Multiple Choice 1. In Ben’s eyes, the pi

Discrete Math Review ? Chapter 3

Name_______________________________

Multiple Choice ? Choose the one alternative that best completes the statement or answers the question.

Angela and Ben want to divide fairly the chocolate-strawberry cake shown below using the divider-chooser method. The total cost of the cake was $6.00. Angela values strawberry three times as much as she values chocolate, while Ben values chocolate twice as much as he values strawberry.

__a___ 1. In Ben's eyes, the piece shown to the right is worth

A. $5.00 B. $4.00 C. $4.50 D. $4.75 E. none of the above

__c___ 2. In Angela's eyes, the piece shown to the right is worth

A. $4.50 B. $5.00 C. $3.75 D. $2.25 E. none of the above

__a___ 3. In Ben's eyes, the piece shown to the right is worth

A. $3.50 B. $3.00 C. $4.00 D. $2.50 E. none of the above

__d___ 4. In Angela's eyes, the piece show to the right is worth

A. $3.75 B. $3.00 C. $3.50 D. $2.25 E. none of the above

Four heirs (A, B, C, and D) must fairly divide an estate consisting of two items ? a house and a cabin ? using the method of sealed bids. The players' bids (in dollars) are:

House Cabin

A 195,000 45,000

B 212,000 36,000

C 201,000 35,000

D 182,000 42,000

__d___ 5. The original fair share of player A is worth

A. $1210,000 B. $240,000 C. $80,000 D. $60,000 E. none of the above

__c___ 6. In the initial allocation, player A

A. gets the house and an additional $55,000 from the estate B. gets the cabin and pays the estate $15,000 C. gets the cabin and an additional $15,000 from the estate D. gets the house and pays the estate $135,000 E. none of the above

__a___ 7. In the initial allocation, player B

A. gets the house and pays the estate $150,000 B. gets $62,000 from the estate C. gets the house and the cabin and pays the estate $186,000 D. gets the house and pays the estate $274,000 E. none of the above

__d___ 8. After the initial allocation to each player is made, there is a surplus of

A. $150,000 B. $0 C. $32,000 D. $20,000 E. none of the above

_c____ 9. After all is said and done, the final allocation to player A is

A. $65,000 in cash B. the cabin plus $5,000 in cash C. the cabin plus $20,000 in cash D. the cabin plus $10,000 in cash E. none of the above

Four players (A, B, C, and D) agree to divide the 12 items below using the method of markers. The players' bids are as indicated.

A1 C1D1 B1

B2 A2C2 D2 B3 C3 A3 D3

_e__ 10. Item 5

A. goes to C B. goes to A C. goes to D D. is left over E. goes to B

__c__ 11. Item 8

A. goes to A B. goes to B C. goes to C D. goes to D E. is left over

_e____ 12. Item 11

A. goes to A B. goes to B C. goes to C D. goes to D E. is left over

Free Response: Show all work for credit

13. Molly buys a chocolate ? strawberry ? vanilla cake for $11.20. Molly values strawberry twice as much as chocolate and values chocolate twice as much as vanilla.

a) Find the value of each flavor.

V: $1.60 C: $3.20 S: $6.40

b) If the cake is cut into six 600 wedges shown in the picture, find the value of each piece.

P1: $0.80 P2: $2

P3: $3.20 P4: $2.4 P5: $1.6 P6: $1.20

14. Three players (Minnie, Mickey, Donald) must divide a cake among themselves. Suppose the cake is divided into three slices: ( s1 s2 s3). The following table shows the percentage of value of the entire cake that each slice represents to each player.

Minnie Mickey Donald

whole cake $12.00 $15.00 $13.50

s1 $3.00 $4.00 $4.50

s2 $5.00 $4.50 $4.50

s3 $4.00 $6.50 $4.50

Minnie ? S2, S3

Mickey ? S3

S1 ? Donald

S2 ? Minnie

S3 ? Mickey

Donald ? S1, S2, S3

15. Three players (Alex, Betty, and Cindy) must divide a cake among themselves. Suppose the cake is divided into three slices: ( s1 s2 s3). The following table shows the percentage of value of the entire cake that each slice represents to each player.

For each player, indicate which of the slices are fair shares.

Alex Betty Cindy

s1 30% 35% 33 1/3 %

s2 40% 25% 50%

s3 30% 40% 16 2/3%

Alex ? S2

S1- Cindy, S2 ? Alex, S3 - Betty

Betty ? S1, S3

Cindy ? S1, S2

16. Five players (Addie, Ben, Callie, Dave, and Erin) must divide a cake among themselves. Suppose the cake is divided into 5 slices (s1, s2, s3, s4, s5). The value of the entire cake and of each of the five slices in the eyes of the players is shown in the table below.

For each player, indicate which of the slices are fair shares.

Addie Ben Callie Dave Erin

whole cake $15.00 $18.00 $12.00 $10.00 $20.00

s1 $3.25 $4.00 $2.40 $2.50 $3.50

s2 $2.50 $4.25 $2.40 $1.75 $3.75

s3 $3.50 $3.25 $2.40 $1.50 $5.00

s4 $2.50 $3.75 $2.40 $2.25 $4.90

s5 $3.25 $2.75 $2.40 $2.00 $2.85

Addie ? S1, S3, S5

Ben ? S1, S2, S4

Callie ? S1, S2, S3, S4, S5

Dave ? S1, S4, S5

Erin ? S3, S4

17. Determine a fair allocation of assets

Coffee Maker Waffle Iron Quesadilla Maker

Total: Fair Share:

Cailey 45 60 50 155

38.75

Ryan 25 70 55 150

37.50

Jason 35 65 65 165

41.25

Priscilla 20 50 30 100 25

a) Determine a fair initial allocation of the items

Cailey ? coffee maker

owe 6.25

Ryan ? Waffle maker

owe 32.50

Jason ? Quesadilla maker

owe 23. 75

Priscilla -

receive 25

b) Determine the surplus funds

37. 50 is surplus

everybody receive extra $9.38

c) Determine the final allocation of assets

Cailey ? coffee maker

receive 2.76

Ryan ? Waffle maker

owe 23.12

Jason ? Quesadilla maker

owe 14.37

Priscilla -

receive 34.38

18. Ana, Billy, and Cara are dividing four pieces of furniture using the method of sealed bids. Their bids on each of the items are given in the following table.

Dresser Desk Vanity Tapestry

Ana $150 $180 $170 $400

Billy $300 $150 $200 $250

Cara $275 $165 $260 $500

a) Determine a fair initial allocation of the items

Ana ? desk Billy ? dresser Cara ? vanity and tapestry

receive 120 owe 360

b) Determine the surplus funds

240

and everybody receives an extra 80

c) Determine the final allocation of assets

Ana ? desk Billy ? dresser Cara ? vanity and tapestry

receive 20 receive 80 owe 280

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