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With a multiple entry Schengen visa the allowed duration of stay is the maximum number of calendar days (midnight to midnight) at least partly spent in the Schengen Area.
Your family vacation (10β24 September) will count as 15 days. Add the days for your business trip. If the sum does not exceed the maximum number of days allowed according to your visa title, you are fine. Having used your visa for its original purpose, you are allowed to what is left of its validity for another visit.