Chapter 1 Review Name ____________________________________
1. An election is
held by the 27 members of the National Football League Executive Committee to
choose the host city for Super Bowl XL.
The finalists are
Number of Voters 10 6 5 4 2
1st choice A B B C D
2nd choice C D C A C
3rd choice B C A D B
4th choice D A D B A
a. Which city would host the Super Bowl if you use the plurality method?
B
b. Is the plurality winner also a majority winner? Why or why not?
No. 
c. Which city would host the Super Bowl if you used the Borda method?
C
d. Which city would host the Super Bowl if you used the runoff method?
A
e. Which city would host the Super Bowl if you used the sequential runoff method?
A
f. Using any of the above methods, could someone vote insincerely and change the outcome of
the election? How?
In
a runoff, C could win if the first preference
Schedule
showed A and C switched.
2. Give an example of each of Arrow’s 5 conditions.
Answers
vary
3. Which voting method violates Arrow’s conditions the least?
Approval
Voting
Chapter 2 Review Name ____________________________________
1.
(afternoon), C (evening), and D (overnight). The number of nurses apportioned to each shift is
based on the average number of patients per shift, given in the following table:
Shift A B C D
# of patients 871 1029 610 190
a. What is the ideal ratio?
12
b. What is the quota for each class?
A: 72.583
B: 85.75
C: 50.83
D: 15.83
c.
Find the number of nurses on each shift using the
A: 72
B: 86
C: 51
D: 16
d.
Find the number of nurses on each shift using the
A: 73
B: 86
C: 51
D: 15
e. Find the number of nurses on each shift using the Webster method.
A: 72
B: 86
C: 51
D: 16
f. Find the number of nurses on each shift using the Hill method.
A: 72
B: 86
C: 51
D: 16
2. Adam, Bill, Cindy, and Dolores inherit an estate from their Grandpa Joe. The estate includes a house, an RV, a boat, and $200,000 in cash. What each person bid on each item is listed below:
House RV boat
Adam $300,000 $72,000 $12,500
Bill $250,000 $80,000 $12,300
Cindy $210,000 $60,000 $10,000
Dolores $280,000 $75,000 $11,500
a. What is each person’s fair share?
Adam: $146,125
Bill: $135,575
Cindy: $120,000
Dolores: $141,625
b. What items did each person receive (if any)?
Adam: House,
Boat
Bill: RV
Cindy: Nothing
Dolores: Nothing
c. How much cash will each person get before any remaining cash is distributed?
Adam: -$166,375
Bill: $55,575
Cindy: $120,000
Dolores: $141,625
d. How much extra cash is left over? How much will each person get?
$49, 175 $12,293.75
e. In their own eyes, what is the value of each person’s inheritance (total cash + items)?
Adam: $158,418.75
Bill: $147,868.75
Cindy: $132,293.75
Dolores: $153,918.75
Chapter 3 Review Name ___________________________________
1. In order to add or subtract matrices, what has to be true of their dimensions? What about
multiplication?
Add:
dimensions must be the same. Multiply:
the inner dimensions must be equal.
2. What is the 3x3 identity matrix?
3. Write the transpose(AT) of A = 
4. Three math classes at NCHS decide to sell candy as a fundraiser. The number of each kind of candy sold by each of the classes is shown below.
Starburst
Power Bars
Butterfingers
M & M’s
The profit for each candy is as follows: M&M’s: 40 cents, Power Bars: 88 cents,
Butterfingers: 25 cents, Starburst: 60 cents. Use matrix multiplication to compute the
profit made by each class on its candy sales. Who made the most money?
Discrete made
$611.00
5. The dimensions of matrices P, Q, R, and S are 3x2, 3x3, 4x3, and 2x3, respectively. If matrix
multiplication is possible, find the dimensions of the following matrix products.
a. QP b. RQ c. QS d. RPS
3 x 2 4
x 3 not possible 4 x 3
6. Use the following chart to answer the questions.
Birth and Survival
rates for
Age (months) 0 – 6 6 – 12 12 – 18 18 – 24 24 – 30
Birthrate 0 0.7 1.4 .7 .3
Survival rate 0.7 0.8 0.9 0.4 0
a. Construct a Leslie matrix for this animal.
b. Given that P0 = [40 36 32 18 4], find the population breakdown after 4 cycles.
c. Given that P0 = [85 43 36 44 23], find the total population after 6 years.
1776.656157 ≈ 1777
d. Using the P0 from part c, what is the long-term growth rate of the population?
19.5%
e. How many years will it take to get a population over 3900?
17 cycles or 8.5 years
Chapter 4 Review Name ____________________________________
1. From the table below, construct a graph to represent the information.
Task Time Prerequisites Task Time Prerequisites
Start 0 A 3 None B 5 None C 3 A, B D 1 A, B E 6 C, D F 5 C, D G 7 E, F
a. What is the earliest start time for each vertex?
A: 0 B:
0 C: 5 D: 5
E:
b. What is the minimum project time?
21
c. What is the critical path?
Start—BCEG—Finish
d. What would the minimum project time be if the time for task A were changed to 1?
No change
2. Using the following graph, determine the minimum project time and the critical path.
Minimum Project Time: 35
Start—ADG—Finish
|
|
KQAA |
KQBB |
KQCC |
KQDD |
KQEE |
KQFF |
|
KQAA |
- |
25 |
202 |
77 |
375 |
106 |
|
KQBB |
25 |
- |
175 |
51 |
148 |
222 |
|
KQCC |
202 |
175 |
- |
111 |
365 |
411 |
|
KQDD |
77 |
51 |
111 |
- |
78 |
297 |
|
KQEE |
375 |
148 |
365 |
78 |
- |
227 |
|
KQFF |
106 |
222 |
411 |
297 |
227 |
- |
3. The Federal Communications
Commission (FCC) monitors radio stations to make sure that their signals do not
interfere with each other. They prevent interference by assigning appropriate
frequencies to each station. How many
different frequencies are needed for the six stations located at the distances
shown in the table, if two stations cannot use the same channel when they are
within 150 miles of each other?_______3_________
a. What will
each vertex represent?
Radio Stations
b. What
will each edge represent?
Distance between stations
c. What will the chromatic number
tell you?
How many frequencies needed
4. What is the chromatic number of any planar graph? Is it possible to draw a graph that has a greater chromatic number than that?
4. Yes it is possible to draw a graph with a
chromatic number greater than 4 but it
will not be planar.
5. Tell whether each
graph has a(n) Euler circuit, Euler path, Hamiltonian
circuit, and/or Hamiltonian path. Give a
reason for each one, and trace the path/circuit if there is one.
In order as they appear on the
page:
Euler
Path Euler
Circuit Euler
Path
Euler
Circuit Euler
Path Euler
Path
Euler
Circuit
Chapter 5 Review Name ____________________________________
1. Determine whether each of the following graphs is a tree. If it’s not, tell why.
(a) (b) (c) (d)
Yes,
connected No,
there is
No, there Yes,
connected and
and no cycles a
cycle is a cycle no
cycles
(e) (f) (g)
No, not
connected Yes, connected No,
there is a cycle
and no cycles
2. For each bipartite graph, list the two distinct sets into which the vertices can be divided.
{A, D} {H} Not
bipartite
{B, C,
E} {F,
I, J, K, G}
3. Use the following picture to answer the
questions.
a. Construct a tree diagram showing all possible circuits that begin at vertex A, visit each vertex exactly once,
and end back at A. Which path(s) is/are the shortest?
Branches:
ABCDA = 125 ABDCA = 150 ACBDA = 115
ACDBA = 150 ADBCA = 115 ADCBA = 125
Shortest:
ADBCA or ACBDA
b. Use the nearest-neighbor algorithm to find the shortest path.
ACDBA =
150
4. Find the shortest path from A to all the other vertices.
AB: 2
ABC: 6
ABCD: 9
ABCDE: 15
ABCDGF: 13
ABCDG: 11
AH: 5
5. Use Kruskal’s algorithm to find a minimum spanning tree. Give the total weight.
Weight 160
6. Use the Breadth-First algorithm to find a spanning tree. Start at G.
Other spanning trees exist
Chapter 6 Review Name ___________________________________
1. An
followed by three more digits (0 through 9).
a. How many different
158,184,000
b. How many different
1,757,600
c. How many different
are different and all the letters are different)?
70,761,600
2. Four men and four women line up at a the checkout line at Best Buy.
a. In how many ways can they line up?
40,320
b. In how many ways can they line up if the first person in line must be a woman?
20,160
c. In how many ways can they line up if they must alternate woman, man, woman, man,
etc.?
576
3. The golf club at
the
committee of 3 members (president, VP, and Treasurer) need to be selected.
a. How many different 3-member committees can be picked?
29,760
b. How many different 3-member committees can be picked if the VP must be a male?
13,950
c. How many different 3-member committees can be picked if the committee must be
made of all males or all females?
6810
4. The board of directors of the XYZ Corporation has 15 members. In how many ways can one
choose…
a. a committee of 4 members (Pres, VP, Treasurer, and Secretary)?
32,760
b. a delegation of 4 members where all members have equal standing?
1365
5. There are 10 horses entered in a race. In how many ways can one pick…
a. the top three finishers regardless of order?
120
b. the first, second, and third place finishers in the race?
720
6. There are 20 singers auditioning for a musical. In how many different ways can the director
choose…
a. a duet?
190
b. a lead singer and a backup?
380
c. a quintet?
15,504
7. There are 117 Division I-A college football teams.
a. How many Top 25 rankings are possible?
3.191674713 x 1050
b. How many ways are there to choose 8 teams for a playoff?
6.819274133 x 1011