Finite Math A: Homework 4.1-4.2
Apportionment & Hamilton’s Method
1. A small country consists of four states. The population of state A is 67,200. The population of state B is 78,300. The population of state C is 73,800. The population of state D is 80,700. The total number of seats in the legislature is 100.
a) What is the standard divisor?
b) Find each state’s standard quota.
2. A small country consists of four states. The total population of the country is 400,000. The standard quotas for each state are the following: A=179.8 , B=129.6, C=79.2, D = 11.4.
a) What is the standard divisor?
b) Find each state’s population.
For Situations # 3 – 6 below a) Determine the Standard Divisor (SD) and what it represents in the given situation.
b) Use Hamilton’s Method to find the apportionment for the given number of seats, M.
3. The Bandana Republic is a small country consisting of four states: Apure, Barinas, Carabobo, and Delores. Suppose there are 160 seats in the Bandana Congress to be apportioned among the four states based on their respective populations. The following table show the population per state.
State / Apure / Barinas / Carabobo / DeloresPopulation / 3,310,000 / 2,670,000 / 1,330,000 / 690,000
4. The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates six bus routes (called A, B, C, D, E, and F) and 130 busses. The number of busses is apportioned to each route based on the number of passengers riding that route. The following table shows the daily average ridership on each route.
Route / A / B / C / D / E / FRidership / 45,300 / 31,070 / 20,490 / 14,160 / 20,260 / 8,720
5. The Placerville General Hospital has a nursing staff of 225 nurses working 4 shifts: A (7am-1pm), B (1pm to 7pm), C (7pm to 1am), and D (1am-7am). The number of nurses apportioned to each shift is based on the average number of patients treated in that shift, which is given in the following table:
Shift / A / B / C / DPatients / 871 / 1029 / 610 / 190
6. The Interplanetary Federation of Fraternia consists of six planets: Alpha, Kappa, Beta, Theta, Chi Omega, Delta Gamma, Epsilon Tau, and Phi Sigma (A, B, C, D, E, F for short). The federation is governed by the Inter-Franternia Congress, consisting of 200 seats apportioned among the planets according to their populations. The following table gives the planet populations.
Planet / A / B / C / D / E / FPopulation / 11,370,000 / 8,070,000 / 38,620,000 / 14,980,000 / 10,420,000 / 16,540,000
Finite Math A: Homework 4.3
Hamilton’s Method Paradoxes
1. a. Mom has a box of 11 candy bars that she decides to apportion among her three youngest children according to the number of minutes each child spent doing homework that week. The number of minutes each child spent are listed in the table below. Use Hamilton’s Method to find an apportionment for each child.
Child / Bob / Peter / RonMinutes spent on Homework / 54 / 243 / 703
b. Supposed that before mom hands out candy bars, the children spend a little “extra” time on homework. Bob puts in an extra 2 minutes, Peter an extra 12 minutes, and Ron an extra 86 minutes. Their new totals are listed in the table below. Use Hamilton’s Method to find an apportionment for each child.
Child / Bob / Peter / RonMinutes spent on Homework / 56 / 255 / 789
c. Which paradox is demonstrated in this example?
2. a. Mom has a box of 10 candy bars that she decides to apportion among her three youngest children according to the number of minutes each child spent doing homework that week. The number of minutes each child spent are listed in the table below. Use Hamilton’s Method to find an apportionment for each child.
Child / Bob / Peter / RonMinutes spent on Homework / 54 / 243 / 703
b. Suppose just before she hands out candy bars, she finds an extra one, bringing the total number of candy bars to 11. Apportion using Hamilton’s method. (This should be the same answer as #1a – just rewrite your answers from that problem)
Bob = _____ Peter = ______Ron = _____
c. Which paradox is demonstrated in this example?
3. a. Mom has a box of 11 candy bars that she decides to apportion among her three youngest children according to the number of minutes each child spent doing homework that week. The number of minutes each child spent are listed in the table below. Use Hamilton’s Method to find an apportionment for each child.
Child / Bob / Peter / RonMinutes spent on Homework / 54 / 243 / 703
(This is the same as #1a – just copy down your answers from that problem)
Bob = _____ Peter = ______Ron = _____
b. Suppose that before the 11 candy bars are given out, cousin Jim shows up. Jim did homework for 580 minutes. To be fair, Mom adds 6 more candy bars to the original 11 bringing the total up to 17. Use Hamilton’s Method to find an apportionment for each child.
Child / Bob / Peter / Ron / JimMinutes spent on Homework / 54 / 243 / 703 / 580
c. Which paradox is demonstrated in this example?
Finite Math A: Homework 4.4-4.6
Hamilton’s Method Paradoxes
For Situations # 1 – 4 below
Use Webster’s Method to Apportion (if you use a modified divisor, be sure to state what it was)
1. The Bandana Republic is a small country consisting of four states: Apure, Barinas, Carabobo, and Delores. Suppose there are 160 seats in the Bandana Congress to be apportioned among the four states based on their respective populations. The following table show the population per state.
State / Apure / Barinas / Carabobo / DeloresPopulation / 3,310,000 / 2,670,000 / 1,330,000 / 690,000
2. The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates six bus routes (called A, B, C, D, E, and F) and 100 busses. The number of busses is apportioned to each route based on the number of passengers riding that route. The following table shows the daily average ridership on each route.
Route / A / B / C / D / E / FRidership / 45,300 / 31,070 / 20,490 / 14,160 / 20,260 / 8,720
3. The Placerville General Hospital has a nursing staff of 225 nurses working 4 shifts: A (7am-1pm), B (1pm to 7pm), C (7pm to 1am), and D (1am-7am). The number of nurses apportioned to each shift is based on the average number of patients treated in that shift, which is given in the following table:
Shift / A / B / C / DPatients / 871 / 1029 / 610 / 190
4. The Interplanetary Federation of Fraternia consists of six planets: Alpha, Kappa, Beta, Theta, Chi Omega, Delta Gamma, Epsilon Tau, and Phi Sigma (A, B, C, D, E, F for short). The federation is governed by the Inter-Franternia Congress, consisting of 2000 seats apportioned among the planets according to their populations. The following table gives the planet populations.
Planet / A / B / C / D / E / FPopulation / 11,370,000 / 8,070,000 / 38,620,000 / 14,980,000 / 10,420,000 / 16,540,000