Probability & Statistics

A mean, a percentage or a proportion hides the quantity you can actually add up. Convert it back: a mean of $\bar{x}$ over $k$ items means the total is $\bar{x}\times k$, and a percentage means a fixed amount of the whole. Once everything is an absolute total (or a whole-number count), you can add, subtract and conserve it directly, which is far safer than juggling fractions.

Trigger: the data is described only by means, ratios or percentages, and you need to combine, exchange or recombine the groups.

Instances:
(i) for a weighted mixture, write each component's contribution as volume times fraction and sum them, then divide by the total volume;
(ii) when several proportions act on one population, choose a convenient total (a common multiple of the denominators) so every proportion becomes a whole-number count;
(iii) when items move between groups, use that the grand total is conserved: set the total before equal to the total after to pin down the unknown size or mean.