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.
B.
C.
D.
E.
$5.00
$4.00
$4.50
$4.75
none of the above
__c___ 2. In Angela’s eyes, the piece shown to the right is worth
A.
B.
C.
D.
E.
$4.50
$5.00
$3.75
$2.25
none of the above
__a___ 3. In Ben’s eyes, the piece shown to the right is worth
A.
B.
C.
D.
E.
$3.50
$3.00
$4.00
$2.50
none of the above
__d___ 4. In Angela’s eyes, the piece show to the right is worth
A.
B.
C.
D.
E.
$3.75
$3.00
$3.50
$2.25
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.
B.
C.
D.
E.
$1210,000
$240,000
$80,000
$60,000
none of the above
__c___ 6. In the initial allocation, player A
A.
B.
C.
D.
E.
gets the house and an additional $55,000 from the estate
gets the cabin and pays the estate $15,000
gets the cabin and an additional $15,000 from the estate
gets the house and pays the estate $135,000
none of the above
__a___ 7. In the initial allocation, player B
A.
B.
C.
D.
E.
gets the house and pays the estate $150,000
gets $62,000 from the estate
gets the house and the cabin and pays the estate $186,000
gets the house and pays the estate $274,000
none of the above
__d___ 8. After the initial allocation to each player is made, there is a surplus of
A.
B.
C.
D.
E.
$150,000
$0
$32,000
$20,000
none of the above
_c____ 9. After all is said and done, the final allocation to player A is
A.
B.
C.
D.
E.
$65,000 in cash
the cabin plus $5,000 in cash
the cabin plus $20,000 in cash
the cabin plus $10,000 in cash
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.
B.
C.
D.
E.
goes to C
goes to A
goes to D
is left over
goes to B
__c__ 11. Item 8
A.
B.
C.
D.
E.
goes to A
goes to B
goes to C
goes to D
is left over
_e____ 12. Item 11
A.
B.
C.
D.
E.
goes to A
goes to B
goes to C
goes to D
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.
s1
30%
35%
33 1/3 %
Alex
Betty
Cindy
Alex – S2
s2
40%
25%
50%
s3
30%
40%
16 2/3%
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
Addie – S1, S3, S5
Ben – S1, S2, S4
Callie – S1, S2, S3, S4, S5
Dave – S1, S4, S5
Erin – S3, S4
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
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
19. Determine a fair allocation of assets
Johnathan
25,000
150,000
1,000
176000
44000
Car
House
Painting
Total:
Fair Share:
Joe
32,000
220,000
950
252950
63237.50
Billy
41,000
190,000
800
231800
57950
Jeremy
27,000
200,000
980
227980
56995
a) Determine a fair initial allocation of the items
Johnathan – painting
Joe – house
Billy – car
Jeremy -
receive 43000
owe 156762.50
receive 16950
receive 56995
b) Determine the surplus funds
39817.50
everybody receives 9954.38
c) Determine the final allocation of assets
Johnathan – painting
Joe – house
Billy – car
Jeremy -
receive 52954.38
owe 146808.13
receive 26904.38
receive 66949.38
20.
Determine the allocation of candy to each player:
A: __12-16________________________
B: ___1-4_______________________
C: ___7-9 _______________________
D: ____17-10______________________
Which items are left over? 5, 6, 10, 11
Three children (A, B, and C) are dividing the array of 12 candy pieces show in the following figure using the
method of markers. The players’ bids are indicated in the figure.
1
2
3
4
5
6
A1
C1
a. Describe the allocation of candy to each child.
A – 1, 2
B – 10 -12
C–4–9
b. Which items are left over? 3 , 8, 9
7
8
A2
B1
C2
10
9
B2
11
12