## Thinking Mathematically (6th Edition)

(a) We can find the standard divisor. $standard~divisor = \frac{total ~enrollment}{number~of~ teaching ~assistants}$ $standard~divisor = \frac{1800}{30}$ $standard~divisor = 60$ We can find each department's standard quota. The standard quota of each department is the department's enrollment divided by the standard divisor. College Algebra: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{978}{60}$ $standard~quota = 16.3$ Statistics: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{500}{60}$ $standard~quota = 8.33$ Liberal Arts Math: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{322}{60}$ $standard~quota = 5.37$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus teaching assistants are given, one at a time, to the departments with the largest decimal parts in their standard quotas until there are no more surplus teaching assistants. Initially, each department is apportioned its lower quota. College Algebra is apportioned 16 teaching assistants. Statistics is apportioned 8 teaching assistants. Liberal Arts Math is apportioned 5 teaching assistants. We can find the total number of teaching assistants which have been apportioned. total = 16 + 8 + 5 = 29 teaching assistants Since there is a total of 30 teaching assistants, there is one surplus teaching assistant. One more teaching assistant is given to Liberal Arts Math because it has the largest decimal part (0.37) in its standard quota. Using Hamilton's method, each department is apportioned the following number of teaching assistants: College Algebra is apportioned 16 teaching assistants. Statistics is apportioned 8 teaching assistants. Liberal Arts Math is apportioned 5 + 1 = 6 teaching assistants. (b) We can find the standard divisor. $standard~divisor = \frac{total ~enrollment}{number~of~ teaching ~assistants}$ $standard~divisor = \frac{1800}{31}$ $standard~divisor = 58.06$ We can find each department's standard quota. The standard quota of each department is the department's enrollment divided by the standard divisor. College Algebra: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{978}{58.06}$ $standard~quota = 16.84$ Statistics: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{500}{58.06}$ $standard~quota = 8.61$ Liberal Arts Math: $standard ~quota = \frac{enrollment}{standard~divisor}$ $standard~quota = \frac{322}{58.06}$ $standard~quota = 5.55$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus teaching assistants are given, one at a time, to the departments with the largest decimal parts in their standard quotas until there are no more surplus teaching assistants. Initially, each department is apportioned its lower quota. College Algebra is apportioned 16 teaching assistants. Statistics is apportioned 8 teaching assistants. Liberal Arts Math is apportioned 5 teaching assistants. We can find the total number of teaching assistants which have been apportioned. total = 16 + 8 + 5 = 29 teaching assistants Since there is a total of 31 teaching assistants, there are two surplus teaching assistants. The first teaching assistant is given to College Algebra because it has the largest decimal part (0.84) in its standard quota. The second teaching assistant is given to Statistics because it has the second largest decimal part (0.61) in its standard quota. Using Hamilton's method, each department is apportioned the following number of teaching assistants: College Algebra is apportioned 16 + 1 = 17 teaching assistants. Statistics is apportioned 8 + 1 = 9 teaching assistants. Liberal Arts Math is apportioned 5 teaching assistants. We can see that the Alabama paradox occurs. Initially, with 30 teaching assistants, Liberal Arts Math was allocated 6 teaching assistants. After the number of teaching assistants increased to 31, Liberal Arts Math was allocated only 5 teaching assistants. Therefore, the Alabama paradox occurs.