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Mathematics #2

Apportionment

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1. A teacher wishes to distribute 10 identical pieces of candy among 4 students, based on how many pages of a book they read last month, using Hamilton’s method. The table below lists the total number of pages read by each student.

ChildPages
Zach405
Aidan280
Bill555
Jordan530

Answer the following questions. Give all answer to at least 4 decimal places as appropriate. Find the divisor:  Find the quota for Zach:  Find the initial apportionment for Zach: 

2. 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 19 tutors, determine how many tutors should be assigned to each subject using Hamilton’s method.

SubjectStudents EnrolledTutors to Assign
Math330
English280
Chemistry130
Biology55

3. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift.  Apportion 12 salespeople using Hamilton’s method given the information below.

ShiftMorningMiddayAfternoonEvening
Average number of customers145330415510
Salespeople to assign

4. Three people invest in a treasure dive, each investing the amount listed below.  The dive results in 35 gold coins.  Using Hamilton’s method, apportion those coins to the investors based on their investment.

InvestorInvestmentAllocation of 35 coins
Samantha$18,350
Kendra$12,290
Zoe$4,360

  Right before the coins are distributed, the divers find one more coin they had misplaced.  Redo the allocation, now with 36 coins

InvestorInvestmentAllocation of 36 coins
Samantha$18,350
Kendra$12,290
Zoe$4,360

  Does this situation illustrate any apportionment issues?        

5. The legislature in a state has 39 seats.  Apportion these seats to the five counties below using Hamilton’s method.

CountyPopulationSeats Received
Adams274,000
Grant385,000
Colton307,000
Davis431,000
Hayes152,000

6. The legislature in a state has 35 seats.  Apportion these seats to the six counties below using Hamilton’s method.

CountyPopulationSeats Received
Adams300,000
Grant333,000
Colton227,000
Davis305,000
Hayes477,000
McKinley80,000

7. A legislator in a state consists of 50 seats.  Apportion the seats to the counties using Hamilton’s method.

CountyPopulationSeats
Jefferson30,000
Clay15,600
Madison34,600
Jackson40,800
Franklin59,000

  Ten years, the populations are recounted.  Reapportion the 50 seats.

CountyPopulationSeats
Jefferson30,000
Clay15,600
Madison36,200
Jackson40,800
Franklin59,200

  Does this situation illustrate any apportionment issues?        

8. A school district has two high schools. The district could only afford to hire 16 guidance counselors.  Determine how many counselors should be assigned to each school using Hamilton’s method.

SchoolStudents EnrolledCounselors to Assign
Lowell6306
Fairview3288

  The next year, a new school is opened, with 3768 students.  Using the divisor from above, determine how many additional counselors should be hired for the new school:   counselors.

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

SchoolStudents EnrolledCounselors to Assign
Lowell6306
Fairview3288
New School3768

Does this situation illustrate any apportionment issues?        

9. 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 18 tutors, determine how many tutors should be assigned to each subject using Jefferson’s method.

SubjectStudents EnrolledTutors to Assign
Math270
English265
Chemistry135
Biology85

What modified divisor did you use?  

10. The legislature in a state has 103 seats.  Apportion these seats to the four counties below using Jefferson’s method.

CountyPopulationInitial AllocationSeats Received
Adams33,400
Grant108,500
Colton10,700
Davis12,200

Does this situation illustrate any apportionment issues?        

11. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift.  Apportion 15 salespeople using Jefferson’s method given the information below.

ShiftMorningMiddayAfternoonEvening
Average number of customers85320405545
Salespeople to assign

What modified divisor did you use?  

12. The legislature in a state has 30 seats.  Apportion these seats to the five counties below using Jefferson’s method.

CountyPopulationSeats Received
Adams106,000
Grant425,000
Colton432,000
Davis405,000
Hayes298,000

13. The legislature in a state has 36 seats.  Apportion these seats to the six counties below using Jefferson’s method.

CountyPopulationSeats Received
Adams213,000
Grant449,000
Colton407,000
Davis241,000
Hayes425,000
McKinley460,000

14. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. The college can only afford to hire 20 tutors.  Use Webster’s method to determine the initial allocations with the standard divisor, then finish the apportionment to determine how many tutors should be assigned to each subject.

SubjectStudents EnrolledInitial AllocationTutors to Assign
Math355
English265
Chemistry110
Biology60

What modified divisor did you use for the final apportionment?  

15. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift.  Apportion 19 salespeople using Webster’s method given the information below.

ShiftMorningMiddayAfternoonEvening
Average number of customers135275370450
Salespeople to assign

What modified divisor did you use?  

16. The legislature in a state has 44 seats.  Apportion these seats to the five counties below using Webster’s method.

CountyPopulationSeats Received
Adams147,000
Grant192,000
Colton492,000
Davis183,000
Hayes115,000

17. The legislature in a state has 59 seats.  Apportion these seats to the six counties below using Webster’s method.

CountyPopulationSeats Received
Adams79,000
Grant400,000
Colton180,000
Davis178,000
Hayes318,000
McKinley330,000

18. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. The college can only afford to hire 13 tutors.  Using Huntington-Hill’s method determine the initial geometric means and the initial allocations with the standard divisor.  You do not need to finish the apportionment.

SubjectStudents EnrolledInitial Geometric MeanInitial Allocation
Math290
English230
Chemistry130
Biology70

Give decimal answers accurate to at least 3 decimal place

19. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift.  Apportion 17 salespeople using Huntington-Hill’s method given the information below.

ShiftMorningMiddayAfternoonEvening
Average number of customers125305425555
Salespeople to assign

What modified divisor did you use?  

20. The legislature in a state has 47 seats.  Apportion these seats to the five counties below using the Huntington-Hill method.

CountyPopulationSeats Received
Adams466,000
Grant238,000
Colton218,000
Davis259,000
Hayes402,000

21. The legislature in a state has 53 seats.  Apportion these seats to the six counties below using the Huntington-Hill method.

CountyPopulationSeats Received
Adams321,000
Grant191,000
Colton334,000
Davis103,000
Hayes130,000
McKinley249,000

22. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. The college can only afford to hire 20 tutors.  Using Lowndes’s method determine the decimal/whole ratios (to 4 decimal places) and the final apportionment. 

SubjectStudents EnrolledRatiosTutors assigned
Math260
English260
Chemistry155
Biology85

23. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift.  Apportion 22 salespeople using Lowndes’ method given the information below.

ShiftMorningMiddayAfternoonEvening
Average number of customers100280380560
Salespeople to assign

24. The legislature in a state has 45 seats.  Apportion these seats to the five counties below using Lowndes’ method.

CountyPopulationSeats Received
Adams248,000
Grant305,000
Colton185,000
Davis273,000
Hayes395,000

25. The legislature in a state has 56 seats.  Apportion these seats to the six counties below using Lowndes’ method.

CountyPopulationSeats Received
Adams469,000
Grant176,000
Colton186,000
Davis177,000
Hayes190,000
McKinley217,000

Math Homework

apportionment

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.

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

7. A small country consists of five states, whose populations are listed below. If the legislature has 119 seats, apportion the seats. D: 594,000 E: 211,00

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 C: 11,586 E: 3,126

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.

Fair Division

1. Chance and Brianna buy a pizza for $10 that is half pepperoni and half veggie. They cut the pizza into 8 slices. If Chance likes veggie three times as much as pepperoni, what is the value of a slice that is half pepperoni, half veggie?

3. Erin, Catherine, and Shannon are dividing a large bag of candy. They randomly split

the bag into three bowls. The values of the entire bag and each of the three bowls in

the eyes of each of the players are shown below. For each player, identify which

bowls they value as a fair share.

Whole Bag Bowl 1 Bowl 2 Bowl 3

Erin $5 $2.75 $1.25 $1.00

Catherine $4 $0.75 $2.50 $0.75

Shannon $8 $1.75 $2.25 $4.00

4. Jenna, Tatiana, and Nina are dividing a large bag of candy. They randomly split the

bag into three bowls. The values of the entire bag and each of the three bowls in the

eyes of each of the players are shown below. For each player, identify which bowls

they value as a fair share.

Whole Bag Bowl 1 Bowl 2 Bowl 3

Jenna $8 $4.50 $0.75 $2.75

Tatiana $4 $1.00 $1.00 $2.00

Nina $6 $1.75 $2.50 $1.75

9. A 6-foot sub valued at $30 is divided among five players (P1, P2, P3, P4, P5) using the

last-diminisher method. The players play in a fixed order, with P1 first, P2 second, and

so on. In round 1, P1 makes the first cut and makes a claim on a piece. For each of the

remaining players, the value of the current claimed piece at the time it is their turn is

given in the following table:

P2 P3 P4 P5

Value of the current claimed piece $6.00 $8.00 $7.00 $6.50

a. Which player gets his or her share at the end of round 1?

b. What is the value of the share to the player receiving it?

c. How would your answer change if the values were:

P2 P3 P4 P5

Value of the current claimed piece $6.00 $8.00 $7.00 $4.50

11. Four heirs (A, B, C, and D) must fairly divide an estate consisting of two items – a

desk and a vanity – using the method of sealed bids. The players’ bids (in dollars) are:

A B C D

Desk 320 240 300 260

Vanity 220 140 200 180

a. What is A’s fair share?

b. Find the initial allocation.

c. Find the final allocation

Math Fair

1. Jeremy and Adrianna buy a pizza for $12 that is half pepperoni and half veggie. They cut the pizza into 8 slices. If Jeremy likes veggie twice as much as pepperoni, what is the value of a slice that is all pepperoni? $ Round to the nearest cent if necessary.

2. Francisco, Michael, and Casey are dividing a large bag of candy. They randomly split the bag into three bowls. The values of the entire bag and each of the three bowls in the eyes of each of the players are shown here:

Whole BagBowl 1Bowl 2Bowl 3
Francisco$5$3.50$0.75$0.75
Michael$8$1.75$1.75$4.50
Casey$6$1.75$2.50$1.75

Which of the three bowls is a fair share to Francisco? Bowl 

3. Dakota and Zoe want to split a bag of fun-sized candy, and decide to use the divider-chooser method. The bag contains 100 Snickers, 100 Milky Ways, and 100 Reese’s, which Dakota values at $2, $1, and $3 respectively. (This means Dakota values the 100 Snickers together at $2, or $0.02 for 1 Snickers) If Zoe is the divider, and in one half puts: 20 Snickers 30 Milky Ways 60 Reese’s What is the value of this half in Dakota’s eyes? $ Is this a fair share?    

4. Roy and Ashley want to split a bag of candy, and decide to use the divider-chooser method. The bag contains 100 Snickers, 100 Milky Ways, and 100 Reese’s, which Roy values at $5, $2, and $4 respectively. If Roy is the divider, find a possible division that is consistent with his value system. In this division, one half contains:  Snickers  Milky Ways  Reese’s

5. Kori, Hillary, Tatiana, and Riley are dividing a piece of land using the lone-divider method. The values of the four pieces of land in the eyes of the each player are shown below

Piece 1Piece 2Piece 3Piece 4
Kori19%30%35%16%
Hillary26%32%22%20%
Tatiana24%15%45%16%
Riley25%25%25%25%

If playing honestly, what will Kori’s declaration be?

· Piece 1

· Piece 2

· Piece 3

· Piece 4

Which piece will Kori receive?       Which piece will Riley receive?      

6. A 6-foot sub valued at $25 is divided among five players, (P1, P2, P3, P4, P5) using the last-diminisher method. The players play in a fixed order, with P1 first, P2 second, and so on. In round 1, P1 makes the first cut and makes a claim on a C-piece. For each of the remaining players, the value of the current C-piece at the time it is their turn is given in the following table:

P2P3P4P5
Value of the current C-piece$3.00$6.00$5.50$3.00

Which player gets his or her share at the end of round 1?        What is the value of the share to the player receiving it? $

7. A 6-foot sub valued at $25 is divided among five players, (P1, P2, P3, P4, P5) using the last-diminisher method. The players play in a fixed order, with P1 first, P2 second, and so on. In round 1, P1 makes the first cut and makes a claim on a C-piece. For each of the remaining players, the value of the current C-piece at the time it is their turn is given in the following table:

P2P3P4P5
Value of the current C-piece$3.50$6.50$6.00$6.50

Which player gets his or her share at the end of round 1?        What is the value of the share to the player receiving it? $

8. Four heirs (A, B, C, and D) must fairly divide an estate consisting of two items – a desk and a vanity – using the method of sealed bids. The players’ bids (in dollars) are:

ABCD
Desk220240200300
Vanity180200140220

The original fair share of A is worth: $  In the initial allocation, player A:       and     the estate $ After all is said and done, in the final allocation, player A:       and     the estate $

9. As part of an inheritance, three children, Abby, Ben and Carla, are dividing four vehicles using Sealed Bids. Their bids (in thousands of dollars) for each item is shown below.

AbbyBenCarla
Motorcycle9116
Car12108
Tractor512
Boat347

In the final allocation, Abby gets which items?  (click none, one, or multiple boxes)

· Motorcycle

· Car

· Tractor

· Boat

In addition, she     to/from the estate: $

Give your answer to the last question to the nearest dollar (careful here – the original amounts were in thousands of dollars)

10. This question illustrates how bidding dishonestly can end up hurting the cheater. Four partners are dividing a million-dollar property using the lone-divider method. Using a map, Danny divides the property into four parcels s1, s2, s3, and s4. The following table shows the value of the four parcels in the eyes of each partner (in thousands of dollars):

s1s2s3s4
Danny$250$250$250$250
Brianna$470$180$200$150
Carlos$300$320$190$190
Greedy$340$300$300$60

Assuming all players bid honestly, which piece will Greedy receive?

· s1

· s2

· s3

· s4

Assume Brianna and Carlos bid honestly, but Greedy decides to bid only for s1, figuring that doing so will get him s1. In this case there is a standoff between Brianna and Greedy. Since Danny and Carlos are not part of the standoff, they can receive their fair shares. Suppose Danny gets s3 and Carlos gets s2, and the remaining pieces are put back together and Brianna and Greedy will split them using the basic divider-chooser method. If Greedy gets selected to be the divider, what will be the value of the piece he receives?  thousand dollars

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