Chapter 4: Mathematics of Appotionment



Honors Discrete Quiz 4.1 – 4.3 Review Guide:

Key Terms: Be able to identify and provide a definition for each of the following terms

o Seats

o Populations

o States

o Standard Divisor

o Standard Quota

o Upper Quotas

o Lower Quotas

o “Good” Apportionment

o Alabama Paradox

o New States Paradox

o Population Paradox

o Quota Rule

Be able to calculate the following:

o Standard Divisor

o Standard Quota

o Upper Quotas

o Lower Quotas

o Hamilton’s Method

Be able to interpret the meaning of the following in a given problem:

o Seats

o Populations

o States

o Standard Divisor

(1) The Google has a janitorial staff of 250 janitors working in five shifts (A, B, C, D, E). The number of workers apportioned to each shift is based on the average number of other Google employees working per shift, given the following table.

|SHIFT |A |B |C |D |E |

|Average number of other Google employees |423 |1071 |1229 |950 |327 |

a. Identify the seats, states, and populations in this problem.

b. Calculate the standard divisor in this problem.

c. What does the standard divisor represent in this problem?

d. Find the standard quota for each shift.

e. Find the lower and upper quota for each shift.

(2) The Richmond has 400 new police officers. There are 6 counties and number of police officers apportioned to each county is based on the number of crimes committed thus far listed below:

County #1 had a 48 crimes, County #2 had a 80 crimes,

County #3 had a 62 crimes, County #4 had a 33 crimes,

County #5 had a 40 crimes, and County #6 had a 37 crimes.

a. Identify the seats, states, and populations in this problem.

b. Calculate the standard divisor in this problem.

c. What does the standard divisor represent in this problem?

d. Find the standard quota for each shift.

e. Find the lower and upper quota for each shift.

(3) 4 students need to share a $2,000 prize they won in the lottery. They decide to apportion money to themselves based on number of service hours accumulated.

|Friend |Joe |Jane |John |Jessica |

|Service Hours |200 |150 |270 |180 |

a. Identify the seats, states, and populations in this problem.

b. Calculate the standard divisor in this problem.

c. What does the standard divisor represent in this problem.

d. Find the standard quota for each shift.

e. Find the lower and upper quota for each shift.

(4) The small nation is made up of four counties and has 125 seats in its Congress, which are apportioned to the islands according to their populations under Hamilton’s Method.

|County |San Andreas |San Bernadino |Santa Clara |San Diego |

|Population |62,400 |26,160 |28,480 |39,120 |

(5) A mother has 30 pieces of candy to share among her 4 children. The mother decides to apportion the candy based on the children’s GPA last school year. .

|Child |Anthony |Bridget |Chris |Dawn |

|GPA |2.9 |3.15 |3.85 |3.3 |

a. Show all steps to find the apportionment under Hamilton’s Method.

b. Anthony, Chris, and Dawn had done extra credit, which wasn’t originally put into their GPAs.

|Child |Anthony |Bridget |Chris |Dawn |

|GPA |3.0 |3.15 |3.9 |3.5 |

Find the apportionment under Hamilton’s Method and state if this is an example of any known PARADOXES.

(6) The populations of 4 states that make up a union are given in percent of actual populations. The union has a legislature of 68 seats.

|States |State #1 |State #2 |State #3 |State #4 |

|Population |230 |600 |4320 |4850 |

a. Show all steps to find the apportionment under Hamilton’s Method.

b. A 69th seat is added to the legislature. Find the apportionment under Hamilton’s Method and state if this is an example of any known paradoxes.

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