Why does sharing representative seats require more than rounding percentages?

A representative body may need to divide a fixed number of seats among regions according to population. Proportional shares are easy to describe as fractions, but seats must usually be assigned as whole numbers.

Imagine three regions with fifty, thirty-three and seventeen percent of a population sharing ten seats. Their exact proportional quotas are five, three point three and one point seven. These add to ten, but two quotas contain fractions.

One possible rule first gives each region the whole-number part, then assigns remaining seats to the largest fractional remainders. Here that gives five, three and two seats. This rule must be stated, because other allocation methods use different calculations.

Real systems can also guarantee a minimum representation and define which population is counted. Those choices affect the calculation. A census supplies evidence about people, while an apportionment rule translates those counts into indivisible seats; neither population measurement nor arithmetic alone chooses the institutional rule.