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A Mathematical View of Our World

A Mathematical View of Our World. 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer. Chapter 5. Apportionment. Section 5.1 Quota Methods. Goals Study apportionment Standard divisor Standard quota Study apportionment methods Hamilton’s method Lowndes’ method. 5.1 Initial Problem.

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A Mathematical View of Our World

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  1. A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

  2. Chapter 5 Apportionment

  3. Section 5.1Quota Methods • Goals • Study apportionment • Standard divisor • Standard quota • Study apportionment methods • Hamilton’s method • Lowndes’ method

  4. 5.1 Initial Problem • The number of campers in each group at a summer camp is shown below.

  5. 5.1 Initial Problem, cont’d • The camp organizers will assign 15 counselors to the groups of campers. • How many of the 15 counselors should be assigned to each group? • The solution will be given at the end of the section.

  6. Apportionment • The verb apportion means • “Assign to as a due portion.” • “To divide into shares which may not be equal.” • Apportionment problems arise when what is being divided cannot be divided into fractional parts. • An example of apportionment is the process of assigning seats in the House of Representatives to the states.

  7. Apportionment, cont’d • The apportionment problem is to determine a method for rounding a collection of numbers so that: • The numbers are rounded to whole numbers. • The sum of the numbers is unchanged.

  8. Apportionment, cont’d • The Constitution does not specify a method for apportioning seats in the House of Representatives. • Various methods, named after their authors, have been used: • Alexander Hamilton • Thomas Jefferson • Daniel Webster • William Lowndes

  9. The Standard Divisor • Suppose the total population is P and the number of seats to be apportioned is M. • The standard divisor is the ratio D = P/M. • The standard divisor gives the number of people per legislative seat.

  10. Example 1 • Suppose a country has 5 states and 200 seats in the legislature. • The populations of the states are given below. • Find the standard divisor.

  11. Example 1, cont’d • Solution: The total population is found by adding the 5 state populations. • P = 1,350,000 + 1,500,000 + 4,950,000 + 1,100,000 + 1,100,000 = 10,000,000. • The number of seats is M = 200

  12. Example 1, cont’d • Solution, cont’d: The standard divisor is D = 10,000,000/200 = 50,000. • Each seat in the legislature represents 50,000 citizens.

  13. The Standard Quota • Let D be the standard divisor. • If the population of a state is p, then Q = p/D is called the standard quota. • If seats could be divided into fractions, we would give the state exactly Q seats in the legislature.

  14. Example 2 • In the previous example, the standard divisor was found to be D = 50,000. • Find the standard quotas for each state in the country.

  15. Example 2, cont’d • Solution: Divide the population of each state by the standard divisor. • State A: Q = 1,350,000/50,000 = 27 • State B: Q = 1,500,000/50,000 = 30 • State C: Q = 4,950,000/50,000 = 99 • State D: Q = 1,100,000/50,000 = 22 • State E: Q = 1,100,000/50,000 = 22

  16. Example 2, cont’d • Solution, cont’d: This solution, with all standard quotas being whole numbers, is not typical. • Note that the sum of the quotas is 27 + 30 + 99 +22 + 22 = 200, the total number of seats. • The standard quotas indicate how many seats each state should be assigned.

  17. Question: A country consists of 3 states with populations shown in the table below. Find the standard quotas if 200 legislative seats are to be apportioned. Round the quotas to the nearest hundredth. a. State A: 0.01, State B: 0.02, State C: 0.02 b. State A: 89.75, State B: 44.31, State C: 65.45 c. State A: 90.91, State B: 42.12, State C: 67.29 d. State A: 90.91, State B: 43.64, State C: 65.45

  18. Apportionment, cont’d • Typically, the standard quotas will not all be whole numbers and will have to be rounded. • The various apportionment methods provide procedures for determining how the rounding should be done.

  19. Hamilton’s Method • Find the standard divisor. • Determine each state’s standard quota. • Round each quota down to a whole number. • Each state gets that number of seats, with a minimum of 1 seat. • Leftover seats are assigned one at a time to states according to the size of the fractional parts of the standard quotas. • Begin with the state with the largest fractional part.

  20. Example 3 • A country has 5 states and 200 seats in the legislature. • Apportion the seats according to Hamilton’s method.

  21. Example 3, cont’d • Solution: the standard divisor is found: • Then the standard quota for each state is found. • For example:

  22. Example 3, cont’d • Solution, cont’d: All of the standard quotas are shown below.

  23. Example 3, cont’d • Solution, cont’d: The integer parts of the standard quotas add up to 26 + 30 + 98 + 22 + 22 = 198. • A total of 198 seats have been apportioned at this point. • There are 2 seats left to assign according to the fractional parts of the standard quotas.

  24. Example 3, cont’d • Solution, cont’d: Consider the size of the fractional parts of the standard quotas. • State C has the largest fractional part, of 0.7 • State A has the second largest fractional part, of 0.4. • The 2 leftover seats are apportioned to states C and A.

  25. Example 3, cont’d • Solution, cont’d: The final apportionment is shown below.

  26. Question: A country consists of 3 states with populations shown in the table below. Hamilton’s method is being used to apportion the 200 legislative seats. The standard quotas are State A: 90.91, State B: 43.64, State C: 65.45. What is the final apportionment?

  27. Question cont’d a. State A: 91 seats, State B: 44 seats, State C: 65 seats b. State A: 91 seats, State B: 43 seats, State C: 66 seats c. State A: 92 seats, State B: 43 seats, State C: 65 seats d. State A: 90 seats, State B: 44 seats, State C: 66 seats

  28. Quota Rule • Any apportionment method which always assigns the whole number just above or just below the standard quota is said to satisfy the quota rule. • Any apportionment method that obeys the quota rule is called a quota method. • Hamilton’s method is a quota method.

  29. Relative Fractional Part • For a number greater than or equal to 1, the relative fractional part is the fractional part of the number divided by the integer part. • Example: The relative fractional part of 5.4 is 0.4/5 = 0.08.

  30. Lowndes’ Method • Find the standard divisor. • Determine each state’s standard quota. • Round each quota down to a whole number. • Each state gets that number of seats, with a minimum of 1 seat.

  31. Lowndes’ Method, cont’d • Determine the relative fractional part of each state’s standard quota. • Any leftover seats are assigned one at a time according to the size of the relative fractional parts. • Begin with the state with the largest relative fractional part. • Note that Lowndes’ method is a quota method.

  32. Example 4 • Apportion the seats from the previous example using Lowndes’ method. • Solution: The standard quotas were already found and are shown below.

  33. Example 4, cont’d • Solution, cont’d: As with Hamilton’s method, 198 seats have been apportioned so far, based on the whole number part of the standard quotas. • To apportion the remaining 2 seats, calculate the relative fractional part of each state’s standard quota.

  34. Example 4, cont’d • Solution, cont’d: States D and A have the largest relative fractional parts for their standard quotas. • States D and A each get one more seat.

  35. Example 4, cont’d • Solution, cont’d: The final apportionment is shown below.

  36. Question: A country consists of 3 states with populations shown in the table below. Lowndes’ method is being used to apportion the 200 legislative seats. The standard quotas are State A: 90.91, State B: 43.64, State C: 65.45. A total of 198 seats are apportioned by the integer parts of the standard quotas. To which state is the first leftover seat assigned? a. State A b. State B c. State C

  37. Method Comparison • Hamilton’s method and Lowndes’ method gave different apportionments in the previous example. • Hamilton’s method is biased toward larger states. • Lowndes’ method is biased toward smaller states.

  38. 5.1 Initial Problem Solution • A camp needs to assign 15 counselors among 3 groups of campers. • The groups are shown in the table below.

  39. Initial Problem Solution, cont’d • First the counselors will be apportioned using Hamilton’s method. • The standard divisor is: • This indicates that 1 counselor should be assigned to approximately every 12.67 campers.

  40. Initial Problem Solution, cont’d • Next, calculate the standard quota for each group of campers.

  41. Initial Problem Solution, cont’d • A total of 3 + 5 + 6 = 14 counselors have been assigned so far. • The 1 leftover counselor is assigned to the 6th grade group.

  42. Initial Problem Solution, cont’d • The final apportionment according to Hamilton’s method is: • 3 counselors for 4th grade • 5 counselors for 5th grade • 7 counselors for 6th grade

  43. Initial Problem Solution, cont’d • Next the counselors will be apportioned using Lowndes’ method. • The standard divisor is D = 12.67. • The standard quotas are: • 4th Grade: 3.31 • 5th Grade: 5.29 • 6th Grade: 6.39

  44. Initial Problem Solution, cont’d • As before, 14 counselors have been apportioned so far.

  45. Initial Problem Solution, cont’d • The standard quota for the group of 4th grade campers has the largest relative fractional part, so they get the leftover counselor. • The final apportionment using Lowndes’ method is: • 4 counselors for 4th grade • 5 counselors for 5th grade • 6 counselors for 6th grade

  46. Section 5.2Divisor Methods • Goals • Study apportionment methods • Jefferson’s method • Webster’s method

  47. 5.2 Initial Problem • Suppose you, your sister, and your brother have inherited 85 gold coins. • The coins will be divided based on the number of hours each of you have volunteered at the local soup kitchen. • How should the coins be apportioned? • The solution will be given at the end of the section.

  48. Apportionment Methods • Section 5.1 covered two quota methods. • Hamilton’s method • Lowndes’ method. • This section will consider two apportionment methods that do not follow the quota rule.

  49. Jefferson’s Method • Suppose M seats will be apportioned. • Choose a number, d, called the modified divisor. • For each state, compute the modified quota, which is the ratio of the state’s population to the modified divisor:

  50. Jefferson’s Method, cont’d • Cont’d: • If the integer parts of the modified quotas for all the states add to M, then go on to Step 2. Otherwise go back to Step 1, part (a) and choose a different value for d. • Assign to each state the integer part of its modified quota.

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