2023 AIME II Problem 5

Let S be the set of all positive rational numbers r such that when the two numbers r and 55r are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of S can be expressed in the form \frac{p}{q}, where p and q are relatively prime positive integers. Find p + q.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.