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

Sums of four and more unit fractions and approximate parametrizations