EXAMPLE 8 A Mathematics Department Meets the Alabama Paradox

A mathematics department has 30 teaching assistants (TAs) to cover recitation sections for College Algebra, Calculus I, Calculus II, Calculus III, and Contemporary Mathematics. The enrollments of these courses are given in Table 14.7. The department will use the Hamilton method to apportion the TAs to the five subjects. in this problem, the house size is 30 (the number of TAs) and the population is the number of students, 750. The states are the five courses to be offered. The standard divisor is , which represents the average number of students per recitation section. Each quota shown in the table was determined by dividing the enrollment of the course by this divisor.

The lower quotas add up to 27, so the three courses whose quotas have the largest fractional parts, Calculus I and III and Contemporary Mathematics, were given their upper quotas.

After the TAs were given their teaching assignments, the graduate school authorized the department to hire an additional TA. To determine which course should get the new TA, the department had to recalculate the apportionment. With 31 TAs, the standard divisor was . The new quotas, determined by dividing each population by this new divisor, are shown in Table 14.8. Now the lower quotas add up to 28, so again three additional TAs go to the subjects whose quotas have the largest fractions. The Calculus III fraction, which had been larger than the College Algebra fraction when there were just 30 teaching assistants, has been surpassed. The new TA was placed in College Algebra, and one of the Calculus III TAs had to be reassigned to Calculus II.

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