  We prove new upper bounds on the number of representations of rational
numbers $\frac{m}{n}$ as a sum of $4$ unit fractions, giving five different
regions, depending on the size of $m$ in terms of $n$. In particular, we
improve the most relevant cases, when $m$ is small, and when $m$ is close to
$n$. The improvements stem from not only studying complete parametrizations of
the set of solutions, but simplifying this set appropriately. Certain subsets
of all parameters define the set of all solutions, up to applications of
divisor functions, which has little impact on the upper bound of the number of
solutions. These "approximate parametrizations" were the key point to enable
computer programmes to filter through large number of equations and
inequalities. Furthermore, this result leads to new upper bounds for the number
of representations of rational numbers as sums of more than $4$ unit fractions.
