Mathematics #2

Apportionment

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.

Child | Pages |

Zach | 405 |

Aidan | 280 |

Bill | 555 |

Jordan | 530 |

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.

Subject | Students Enrolled | Tutors to Assign |

Math | 330 | |

English | 280 | |

Chemistry | 130 | |

Biology | 55 |

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.

Shift | Morning | Midday | Afternoon | Evening |

Average number of customers | 145 | 330 | 415 | 510 |

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.

Investor | Investment | Allocation 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

Investor | Investment | Allocation 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.

County | Population | Seats Received |

Adams | 274,000 | |

Grant | 385,000 | |

Colton | 307,000 | |

Davis | 431,000 | |

Hayes | 152,000 |

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

County | Population | Seats Received |

Adams | 300,000 | |

Grant | 333,000 | |

Colton | 227,000 | |

Davis | 305,000 | |

Hayes | 477,000 | |

McKinley | 80,000 |

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

County | Population | Seats |

Jefferson | 30,000 | |

Clay | 15,600 | |

Madison | 34,600 | |

Jackson | 40,800 | |

Franklin | 59,000 |

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

County | Population | Seats |

Jefferson | 30,000 | |

Clay | 15,600 | |

Madison | 36,200 | |

Jackson | 40,800 | |

Franklin | 59,200 |

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.

School | Students Enrolled | Counselors to Assign |

Lowell | 6306 | |

Fairview | 3288 |

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.

School | Students Enrolled | Counselors to Assign |

Lowell | 6306 | |

Fairview | 3288 | |

New School | 3768 |

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.

Subject | Students Enrolled | Tutors to Assign |

Math | 270 | |

English | 265 | |

Chemistry | 135 | |

Biology | 85 |

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.

County | Population | Initial Allocation | Seats Received |

Adams | 33,400 | ||

Grant | 108,500 | ||

Colton | 10,700 | ||

Davis | 12,200 |

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.

Shift | Morning | Midday | Afternoon | Evening |

Average number of customers | 85 | 320 | 405 | 545 |

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.

County | Population | Seats Received |

Adams | 106,000 | |

Grant | 425,000 | |

Colton | 432,000 | |

Davis | 405,000 | |

Hayes | 298,000 |

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

County | Population | Seats Received |

Adams | 213,000 | |

Grant | 449,000 | |

Colton | 407,000 | |

Davis | 241,000 | |

Hayes | 425,000 | |

McKinley | 460,000 |

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.

Subject | Students Enrolled | Initial Allocation | Tutors to Assign |

Math | 355 | ||

English | 265 | ||

Chemistry | 110 | ||

Biology | 60 |

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.

Shift | Morning | Midday | Afternoon | Evening |

Average number of customers | 135 | 275 | 370 | 450 |

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.

County | Population | Seats Received |

Adams | 147,000 | |

Grant | 192,000 | |

Colton | 492,000 | |

Davis | 183,000 | |

Hayes | 115,000 |

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

County | Population | Seats Received |

Adams | 79,000 | |

Grant | 400,000 | |

Colton | 180,000 | |

Davis | 178,000 | |

Hayes | 318,000 | |

McKinley | 330,000 |

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.

Subject | Students Enrolled | Initial Geometric Mean | Initial Allocation |

Math | 290 | ||

English | 230 | ||

Chemistry | 130 | ||

Biology | 70 |

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.

Shift | Morning | Midday | Afternoon | Evening |

Average number of customers | 125 | 305 | 425 | 555 |

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.

County | Population | Seats Received |

Adams | 466,000 | |

Grant | 238,000 | |

Colton | 218,000 | |

Davis | 259,000 | |

Hayes | 402,000 |

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

County | Population | Seats Received |

Adams | 321,000 | |

Grant | 191,000 | |

Colton | 334,000 | |

Davis | 103,000 | |

Hayes | 130,000 | |

McKinley | 249,000 |

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.

Subject | Students Enrolled | Ratios | Tutors assigned |

Math | 260 | ||

English | 260 | ||

Chemistry | 155 | ||

Biology | 85 |

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.

Shift | Morning | Midday | Afternoon | Evening |

Average number of customers | 100 | 280 | 380 | 560 |

Salespeople to assign |

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

County | Population | Seats Received |

Adams | 248,000 | |

Grant | 305,000 | |

Colton | 185,000 | |

Davis | 273,000 | |

Hayes | 395,000 |

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

County | Population | Seats Received |

Adams | 469,000 | |

Grant | 176,000 | |

Colton | 186,000 | |

Davis | 177,000 | |

Hayes | 190,000 | |

McKinley | 217,000 |

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 Bag | Bowl 1 | Bowl 2 | Bowl 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 1 | Piece 2 | Piece 3 | Piece 4 | |

Kori | 19% | 30% | 35% | 16% |

Hillary | 26% | 32% | 22% | 20% |

Tatiana | 24% | 15% | 45% | 16% |

Riley | 25% | 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:

P2 | P3 | P4 | P5 | |

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:

P2 | P3 | P4 | P5 | |

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:

A | B | C | D | |

Desk | 220 | 240 | 200 | 300 |

Vanity | 180 | 200 | 140 | 220 |

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.

Abby | Ben | Carla | |

Motorcycle | 9 | 11 | 6 |

Car | 12 | 10 | 8 |

Tractor | 5 | 1 | 2 |

Boat | 3 | 4 | 7 |

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):

s1 | s2 | s3 | s4 | |

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