An arbitrator’s finding in favor of the School District of Philadelphia should wind up saving the financially strapped system more than $30 million in a pay dispute with the city’s teachers’ union.

The Philadelphia Federation of Teachers claimed that at the beginning of the 2006-07 school year, the district altered its payment methods in such a way that the union’s roughly 15,000 “10-month” district employees — the majority of the federation’s membership, and most of them teachers — had their annual salaries improperly reduced by nearly 4 percent.