Means do not add, but absolute totals do. Whenever a question gives the mean of a group, immediately multiply by the count to find the total. Once everything is an absolute total, you can add, subtract, and equate them safely.
Trigger: A question modifying a group (adding, removing, merging) or giving algebraic constraints, using means instead of totals.
Instances:
(i) adding/removing items: new total = old total ± added items;
(ii) merging groups: combined total = sum of the totals of the sub-groups, which you divide by the total count for the new mean;
(iii) algebraic constraints: mean of n items =x immediately becomes sum =nx.
