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# 4.8: Exercises

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In exercises 1-8, determine the apportionment using

1. Hamilton’s Method
2. Jefferson’s Method
3. Webster’s Method
4. Huntington-Hill Method
5. Lowndes’ method

1. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. If the college can only afford to hire 15 tutors, determine how many tutors should be assigned to each subject.

Math: 330 English: 265 Chemistry: 130 Biology: 70

2. Reapportion the previous problem if the college can hire 20 tutors.

3. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 20 salespeople given the information below.

$$\begin{array}{|l|l|l|l|l|} \hline \text { Shift } & \text { Morning } & \text { Midday } & \text { Afternoon } & \text { Evening } \\ \hline \begin{array}{l} \text { Average number of } \\ \text { customers } \end{array} & 95 & 305 & 435 & 515 \\ \hline \end{array}$$

4. Reapportion the previous problem if the store has 25 salespeople.

5. Three people invest in a treasure dive, each investing the amount listed below. The dive results in 36 gold coins. Apportion those coins to the investors.

Alice: $7,600 Ben:$5,900 Carlos: \$1,400

6. Reapportion the previous problem if 37 gold coins are recovered.

7. A small country consists of five states, whose populations are listed below. If the legislature has 119 seats, apportion the seats.

A: 810,000 B: 473,000 C: 292,000 D: 594,000 E: 211,000

8. A small country consists of six states, whose populations are listed below. If the legislature has 200 seats, apportion the seats.

A: 3,411 B: 2,421 C: 11,586 D: 4,494 E: 3,126 F: 4,962

9. A small country consists of three states, whose populations are listed below.

A: 6,000 B: 6,000 C: 2,000

a. If the legislature has 10 seats, use Hamilton’s method to apportion the seats.

b. If the legislature grows to 11 seats, use Hamilton’s method to apportion the seats.

c. Which apportionment paradox does this illustrate?

10. A state with five counties has 50 seats in their legislature. Using Hamilton’s method, apportion the seats based on the 2000 census, then again using the 2010 census. Which apportionment paradox does this illustrate?

$$\begin{array}{|l|l|l|} \hline \textbf { County } & \mathbf{2 0 0 0} \textbf { Population } & \mathbf{2 0 1 0} \textbf { Population } \\ \hline \text { Jefferson } & 60,000 & 60,000 \\ \hline \text { Clay } & 31,200 & 31,200 \\ \hline \text { Madison } & 69,200 & 72,400 \\ \hline \text { Jackson } & 81,600 & 81,600 \\ \hline \text { Franklin } & 118,000 & 118,400 \\ \hline \end{array}$$

11. A school district has two high schools: Lowell, serving 1715 students, and Fairview, serving 7364. The district could only afford to hire 13 guidance counselors.

a. Determine how many counselors should be assigned to each school using Hamilton's method.

b. The following year, the district expands to include a third school, serving 2989 students. Based on the divisor from above, how many additional counselors should be hired for the new school?

c. After hiring that many new counselors, the district recalculates the reapportion using Hamilton's method. Determine the outcome.

d. Does this situation illustrate any apportionment issues?

12. A small country consists of four states, whose populations are listed below. If the legislature has 116 seats, apportion the seats using Hamilton’s method. Does this illustrate any apportionment issues?

A: 33,700 B: 559,500 C: 141,300 D: 89,100

## Exploration

13. Explore and describe the similarities, differences, and interplay between weighted voting, fair division (if you’ve studied it yet), and apportionment.

14. In the methods discussed in the text, it was assumed that the number of seats being apportioned was fixed. Suppose instead that the number of seats could be adjusted slightly, perhaps 10% up or down. Create a method for apportioning that incorporates this additional freedom, and describe why you feel it is the best approach. Apply your method to the apportionment in Exercise 7.

15. Lowndes felt that small states deserved additional seats more than larger states. Suppose you were a legislator from a larger state, and write an argument refuting Lowndes.

16. Research how apportionment of legislative seats is done in other countries around the world. What are the similarities and differences compared to how the United States apportions congress?

17. Adams’s method is similar to Jefferson’s method, but rounds quotas up rather than down. This means we usually need a modified divisor that is smaller than the standard divisor. Rework problems 1-8 using Adam’s method. Which other method are the results most similar to?

4.8: Exercises is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Mike Kenyon & David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.