \ne 0$.
I would then define equality for rational numbers
by $(a, b) = (c, d)$ if and only if
$ad = bc$.
With the interpretation of $(a, b)$
as the rational number $a/b$,
this means that $a/b = c/d$
if and only if $ad = bc$.
In other words,
our previous knowledge of how
rational number should behave
drives all our definitions relating to
these rational numbers.
For example, since we want
$a/b + c/d = (ad+bd)/bd$,
we would $define$ addition by
$(a, b)+(c,d) = (ad+bc, bd)$.
We would then show that
these "rational numbers"
behave exactly like we want rational numbers to behave.
Finally, we would say
"From now on, we will write
$(a, b)$ as $a/b$",
and then forget all the constructions
since we have laboriously constructed
something we we already quite familiar with.
Another example of this is defining all integers
in terms of positive integers.
We can write an integer as
an ordered pair of positive integers
with equality being defined by
$(a, b) = (c, d)$ if and only if
$a+d = b+c$.
This is because we want $(a, b)$ to mean $a-b$,
and $a-b = c-d$ is the same as
$a+d = b+c$.
This reason we use this definition of equality
is that, in our definition
of "integers", we can only use positive integers,
where addition is always possible, but subtraction is not always possible.
Similarly, in our definition of
rational numbers in terms of integers,
multiplication is always possible,
but division is not always possible.
For more than you probably want to know
about this process,
read "Foundations of Analysis" by Edmund Landau.
This starts with constructing the integers
and ends up with the complex numbers.