Round dates from 1to 15 to 1st and 16 to 31 to 16th of same month

This should work (where A1 is the date you want to change):

=IF(DAY(A1)<16,DATE(YEAR(A1),MONTH(A1),1),DATE(YEAR(A1),MONTH(A1),16))

Hope this helps! :)
 

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