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# 8.5: Exercises

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1. Three players are dividing a business. The assets are divided into three shares, S1, S2, and S3. The following table shows how each player sees each share. For each player, list the shares that the player considers a fair share.

S1

S2

S3

Doug

40%

30%

30%

Eddie

33 1/3%

33 1/3%

33 1/3%

Fred

35%

30%

35%

1. Four cousins are dividing a pizza. The pizza has been divided into four pieces, S1, S2, S3, and S4. The following table shows how each cousin sees each piece. For each cousin, list the pieces that the cousin considers a fair share.

S1

S2

S3

S4

Anne

0%

0%

50%

50%

Bob

30%

30%

30%

10%

Cathy

20%

30%

20%

30%

Don

25%

25%

25%

25%

1. A three-flavored cake is one-third chocolate, one-third vanilla, and one-third strawberry. If the chocolate part is worth $4, the vanilla part is worth$6 and the strawberry part is worth $12 to Francis, find the value of each of the following slices. 1. A three-flavored cake is one-third chocolate, one-third vanilla, and one-third strawberry. If the chocolate part is worth$6, the vanilla part is worth $8 and the strawberry part is worth$10 to George, find the value of each of the following slices.

1. A 12-inch sandwich worth $6 is half turkey and half meatball. To Jack, the turkey half is worth$4 and the meatball half is worth $2. Find the value of the following slices of the sandwich. 1. A 12-inch sandwich worth$9 is half turkey and half meatball. To Jack, the turkey half is worth $3 and the meatball half is worth$6. Find the value of the following slices of the sandwich.

1. Alice wants to divide a half-strawberry half-vanilla cake worth $12 into two pieces of equal value. She likes strawberry three times as much as vanilla. How should she cut the cake so that each piece is a fair share to her? 1. Sam has a pizza that is one-third pepperoni, one-third mushrooms, and one-third sausage. He likes both pepperoni and mushrooms twice as much as sausage. He wants to split the pizza into two pieces to share with his roommate. How should Sam cut the pizza so that each piece is a fair share to him? 1. Luke wants to split a twelve-inch half turkey and half veggie sub sandwich worth$12 with a friend. Luke likes turkey twice as much as veggies. How should he cut the sandwich so that each piece is a fair share to him?

1. Alice and Betty want to divide a half-strawberry half-vanilla cake worth $12 by the divider/chooser method. Alice likes strawberry three times as much as vanilla and Betty likes vanilla twice as much as strawberry. A coin is tossed and Alice is the divider. 1. How should Alice cut the cake? 2. Which piece should Betty choose and what is its value to her? 1. Sam and Ted have a pizza that is one-third pepperoni, one-third mushrooms, and one-third sausage. Sam likes pepperoni and sausage equally well but does not like mushrooms. Ted likes pepperoni twice as much as sausage and likes mushrooms twice as much as pepperoni. They want to split the pizza by the divider/chooser method. After drawing straws, Sam is the divider. 1. How should Sam cut the pizza? 2. Which piece should Ted choose and what is its value to him? 1. Luke and Mark want to use the divider/chooser method to split a twelve-inch half turkey and half veggie sub sandwich worth$12. Luke likes turkey three times as much as veggies and Mark like veggies twice as much as turkey. They draw cards and Mark is the divider.
1. How should Mark cut the sandwich?
2. Which piece should Luke choose and what is its value to him?

1. Two brothers want to divide up a piece of land their grandfather left them. The piece of land, valued at $300,000 is made up of two distinct parts as shown in the following figure. Joseph likes the woods twice as much as the fields. Kevin likes the fields but does not like the woods at all. The brothers decide to use the divider/chooser method to divide the land. They toss a coin and Joseph is the divider. 1. How should Joseph divide the land if he makes one horizontal cut on the map: Which piece of land should Kevin choose: What is its value to him? 2. How should Joseph divide the land if he makes one vertical cut on the map: Which piece of land should Kevin choose: What is its value to him? 3. If Joseph can make more than one cut (i.e. cut out a rectangle or a triangle) how should he cut the piece of land: Hint: there is more than one answer. 1. Amy, Becky, Charles and Doug want to use the lone divider method to split a piece of land they inherited from their father. They draw cards to determine that Doug is the divider. After Doug divides the land, Amy bids {S2, S3}, Becky bids {S1, S2}, and Charles bids {S3}. Describe the fair division. 1. Edward, Frank, George, and Harold want to use the lone divider method to split a piece of land they inherited from their Grandfather. They draw cards to determine that George is the divider. After George divides the land, Edward bids {S3, S4}, Frank bids {S2}, and Harold bids {S3, S4}. Describe the fair division. 1. Inez, Jackie, Kelly, and Louise want to use the lone divider method to split a piece of land they inherited from their father. They draw cards to determine that Jackie is the divider. 1. If Inez bids {S2}, Kelly bids {S1, S2}, and Louise bids {S2, S4} describe the fair division. 2. At the last minute Louise changes her bid to {S2}. If Inez and Kelly do not change their bids, describe the fair division. 1. Frank, Greg, and Harriet want to divide a cake worth$24 that is half chocolate and half strawberry. Frank likes all cake equally well. Greg likes chocolate twice as much as strawberry and Harriet like strawberry three times as much as chocolate. They decide to use the lone chooser method and draw cards to determine that Greg will cut the cake first and Harriet will be the chooser.
1. How should Greg cut the cake?
2. Which piece will Frank choose and what is its value to him?
3. How will Greg subdivide his piece of the cake?
4. How will Frank subdivide his piece of the cake?
5. Which pieces of cake will Harriet choose?
6. Describe the final division of the cake. Which pieces does each player receive and what are their values?

1. Paul, Rachel and Sally want to divide a four-topping pizza using the lone chooser method. The draw straws to determine that Rachel is the chooser and Paul will make the first cut. Paul likes pepperoni twice as much as mushrooms, likes sausage three times as much as mushrooms and does not like pineapple. Rachel likes mushrooms and pineapple equally well but does not like pepperoni or sausage. Sally likes all pizza equally well.
1. How should Paul cut the pizza?
2. Which piece should Sally choose and what is its value to her?
3. How will Paul subdivide his piece of pizza?
4. How will Sally subdivide her piece of pizza?
5. Which pieces of pizza will Rachel choose?
6. Describe the final division of the pizza. Which pieces does each player receive and what are their values?

1. Eight heirs inherit a large piece of property. They decide to use the last diminisher method to divide the property. They draw straws to choose an order. Assume the order of the players is P1, P2, P3, P4, P5, P6, P7, and P8. In round one, P3, P4 and P7 are the only diminishers. In round two, no one diminishes the piece. In round three, P3 and P4 are the only diminishers. No one diminishes the piece in rounds four and five. Every player diminishes the piece in round 6.
1. Who keeps the piece at the end of round one?
2. Who cuts the piece at the beginning of round three?
3. In round six, who cuts the piece and who keeps the piece?
4. Which players are left after round six and how do they finish the division?
1. Seven heirs inherit a large piece of property. They decide to use the last diminisher method to divide the property. They draw straws to choose an order. Assume the order of the players is P1, P2, P3, P4, P5, P6, and P7. In round one, P3, P4 and P7 are the only diminishers. In round two, every player diminishes the piece. In round three, P3 and P4 are the only diminishers. No one diminishes the piece in rounds four or five. Every player diminishes the piece in round 6.
1. Who keeps the piece at the end of round two?
2. In round three, who cuts the piece and who keeps the piece?
3. Who cuts the piece at the beginning of round five?
4. Which players are left after round five and how do they finish the division?
1. Three heirs are dividing an estate consisting of a house, a lakeside cabin, and a small business using the method of sealed bids. The bids are listed in the following table.
 Mary Nancy Olivia House $350,000$380,000 $362,000 Cabin$280,000 $257,000$270,000 Business $537,000$500,000 $520,000 Describe the final settlement including who gets each item and how much money they pay or receive. 1. Five heirs are dividing an estate using the method of sealed bids. The bids are listed in the following table.  A B C D E Item 1$352 $295$395 $368$324 Item 2 $98$102 $98$95 $105 Item 3$460 $449$510 $501$476 Item 4 $852$825 $832$817 $843 Item 5$513 $501$505 $505$491 Item 6 $725$738 $750$744 $761 Describe the final settlement including who gets each item and how much money they pay or receive. 1. Albert, Brett and Carl own a hot dog stand together. Unfortunately, circumstances are forcing them to dissolve the partnership. They decide to use the method of sealed bids with the understanding that one of them will get the hot dog stand and the other two will get cash. Albert bids$81,000, Brett bids $78,000, and Carl bids$87,000. Who gets the hot dog stand and how much does he pay each of the other two partners?
1. The method of sealed bids can be used to divide up negative items like a list of chores that must be done. The main difference in the method is that the item or chore is given to the lowest bidder rather than the highest. You also need to be careful in the “owes to estate”/”estate owes” step. Three roommates need to divide up four chores in order to get their security deposit back. They use the method of sealed bids to divide the chores. The bids are summarized in the following table.
 Harry Ingrid Jeff Clean Bathrooms $65$70 $55 Patch and Paint Wall$100 $85$95 Scrub Baseboards $60$50 $45 Wash Windows$75 $80$90

Describe the final outcome of the division. State which chores each roommate does and how much money each roommate gets or pays.

1. Albert, Bernard, and Charles want to divide up a collection of 17 small objects using the method of markers. The objects are laid out in a straight line and the players place their markers as shown in the following figure. Describe the final division, including which objects each player gets and how they deal with any leftover objects.

1. Alex, Bobby, Carrie, and Doug want to divide a collection of 25 small objects using the method of markers. The objects are laid out in a straight line and the players place their markers as shown in the following figure. Describe the final division, including which objects each player gets and how they deal with any leftover objects.

1. Jack, Kelly, and Larry want to divide a collection of 25 small objects using the method of markers. The objects are laid out in a straight line as shown in the following figure. Jack values each red object at $2, each blue object at$1, each green object at $0.50, and each yellow object at$0. Kelly values each red object at $0, each blue object at$1.50, each green object at $1, and each yellow object at$2. Larry values each red object at $1.50, each blue object at$1.50, each green object at $2, and each yellow object at$0.50. Determine where each player should place his markers. Draw the figure placing each player’s markers in the correct places. Do not determine the division of objects.

Note: The numbers do not come out evenly so you might have to round off a bit.