concordia college mathematics department spring 2012faculty.cord.edu/andersod/textspring2012.pdf ·...

Math 105 WorkbookExploring Mathematics

Concordia College Mathematics Department

Spring 2012

Acknowledgment

First we would like to thank all of our former Math 105 students. Their successes,struggles, and suggestions have shaped how we teach this course in many importantways.

We also want to thank our departmental colleagues and several Cobber mathematicsmajors for many fruitful discussions and resources on the content of this course andthe makeup of this workbook.

Some of the topics, examples, and exercises in this workbook are drawn from otherworks. Most significantly, we thank Samantha Briggs, Ellen Kramer, and Dr. JessieLenarz for their work in Exploring Mathematics, as well as Dr. Dan Biebighauserand Dr. Anders Hendrickson. We have also used:

• Introductory Graph Theory by Gary Chartrand,

• The Heart of Mathematics: An invitation to effective thinking by Edward B.Burger and Michael Starbird,

• Applied Finite Mathematics by Edmond C. Tomastik.

Finally, we want to thank (in advance) you, our current students. Your suggestionsfor this course and this workbook are always encouraged, either in person or overe-mail. Both the course and workbook are works in progress that will continue toimprove each semester with your help.

Let’s have a great semester this spring exploring mathematics together and fulfillingConcordia’s math requirement in 2012.

i

Contents

1 Voting Theory 3

1.1 Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Voting Paradoxes and Problems . . . . . . . . . . . . . . . . . . . . . 15

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 Weighted Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . 26

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Banzhaf Power Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 Voting Theory Finale . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 Chapter Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Counting and Probability 45

2.1 Introduction to Counting . . . . . . . . . . . . . . . . . . . . . . . . . 45

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3 Introduction to Probability . . . . . . . . . . . . . . . . . . . . . . . . 58

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.4 Complements, Unions, and Intersections . . . . . . . . . . . . . . . . 64

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

iii

2.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


2.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 Graph Theory 87

3.1 Introduction to Graph Theory . . . . . . . . . . . . . . . . . . . . . . 87

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Paths and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3 Subgraphs and Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.4 Graph Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.6 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126


3.8 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4 Number Theory and Infinity 139

4.1 Pigeonhole Principle and Divisors . . . . . . . . . . . . . . . . . . . . 139

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.3 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.4 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . 152

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

iv

1

4.5 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


4.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5 Consumer Mathematics 165

5.1 Percentages and Simple Interest . . . . . . . . . . . . . . . . . . . . . 165

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.3 Effective Annual Yield . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.4 Ordinary Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.5 Mortgages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190


5.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Chapter 1

Voting Theory

1.1 Voting Systems

1. The Sugar Beet Company is preparing to elect its president. Three membersare considering running for president: Alice, Bob, and Charles. Each of the fivevoters has a first, second, and third choice, as listed in the following table:

Voter: Voter 1 Voter 2 Voter 3 Voter 4 Voter 5

1st choice: Alice Bob Bob Charles Bob2nd choice: Charles Alice Charles Alice Alice3rd choice: Bob Charles Alice Bob Charles

Depending on who decides to run, the ballot the voters see in the booth willlook different. For each of the following ballots, how many votes will eachcandidate receive? Who will win?

(a)

Who should bepresident?

� Alice� Bob� Charles

(c)


� Alice� Charles

(b)


� Alice� Bob

(d)


� Bob� Charles

3

4 CHAPTER 1. VOTING THEORY

2. A company is electing a manager. Four candidates are running for the position:Micah, Darba, Jim, and Pam. The voters have the following preferences:

Number of BallotsRanking 3 2 4 3 2

1st choice M M D J P2nd choice P P P P D3rd choice D J J M J4th choice J D M D M

(a) Here is the ballot:

Who should be thenew manager?

� Micah� Darba� Jim� Pam

Who will be elected manager? What percent of the vote will he or shehave?

(b) As you look at the voters’ preferences, who seems like the best choice tomake the most voters happy?

(c) If your answers to parts (a) and (b) are different, can you explain why?

(d) Can you think of a different voting system that would help the voterselect your choice from part (b)?

1.1. VOTING SYSTEMS 5

3. Plurality voting:

4. Borda count:


5. If the company above runs their election as a Borda count, who will win?


1st choice M M D J P2nd choice P P P P D3rd choice D J J M J4th choice J D M D M


6. A company warehouse is voting for its new warehouse manager, and the can-didates are Dante, Creed, Phyllis, and Angela.

# of BallotsRanking 3 10 9 8

1st choice D C A P2nd choice P P D A3rd choice C D P D4th choice A A C C

(a) Each League member votes for his favorite candidate. What percentageof votes did the first-place candidate receive?

(b) Rather than elect a candidate who didn’t receive a majority vote, theLeague holds a run-off election between the top two candidates. Whowill win the run-off election?

(c) Was anyone eliminated from the run-off who perhaps shouldn’t havebeen?

(d) Can you think of a better runoff system for the League than a runoffbetween just the top two candidates? What result does it give?


7. Plurality with elimination:


8. The Army Corps of Engineers is trying to decide on a flood prevention projectfor the Fargo-Moorhead metro area. The options are

(D) A diversion on the North Dakota side(R) Retention and reservoir system along the Red River valley(F) Building floodwalls to 50′ throughout both cities(C) Relocating the community to higher ground in, say, Colorado(M) A diversion on the Minnesota side

The eighteen members of the Corps have the following preferences:

Number of BallotsRanking 5 4 2 3 1 3

1st choice F M R D C R2nd choice D D F M D F3rd choice R R C R F D4th choice M C D F M M5th choice C F M C R C

The ACE’s voting rules employ plurality with elimination.

(a) What will the vote tallies be after the first round of voting? Is there awinner yet? If not, who should be eliminated?

(b) What will the vote tallies be after the second round of voting? Is there awinner yet? If not, who should be eliminated?

(c) What will the vote tallies be after the third round of voting? Is there awinner yet? If not, who should be eliminated?

9. If there are n candidates in an election conducted by plurality with elimination,what is the most rounds of voting that could be required?


10. A company is voting on the Accountant-of-the-Month. The candidates are:

• Olivia

• Kevin

• Anja

• Toby

The preference table is the following:


1st choice A T K O O2nd choice O O A K A3rd choice K K T A T4th choice T A O T K

(a) Who would win in a head-to-head race between Olivia and Kevin?

(b) Who would win in a head-to-head race between Olivia and Anja?

(c) Fill in the following table of who would win head-to-head races; if thereare any ties, put both names in the box.

Toby Anja Kevin

Olivia

Kevin

Anja

(d) Who won the most head-to-head races?(Count any ties as 1

2a win for each contender.)


11. Pairwise Comparison:

12. A tip for setting up the head-to-head races:


13. Our Math 105 classes are voting on where to take a math field trip. Theoptions are:

(E) The mudflats of Egypt, birthplace of geometry

(F) Florence, home of Renaissance perspective

(W) West Acres in Fargo

(K) Konigsberg, for a walking tour of the bridges

(P) Papa John’s Pizza, home of yummy garlic sauce

The preference table is the following.


1st choice K W W E P K2nd choice E K E P K F3rd choice F F F K E W4th choice P E K W W P5th choice W P P F F E

We decide to use the pairwise comparison method to decide. Where will wego?


Homework

Show all your work. In particular, for plurality with elimination, show the results ofeach round of voting; for pairwise comparison, show the result of each head-to-headvote.

1. Consider the following preference matrix:


1st choice D D A E B A2nd choice C C B C E B3rd choice E E E B D C4th choice B A D A A D5th choice A B C D C E

This race is being decided by plurality vote.

(a) How many voters are voting in this election?

(b) How many votes would be required to win a majority in this election?

(c) Who will win this election? How many votes will that person get? Doesthat candidate have a majority?

(d) Suppose candidate B drops out of the race. Who would win the election?

(e) Suppose everyone drops out of the race except A and C. Who will winthe election?



1st choice B C D B A2nd choice D A C C B3rd choice C D A A C4th choice A B B D D

Find the winner of an election held using each of the following voting schemes.(If it is a tie, say who tied.)

(a) Using plurality vote.

(b) Using Borda Count.


(c) Find the winner using plurality with elimination.

(d) Find the winner using pairwise comparison.



1st choice B B C A C A2nd choice C A B C A B3rd choice A C A B B C







# of BallotsRanking 3 5 4

1st choice C B D2nd choice E E C3rd choice A A E4th choice D C A5th choice B D B






1.2. VOTING PARADOXES AND PROBLEMS 15

1.2 Voting Paradoxes and Problems

1. Of the four voting systems we’ve studied, which is best? Why?

2. The Fargo Running Club is electing a new team captain. There are threecandidates: Ira, Jannika, and Kelsey. The preference table is the following.


1st choice J I K2nd choice I J I3rd choice K K J

(a) If the club uses the plurality method, who will win the election?

(b) If it were a head-to-head race between Ira and Jannika, who would win?

(c) If it were a head-to-head race between Ira and Kelsey, who would win?

(d) If it were a head-to-head race between Jannika and Kelsey, who wouldwin?

(e) Why does the plurality method’s result seem unfair in this election?


3. Head-to-Head Criterion:

4. The school board is voting whether to award the contract for building a newelementary school to Aasgaard Architects, Bob’s Builders, Cobber Construc-tion, or Dora’s Developers.


1st choice B B C A2nd choice C C A D3rd choice D A D C4th choice A D B B

(a) The school board votes using a Borda count.Which company will get the contract?

(b) Does this seem fair? Why or why not?


5. Majority Criterion:


6. The International Olympic Committee has to select one of four cities for theWinter Olympics: O, T, M, F. As of this moment, the voters’ preferences areas follows:


1st choice F M T O2nd choice O F M T3rd choice T T O F4th choice M O F M

The IOC uses the plurality with elimination method.

(a) If the vote is held today, which city will be chosen as the host city?


(b) After receiving a very nice bribe, the single voter in the last column de-cides that F really is the best candidate after all, and so his preferenceschange to F,O,T,M. Thus when the election is held tomorrow, the pref-erence table is


1st choice F M T ×O F2nd choice O F M ×T O3rd choice T T O ×F T4th choice M O F ×M M

Which city will be chosen?

(c) What is odd about your answers to (a) and (b)?

7. Monotonicity Criterion:


8. Voters have 4 choices: A, B, C, D. Here is the preference table.

# of BallotsRanking 10 6 5 4 2

1st choice D C C A B2nd choice A B A D A3rd choice C A D B C4th choice B D B C D

(a) Find the winner of the election under the plurality with eliminationmethod.

(b) Find the winner of the election under the pairwise comparison method.

(c) What is strange about your answers to (a) and (b)?


9. Voters are trying to decide what to do with a $50 million surplus in the statebudget (clearly not in Minnesota). Their options are:

(T) Give the money back to taxpayers as property tax relief.

(R) Spend it on road construction and other infrastructure.

(H) Build a new state historical building.

Here is the preference table:

Number of BallotsRanking 70,000 50,000 40,000

1st choice T R H2nd choice R H T3rd choice H T R

(a) The election is run by pairwise comparison. What is the result?

(b) What is odd about your results from part (a)?


10. Irrelevant Alternatives Criterion:

11. Arrow’s Impossibility Theorem:


Homework



1st choice C D A C A2nd choice D C C B B3rd choice B A B A D4th choice A B D D C

(a) Does the Borda Count violate the Majority Criterion for this particularpreference matrix?

(b) Does the Borda Count violate the Head-to-Head Criterion for this par-ticular preference matrix?



1st choice C D C D A B2nd choice A A A A D D3rd choice D C B B C C4th choice B B D C B A

(a) Does the Borda Count violate the Majority Criterion for this particularpreference matrix?

(b) Does the Borda Count violate the Head-to-Head Criterion for this par-ticular preference matrix?



1st choice B B D D2nd choice D D B B3rd choice C A A C4th choice A C C A

(a) How many points will each candidate receive in a Borda count? Who willwin?


(b) The five voters in the last column really do think D is the best candidateand B is the second-best. However, they decide to be sneaky and lie ontheir Borda count ballots, claiming they think B is the worst candidate;in other words, they say they prefer D, C, A, and B in that order. Nowhow many points will each candidate receive in a Borda count? Who willwin?

(c) Explain in a few complete sentences how these voters manipulated theBorda count and why it is unfair.

4. Use a Borda count to determine the preference of the voters.


1st choice A C E2nd choice B D C3rd choice C E D4th choice D B B5th choice E A A

Suppose the last two voters change their minds with respect to B and D:


1st choice A C E2nd choice B B C3rd choice C E B4th choice D D D5th choice E A A

What is the preference now? Explain in a few sentences what you see happen-ing in these results.

5. Use a Borda count and straight plurality to determine the following election.Explain the results.


1st choice A B C2nd choice B C B3rd choice C A A


6. Use plurality with elimination to determine the winner of the following elec-tion.


1st choice A C B2nd choice B B A3rd choice C A C

Now suppose voters in the middle column change their minds about C:


1st choice A B B2nd choice B C A3rd choice C A C

Has the outcome changed? Explain the results in a sentence or two.


1.3 Weighted Voting Systems

1. Notes on weighted voting systems:

(a) Weight:

(b) Example: Four partners start a new business. Partner A owns 8 shares,Partner B owns 7 shares, Partner C owns 3 shares, and Partner D owns2 shares. If

1 share = 1 vote

represent this weighted voting system.

(c) Motion:

(d) Quota:

(e) Example: If the quota for [8, 7, 3, 2] is a simple majority, find the quota.What is the quota if a two-thirds majority is required?

(f) Weight/quota notation:

1.3. WEIGHTED VOTING SYSTEMS 27

2. In a weighted voting system with weights [30, 29, 16, 8, 3, 1], if a two-thirdsmajority of votes is needed to pass a motion, what is the quota?

3. Consider the weighted voting system [14, 9, 8, 5].

(a) What is the largest reasonable quota for this system?

(b) What is the smallest reasonable quota for this system?

4. Consider the weighted voting system [20|7, 5, 4, 4, 2, 2, 2, 1, 1].

(a) How many voters are there?

(b) What is the quota?

(c) What is the weight for voter P2?

(d) If the first 4 voters vote for a motion and the rest vote against, does themotion pass?

(e) If P1 and P2 vote against a motion, will the motion pass?


5. What is peculiar about each of the following weighted voting systems?

(a) [20|10, 10, 9]

(b) [7|4, 2, 1]

(c) [51|50, 49, 1]

(d) [6|6, 2, 1, 1]

(e) [21|10, 8, 5, 3, 2]

1.3. WEIGHTED VOTING SYSTEMS 29

6. A dummy is . . .

7. A dictator is . . .

8. A voter has veto power if . . .

9. Is weight the same as power? Consider the weighted voting systems [12|13, 7, 2]and [19|8, 7, 3, 2].

10. In the weighted voting system [12|9, 5, 4, 2], are there any dummies or dicta-tors?

11. In designing a weighted voting system [q|6, 5, 4, 3, 2, 1], what is the largestquota q you could pick without giving veto power to anyone?

12. In the weighted voting system [q|8, 5, 4, 1], if every voter has veto power, whatis the quota q?


Homework

1. Anton, Bella, Chaz, and Debra are the stockholders in Red River Industries.Anton owns 252 shares, Bella owns 741 shares, Chaz inherited 637 shares, and412 shares are in Debra’s hands. As usual, each share corresponds to a votein the stockholder’s meeting.

(a) If a certain type of motion requires a majority vote, what is the smallestnumber of votes needed to pass the motion?

(b) A different type of motion requires a 2/3 vote to pass. What is thesmallest number of votes needed to pass this motion?

(c) Using the quota you found in part (b), express the weighted voting systemin the correct notation (with brackets and quota).

2. Find all dictators, dummies, and voters with veto power in the followingweighted voting systems:

(a) [51|20, 20, 20]

(b) [51|36, 34, 23, 6]

(c) [25|27, 11, 7, 2]

(d) [31|15, 13, 6, 4, 2]

3. Which voters have veto power in the system [51|29, 21, 8, 3, 1]?

4. In 1958, the Treaty of Rome established the European Economic Community(EEC) and instituted a weighted voting system for the EEC’s governance. Themembers at that time were France, Germany, Italy, Belgium, the Netherlands,and Luxembourg. The three largest countries (France, Germany and Italy)were each given a vote with weight 4, Belgium and the Netherlands had votesof weight 2 and Luxembourg’s vote had weight 1. The quota was 12.

What is unusual or interesting about this weighted voting system?

1.4. BANZHAF POWER INDEX 31

1.4 Banzhaf Power Index

1. A coalition is . . .

2. (a) Consider a weighted voting system with three voters P1, P2, and P3. Listall the coalitions. How many are there?

(b) Consider a weighted voting system with four voters P1, P2, P3, and P4.List all the coalitions. How many are there?

(c) If a weighted voting system has n voters P1, P2, . . . , Pn, how manycoalitions are there?


3. A winning coalition is . . .

4. List all the winning coalitions in the weighted voting system [10|5, 4, 3, 2, 1].

5. In the weighted voting system [10|6, 4, 3, 2, 1], consider the winning coalition{P1, P2, P3, P4}. Which voter(s) could change their minds and vote “no” with-out changing the outcome of the vote? Which voter(s) need to keep voting“yes” in order for the motion to pass?

6. A critical voter in a winning coalition is . . .


7. Consider the voting system [19|11, 9, 8, 5].

(a) List all the winning coalitions.

(b) In each winning coalition above, circle the critical voters.

(c) Count the number of times each voter is a critical voter. This is calledthat voter’s Banzhaf power.

Voter Banzhaf power

P1

P2

P3

P4

(d) Add up all the voters’ Banzhaf powers; this sum is called the totalBanzhaf power of the voting system.

(e) Finally, divide each voter’s Banzhaf power by the total Banzhaf power.The percentage that results is called the voter’s Banzhaf power index.

Voter Banzhaf power index

P1

P2

P3

P4


8. Calculate the Banzhaf Power Index for each voter in the weighted voting sys-tem [51|32, 22, 12].

9. Make up a weighted voting system with a dummy, and calculate the BanzhafPower Index for the dummy.

10. Make up a weighted voting system with a dictator, and calculate the BanzhafPower Index for the dictator.

11. Make up a weighted voting system in which several voters have veto power.Calculate the Banzhaf Power Index for the voters with veto power. What doyou notice?


Homework

1. Calculate the Banzhaf Power Index for each voter in the weighted voting sys-tem [34|12, 10, 7, 6].

2. Consider the voting system [25|24, 20, 1].

(a) Calculate the percentage of the total weight that each voter holds.

(b) Calculate the Banzhaf Power Index for each voter.

(c) Comparing your answers to parts (a) and (b), explain in complete sen-tences why the weight controlled by the voter is not the same thing asthe power held by each voter.



5. Nassau County, New York used to be governed by a Board of Supervisors.The county had six districts, each of which one delegate to vote on countyissues. The delegates’ votes were weighted proportionately to the districts’population in 1964:

District WeightHempstead #1 31Hempstead #2 31Oyster Bay 28North Hempstead 21Long Beach 2Glen Cove 2

A simple majority was needed to pass a motion.

(a) Express this weighted voting system in our usual notation.

(b) Calculate the Banzhaf power of each district.

(c) What percentage of the county population lived in districts that are dum-mies?

In 1965 John F. Banzhaf III argued in court that even though the weightswere proportionate to population, this system of government was unfair. Hewon!


1.5 Voting Theory Finale

The following problems are to be worked on in class, with the answers written upon a separate sheet of paper to be turned in as homework for this section.

1. The Coombs Method: This method is just like the plurality-with-eliminationmethod except that in each round we eliminate the candidate with the largestnumber of last-place votes (instead of the one with the fewest first-place votes).


1st choice W Z Z Y X2nd choice X W X Z Z3rd choice Y Y W X W4th choice Z X Y W Y

(a) Find the winner of the election above using the Coombs method.

(b) Find the winner of the election above using the pairwise comparison(head-to-head) method.

(c) What do you notice when comparing the two methods for this data?Mention any fairness criteria that are violated.

2. The Coombs method continued: Recall that in each round we eliminate thecandidate with the largest number of last-place votes.


1st choice B C C A2nd choice A A B B3rd choice C B A C

(a) Find the winner of the election above using the Coombs method.

(b) Suppose all 8 voters in the second column switch A and B; that is, theyput B in 2nd and A in 3rd place. Use the Coombs method to determineif the overall results of the election are the same.

(c) What do you notice when comparing the outcomes for this method? Men-tion any fairness criteria that are violated.

1.5. VOTING THEORY FINALE 37

3. Consider the weighted voting system [8∣∣5, 3, 1, 1, 1].

(a) Make a list of all winning coalitions.

(b) Using (a), find the Banzhaf power distribution of this weighted votingsystem.

(c) Suppose that P1, with 5 votes, sells one of her votes to P2, resultingin the weighted voting system [8

∣∣4, 4, 1, 1, 1]. Find the Banzhaf powerdistribution of this system.

(d) Compare the power index of P1 in (b) and (c). Describe in your ownwords the paradox that occurred.

4. A business firm is owned by 4 partners, P1, P2, P3, P4. When making decisions,each partner has one vote and the majority rules, except in the case of a 2− 2tie; in that case, the coalition that contains P4–the partner with the leastseniority–loses. What is the Banzhaf power distribution in this partnership?

5. A panel of sportswriters and broadcasters every year selects an NBA rookieof the year using a variation of the Borda count method. The following tableshows the results of the balloting for the 2003− 2004 season.

Player 1st Place 2nd Place 3rd Place Total PtsLeBron James 78 39 1 508Carmelo Anthony 40 76 2 430Dwayne Wade 0 3 108 117

Using the information from the table, figure out how many points are awardedfor each 1st-place, 2nd-place, and 3rd-place vote in this election. Assume thatthese numbers are different positive integers.

6. An NBA team has a head coach H and 3 assistant coaches C1, C2, C3. Playerpersonnel decisions require at least 3 of the coaches to vote yes, one of whichmust be H to pass. If we use [q

∣∣h, c, c, c] to describe this weighted votingsystem, find q, h, c.


1.6 Chapter Projects

1. Background and applications of the Banzhaf power index. John Banhaf wasa lawyer in Nassau County (NY) who studied the Nassau County Board andfound that their weighted voting system essentially consigned the smaller dis-tricts in the county to the role of dummies! Explain their weighted votingsystem in the 1960s vs. the 1990s and explain the weaknesses of the 1960sversion. Include any interesting information on the subsequent legal fight toget the voting system changed.

2. Present the Shapley-Shubik power index. (Who are Shapley and Shubik?)Compare this index to the Banzhaf power index that we discussed in Sec-tion 7.3. Give some examples, including real examples, such as the ElectoralCollege. There are many good introductions to this index online, along withplenty of small examples. Some tables including real examples of the indexcan be found at

http://www.cut-the-knot.org/Curriculum/SocialScience/ShapleyShubikIndex.shtml

3. Discuss the Single Transferable voting system. Explain what it is, how thevotes are counted, what quotas are set, and how winners are determined.Give examples of simple elections using this system (for hints, see wikipedia’sexample on single transferable voting, for starters). Who uses this methodtoday? Mention governments and the Academy Awards.

4. Introduce the Bucklin voting system, named after James Bucklin of GrandJunction, Colorado. Explain how the Bucklin system works, give examples ofsimple elections using this system (for hints, see wikipedia’s example on singletransferable voting, for starters). Who uses this method today?

5. Antagonists: In a weighted voting system, suppose there is a player A thathates player B so much that A always votes opposite of B, regardless of theissue. Call player A an antagonist of B. Explore this idea in various weightedvoting system scenarios.

6. Present the Johnston power index. Include a mathematical description of theprocedure for computing Johnston power, give examples, and compare theresults with the ones obtained using the Banzhaf method. Include a personalanalysis of the merits of the Johnston method compared with the Banzhafmethod.

1.7. CHAPTER REVIEW 39

1.7 Chapter Review

To prepare for the exam over this chapter, you should review the in-class worksheetsand homework. Be ready to do the kind of problems you faced on the homework.

1. Know how to read a table of voters’ preferences

2. Calculate the winner according to

(a) Plurality

(b) Borda count

(c) Plurality with elimination

(d) Pairwise comparison

3. Devise preference tables that satisfy given conditions (e.g., “Come up with apreference table where the pairwise comparison test produces no winner.”)

4. Weighted voting

(a) Know how weighted voting on yes/no motions works.

(b) Understand the notation [q|w1, . . . , wn].

(c) Given a weighted voting system, find any dictators, dummies, or voterswith veto power.

5. Construct weighted voting systems that satisfy given conditions (e.g., “Comeup with a weighted voting system where two people have veto power.”)

6. Calculate the Banzhaf Power Index for the voters in a weighted voting system.


Review Exercises:

1. Four candidates (A, B, C, and D) run in an election with the following results:


1st choice B C A D2nd choice C A C C3rd choice A D B B4th choice D B D A

(a) Determine the winner of the election using the

i. plurality method

ii. Borda count method

iii. plurality-with-elimination method

iv. pairwise comparison method

(b) Do any of the methods violate the majority criterion?

(c) Do any of the methods violate the head-to-head criterion?

2. Three candidates (A, B, and C) run in an election with the following results:


1st choice A B B C2nd choice B C A A3rd choice C A C B


i. plurality method


iii. plurality with elimination method


(b) Do any of the methods violate the majority criterion?

(c) Do any of the methods violate the head-to-head criterion?


3. Three candidates (A, B, and C) run in an election with the following results:


1st choice A B C2nd choice B C B3rd choice C A A


i. plurality method




(b) Suppose candidate C drops out of the election. Determine the winner ofthe election using the

i. plurality method




(c) Do any of the methods violate the irrelevant alternatives criterion?

4. Voters have 4 choices: A, B, C, D. Here is the preference table.


1st choice D A C B A2nd choice B B A C C3rd choice A C B A B4th choice C D D D D


i. plurality method




(b) Do any of the methods violate any of the fairness criteria?


5. Determine the quota if a simple majority is needed to pass a motion. Useproper notation to express the voting system.

(a) [7, 4, 3, 3, 2, 1]

(b) [10, 6, 5, 4, 2]

(c) [6, 4, 3, 3, 2, 2]

6. Determine the quota if the given percentage is needed to pass a motion. Useproper notation to express the voting system.

(a) [7, 4, 3, 3, 2, 1], 67%

(b) [10, 6, 5, 4, 2], 75%

(c) [6, 4, 3, 3, 2, 2], 80%

7. Consider the weighted voting system [q∣∣5, 3, 1]. Find the Banzhaf power dis-

tribution of this weighted voting system when (a) q = 6 (b) q = 8.

8. In the weighted voting system [q∣∣8, 5, 4, 1], if every voter has veto power what

is q?

9. In the weighted voting system [q∣∣6, 5, 4, 3, 2, 1], what is the largest value of q

so that no voter has veto power?

10. Describe anything you find unusual or interesting about the following weightedvoting systems.

(a) [22∣∣11, 11, 10]

(b) [10∣∣5, 3, 2]

(c) [41∣∣40, 39, 1]


Some Review Answers:

1. (a) D, C, D, D

(b) Borda count method violates the majority criterion.

(c) Borda count method violates the head-to-head criterion.

2. (a) B, B, A, A

(b) No

(c) Plurality method and the Borda count method violate the head-to-headcriterion.

3. (a) A, B, B, B

(b) B, B, B, B

(c) Plurality method violates the irrelevant alternatives criterion.

4. (a) D, B, C, A

(b) Plurality, plurality with elimination, and Borda count all violate the head-to-head criterion.

5. (a) [11∣∣7, 4, 3, 3, 2, 1]

(b) [14∣∣10, 6, 5, 4, 2]

(c) [11∣∣6, 4, 3, 3, 2, 2]

6. (a) [14∣∣7, 4, 3, 3, 2, 1]

(b) [21∣∣10, 6, 5, 4, 2]

(c) [16∣∣6, 4, 3, 3, 2, 2]

7. (a) P1 has 3/5=60%; P2 and P3 have 1/5=20%. (b) P1 and P2 have 1/2=50%and P3 has none=0%.

8. q = 18, because the only way to guarantee P4 has veto power is to set q =w1 + w2 + w3 + w4 = 8 + 5 + 4 + 1 = 18.

9. q = 15, because in order to prevent P1 from having veto power, we must setthe quota equal to the sum of the other weights: q = w2 +w3 +w4 +w5 +w6 =5 + 4 + 3 + 2 + 1 = 15.

10. (a) [22∣∣11, 11, 10]: P3 is a dummy and has no power.

(b) [10∣∣5, 3, 2]: All voters have the same power, and this is equivalent to

[3∣∣1, 1, 1].

(c) [41∣∣40, 39, 1]: P2 and P3 have the same power, and this is equivalent to

[3∣∣2, 1, 1].

Chapter 2

Counting and Probability

2.1 Introduction to Counting

1. We flip a fair coin 3 times. How many different sequences of Heads and Tailsare possible?

45

46 CHAPTER 2. COUNTING AND PROBABILITY

Fundamental Principle of Counting:

Suppose a task can be divided into m consecutive subtasks.

If

Subtask 1 can be completed in n1 ways, and then

Subtask 2 can be completed in n2 ways, and then

...

Subtask m can be completed in nm ways,

then the overall task can be completed in ways.

2. If we flip a fair coin 5 times, how many sequences of Heads and Tails arepossible? What about 10 times?

3. There are 26 letters (A-Z) and 10 digits (0-9). If a license plate must contain 3letters followed by 3 digits, how many license plates are possible if repetitionsare allowed? What if repetitions are not allowed?

2.1. INTRODUCTION TO COUNTING 47

Definition: A permutation of a collection of different objects is an orderedarrangement of the objects.

Definition: For any positive integer n, we define n factorial to be

n! = n · (n− 1) · (n− 2) · · · · · 3 · 2 · 1.

We also define 0! = 1.

Definition: A permutation of r objects from a collection of n different ob-jects is an ordered arrangement of r of the n objects. The number of thesepermutations is denoted by P (n, r).

4. If we have a division of 8 basketball teams and we decide to rank our top 2teams (first and second place), how many choices are possible? What if wewant to rank the top 4 teams? All 8 teams?


Formulas for P (n, r):

5. If we have a popularity contest in this class and choose a first, second, andthird place person, how many choices are possible?


1. Suppose 3 students are on a road trip, and all 3 are willing to drive. In howmany ways can they be seated in the car?

2. Four students are returning from break. If all are willing to drive, in how manyways can they be seated in the car?

3. Definition: The Sample Space of an experiment is the set of all possible out-comes.

4. Multiplication Principal of Counting:

If one event can occur in a ways, and for each of those a ways another eventcan occur in b ways, then the total number of events is the multiplication a×b.

5. A 6 member board sits around a table. How many different seating arrange-ments are possible?

6. A 6 member board self selects a president and a treasurer. In how many wayscan this be done?

7. A true/false quiz has 5 questions. In how many ways can the quiz be com-pleted?

8. A softball team with 9 players needs a batting order. How many different waysare possible if:

• there are no restrictions;

• the pitcher bats last;

• the catcher bats last and the pitcher bats anywhere but first?

9. An identification tag has 2 letters followed by 4 numbers. How many differenttags are possible if:

• repetition of letters and numbers is allowed?

• repetition of neither letters nor numbers is allowed?

10. The athletic department wants a picture for a brochure that includes 4 of the11 starting offensive players from the football team on the left, 3 of the 6volleyball players in the center, and 2 of the 5 starting basketball players onthe right. How many possible pictures are there?


Homework

1. Simplify the following fractions as much as possible (you will get a wholenumber as the answer in each case):

(a)8!

4!

(b)7!

4! · 3!

(c)100!

98!

(d)9!

5! · 4!

(e)n!

(n− 1)!, where n is any integer greater than or equal to one

2. An urn holds 6 balls: a red ball, an orange ball, a yellow ball, a green ball,a blue ball, and a purple ball. John selects one ball from the urn, and then,without replacing the first ball, he selects a second ball. How many choicesare possible for John?

3. Suppose your debit card PIN must be four digits (0-9).

(a) If there are no restrictions on the digits for your PIN, how many choicesare possible?

(b) If you are not allowed to repeat digits for your PIN, how many choicesare possible?

(c) If your PIN must contain the digits 5 and 1 (in some order, and notnecessarily as consecutive digits) and you are not allowed to repeat digits,how many choices are possible?

4. Suppose Adam’s e-mail password must be exactly 8 characters long.

(a) If the password must be 6 upper-case letters (A-Z) followed by 2 digits(0-9), how many choices are possible?

(b) If the password must be 7 upper-case letters followed by a single digit,how many choices are possible?

(c) If the password must contain 7 upper-case letters and one digit, but thedigit can be in any position, how many choices are possible?


5. A large kindergarten class of of 51 students lines up single-file to walk to recess.How many different lines are possible?

6. Find the number of ways to rearrange the letters in the following words:

(a) MATH

(b) IVERS

(c) DOOR

7. Suppose Alice, Bob, Connie, Daniel, Erika, and Fred are the six membersof a committee, and they need to elect the following officers: Chairperson,Secretary, and Treasurer. (The officers are members of the committee.)

(a) How many different ways are there to select the officers?

(b) How many selections are there in which Daniel is not an officer?

(c) How many selections are there in which neither Bob nor Daniel is anofficer?

(d) How many selections are there in which Bob and Daniel are both officers?

(e) How many selections are there in which Bob or Daniel may or may notbe officers, but Bob and Daniel are definitely not both officers?


2.2 Combinations

1. Suppose we have three people: Angela, Bob, and Creed.

(a) If we need to select one to be president and one to be vice-president, howmany choices are possible?

(b) If we instead only need to select 2 of the 3 people to be “leaders,” thenhow many choices are possible?

Definition: A combination of r objects from a collection of n different objectsis a selection of r of the objects where the order of the objects selected is notimportant. The number of these combinations is denoted by C(n, r).

2. Suppose we have four people: Angela, Bob, Creed, and David. How manyways are there to select three of these four people:

(a) If order does matter

(b) If order doesn’t matter

2.2. COMBINATIONS 53

Finding a formula for C(n, r):

3. If we pick 2 teams from 8 teams to go to the playoffs, where the order of theteams is irrelevant, how many choices are possible?

4. A small company has 12 employees. They will send 3 to a meeting in WhiteBear Lake, another 1 to a meeting in St. Louis, another 1 to a meeting inNashville, and another 1 to a meeting in Moorhead. How many differentchoices are possible?


5. Suppose we flip a coin 10 times.

(a) How many sequences of Heads and Tails are possible?

(b) How many sequences have exactly 7 Heads?

(c) How many sequences have at least 7 Heads?

(d) How many sequences have at least 1 Head?


1. A 10-member board self selects a president, a vice president, and a treasurer.In how many ways can this be done? (Order matters)

P (10, 3) =10!

(10− 3)!=

10!

7!= 10× 9× 8 = 720.

2. A 10-member board self selects a subcommittee of 3. In how many ways canthis be done? (Order no longer matters).

P (10, 3)

3!=

10!

(10− 3)!3!=

10!

7!3!=

10× 9× 8

3× 2× 1= 120.

3. Evaluate C(9, 5)

4. Evaluate C(5, 3)

5. How many different 5-card poker hands can be dealt from a standard deck of52 cards?

6. In how many different ways can the 9-member US Supreme Court reach a 6-3decision?

7. A quiz contains 5 true/false questions.

• In how many ways can the quiz be completed?

• How many of the ways from (a) contain exactly 3 correct answers?

• How many of the ways from (a) contain at least 3 correct answers?

8. A committee of 13 has 7 women and 6 men. In how many ways can a sub-committee of 5 be formed if it consists of:

• all women?

• any 5 people?

• exactly 2 men and 3 women?

• at least 3 men?

• at least 1 woman?

9. A telemarketer makes 15 phone calls in 1 hour. In how many ways can theoutcomes of the calls be 3 sales, 8 no-sales, and 4 answering machines?


Homework

1. Answer the following counting questions. Answers with P s and Cs are fine, ifyou do not want to calculate the actual numbers.

(a) In a regional track meet, 16 runners are competing in the 10000-meterrun. The top eight runners qualify for the national meet. How manydifferent ways are there for the runners to qualify for the national meet?(The order of the qualification doesn’t matter.)

(b) In how many ways can we choose 17 flowers from a collection of 51 flowersto use in a vase?

(c) A couple is planning a menu for a rehearsal dinner. The restaurant makesthem choose 4 of the 7 possible appetizers, 3 of the possible 6 maincourses, and 2 of the possible 5 desserts. How many different menus arepossible?

(d) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have three boys and two girls?

(e) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have two boys and three girls?

(f) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have at most two boys and at least three girls?

(g) In how many ways can 15 teams be divided up so that 5 are in the NorthDivision, 5 are in the Central Division, and 5 are in the South Division?

(h) Given 10 points on a sheet of paper, where no three points lie on the sameline, how many triangles can be drawn that use three of these points asvertices?

(i) In a group of 12 people, we select a committee of 5 members. One memberof the committee is the chairperson, and another is the secretary. Howmany different choices for this committee are possible?

(j) Michael, Dwight, Jim, and Pam decide to play a game of cards during aslow day in the paper business. In how many different ways can the 52cards in a deck be dealt to the four players, where each player gets 13cards? Assume that the order of the cards in each player’s hand does notmatter.

(k) Suppose there are three divisions of five teams each. One team from eachdivision will go to the playoffs, along with one other team that can comefrom any of the divisions. How many different combinations of teams cango to the playoffs, where the “wild-card” team is counted just like any


other team? In other words, in how many ways can we select four teamsto go to the playoffs where the order of the teams is irrelevant, but everydivision sends at least one team?


2.3 Introduction to Probability

Definition: An experiment is a process that can be repeated and has observableresults, which are called outcomes. A sample space for an experiment is a set of allof the possible outcomes.

Definition: A sample space is uniform if all of the outcomes in the sample spaceare equally likely to occur.

Definition: An event is a subset of the sample space.

2.3. INTRODUCTION TO PROBABILITY 59

Experimental Probability:

To find the experimental probability of an event, we perform the experiment for alarge number of times and count the number of times our event occurs. Then wedivide the number of times our event occurred by the total number of experiments.

Theoretical Probability:

Suppose the sample space for our experiment is called S, and that S is a uniformsample space (so every outcome is equally likely, in theory, to occur). Then if E isan event (a subset of S), the theoretical probability of E is denoted by P (E) and isgiven by

P (E) =Number of outcomes in E

Number of outcomes in S.


1. Suppose we roll 2 fair dice (each numbered 1 through 6). What are the theo-retical probabilities of the following events?

(a) A = “Roll doubles (each die is the same)”

(b) B = “Roll a total of exactly 9”

(c) C = “Roll at least one 5”


Cards:

A standard deck of cards contains 52 cards. These cards are divided into 4Suits :

Diamonds, Hearts, Clubs, Spades

Diamonds and Hearts are Red, and Clubs and Spades are Black.

Each Suit is divided into 13 Ranks :

Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King

Jacks, Queens, and Kings are Face Cards.

2. Suppose we select 5 cards from a standard 52-card deck.

(a) What is the probability that we get 5 face cards?

(b) What is the probability that we get exactly 2 Spades?


Homework

1. Here’s a fraction review. Simplify the following fractions:

(a)5

8+

1

4

(b)1

6+

1

2− 1

12

(c)1

2· 2

3· 3

4

(d) 1− 3

7

(e)2

5· 1

3+

3

5· 1

6

(f)51

17

2. A standard six-sided die (1-6) is rolled 100 times. The results of the rolls are

in the following table.

Outcome Frequency1 142 193 184 165 176 16

Find the experimental probability of:

(a) Rolling a 5

(b) Rolling an odd number

(c) Rolling a number less than 3

(d) Rolling a number at least as big as 3

3. A fair six-sided die is rolled once. Find the probability of:

(a) Rolling a 5

(b) Rolling an odd number

(c) Rolling a number less than 3

(d) Rolling a number at least as big as 3

4. A card is randomly selected from a standard 52-card deck. What is the prob-ability that the card is:


(a) A Heart?

(b) A Queen?

(c) A Club or a Spade?

(d) A face card?

5. Two fair six-sided dice are rolled once.

(a) What is the probability that the sum of the dice will be 5?

(b) What is the probability that the sum of the dice will be 12?

(c) What is the probability that the sum of the dice will be 51?

6. A Math 105 class consists of 14 females and 10 males. The dapper instructorrandomly selects 8 of the students to each earn 51 bonus points. What is theprobability that all 8 of the students selected are females?

7. The English alphabet has 26 letters, 5 of which are vowels. If John randomlyselects 4 different letters from the alphabet, what is the probability that all4 letters are vowels? What is the probability that none of the 4 letters arevowels?

8. An urn contains 10 identical balls except that 1 is white, 4 are red, and 5 areblack.

(a) An experiment consists of selecting a ball from the urn and observing itscolor. What is a sample space for this experiment?

(b) Find the probability of the event “the ball is not white.”

(c) Suppose we have the same urn with the same 10 balls, but our experimentnow is to select 2 balls without replacement (so we select a ball, remove it,and then select another ball). What is a sample space for this experiment?(Your sample space does not need to be a uniform sample space.)

(d) Write all of the outcomes in the event “the white ball is not selected.”

9. A Math 105 class consists of 19 females and 15 males. The friendly instructorselects 6 of these students to form a Math 105 Dance Planning Committee.

(a) What is the probability that the committee consists of exactly 3 femalesand exactly 3 males?

(b) What is the probability that the committee consists of at least 5 females?


2.4 Complements, Unions, and Intersections

Definition: Let E be an event is a sample space S. The complement of E is denotedby E and represents the outcomes that are in S but not in E.

Definition: Let A and B be events. The union of A and B is denoted by A ∪ Band contains the outcomes that are in A or B, or both.

The intersection of A and B is denoted by A ∩ B and contains the outcomes thatare in A and B.

2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 65

1. Suppose we have an experiment where we select one Math 105 student. Soour sample space S is the set of all Math 105 students. Consider the followingthree events.

• M = “The student likes Michael”

• D = “The student likes Dwight”

• P = “The student likes Pam”

(a) Describe the following events in words:

i. M ∩ P

ii. P ∪D

iii. P ∩D

(b) Describe the events below using the set-theoretic notation we just dis-cussed in class:

i. The student does not like Pam

ii. The student likes Michael but not Dwight

iii. The student likes only Dwight (out of the three options)

iv. The student likes Pam or Dwight (or both), but not Michael


Probability Formulas:

• If E is an event, then P (E) ≥ 0 and P (E) ≤ 1

• If E is impossible, then P (E) = 0

• IF E must happen, then P (E) = 1

• P (E) + P (E) = 1 (or, equivalently, P (E) = 1− P (E) )

• If A and B are events, then P (A ∪B) = P (A) + P (B)− P (A ∩B)

• In the special case when A and B do not overlap, we say that A andB are mutually exclusive (they cannot both simultaneously happen). In

this special case, P (A ∪B) = P (A) + P (B)


2. Suppose we draw a card from a standard 52-card deck. Find the followingprobabilities.

(a) The probability that the card is not a 6

(b) The probability that the card is an Ace or a black card (or both)

(c) The probability that the card is a face card or a Spade (or both)

(d) The probability that the card is a red card or a black card (or both)


Homework

1. 17% of American children have blue eyes. 6.6% of American children haveType O- blood. 1.1% of American children have both blue eyes and Type O-blood. What is the probability that a randomly selected American child willhave blue eyes or Type O- blood (or both)?

2. In 2009, there were roughly 21,500,000 teenagers (ages 15-19) in the UnitedStates. 11,000,000 of these teenagers were males and 10,500,000 of theseteenagers were females. 130,000 of the male teenagers had ever been mar-ried, and 240,000 of the female teenagers had ever been married. What is theprobability that a randomly selected teenager in 2009 was female or had everbeen married (or both)?

3. Suppose we select a Concordia student at random. Let A be the event that thestudent is a math major and let B be the event that the student is a junior.Write what the following probabilities represent in words:

(a) P (A ∪B)

(b) P (A ∩B)

(c) P (A)

(d) P (A ∪B)

4. The probability of winning a Math 105 collector’s edition beanie is 0.49. Whatis the probability of not winning the beanie?

5. Two fair six-sided dice are rolled once.

(a) What is the probability that the sum of the dice is not 12?

(b) What is the probability that the sum of the dice is not 10?

(c) What is the probability that the sum of the dice is not an even number?

6. If P (A) = 0.5, P (B) = 0.6, and P (A ∩B) = 0.3, find P (A ∪B).

7. If P (A ∪B) = 0.9, P (A) = 0.6, and P (A ∩B) = 0.2, find P (B).


8. A card is randomly selected from a standard 52-card deck. What is the prob-ability that the card is:

(a) A Heart or a Spade?

(b) A Heart or a King?

(c) A Heart or a face card?

(d) A Red card or a face card?

9. Suppose A and B are events in a sample space S with P (A) = 12

and P (B) =710

. What is the smallest possible value of P (A ∩ B)? What is the largestpossible value of P (A ∩B)?

10. An experiment consists of selecting a car at random from the Hvidsten parkinglot and observing the color and make of the car. Let R be the event “The caris red,” let F be the event “The car is a Ford,” let G be the event “The car is agreen Saturn,” and let B be the event “The car is blue or a Buick (or both).”

Which of the following pairs of events are mutually exclusive?

(a) R and F

(b) R and G

(c) F and G

(d) R and B

(e) F and B

(f) G and B

(g) R and G

(h) F and B


2.5 Conditional Probability

Key Idea: Sometimes learning new information about a situation can change theprobabilities.

Example:

Definition: If A and B are events, then the probability that A will occur giventhat B has occurred is called the conditional probability and is denoted by P (A|B).

P (A|B) =P (A ∩B)

P (B)

2.5. CONDITIONAL PROBABILITY 71

1. In a survey of Cobbers:

• 70% read The Forum

• 80% read The Concordian

• 90% read at least one of the two papers

If a Cobber reads The Concordian, what is the probability that they read TheForum?

P (A ∩B) = P (A) · P (B|A) and P (A ∩B) = P (B) · P (A|B)


2. Suppose we draw 2 cards from a standard deck without replacement. What isthe probability that both are Kings?

3. Suppose we flip a fair coin 3 times.

(a) What is the probability that we get HHH?

(b) What is the probability that we get HHH given that at least one of thefirst two flips was Heads?


Homework

1. Suppose we select a Concordia student at random. Let A be the event that thestudent is a math major and let B be the event that the student is a junior.Write what the following probabilities represent in words:

(a) P (A|B)

(b) P (B|A)

(c) P (A|B)

(d) P (A|B)

2. Two cards are selected at random in order from a standard 52-card deck. Findthe probability that the first is a Jack and the second is a Queen:

(a) With replacement

(b) Without replacement

3. Two cards are selected at random in order from a standard 52-card deck. Findthe probability that the first is a Jack or the second is a Queen (or both):

(a) With replacement

(b) Without replacement

4. An urn contains 4 white and 6 red balls.

(a) If three balls are drawn from this urn with replacement, what is theprobability that the last ball is red?

(b) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red?

(c) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that the first two balls were red?

(d) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that the first two balls werewhite?

(e) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that at least one of the first twoballs were white?


(f) Suppose we follow these rules for our urn: Two balls are randomly drawn.If the first ball is white, it is replaced before we make our second draw.If the first ball is red, it is not replaced before we make our second draw.What is the probability of drawing at least one white ball?

5. Suppose we flip a fair coin 20 times. What is the probability that:

(a) We obtain no Heads

(b) We obtain Heads exactly once

(c) We obtain Heads at most once

(d) We obtain Heads at least once

(e) We obtain Heads exactly 10 times

6. From a standard 52-card deck, we deal a random 5 card hand. What is theprobability that:

(a) The hand contains only Black cards

(b) The hand has exactly two Spades

(c) The hand has at most two Spades

(d) The hand has more Black cards than Red cards

(e) The hand has no pairs

(f) The hand has exactly one pair

(g) The hand has exactly two pairs

(h) The hand is a straight (5 cards with consecutive ranks, where Aces mustbe low)

(i) The hand is a flush (all cards have the same suit)

(j) The hand contains no cards above a Nine (Aces are allowed)

(k) The hand is a full house (3 cards in one rank, 2 cards in another)

(l) The hand has exactly three in one rank, and no other pairs

(m) The hand has exactly three cards in one suit, and the other cards aredifferent suits

(n) The hand contains all four suits


7. The English alphabet contains 26 letters, 5 of which are vowels. What is theprobability that a random 5 letter word:

(a) Has no repetitions

(b) Begins with J, ends with F, and has no repetitions

(c) Begins with J, ends with F, contains L, and has no repetitions

(d) Contains J, and has no repetitions

(e) Contains J and F, and has no repetitions

(f) Contains exactly two vowels, and has no repetitions

(g) Contains at most two vowels, and has no repetitions

(h) Contains more consonants than vowels, and has no repetitions

(i) Has consonants and vowels alternating, and has no repetitions

(j) Contains exactly two L’s, and has no other repetitions

(k) Has exactly one letter appearing exactly twice, and no other letter repeats

(l) Has at most one letter that repeats, and no letter appears more thantwice

(m) Has at most one letter that repeats

(n) Has J appearing exactly twice and F appearing exactly twice

(o) Has exactly two letters appearing exactly twice each



1. The French mathematicians Pierre de Fermat and Blaise Pascal are usuallygiven credit for originating the theory of probability. The first problems inprobability were posed to Pascal by the famous gambler Chevalier de Mere,and Pascal and Fermat exchanged letters developing the theory of probabilityin order to answer de Mere’s questions. Provide some historical backgroundabout these men and the letters, and present a complete solution to at leastone of these first problems in probability.

2. Describe Pascal’s Triangle. Include its connections to counting theory andpresent how it relates to the binomial theorem, which allows us to expandexpressions like (x+ y)2, (x+ y)3, . . . . Also explain how it helps us count thenumber of paths from one corner to another corner in a rectangular grid. Feelfree to include other applications of this triangle as well!

3. Discuss the Monty Hall Problem: Suppose you’re on a game show, and you’regiven the choice of three doors. Behind one door is a car, behind the others,goats. You pick a door, say #1, and the host, who knows what’s behind thedoors, opens another door, say #3, which has a goat. He says to you, “Doyou want to pick door #2?” Is it to your advantage to switch your choice ofdoors?

This problem was posed to Marilyn vos Savant in 1990, and she answered itcorrectly, but thousands of people, including quite a few Ph.D. mathemati-cians, wrote to her and told her she was wrong. Explain the problem, presentsome of the letters, and use probability to give a correct solution to the prob-lem.


2.7 Chapter Review

Concepts:

• Permutations are where order matters, and the number of permutations isdenoted by P (n, r)

• P (n, r) =n!

(n− r)!= n · (n− 1) · (n− 2) · · · · · (n− (r − 1))

• Combinations are where order does not matter, and the number of combina-tions is denoted by C(n, r)

• C(n, r) =P (n, r)

r!=

n!

(n− r)! · r!

• Sample space

• Uniform sample space

• If E is an event in a uniform sample space, then

P (E) =number of outcomes in E

number of outcomes in S

• A ∪B denotes the union of A and B (Or)

• A ∩B denotes the intersection of A and B (And)

• P (A ∪B) = P (A) + P (B)− P (A ∩B)

• P (A ∪ B) = P (A) + P (B) if A and B are mutually exclusive (they do notoverlap at all)

• C denotes the complement of C (Not)

• P (C) = 1− P (C)

• The conditional probability of A given B is denoted by P (A|B)

• P (A|B) =P (A ∩B)

P (B)(this comes from an outstanding Venn diagram)

• P (A∩B) = P (A) ·P (B|A) and P (A∩B) = P (B) ·P (A|B) (in particular, wecan multiply along a tree diagram)


• Coin questions

• Card questions

• Word questions


Review Exercises:

1. A state makes license plates with three letters followed by three digits. Thereare 26 possible letters (A through Z) and 10 possible digits (0 through 9).Letters are not allowed to repeat on a license plate, but the digits are allowedto repeat.

Assuming that the state makes license plates randomly according to theserules, what is the probability that a license plate made in this state beginswith the letter H and ends with the digit 8?

2. An urn holds 10 identical balls except that 4 are red and 6 are white. Anexperiment consists of selecting two balls in succession from the urn withoutreplacing the first ball selected.

(a) What is a sample space for this experiment?

(b) What is the probability that we select two red balls?

(c) What is the probability that we select a red ball if we know that the firstball we select is white?

3. Suppose A and B are events in a sample space S with P (A ∪ B) = 0.7,P (B) = 0.2, and P (A ∩B) = 0.1. What is the value of P (A)?

4. A single card is drawn from a standard 52-card deck. What is the probabilitythat the card is a Jack or Red (or both)?

5. On a children’s baseball team, there are six players who can play any of thefive following infield positions: catcher, first base, second base, third base, andshortstop. There are four possible pitchers, none of whom can play any otherposition. And there are five players who can play any of the three outfieldpositions: left field, center field, or right field. In how many ways can thecoach assign these players to positions?

6. A chef can make 10 main courses. Every day a menu is formed by selecting 6of the main courses and listing them in order. How many different such menuscan be made?

7. A chef can prepare 10 different entrees. In how many ways can the chef select6 entrees for today’s menu?

8. A license plate has 6 digits with repetitions permitted. How many possiblelicense plates of such type are there?


9. At an awards ceremony, 3 men and 4 women are to be called one at a timeto receive an award. In how many ways can this be done if women and menmust alternate?

10. An executive is scheduling meetings with 10 people in succession. The first2 meetings must be with 2 directors on the board, the second 3 with 3 vicepresidents, and the last 5 with 5 junior executives. How many ways can thisschedule be made out?

11. In a certain lotto game, 5 numbered ping pong balls are randomly selectedwithout replacement from a set of balls numbered from 1 to 35 to determine awinning set of numbers (without regard to order). Find the number of possibleoutcomes.

12. The Pi Mu Epsilon honor society consists of 9 men and 7 women. If the societyforms a committee with 3 women and 2 men, how many different ways canthis be done?

13. In a new group of 10 employees, 4 are to be assigned to production, 2 to sales,and 1 to advertising. In how many ways can this be done?

14. A fair (each side is equally likely to land up) 10-sided die is rolled 100 timeswith the following results:

Outcome Frequency

1 82 83 124 75 156 87 88 139 910 12

(a) What is the experimental probability of rolling a 3?

(b) What is the theoretical probability of rolling a 3?

(c) What is the experimental probability of rolling a multiple of 3?

(d) What is the theoretical probability of rolling a multiple of 3?


15. One card is drawn from a standard 52 card deck. What is the probability ofdrawing:

(a) the queen of hearts?

(b) a queen?

(c) a heart?

(d) a face card (J, Q, or K)?

16. Two 6-sided dice are rolled. What is the probability of rolling

(a) a total of 6?

(b) not a total of 6?

(c) a total of 6 or 7?

(d) a total of 6 or more?

17. A gumball machine has gumballs of four flavors: apple, berry, cherry, andpumpkin. When a quarter is put into the machine, it dispenses 5 gumballs atrandom. What is the probability that

(a) each gumball is a different flavor?

(b) at least two gumballs are the same flavor?

18. A coin is flipped ten times in a row. Find the probability that

(a) no tails show.

(b) exactly one tail shows.

(c) exactly twice as many heads as tails occur.

19. A group of 15 students is to be split into 3 groups of 5. In how many wayscan this be done?

20. Five cards are drawn from a standard 52 card deck. What is the probabilityof drawing 5 cards of the same color?

21. Suppose a jar has 4 coins: a penny, a nickel, a dime and a quarter. You removetwo coins at random without replacement. Let A be the event you remove thequarter. Let B be the event you remove the dime. Let C be the event youremove less than 12 cents.

(a) List the sample space.

(b) Draw a probability tree diagram to represent the possible scenarios.


(c) Find P (A); P (C); and P (B).

(d) Compute and interpret P (A ∪B) and P (B ∪ C).

22. The probability is 0.6 that a student will study for an exam. If the studentstudies, she has a 0.8 chance of getting an A on the exam. If she does notstudy, she has a 0.3 probability of getting an A. Make a probability tree forthis situation. What is the probability that she gets an A? If she gets an A,what is the conditional probability that she studied?

23. If P (A) = 0.6, P (B) = 0.3, and P (A ∩B) = 0.2, find P (A ∪B) and P (A|B).

24. If P (A) = 0.4, P (B) = 0.5, and P (A ∪B) = 0.7, find P (A ∩B) and P (B|A).

25. A dental assistant randomly sampled 200 patients and classified them accord-ing to whether or not they had a least one cavity in their last checkup andaccording to what type of tooth decay preventative measures they used. Theinformation is as follows

At least one cavity No CavitiesBrush only 69 2

Brush and floss only 34 11Brush and tooth sealants only 22 13Brush, floss and tooth sealants 3 46

If a patient is picked at random from this group, find the probability that

(a) the patient had at least one cavity;

(b) the patient brushes only;

(c) the patient had no cavities, given s/he brushes, flosses and has toothsealants;

(d) the patient brushes only, given that s/he had at least one cavity.



1.1 · 25 · 24 · 10 · 10 · 1

26 · 25 · 24 · 10 · 10 · 10

2. (a) {RR,RW,WR,WW} is one possibility

(b)4

10· 3

9or

C(4, 2)

C(10, 2)

(c)4

9

3. 0.6

4.28

52

5. P (6, 5) · P (4, 1) · P (5, 3)

6. P (10, 6) =10!

4!= 151, 200

7. C(10, 6) =10!

4!6!= 210

8. 106 = 1, 000, 000

9. 4 3 3 2 2 1 1= 4!3! = 144

10. 2!3!5! = 1440

11. C(35, 5) =35!

30!5!= 324, 632

12. C(7, 3) · C(9, 2) = 1260

13. C(10, 4) · C(6, 2) · C(4, 1) = 12, 600

14. (a) 12/100 = 3/25

(b) 1/10

(c)12 + 8 + 9

100=

29

100

(d)1 + 1 + 1

10=

3

10

15. (a) 1/52

(b) 4/52 = 1/13


(c) 13/52 = 1/4

(d) 12/52 = 3/13

16. (a) 5/36

(b) 1− 5/36 = 31/36

(c) 11/36

(d) 26/36

17. (a) 0, since there are only 4 flavors.

(b) 1− 0 = 1

18. (a)1

210

(b)C(10, 1)

210=

10

210= 0.009765

(c) 0, since it is not possible. 6 heads and 3 tails is only 9 flips, and 8 headsand 4 tails is 12 flips, which is too many.

19.C(15, 5) · C(10, 5) · C(5, 5)

3!= 126, 126

20.2C(26, 5)

C(52, 5)=

253

4998= 0.05062

21. (a) The sample space is {PN,PD, PQ,NP,ND,NQ,DP,DN,DQ,QP,QN,QD}.(b) Probability tree

(c) P (A) = 1/2; P (C) = 4/12 = 1/3; and P (B) = 1/2.

(d) P (A∪B) = P (A)+P (B)−P (A∩B) = 1/2+1/2−1/6 = 5/6 representsthe probability that we remove either a quarter or a dime, and P (B∪C) =1/2 + 1/3− 1/6 = 2/3 is the probability that we remove either a dime orless than 12 cents.

22. What is the probability that she gets an A? 0.6× 0.8 + 0.4× 0.3 = 0.6.

If she gets an A, what is the conditional probability that she studied?0.48

0.6=

0.8.

23. If P (A) = 0.6, P (B) = 0.3, and P (A ∩B) = 0.2, then

P (A ∪B) = P (A) + P (B)− P (A ∩B) = 0.6 + 0.3− 0.2 = 0.7, and

P (A|B) =P (A ∩B)

P (B)=

0.2

0.3=

2

3.


24. If P (A) = 0.4, P (B) = 0.5, and P (A ∪B) = 0.7, then

P (A ∩B) = P (A) + P (B)− P (A ∪B) = 0.4 + 0.5− 0.7 = 0.2, and

P (B|A) =P (A ∩B)

P (A)=

0.2

0.4=

1

2.

25. (a) 128/200 = 0.64

(b) 71/200 = 0.355

(c) 46/49 = 0.938

(d) 69/128 = 0.539

Chapter 3

Graph Theory

3.1 Introduction to Graph Theory

Definition: A graph is a collection of vertices (points) connected by edges (lines).

Examples:

Graph Applications:

87

88 CHAPTER 3. GRAPH THEORY

Definition: A loop is an edge from a vertex to itself.

Definition: If more than one edge joins two vertices, these edges are called multipleedges.

Definition: A graph is simple if it has no loops and no multiple edges.

Definition: A graph is connected if you can get from any vertex to any other vertexby following edges in the graph. If a graph is disconnected, the connected pieces ofthe graph are called the components of the graph.

Definition: The degree of a vertex is the number of edges at that vertex. A loopat a vertex counts as 2 for the degree of the vertex.

3.1. INTRODUCTION TO GRAPH THEORY 89

1. For each of the following lists of numbers:

I. If possible, draw a graph that has the list as its list of vertex degrees.

II. If possible, draw a simple graph that has the list as its list of vertex degrees.

(a) 1, 1, 2, 2, 4

(b) 1, 1, 1, 2, 2, 4

(c) 1, 1, 2, 2, 6

If some of the lists are impossible, can you explain why?


Definition: If n is a positive integer, the complete graph on n vertices isdenoted by Kn and is the simple graph with n vertices and all possible edges.

3.1. INTRODUCTION TO GRAPH THEORY 91

Homework

1. For each of the graphs below:

(a) Write the vertex set

(b) Write the edge set

(c) List the degrees of each vertex

2. For each of the given vertex sets and edge sets, draw two different pictures ofa graph with those sets.

(a) V = {A,B,C,D}, E = {AB,AC,AD,BC}(b) V = {M,A, T,H, Y }, E = {HM,HT,HY,HY,MM,MT,MY }

3. (a) Draw a connected graph with five vertices where each vertex has degree 2.

(b) Draw a disconnected graph with five vertices where each vertex has de-gree 2.

(c) Draw a graph with five vertices where four of the vertices have degree 1and the other vertex has degree 0.

4. Give an example of a graph with four vertices, each of degree 3, with:

(a) No loops and no multiple edges

(b) Loops but no multiple edges

(c) Multiple edges but no loops

(d) Loops and multiple edges.


5. Suppose our city has 51 city council members, and that these members serveon 10 city committees. Each council member serves on at least one committee,and some members serve on multiple committees.

(a) If we wanted to know which pairs of members are on the same committee,would it be better to draw a graph where the vertices represent membersor a graph where the vertices represent committees?

(b) If we wanted to know which committees have members in common, wouldit be better to draw a graph where the vertices represent members or agraph where the vertices represent committees?

6. (a) Draw K6 and K7.

(b) Find a formula, which depends on the positive integer n, for the numberof edges in Kn.

7. For each of the following sets of conditions, explain why no graph satisfiesthose conditions:

(a) A graph with exactly five vertices each of degree 3

(b) A graph with exactly four edges, exactly four vertices, and the verticeshave degrees of 1, 2, 3, and 4

(c) A simple graph with exactly four vertices, where the vertices have degreesof 1, 2, 3, and 4

(d) A simple graph with exactly six vertices, where the vertices have degreesof 1, 2, 3, 4, 5, and 5

3.2. PATHS AND CIRCUITS 93

3.2 Paths and Circuits

Definition: A path is a sequence of consecutive vertices that are joined by edgesin a graph. The length of a path is the number of edges used in the path. Verticescan be repeated in a path, but edges cannot be repeated.

Example:

Definition: A circuit is a path that starts and ends at the same vertex.


1. In the following graphs, is there a path that uses every edge exactly once? Isthere a circuit that uses every edge exactly once?

(a)

A B

C D

E F

(b)

A B

C D

E F

(c)

A B

C D

E F

(d)

A B

C D

E F


Definition: Suppose we have a connected graph. A path that uses every edgeof the graph exactly once is called an Euler path. A circuit that uses everyedge exactly once is called an Euler circuit.

Konigsberg, from Leonhard Euler, Solutio problematis ad geometriam situs perti-nentis, 1736


Main Theorem:

Examples:


Homework

1. Consider the following graph:

(a) Find a path of length 5 from A to H.

(b) Find a path of length 7 from A to H.

(c) How many different paths are there from A to D?

(d) How many different paths are there from D to H?

(e) How many different paths are there from A to H?

2. For each of the following graphs, determine if the graph has an Euler circuit,an Euler path but no Euler circuit, or no Euler path and no Euler circuit.Explain your answer. You do not need to show an Euler path and/or Eulercircuit if they exist.

(a)

(b)

(c)


(d)

(e)

(f)

(g)

(h)


3. Find an Euler circuit in the following graph.

4. Find an Euler circuit in the following graph.

5. Find an Euler path in the following graph.



7. In the map below, is it possible to travel a route that crosses each bridgeexactly once? If so, give such a route. If not, explain why not.

8. A letter carrier is responsible for delivering mail to all of the houses on bothsides of the streets shown in the figure below. (The streets are white, and thehouses are in the gray regions.) If the letter carrier does not keep crossing astreet back and forth to get to houses on both sides of a street, then she willneed to walk along a street at least twice, once on each side, to deliver themail.

(a) Is it possible for the letter carrier to construct a round trip so that shewalks on each side of every street exactly once?

(b) If the street diagram was different, would you arrive at the same conclu-sion?

3.3. SUBGRAPHS AND TREES 101

3.3 Subgraphs and Trees

Definition: A subgraph of a graph G is a graph contained in G. A subgraph of Gis spanning if it contains all of the vertices of G.

Examples:

Weighted Graphs:


1. For each of the following graphs, find a connected spanning subgraph with thesmallest possible total weight.


Minimal Spanning Trees

Definition: A connected graph with no circuits is called a tree.

Fact: If a tree has n vertices, it must have n− 1 edges. Also, any connectedgraph with n vertices and n− 1 edges is a tree.


2. How many different spanning trees does the following graph contain?

Greedy Algorithms:


Homework

1. For the following graphs, determine whether or not the graph is a tree. Explainyour answers.

(a)

(b)

(c)

(d)

2. Draw all of the different spanning trees of the following graph.


3. How many different spanning trees does the following graph have?

4. How many different spanning trees does the following graph have?

5. Find a minimal spanning tree, and give its total weight.


6. Find a minimal spanning tree, and give its total weight.

7. Below is a mileage chart between some cities in Minnesota. Draw a weightedgraph that reflects the mileages and find a minimal spanning tree that connectsall of these cities in your graph.

Bem

idji

Frazee

International

Falls

Moorhead

TwoHarbors

WhiteBearLake

Bemidji — 82 112 127 180 224

Frazee 82 — 198 55 223 194

International Falls 112 198 — 242 177 280

Moorhead 127 55 242 — 277 239

Two Harbors 180 223 177 277 — 167

White Bear Lake 224 194 280 239 167 —

8. For each of the following statements, if the statement is true, explain why. Ifthe statement is false, give an example of a graph where the statement is false.

(a) If G is a connected simple graph with weighted edges, and all of theweights are different, then different spanning trees of G have differenttotal weights.

(b) If G is a connected simple graph with weighted edges, and e is an edgeof G with a smaller weight than any other edge of G, then e must beincluded in every minimal spanning tree of G.


9. Suppose we have a large collection of 1-cent, 8-cent, and 10-cent stamps avail-able. We want to select the minimum number of stamps needed to make agiven amount of postage. Consider a greedy algorithm that selects stamps byfirst choosing as many of the 10-cent stamps as possible, then as many of the8-cent stamps as possible, then as many 1-cent stamps as possible. Does thisgreedy algorithm always produce the fewest number of stamps needed for eachpossible amount of postage? Why or why not?

3.4. GRAPH COLORINGS 109

3.4 Graph Colorings

Definition: A (vertex) coloring of a graph is an assignment of colors to the verticesof the graph so that vertices that are joined by an edge (adjacent vertices) havedifferent colors.

Example:

Definition: For a graph G, the smallest number of colors needed to color G iscalled the chromatic number of G and is denoted χ(G). Note that χ is the Greekletter chi, for chromatic.

Example: If G is , then χ(G) = 3. It is possible to color G using

just 3 colors, and we need at least 3 colors because G has a triangle.

Example: Color each of the vertices of the following graph red (R), white (W), orblue (B) in such a way that no adjacent vertices have the same color.

Example: If G is find χ(G).


1. What is the coloring number of the following complete graphs?

Ki χ(Ki)

K1 =

K2 =

K3 =

K4 =

K5 =

Kn


2. What is the coloring number of the following paths?

Pi χ(Pi)

P1 =

P2 =

P3 =

P4 =

P5 =

Pn

3. (a) Find the chromatic number of the following tree:

(b) Draw a tree with 11 vertices, and find its chromatic number.


4. Corncob College elects 10 students to serve as officers on 8 committees. Thelist of the members of each of the committees is:

• Corn Feed Committee: Darcie, Barb, Kyler

• Dorm Policy Committee: Barb, Jack, Anya, Kaz

• Extracurricular Committee: Darcie, Jack, Miranda

• Family Weekend Committee: Kyler, Miranda, Jenna, Natalie

• Homecoming Committee: Barb, Jenna, Natalie, Skye

• Off Campus Committee: Kyler, Jenna, Skye

• Parking Committee: Jack, Anya, Miranda

• Student Fees Committee: Kaz, Natalie

They need to schedule meetings for each of these committees, but two com-mittees cannot meet at the same time if they have any members in common.How many different meeting times will they need?


An example where Greedy Coloring fails :


Homework

1. Find the chromatic number for each of the following graphs.

(a) (b) (c)

(d) (e)

2. Consider the following graph classes:

Cycles

· · ·

C1 C2 C3 C4 C5

Wheels

· · ·

W3 W4 W5 W6


Find χ(Cn) and χ(Wn) for every integer n ≥ 3, where Cn denotes the cyclewith n vertices and Wn denotes the wheel with n spokes (n+1 vertices). Youranswers will change as n changes.

3. The mathematics department at Cornucopia College will offer seven coursesnext semester: Math 105 (M), Numerical Analysis (N), Linear Operators (O),Probability (P), Differential Equations (Q), Real Analysis (R), Statistics (S).The department has twelve students, who will take the following classes:

Alice: N, O, Q Dan: N, O Greg: M, P Jill: N, S, QBob: N, R, S Emma: O, M Homer: R, O Kate: P, SChaz : R, M Fonz: N, R Inez: N, Q Lara: P, Q

The department needs to schedule class times for each of these courses, but twocourses cannot meet at the same time if they have any students in common.How many different class times will they need?


3.5 Planar Graphs

Definition: A graph is planar if we can draw it in the plane (a flat surface) withno edges crossed. If a graph is drawn in the plane with no edges crossed, we saythat the drawing is a plane graph.

Examples:

1. Which complete graphs are planar graphs?

3.5. PLANAR GRAPHS 117

Definition: A connected plane graph divides the plane into different regionscalled faces.

Faces are only defined when the drawing is a plane graph — when the edgesdon’t cross.

2. For each of the following graphs, count the number of vertices, the number ofedges, and the number of faces. Do you see a relationship between these threenumbers?


Euler’s Formula:

Explanation:

Fact: If G is a connected simple plane graph with at least three vertices, then3V − 6 ≥ E.

3.5. PLANAR GRAPHS 119

Homework

1. For the following plane graph, count the number of vertices, the number ofedges, and the number of faces. Verify that Euler’s formula holds.

2. For each of the following graphs, if the graph is planar, draw it in the planeso that no edges cross. Otherwise, state that the graph is not planar.

(a)A

B

C D

E

(b)

A

BC

D

E F

(c)

AB

C

DE

F

(d)A

B

C D

EFG

H I

J

3. Let G be a connected planar graph. Explain why, no matter how we draw Gas a plane graph, the number of faces of G will always be the same.

4. Suppose G is a connected plane graph with nine vertices, where the degreesof the vertices are 2, 2, 2, 3, 3, 3, 4, 4, and 5. How many edges does G have?How many faces does G have?


5. Suppose you have a graph G that is not planar. Is there a way to make itplanar by adding more vertices and edges? If so, describe your method; if not,why not?

6. Suppose you have a graph G that is planar. Is there a way to make it notplanar by adding more vertices and edges? If so, describe your method; if not,why not?

3.6. DIRECTED GRAPHS 121

3.6 Directed Graphs

Definition: A directed graph is a graph where every edge in the graph is assigneda direction between its two vertices.

For an edge , we say that the edge is directed from X to Y . Alternatively,Y is the head of the edge and X is the tail.

Applications:


Suppose we have the following tasks as we build a house:

Task Time RequiredStart: Decide to build house No time

A: Clear land 1 dayB: Build foundation 3 daysC: Build frame and roof 15 daysD: Electrical work 9 daysE: Plumbing work 5 daysF: Complete exterior work 12 daysG: Complete interior work 10 daysH: Landscaping 6 days

Finish: Move in! No time

The activity directed graph is given below. We draw an arrow from task X to taskY if and only if task Y must be completed directly after completing task X. Welabel each vertex with the amount of time needed to complete that task.

The longest path following the arrows from Start to Finish (in terms of time) is acritical path. The length of the critical path is the length of time needed to completethe overall task.

1. Find the critical path and give its length (in days).


Definition: A tournament is a complete graph where every edge is directed.

Definition: Two tournaments are isomorphic if they can be redrawn to looklike each other.


Definitions: Given a vertex v in a directed graph, the number of arrows goinginto v is called the in-degree of v, and the number of arrows going out of v iscalled the out-degree of v.

In a tournament, a vertex with only arrows going out (in-degree = 0) is calleda source, and a vertex with only arrows coming in (out-degree = 0) is called asink.

Fact 1: In any tournament, a longest path that does not repeat any verticeswill visit every vertex exactly once.

Fact 2: Every tournament has at least one king chicken.

A king chicken is a team v such that, for any other team w:

• v beat w, or

• v beat a team that beat w.


2. In each of the tournaments below, find a longest path that doesn’t repeatvertices, and find a king chicken.


Homework

1. Suppose we have to complete the following tasks as we produce a film. Theamount of time each intermediate task requires is listed in the table.

Task Time RequiredStart: Obtain script No time

A: Acquire funding 4 weeksB: Sign director 2 weeksC: Hire actors 4 weeksD: Choose filming locations 2 weeksE: Build sets 3 weeksF: Film scenes 10 weeksG: Edit film 5 weeksH: Create soundtrack 3 weeks

Finish: Release film No time

The activity directed graph is given below. Find the critical path and give itslength (in weeks).


2. Michael and Jan have invited some friends over for a dinner party and needto set the dinner table. Estimate the time required (in minutes) needed tocomplete the following tasks.

Task Time RequiredStart: Finish playing charades No time

A: Put tablecloth on tableB: Fold napkins and place on tableC: Place dishes and silverware on tableD: Put water and ice in water glassesE: Pour wine into wine glassesF: Put food on table

Finish: Begin eating No time

Now draw a possible activity directed graph for these tasks, and find theminimum amount of time needed to complete the project by finding a criticalpath in your graph. Assume that only Dwight has volunteered to help Michaeland Jan set the table, so that at most three tasks can occur at any given time.

3. Draw all of the different (non-isomorphic) types of tournaments with exactlyfour vertices.

4. If an undirected graph has E edges and we add all of the vertex degrees in thegraph, we know that the sum is 2E (by Euler’s Handshake Theorem).

(a) If we have a directed graph with E edges and we add all of the in-degrees,what will the sum be?

(b) If we have a directed graph with E edges and we add all of the out-degrees,what will the sum be?

5. Explain why a tournament can have at most one source and at most one sink.

6. (a) Give an example where five teams play in a round robin tournament andall five teams tie for first place.

(b) Explain why, if six teams play in a round robin tournament, it is impos-sible for all six teams to tie for first place.


7. In each of the following tournaments:

(a) Find a longest path that doesn’t repeat vertices (one that ranks all of theteams)

(b) Find a king chicken

3.7. CHAPTER PROJECTS 129


1. Present the idea of a hamiltonian circuit in a graph (a circuit that uses everyvertex other than the starting and ending vertex exactly once). Give examplesof graphs that do and graphs that do not have hamiltonian circuits. Discusssome of the history of hamiltonian circuits, including some information aboutHamilton himself, and his game “Around the World.” Describe the connectionto the famous Traveling Salesperson Problem.

2. Present the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron,icosahedron). Use Euler’s formula to show that these five solids are the onlypossible Platonic solids. If you want, describe some of the historical connec-tions between the solids and religion, philosophy, and astronomy.

3. Present the idea of coloring maps. The famous Four Color Theorem was provedin 1976 — the proof is extremely complicated and has never been checked byhand, only by computers. So you don’t need to prove the theorem! At leastpresent what the theorem says, and discuss maps that can be colored withfewer than four colors. Use plenty of examples. If you want, present someof the historical background involving map colorings, including the fact thatthere were incorrect proofs that were believed to be correct for years. If youwant, discuss whether or not a proof that no person has ever checked is actuallyvalid.


3.8 Chapter Review

Concepts:

• Graph

• Vertices, Edges

• Loops, Multiple edges

• Simple graphs

• Connected

• Degrees

• Handshake Theorem (the sum of the degrees in a graph must be even)

• Path, Circuit

• Euler paths and circuits, and the connection to vertex degrees being even

• Network

• Tree

• Spanning Tree

• Kruskal’s Algorithm for Minimal Spanning Trees

• Greedy algorithms

• Complete Graphs: K1, K2, K3, K4, . . .

• Coloring, Chromatic number χ(G)

• Planar graphs, Plane graphs

• Faces

• Euler’s Formula (for connected plane graphs): V − E + F = 2

• Directed graphs

• Activity directed graphs and Critical paths

• Tournaments

• Source and Sink

• Longest paths and King chickens in tournaments


Review Exercises:


2. Find the number of different spanning trees in the following graph.

3. In the following tournament:

(a) Find any sources or sinks

(b) Find a longest path that doesn’t repeat vertices

(c) Find all of the king chickens

4. Draw K6.


5. (a) For each of the following graphs, determine whether the graph is planaror not.

(b) Find the chromatic number for each of the graphs above.

6. Explain why we cannot have a simple graph with exactly six vertices, wherethe vertices have degrees of 1, 1, 1, 3, 5, and 5.

7. For each of the following graphs, answer the following questions or do therequested task.

(i) Does the graph have an Euler path? If so, find an Euler path in the graph.If not, explain why not.

(ii) Does the graph have an Euler circuit? If so, find an Euler circuit in thegraph. If not, explain why not.

(iii) What is the chromatic number χ of the graph?

(iv) If the graph has no Euler path, find an optimal semi-eulerization for it.

(v) If the graph has no Euler circuit, find an optimal eulerization for it.

(a)

A B C

D E F (b)

A B C

D

EFG

H


(c)

A B

CD

E

(d)

A B

CD

E F

8. Find the number of spanning trees in the following graphs:

(a) (b)

9. Find a minimal spanning tree for each of the following graphs.

150

200

210

125

50

30075

100


3.5

1.0

1.3 1.6

1.8

2.1

2.51.5

3.6 2.3

1.12.8

10. Are the following graphs planar? If so, draw a plane graph version of the graphand verify Euler’s formula for it; if not, state that it is not planar.

11. Show that the following graph is planar, find its chromatic number, and verifyEuler’s formula for it.

A

B

C

D

E

F

12. Find an eulerization of the following graph.


13. A connected simple planar graph has six vertices with degrees 1, 2, 3, 3, 4,and 5.

(a) How many edges does the graph have?

(b) If the graph is drawn in the plane without edges crossing, how many facesare there?

14. For each of the following sets of conditions, either draw a graph that satisfiesthose conditions, or explain why such a graph is impossible:

(a) A graph with exactly 4 vertices, where the vertices have degrees of 1, 1,2, 3

(b) A graph with exactly 4 vertices, where the vertices have degrees of 1, 3,4, 4

(c) A simple graph with exactly 4 vertices, where the vertices have degreesof 2, 2, 3, and 3

15. Draw an example of a tree that has exactly seven vertices, where exactly threeof the vertices have a degree of 2.



1. One example is CDHGKJIEABCGBFGJFE. Any example must use all17 edges exactly once, and start and end at C or E.

2. 3 · 2 · 4 = 24

3. (a) No source, no sink

(b) One example is AFEBCD

(c) A, D, and F

4.

5. (a) The first is planar, the second is not (M,H, I, J, L forms a K5 in thesecond graph)

(b) 4, 5

6. The two vertices of degree 5 must both be connected to all other vertices ofthe graph, which does not allow for any degree 1 vertices

7. (a) (i) Yes, two odd vertices B and F : BCFBEADEF

(ii) No, not all of the vertices are even

(iii) χ(a) = 3

(iv) Not applicable

(v) Add edge BF

(b) (i) Yes, all vertices are even: ABCDEFGHBDFHA; same for (ii)

(iii) χ(b) = 3

(iv) Not applicable

(v) Not applicable

(c) (i) No, four odd vertices is two too many

(ii) No, see (i)

(iii) χ(c) = 3

(iv) Add edge AB: DABAEBCEDC


(v) Add edges AB and DC: ABAEBCDCEDA

(d) (i) Yes, two odd vertices E and F : EADEBAFECDFCBF

(ii) No, not all of the vertices are even

(iii) χ(d) = 4

(iv) Not applicable

(v) Add edge EF

8. (a) 3× 3 = 9; (b) 3× 4× 3 = 36

9. (a) 425 = 50 + 75 + 100 + 200; (b) 10.4 = 1.0 + 1.3 + 1.6 + 1.8 + 2.1 + 1.1 + 1.5

10. (a) Planar; V = 6, E = 7, F = 3, so V − E + F = 6− 7 + 3 = 2. (b) Planar;V = 6, E = 8, F = 4, so V − E + F = 6− 8 + 4 = 2. (c) Not planar

11. Stretch BD out and FD way out, and stretch AD way out, and move B abovethe AC edge. χ = 4. V = 6, E = 11, F = 7: V − E + F = 6− 11 + 7 = 2, soEuler’s formula checks.

12. Add 4 multiple edges.

13. (a) 1 + 2 + 3 + 3 + 4 + 5 = 18, so there are 18/2 = 9 edges: E = 9.

(b) V = 6, E = 9, V −E+F = 2 =⇒ F = 2−V +E =⇒ F = 2−6+9 = 5.F = 5.

14. (a) Impossible, as a graph cannot contain 3 odd vertices (no graph can containan odd number of odd vertices, since every edge counts toward the degree oftwo vertices).

(b) Possible

(c) Recall that a simple graph has no loops or multiple edges. Here it ispossible.

15. Start with a degree 2 vertex at the top, followed by a degree 3 vertex on theleft and another degree 2 vertex on the right...

Chapter 4

Number Theory and Infinity

4.1 Pigeonhole Principle and Divisors

1. Are there two nonbald people in the world with exactly the same number ofhairs on their heads?

Pigeonhole Principle: If you put m pigeons into n pigeonholes, and n < m,then some pigeonhole contains at least 2 pigeons.

139

140 CHAPTER 4. NUMBER THEORY AND INFINITY

Strong Pigeonhole Principle: If you put m pigeons into n pigeonholes, andn < m, then some pigeonhole contains at least m/n (rounded up) pigeons.

2. Suppose you have 100 pennies and 9 piggybanks. If you put the pennies intothe piggybanks, explain why at least one of the piggybanks will get at least12 pennies.

Definition: If n and d are integers with d 6= 0, we say that d divides n ifthere is an integer q such that n = dq. q is called the quotient and d is calleda divisor or factor of n. If d divides n, we write d | n. Otherwise, we writed - n.

3. a. Does 6 | 12?

b. Does 12 | 6?

c. Does 4 | 9?

d. Does 3 | 51? (Careful!)

4.1. PIGEONHOLE PRINCIPLE AND DIVISORS 141

Theorem: Let m, n, and d be integers where d 6= 0.

• If d | m and d | n, then d | (m+ n).

• If d | m and d | n, then d | (m− n).

• If d | m, then d | mn.

Proof of first part:

4. Consider the following division problems:

a. 3)8 b. 4

)-11 c. 8

)-17

Division Algorithm: Let n be any integer and let d be any positive integer.Then there is exactly one pair of integers q and r such that n = dq + r where0 ≤ r < d.

q is called the quotient and r is called the remainder.


Homework

1. A Math 105 student is paid on every other Friday. Explain why, in each year,there is some month when the student is paid more than two times.

2. Put the following numbers in order from smallest to largest:

• Number of cell phones in the United States

• Number of bricks in the buildings on Concordia’s campus

• Number of grains of sand on the Earth

• Number of desks in Jones 340

• Number of people in the world

3. Suppose you have 16 identical socks except that 8 are blue and 8 are black,and that these socks are mixed up in a pile on your floor. If it’s early in themorning and you don’t want to turn on the lights, how many socks do youneed to take in order to guarantee that you have two socks with the samecolor? How many socks do you need to take to guarantee that you have twoblue socks?

4. Some rich guy offers you $50 million dollars in 5-dollar bills. In order to receivethe money, though, you have to lie down on the floor and have all of the 5-dollar bills stacked on top of your stomach. If you can keep the bills on yourstomach for 20 minutes, then you get to keep the money. Do you want toaccept this offer? Explain your answer carefully, using estimation like we didin class.

5. A positive integer is called a perfect number if it is the sum of all of its divisorsother than itself. For example, the divisors of 6 other than 6 itself are 1, 2,and 3, and 1 + 2 + 3 = 6. Find the next biggest positive integer that is aperfect number.

6. For each of the following pairs of integers d and n, determine if d | n or d - n:

(a) d = 12, n = 54

(b) d = 7, n = 100

(c) d = 3, n = 153

7. Explain why d | 0 no matter what integer, other than 0, we pick for d.


4.2 Modular Arithmetic

1. a. An earthquake occurs at 8 am. A survivor is found 30 hours later. Whattime of day was the survivor found?

b. Iron Chef has a special stew that requires 52 hours to brew. If he startsbrewing the stew at 1 am, when will the stew be finished?

Modular Arithmetic (Clock Arithmetic):

Given a positive integer m as the modulus, the integers a and b are equivalentmodulo m if a− b is a multiple of m. This is the same as a and b having thesame remainder when we divide by m.

We write a ≡ b (mod m).

Examples:

• 30 ≡ 6 (mod 12)

• 51 ≡ 3 (mod 12)

• 22 ≡ 62 (mod 10)

• 3 6≡ 17 (mod 8)

Principal Representatives: Given an integer a and a modulus m, the prin-cipal representative of a is the remainder when we divide a by m.

4.2. MODULAR ARITHMETIC 145

2. Find the principal representatives:

• 17 (mod 12)

• −5 (mod 8)

• −20 (mod 6)

3. Complete the following addition and multiplication tables for principal repre-sentatives modulo 6:

+ 0 1 2 3 4 50 212 534 25 1

× 0 1 2 3 4 50 012 43 345 2

4. Solve the following equations modulo 6:

• x+ 5 ≡ 4 (mod 6)

• 5x ≡ 3 (mod 6)

• 2x+ 3 ≡ 5 (mod 6)

• 4x+ 1 ≡ 4 (mod 6)


5. a. If May 1 is a Sunday, what day of the week is June 1?

b. January 1, 2012 was on a Sunday. What day of the week is January 1,2013?

International Standard Book Number (ISBN):

As of 2007, every ISBN is 13 digits long. The 13th digit is a check digit thatis based on the previous 12 digits. To find the check digit, if the first 12 digitsare d1, d2, d3, . . . , d12, then the check digit is the principal representative of

[10− (d1 + 3d2 + d3 + 3d4 + d5 + 3d6 + d7 + 3d8 + d9 + 3d10 + d11 + 3d12)] (mod 10).

Example: 978-0-13-159318-3

6. What are some shortcuts for finding principal representatives modulo 2? Mod-ulo 10? Modulo 9?

4.2. MODULAR ARITHMETIC 147

Homework

1. Make the addition and multiplication tables for principal representatives mod-ulo 7. (There are seven principal representatives: 0, 1, 2, 3, 4, 5, 6.)


(a) x+ 6 ≡ 2 (mod 7)

(b) 4x ≡ 1 (mod 7)

(c) 3x+ 2 ≡ 5 (mod 7)

(d) 4x+ 6 ≡ 3 (mod 7)

3. (a) What day of the week will it be 93 days after Wednesday?

(b) What day of the week will it be 193 days after Wednesday?

4. (a) In an ordinary 12-hour clock, what time will it show 93 hours after 4:00?

(b) What time will it show 193 hours after 4:00?

5. (a) If June 4 is a Sunday, what day of the week is July 4?

(b) If November 25 is a Thursday, what day of the week is December 25?

6. Find the principal representatives:

(a) −51 (mod 20)

(b) −8 (mod 51)

(c) 123456789 (mod 10)

(d) 76543 (mod 9)

7. An ISBN starts with these 12 digits: 978-0-06-204964. What is the checkdigit?

8. Find a book with a 13-digit ISBN (from 2007 or later). Verify that its checkdigit is correct.


4.3 Prime Numbers

Definition: An integer n > 1 is a prime number if its only positive divisors are 1and n itself. If n > 1 and n is not prime, we say that n is composite.

The number 1 is neither prime nor composite.

Examples:

Fundamental Theorem of Arithmetic:

Every integer n > 1 is either a prime number or a product of two or more primenumbers. If n is a product of two or more primes, it can only be written as a productof prime numbers in exactly one way (not counting different ways to arrange theprime numbers).

Examples:

Why is this theorem true?

4.3. PRIME NUMBERS 149

Prime numbers become less common as the numbers get bigger. Are there infinitelymany prime numbers, or is there a biggest prime number?

Famous unsolved problems with prime numbers:

• Twin Prime Question: Two prime numbers are called twin primes if onediffers from the other by two. Some examples are 3 and 5, 5 and 7, and 41and 43. Are there infinitely many twin primes?

• Goldbach Conjecture (1742): Can every even number greater than 2 bewritten as the sum of exactly two prime numbers? (The conjecture is yes, thisis always possible.)


Homework

1. Find the prime factorizations for the following integers:

(a) 770

(b) 132

(c) 391

2. If you multiply two prime numbers, can you ever get a prime number as theresult? If you multiply two composite numbers, can you ever get a primenumber as the result? Explain your answers.

3. In class, we proved that there are infinitely many prime numbers. Are thereinfinitely many composite numbers? Why or why not?

4. Consider the following sequence of numbers: 11, 101, 1001, 10001, 100001, . . . .Are all of these number prime? If so, explain why. If not, find the smallestnumber in this sequence that is composite and write it as a product of primefactors.

5. When we proved that there are infinitely many prime numbers, we considerednumbers like

N = (p1 · p2 · p3 · · · · · pn) + 1

where p1, p2, p3, . . . , pn were prime numbers.

There were two cases in our proof, depending on whether N was prime or com-posite. Starting with p1 = 2, give an example of consecutive prime numbersp1 = 2, p2, p3, . . . , pn so that N is composite.

6. Recall that the Goldbach Conjecture states that every even number biggerthan 2 can be written as the sum of exactly two primes. Nobody knows ifthis conjecture is true for every even number bigger than 2, but we can checkit for small even numbers. Write all of the even numbers from 4 through 30(inclusive) each as a sum of exactly two prime numbers.

4.3. PRIME NUMBERS 151

7. The Sieve of Eratosthenes is an old procedure for finding prime numbers.(Eratosthenes was an ancient Greek mathematician.) We’ll use the Sieve ofEratosthenes to find all of the prime numbers up to 100. In the table below:

• Circle 2 because it is the smallest prime number. Cross out every multipleof 2.

• Circle the next open number, 3. Now cross out every multiple of 3.



• Continue this process until all numbers in the table have been circled orcrossed out.

2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

The circled numbers are the prime numbers between 2 and 100!

8. Find 7 consecutive numbers that are all composite.

9. Explain why it is impossible to find 3 consecutive numbers that are all prime.


4.4 Rational and Irrational Numbers

A Geometry Question: Consider two line segments A and B. Is it always possibleto find a line segment C such that the length of C evenly divides both the length ofA and the length of B?

Definitions: A real number is rational if it can be written as a fraction a/b wherea and b are both integers and b 6= 0. If a real number cannot be written as such afraction, then it is irrational.

Examples:

Main Theorem:

4.4. RATIONAL AND IRRATIONAL NUMBERS 153

π is irrational, but there are many numbers where we do not know if they are rationalor irrational, like 2π, ππ, π

√2.

Some Numbers:

• 2

• 3

• 57

• 512398709512309574019523871555519238709891070970982412867500045273709479753476089129238069028648293846800927659800796342762332252100125971260587123828970000351551255439000043221239816873728997834949051

One-to-one Correspondences

• Focus on 2:

• What about 5?

• Filling a large theater:

Conclusion: Two collections of objects are the same size if and only if there is aone-to-one correspondence between the objects in the collections.


Homework

1. Carefully modify the details of the proof that we gave in class that√

2 isirrational in order to show that

√3 is irrational.

2. For each of these parts, you may use the fact that√

2 is irrational. You donot need to repeat that argument (or a nearly identical argument) from class.

(a) Prove that −√

2 is irrational.

(b) Prove that√22

is irrational.

3. (a) Give an example of two irrationals whose sum is rational.

(b) Give an example of two irrationals whose product is rational.

4. Consider the following lists of the symbols ♥ and ♦:

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

Are there more ♥’s than ♦’s? Describe how you could quickly answer thequestion without actually counting, and explain the connection with the ideaof a one-to-one correspondence.

5. Consider the following lists of the symbols ♥ and ♦:

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

Are there more ♥’s than ♦’s? Describe how you could quickly answer thequestion without actually counting, and explain the connection with the ideaof a one-to-one correspondence.

6. Do there exist two nonbald people in the world having the property that thereis a one-to-one correspondence between the collection of hairs on one person’shead and the collection of hairs on the other person’s head?

4.4. RATIONAL AND IRRATIONAL NUMBERS 155

7. Every student at Cornucopia College is assigned to one and only one dormroom. Does this mean there is a one-to-one correspondence between dormrooms and students at this college? Explain your answer.

8. If every child has one and only one birth mother, does this imply that there isa one-to-one correspondence between the set of all children and the set of allbirth mothers? Explain your answer.


4.5 Infinity

Definition: The positive integers 1, 2, 3, 4, . . . are called the counting numbers.

Question: Is the collection of integers 2, 3, 4, . . . bigger than the collection of count-ing numbers? Smaller? The same size?

Definitions: The size of an infinite collection is called its cardinality. Two infinitecollections have the same cardinality if and only if there is a one-to-one correspon-dence between the objects of the two collections. If an infinite collection has thesame cardinality as the counting numbers, we say that the collection is countablyinfinite.

So an infinite collection is countably infinite if and only if we can find a one-to-onecorrespondence between the collection and the counting numbers.

Examples:

1. Is 2, 3, 4, 5, 6, . . . countably infinite?

2. Is the entire collection of integers . . . ,−3,−2,−1, 0, 1, 2, 3, . . . countably infi-nite?

4.5. INFINITY 157

3. Is the entire collection of rational numbers (all of the fractions) countablyinfinite?

......

......

...1/5 2/5 3/5 4/5 5/5 · · ·

1/4 2/4 3/4 4/4 5/4 · · ·

1/3 2/3 3/3 4/3 5/3 · · ·

1/2 2/2 3/2 4/2 5/2 · · ·

1/1 2/1 3/1 4/1 5/1 · · ·

0

· · · -5/1 -4/1 -3/1 -2/1 -1/1

· · · -5/2 -4/2 -3/2 -2/2 -1/2

· · · -5/3 -4/3 -3/3 -2/3 -1/3

· · · -5/4 -4/4 -3/4 -2/4 -1/4

· · · -5/5 -4/5 -3/5 -2/5 -1/5...

......

......


Georg Cantor (1872): The collection of all of the real numbers is not countablyinfinite!

Suppose there was some one-to-one correspondence between the counting numbersand the real numbers, like:

Counting Numbers Real Numbers1 51.11348709872345234509. . .2 -7.22345237094237773475. . .3 0.345515151687909889709. . .4 743.9525001930034999439. . .5 -48.5858585858585858585. . .6 1.123456789012345678901. . .7 50.50515253545556575859. . .8 51.00000000000000000000. . ....

...

Cantor’s Theorem (1891): There are infinitely many different infinite cardinali-ties!

Continuum Hypothesis: There is no cardinality between the cardinality of thecounting numbers and the cardinality of the real numbers. (The real numbers aresometimes called the continuum.)

Is the Continuum Hypothesis true or false?

4.5. INFINITY 159

Homework

1. Show that the collection of positive even numbers

2, 4, 6, 8, 10, 12, 14, 16, . . .

is countably infinite by describing a one-to-one correspondence between thesenumbers and the counting numbers.

2. Show that the collection of positive square numbers

1, 4, 9, 16, 25, 36, 49, 64, . . .

is countably infinite by describing a one-to-one correspondence between thesenumbers and the counting numbers.

3. Consider the collection of positive numbers that end in 51:

51, 151, 251, 351, 451, 551, 651, 751, 851, . . . .

Show that this collection is countably infinite by describing a one-to-one cor-respondence between these numbers and the counting numbers.

4. Suppose that the city grid goes on forever — so we have a north-south streetfor every integer and an east-west street for every integer.

Consider the collection of street corners in this infinite grid. Is this collectioncountably infinite? If so, describe a one-to-one correspondence between thiscollection and the counting numbers. If not, explain why not.

5. Show that the cardinality of the positive real numbers is the same as thecardinality of the negative real numbers.

(Warning: You will need to describe a one-to-one correspondence betweenthese two collections of numbers, but since the real numbers are not countablyinfinite, you will not be able to list the numbers in either collection.)


6. Consider an infinite path graph (the path is infinite in one direction):

Consider all of the different ways to color the vertices of this path using eitherBlack or White colors. We don’t need to follow the usual rules of graph coloring— adjacent vertices could be assigned the same color.

Show that the cardinality of the collection of all such colorings is greaterthan that of the counting numbers — in particular, the collection of all suchcolorings is not countably infinite.

7. Consider a (huge!) hotel with an infinite number of rooms, one for each count-ing number:

Room 1, Room 2, Room 3, Room 4, . . . .

Every room can hold exactly one person. Suppose that, one night, every roomis completely full. If you then arrive at the hotel and ask for a room, is itpossible for the hotel to give you a private room (not sharing with anyone else)without having anyone leave the hotel? If so, explain how this accommodationcould be made (it’s OK to have people change rooms). If not, explain why itis impossible.

4.6. CHAPTER PROJECTS 161


1. Discuss the Fibonacci numbers and some of their properties. Who was Fi-bonacci, and why was he investigating these numbers? Describe how theFibonacci numbers can be used to find the Golden Ratio. What is the sig-nificance of the Golden Ratio? Give some of the contexts where we can findFibonacci numbers and the Golden Ratio — for example, in nature or archi-tecture.

2. Describe cryptology (the making and breaking of codes). Give at least oneexample of a simple substitution code (using what’s called a private key).Discuss the idea behind a public key system. The RSA system is the mostwell-known of these systems — give some historical background and a smallexample of the system so that we can see how it works. Feel free to includeany additional historical background for cryptology.


4.7 Chapter Review

Concepts:

• Pigeonhole Principle

• Strong Pigeonhole Principle

• Estimation of large numbers

• Divides, divisor (Notation: d | n)

• Division Algorithm with quotients and remainders

• Modular Arithmetic (including + and × tables)

• Principal Representatives

• Prime and Composite numbers

• Fundamental Theorem of Arithmetic (factorizations into primes)

• Rational and Irrational numbers

•√

2 is irrational (but you don’t need to know the proof)

• One-to-one correspondences

• Counting numbers

• Cardinality

• Countably Infinite


Review Exercises:

1. Recall that, in a standard deck of 52 cards, the cards are divided into 4 suits,each with 13 ranks.

(a) What is the smallest number of cards we would need to take from thedeck to guarantee that we will have at least one pair (2 with the samerank)?

(b) What is the smallest number of cards we would need to take from thedeck to guarantee that we will have at least 5 cards with the same suit?

2. Let m, n, and d be integers where d 6= 0. If d | m and d | n, prove thatd | (m− n).

3. Estimate how many hockey pucks would fit in this classroom. Explain yourreasoning.

4. What is the remainder when we divide 67 by 9? When we divide −51 by 7?When we divide −12 by 5?

5. Make the addition and multiplication tables for principal representatives mod-ulo 5. (There are five principal representatives: 0, 1, 2, 3, 4.)


(a) x+ 4 ≡ 3 (mod 5)

(b) 3x ≡ 1 (mod 5)

(c) 2x− 6 ≡ 4 (mod 5)

7. What day of the week will it be 212 days after Thursday?

8. If p is a prime number bigger than 3, can p+ 1 ever be a prime number? Whyor why not?

9. Is 1309 a prime number? If not, give its factorization into prime numbers.

10. If x is an irrational number, must 1/x be irrational too? Explain your answer.(Notice that x couldn’t be zero, so I didn’t divide by zero. Wow, that wasclose!)

11. Show that the collection of integers

. . . ,−3,−2,−1, 0, 1, 2, 3, . . .

is countably infinite by describing a one-to-one correspondence between theintegers and the counting numbers.


Review Answers:

1. a. 14 b. 17

2. Since d | m, m = dk1 for some integer k1. Since d | n, n = dk2 for some integerk2. Therefore m − n = dk1 − dk2 = d(k1 − k2). Since k1 − k2 is an integer,d | (m− n).

3. A hockey puck is roughly 1 inch thick and 3 inches in diameter, which we’llestimate as roughly a 2.5 inch by 2.5 inch square. Thus the volume of ahockey puck is roughly 2.5 · 2.5 · 1 = 6.25 cubic inches. Now our classroom isroughly 25 feet by 20 feet by 15 feet, which is 300 inches by 240 inches by 144inches. So the volume of our classroom is roughly 300 · 240 · 144 = 10, 368, 000cubic inches. So we would expect to fit roughly 10, 368, 000/6.25 = 1, 658, 880hockey pucks into our classroom.

4. 4, 5, 3

5.

+ 0 1 2 3 40 0 1 2 3 41 1 2 3 4 02 2 3 4 0 13 3 4 0 1 24 4 0 1 2 3

× 0 1 2 3 40 0 0 0 0 01 0 1 2 3 42 0 2 4 1 33 0 3 1 4 24 0 4 3 2 1

6. a. x = 4 b. x = 2 c. x = 0

7. Saturday

8. No, because p must be odd, so p+ 1 would be even and bigger than 2, whichmeans p+ 1 has 2 as a prime factor and hence is composite.

9. No, 1309 = 7 · 11 · 17

10. Yes. If x is irrational but 1/x was rational, then we would have 1/x = a/b asa fraction. But then x = b/a would be a fraction, and this is impossible sincex is irrational. So 1/x must be irrational if x is irrational.

11.1 2 3 4 5 6 7 8 9 10 · · ·l l l l l l l l l l0 1 -1 2 -2 3 -3 4 -4 5 · · ·

Chapter 5

Consumer Mathematics

5.1 Percentages and Simple Interest

Percentages:

Definition: If you lend someone a sum of money, the amount you lend them iscalled the principal or present value. You then could charge the person an additionalamount, called interest, and the total of the two amounts, which is what you collectfrom the person after some time, is called the future value.

P = principal or present value

I = interest

F = future value

F = P + I

165

166 CHAPTER 5. CONSUMER MATHEMATICS

Definition: To determine interest using simple interest, we charge a fixed percent-age of P , called the interest rate, and multiple this amount by the length of time ofthe loan.

r = annual interest rate

t = time in years

I = P · r · t

Simple Interest Formula:

F = P · (1 + rt)

Examples:

5.1. PERCENTAGES AND SIMPLE INTEREST 167

1. If a bank offers a certificate of deposit (CD) with 2% annual simple interestand you want $5100 in the CD after 3 years, how much should you deposittoday?


2. If a street vendor promises to turn your $600 now into $1080 after 4 years,and he is offering simple annual interest, what interest rate is he promising?

3. Suppose we can earn 8% simple annual interest in a savings account. If wedeposit $2000 today, how much time will it take until we have $2680 in theaccount?

5.1. PERCENTAGES AND SIMPLE INTEREST 169

Homework

1. Convert the following percentages to decimal form:

(a) 73%

(b) 0.5%

(c) 2.25%

2. Convert the following decimals to percents:

(a) 0.03

(b) 0.0015

(c) 1.0008

3. In 2012, 6.2% of taxable earnings (up to $110,100) is deducted from workers’paychecks for Social Security. If a worker earns $53,000 in 2012, how muchmoney will the worker pay for the Social Security tax?

4. Given the principal P , the annual simple interest rate r, and the time t, findthe amount F that must be repaid.

(a) P = $15,000, r = 6%, t = 5 years

(b) P = $5,300, r = 2%, t = 3 months

(c) P = $9,000, r = 4.5%, t = 50 days

5. Of the four values F , P , r, and t in the formula for simple interest, three aregiven to you. Use the simple interest formula to find the fourth one.

(a) F = $12,000, r = 3%, t = 3 years

(b) F = $8,500, t = 6 months, P = $8,200

(c) F = $4,250, r = 7%, P = $3,500

6. You borrow $2,000 from Havelock Bank to pay for sidewalk repairs. Youpromise to repay the loan in three years at 5% simple interest. How much willyou pay the bank then?

7. Rafe knows he will inherit $10,000 from his dying aunt within five months.The bank will lend him money at 4% simple interest. What is the largestamount of money he could borrow now, if he plans to use his inheritance torepay it in five months?


8. Andy’s friend Falco wants to borrow $220 from him for three months. If Andywants to earn $20 in interest on the loan, what percent simple interest shouldhe charge?

9. Ryan has borrowed $800 from his friend Kelly, who is charging him 2% simpleinterest. Eventually Ryan repaid the loan, but it cost him $1000 to do so. Forhow long did Ryan borrow Kelly’s money?

10. A coat is marked down 10%. During a special doorbuster sale, the customeris given an additional 15% off. Is this the same as receiving a 25% markdown?If not, which is a better deal for the customer?

5.2. COMPOUND INTEREST 171

5.2 Compound Interest

Examples:


i = interest rate per time period

m = number of time periods

Compound Interest Formula:

F = P · (1 + i)m

In particular, if r is an annual interest rate and we compound n times a year for tyears, then

i =r

nm = nt

So we get

F = P ·(

1 +r

n

)nt1. Suppose we deposit $2000 into an account that earns 8% annual interest. If

the interest is compounded quarterly, how much will be in the account after10 years?


2. If an account earns 6% annual interest, compounded monthly, and we want$10,000 in the account after 5 years, how much do we need to deposit today?

3. We are going to experiment with the compound interest formula by changingthe value of n, which is how many times we compound during the year. Tohelp us focus, let’s pick really simple values for the other variables.

Suppose we invest $1 in an account that pays 100% annual interest for 1 year.

(a) Write the values of P , r, and t.

(b) What is the value of F if n = 1 (we compound annually)?

(c) What is the value of F if n = 2 (we compound semiannually)?


(d) What is the value of F if n = 4 (we compound quarterly)?

(e) What is the value of F if n = 12 (we compound monthly)?

(f) What is the value of F if n = 365 (we compound daily)?

(g) Now pick a large value of n (bigger than 365, but small enough for yourcalculator to handle), and find the value of F for your n.

(h) What will happen to F as n gets really big? (To try to see the pattern,don’t round your answers to the nearest cent.)


Compounding Continuously Formula:

F = Pert where e ≈ 2.718281828

4. If we invest $51 dollars for 17 years in an account that pays 3% annual interestcompounded continuously, how much will we have after 17 years?

5. If an account earns 6% annual interest, compounded continuously, and wewant $10,000 in the account after 5 years, how much do we need to deposittoday?


Homework

1. Given the principal P , the annual interest rate r, the time t, and the frequencyof compounding, find the future value F .

(a) P = $15,000, r = 6.5% compounded quarterly, t = 2 years

(b) P = $100,000, r = 3% compounded monthly, t = 30 years

(c) P = $1,500, r = 4% compounded daily, t = 60 days

(d) P = $6,000, r = 5% compounded monthly, t = 6 months

2. When Phyllis was born, her parents deposited $4,000 into a bank accountearning 5% annual interest, compounded monthly. When she reached age 18,how much was in her account?

3. Scranton Bank is lending $10,000 to Creed Bratton for three years. The bankcompounds interest quarterly. If the bank needs to receive $2,300 in interestfrom Creed to cover its expenses, what annual interest rate should it charge?

4. Angela deposited some money in a bank account earning 3% annual interest,compounded daily. Twelve years later, there was $1,720 in the account. Howmuch money did Angela originally deposit?

5. How much money must Meredith deposit today in order to have $50,000 intwenty years, if her account earns

(a) 8% annual interest, compounded annually?

(b) 8% annual interest, compounded quarterly?

(c) 8% annual interest, compounded monthly?

(d) 8% annual interest, compounded daily?

(e) 8% annual interest, compounded continuously?

6. Suppose Oscar has $5,000 to invest. For each pair of investments, determinewhich will yield the greater return after 3 years:

(a) Investment 1: 6% annual interest, compounded monthly

Investment 2: 5.75% annual interest, compounded continuously

(b) Investment 1: 8% annual interest, compounded quarterly

Investment 2: 7.95% annual interest, compounded continuously


7. Suppose you deposit $10,200 into an account today that pays 10% annualinterest.

(a) If this interest is simple interest, how much will be in your account after40 years?

(b) If this interest is compounded annually, how much will be in your accountafter 40 years?

(c) What is the difference in the amounts from parts (a) and (b)? Why isthere such a significant difference?


5.3 Effective Annual Yield

Example:

Definition: The effective annual yield of a compound interest account at a givennamed rate, called the nominal rate, is the simple interest rate that gives the samefuture value as the nominal interest rate would give after compounding for one year.

r = nominal annual interest rate

n = number of times we compound each year

Y = effective annual yield

Y =(

1 +r

n

)n− 1

5.3. EFFECTIVE ANNUAL YIELD 179

Formula Explanation:

1. Find the effective annual yield for each of the following 2 certificates of deposit(CDs):

(a) CD 1: 3.06% annual interest compounded monthly

(b) CD 2: 3.15% annual interest compounded quarterly


Homework

1. What is the maximum amount you can borrow today if it must be repaid in6 months with an annual simple interest rate of 5% and you know that, 6months from now, you will be able to repay at most $1,500?

2. Find the effective annual yields for the given nominal rates and compoundingfrequencies:

(a) 4.5% annual interest, compounded quarterly

(b) 3.75% annual interest, compounded monthly

(c) 5.1% annual interest, compounded daily

3. Bank 1 offers a nominal annual rate of 4%, compounded daily. Bank 2 pro-duces the same effective annual yield as Bank 1 but only compounds interestquarterly. What nominal annual rate does Bank 2 offer?

4. A 30-year-old worker inherits $250,000. If the worker deposits this amountinto an account that earns 5.5% annual interest, compounded quarterly, howmuch money will be in the account when the worker turns 65 (which is whenthe worker plans to retire)?

5. State Bank offers a savings account that pays 4% annual interest, compoundedquarterly. You have three options:

• Option A: Deposit $1000 on March 17, 2012 and deposit another $1000on March 17, 2013.

• Option B: Deposit $2000 on March 17, 2012.

• Option C: Deposit $1950 on March 17, 2012.

(a) Without doing any calculations, determine whether Option A or OptionB will give a higher account balance on March 17, 2014. Explain youranswer.

(b) Determine whether Option A or Option C will give a higher accountbalance on March 17, 2014. (Feel free to do some calculations here.)

5.3. EFFECTIVE ANNUAL YIELD 181

6. State Bank offers two different certificates of deposit (CDs):

• CD 1: This is a 5-year CD that offers 3% annual interest, compoundedsemiannually (twice each year).

• CD 2: This is a 5-year CD that offers 3.5% annual interest, compoundedsemiannually (twice each year), but charges an initial $50 fee to earn thishigher interest rate.

So, for example, if you deposit $500 into CD 2, only $450 is allowed to earninterest for the 5 years.

(a) If you have $1500 to deposit into one of these CDs, which CD will havethe higher balance after 5 years?

(b) If you have $3000 to deposit into one of these CDS, which CD will havethe higher balance after 5 years?

7. State Bank offers two 3-year variable interest rate accounts.

• Account A: This account offers 3% annual interest the first year, 4%annual interest the second year, and 5% annual interest the third year.

• Account B: This account offers 5% annual interest the first year, 4%annual interest the second year, and 3% annual interest the third year.

(a) If the interest earned is simple annual interest, and you make a depositthat stays in the account for all three years, would you prefer one accountto the other? Why or why not?

(b) If, instead, the interest earned is compounded annually, and you make adeposit that stays in the account for all three years, would you prefer oneaccount to the other? Why or why not?

(c) If the account earns either simple interest or interest compounded annu-ally and you had to withdraw your money at the end of the second year,which account would you prefer?

8. Havelock Bank offers three special savings accounts. Account 1 offers 5.2%annual interest, compounded monthly. Account 2 offers 5.1% annual inter-est, compounded daily. Account 3 offers 5.0% annual interest, compoundedcontinuously.

Find the effective annual yield for each of the three accounts, and use theseyields to determine which account offers the best return on deposits.


9. The United States paid about 4 cents an acre for the Louisiana Purchase in1803.

(a) Suppose the value of this property grew at an annual rate of 5.5% com-pounded annually. What would an acre be worth in 2012?

(b) What would an acre be worth in 2012 if the annual rate was 6% com-pounded annually?

(c) Do these numbers seem realistic?

5.4. ORDINARY ANNUITIES 183

5.4 Ordinary Annuities

1. What is 1 + 2 + 3 + 4 + · · ·+ 99 + 100?

Definition: An ordinary annuity is a sequence of equal payments made atequal time periods, where the payments are made at the end of each timeperiod. Interest is also compounded at the end of each time period.

The term of an annuity is the total length of time from the beginning of thefirst time period to the end of the last time period.

The future value is the total amount in the annuity at the end of the term.


Example: Suppose we deposit $50 a month into an ordinary annuity thatpays 12% annual interest, compounded monthly, for a total of one year. Whatis the future value of the annuity? (In other words, how much is in the annuityafter one year?)

5.4. ORDINARY ANNUITIES 185

F = future value

PMT = payment made each period



F = PMT ·(

(1 + i)m − 1

i

)

2. Suppose we pay $250 a quarter into an ordinary annuity for 7 years, wherethe annual interest is 8%, compounded quarterly.

(a) Find the future value of the annuity.

(b) How much interest did we earn over the course of these 7 years?


Homework

1. Find the future value of each of the following ordinary annuities.

(a) Payments of $1200 made at the end of each year for 10 years, where 7%annual interest is compounded annually

(b) Payments of $300 made at the end of each quarter for 10 years, where8% annual interest is compounded quarterly

(c) Payments of $50 made at the end of each month for 20 years, where 6%annual interest is compounded monthly

(d) Payments of $100 made at the end of each week for 2 years, where 8%annual interest is compounded weekly

2. How much total interest was earned in the annuity in 1. (b)?

3. An uncle said he would set up an ordinary annuity for a newly born niece andpay $100 a month, with the last payment to occur on her 18th birthday. Thepayments would earn 6% annual interest, compounded monthly. The auntsaid they should just give the niece a lump sum of money now that wouldgrow to the same amount (at 6% annual interest, compounded monthly) asthe annuity would by the 18th birthday. If they go with the aunt’s plan, howmuch should they give to the niece now?

4. Mr. Smith decides to pay $300 at the end of each month into an ordinaryannuity that pays 8% annual interest, compounded monthly, for five years.He decides to calculate the future value of this annuity at the end of these fiveyears, but he makes a mistake in his calculations. What was his mistake? Ishis answer too big or too small?

F = PMT ·(

(1 + i)m − 1

i

)F = 300 ·

((1 + .08)60 − 1

.08

)F = 300 · 1253.213296

F = $375963.99

5. What is 1 + 2 + 3 + · · ·+ 48 + 49?

5.5. MORTGAGES 187

5.5 Mortgages

Definition: A mortgage is a loan for a house.

Qualifying for a mortgage:

In order to determine if a buyer qualifies for a mortgage, we compute the buyer’sadjusted monthly income. The adjusted monthly income is the gross monthly in-come minus any unchanging monthly payments that still have more than 10 monthsremaining.

The maximum that a person can spend on housing expenses each month is 28% oftheir adjusted monthly income.

Example:


Example:

Principal and Interest for Mortgages:

The following table records the Monthly Principal and Interest Payment per $1000of Mortgage.

Rate % 10-year 20-year 30-year5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34

Table 5.1: Monthly principal and interest per $1000 of mortgage.

5.5. MORTGAGES 189

Definition: The amount that a buyer pays at closing, called the closing costs, isthe sum of the down payment plus any points the lender charges. A point is 1% ofthe amount borrowed.

Followup:


Homework

1. Calculate what a 20% down payment would be for the following house costs:

(a) $249,900

(b) $119,900

(c) $154,500

(d) $255,000

2. The Zhangs have a gross monthly income of $5,900. They have 13 payments of$160 a month remaining on a student loan and 20 payments of $310 a monthon a car loan.

(a) What is 28% of their adjusted monthly income?

(b) The Zhangs want a 20-year fixed-rate mortgage. They wish to buy ahouse at a price of $189,900. If they make a 25% down payment, thenthey can find a mortgage with an interest rate of 5%. Use Table 5.1 tocalculate their monthly principal and interest payment for the remaining75% of the house price.

(c) If insurance and taxes sum to $240 each month, calculate the Zhangs’total monthly mortgage payment.

(d) Do the Zhangs earn enough money to qualify for this mortgage?

3. Stanley wants to purchase a new home with a price of $299,900. His bankrequires a 20% down payment and a payment of 3 points on the mortgageamount to earn a lower interest rate.

(a) What is the amount of Stanley’s down payment?

(b) What is his mortgage amount?

(c) What will the cost of the 3 points on the mortgage amount be?

5.5. MORTGAGES 191

4. Dwight Schrute, Assistant (to the) Regional Manager at the Scranton branchof Dunder Mifflin, has a gross monthly income of $3,150. He has 11 remainingmonthly payments of $200 for supplies for his next beet crop and 48 remainingmonthly payments of $5 on a loan for his impressive collection of mustard-colored dress shirts.

Dwight is (heretically!) considering abandoning Mose at Schrute Farms andbuying a house that is selling for $129,500. The insurance and taxes on theproperty are $125 and $145 per month, respectively. Dwight’s bank requires a20% down payment, payable to the seller, and a payment of 2 points, payableto the bank, at closing. The bank will approve a loan with a total monthlymortgage payment of principal, interest, property taxes and insurance that isless than 28% of Dwight’s adjusted monthly income.

(a) What is Dwight’s down payment?

(b) What is the mortgage amount?

(c) Determine the closing costs (down payment and points).

(d) What is 28% of Dwight’s adjusted monthly income?

(e) If Dwight wants a 30 year mortgage and the annual interest rate is 5.5%,determine the total monthly payment for the mortgage by first findingthe payment for principal and interest using Table 5.1, and then addingthe amounts for insurance and taxes.

(f) Does Dwight qualify for the mortgage?



1. Discuss how to calculate payments for loans (other than payments for homemortgages). In class, we found a formula for the future value of ordinaryannuities. Explain how to find the formula for the present value of ordinaryannuities. Apply this formula to a variety of situations, which may includelottery payments, comparing investments, retirement accounts, car payments,or comparing business expenses. You could also discuss amortization tables,and how we calculate how payments are divided between principal and interest.

2. Give an introduction to inflation and deflation. Explain the Consumer PriceIndex. Use compound interest to calculate the changing costs of items due toinflation. Compare changes in costs of certain items to actual inflation rates.

3. Present the concept of open-end credit loans, the type of loans used by creditcards. Present different methods credit card companies use to compute interestcharges and minimum monthly payments.


5.7 Chapter Review

Concepts:

This page and the next page will be provided for the test!

P = principal or present value

r = annual interest rate

t = time in years

F = future value

n = number of times we compound each year

Y = effective annual yield

PMT = payment made each period



• Simple InterestF = P · (1 + rt)

• Compound Interest

F = P ·(

1 +r

n

)nt• Compounding Continuously

F = Pert where e ≈ 2.718281828

• Effective Annual Yield

Y =(

1 +r

n

)n− 1

• Future Value of an Ordinary Annuity

F = PMT ·(

(1 + i)m − 1

i

)• Adjusted Monthly Income

The adjusted monthly income is the gross monthly income minus any un-changing monthly payments that still have more than 10 months remaining.To qualify for a mortgage, the maximum that a person can spend on housingexpenses is 28% of their adjusted monthly income.


• Closing Costs

The closing costs are the amount a buyer pays at closing on the day they buythe house. Closing costs include the down payment and any points chargedby the lender. A point is 1% of the amount borrowed.

• Principal and Interest for Mortgages

The following table records the Monthly Principal and Interest Payment per$1000 of Mortgage.

Rate % 10-year 20-year 30-year5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34


Review Exercises:

1. Liz has a gross monthly income of $3,500. She has 17 remaining monthlypayments of $220 for a car loan and 3 remaining monthly payments of $95 onher student loans.

Liz is looking at buying a house that is selling for $155,000. The insuranceand taxes on the property are $110 and $135 per month, respectively. Liz’sbank requires a 20% down payment. The bank will approve a loan with a totalmonthly mortgage payment of principal, interest, property taxes and insurancethat is less than 28% of Liz’s adjusted monthly income.

(a) What is the mortgage amount?

(b) If Liz wants a 20 year mortgage and the annual interest rate is 6%, de-termine the total monthly payment for the mortgage by first finding thepayment for principal and interest, and then adding the amounts for in-surance and taxes.

(c) Does Liz qualify for the mortgage?

2. How much money needs to be deposited into a bank account today so that itgrows to $51,000 after 12 years if:

(a) The account earns 2% annual simple interest

(b) The account earns 2% annual interest, compounded monthly

(c) The account earns 2% annual interest, compounded continuously

3. The same amount of money was invested in each of two different accounts onJanuary 1, 2008. Account A increased by 2.5% in 2008, then by 4% in 2009,and then by 1.5% in 2010. Account B increased by the exact same x% for eachof those three years. The interest is compounded annually in both accounts.At the end of 2010, both accounts had the same amount of money. What isthe value of x?

4. Locksafe Bank offers three special savings accounts. Account 1 offers 4.7%annual interest, compounded quarterly. Account 2 offers 4.6% annual interest,compounded monthly. Account 3 offers 4.5% annual interest, compoundedcontinuously.

Find the effective annual yield for each of the three accounts, and use theseyields to determine which account offers the best return on deposits.


5. Suppose we put $51 at the end of each month into an account earning 3%annual interest, compounded monthly.

(a) How much is in the account after 10 years?

(b) How much interest did this account earn during the 10 years?

6. You borrow $1290 on a credit card that charges simple interest at an annualrate of 11%. What is your interest after two months?

7. If $2500 is invested at a rate of 3.2% for 4 years, find the balance if the interestis compounded (a) annually; (b) monthly; (c) continuously.

8. Suppose you deposit $2300 into an account earning simple interest. Whatsimple interest rate is being charged if the amount at the end of 8 months is$2331?

9. Suppose you deposit $1500 into an account earning simple interest of 3.33%.How long do you have to wait until the account doubles in value?

10. First Bank is advertising a savings account with a 5% interest rate, com-pounded daily. National Bank advertises a savings account with a 5.25% in-terest rate, compounded monthly. Which bank would you deposit your moneywith? Justify your answer.

11. A mid-life worker wants to retire in 15 years. If the worker needs to have$125,000 at retirement to live comfortably, how much should be invested nowat 6% interest compounded weekly?

12. The cost of a laptop is $1199. We can finance this by paying $550 down and$35 per month for 22 months. Determine

(a) the amount financed,

(b) the total installment price,

(c) the finance charge.

13. We purchase a new refrigerator for $2180, financed over an 30-month periodwith 11.5% add-on interest. What is the monthly payment?

14. Find the annual add-on interest rate for a principal of $650 for 20 months witha monthly payment of $38.72.


15. Bella has a gross monthly income of $3,500. She has 17 remaining monthlypayments of $220 for a car loan and 3 remaining monthly payments of $95 onher student loans.

Bella is looking at buying a house that is selling for $155,000. The insuranceand taxes on the property are $110 and $135 per month, respectively. Bella’sbank requires a 20% down payment. The bank will approve a loan with a totalmonthly mortgage payment of principal, interest, property taxes and insurancethat is less than 28% of Bella’s adjusted monthly income.

(a) What is the mortgage amount?

(b) If Bella wants a 20-year mortgage at 6%, determine the total monthlypayment for the mortgage by first finding the payment for principal andinterest, and then adding the amounts for insurance and taxes. Use thetable below.

(c) What is 28% of her adjusted monthly income?

(d) Does she qualify for the loan?

Monthly Principal and Interest Payment per $1000 of MortgageRate % 10-year 20-year 30-year

5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34

16. The Pella family is considering a house priced at $144,500. The taxes on thehouse will be $3200 per year and the homeowners’ insurance will be $450 peryear. They have applied for a mortgage from their bank. The bank is requiringa 15% down payment, and the interest rate on the loan is 7.5%. Their annualgross income is $86,500. They have more than 10 monthly payments remainingon each of the following: $220 on a car, $170 on new furniture, and $210 on astudent loan. Their bank will approve the mortgage if their monthly housingexpenses (principal and interest for mortgage, insurance and taxes) are lessthan 28% of their adjusted monthly income.

(a) What is their down payment?

(b) What is the mortgage amount?

(c) What is 28% of their adjusted monthly income?


(d) If they want a 30-year mortgage, determine the monthly principal andinterest for the mortgage using the table above.

(e) What are their monthly housing expenses (principal, interest, insuranceand taxes)?

(f) Do they qualify for the mortgage?



1. (a) $124000

(b) $1132.84

(c) No

2. (a) $41129.03

(b) $40126.04

(c) $40118.02

3. 2.66%

4. Account 1: 4.783%

Account 2: 4.698%

Account 3: 4.603%

Account 1 offers the best return.

5. (a) $7126.81

(b) $1006.81

6. I = Prt =⇒ I = $1290(0.11)(

212

)= $23.65

7. (a)A = 2500(1 + .032

1

)1(4)=$2835.69 (b)A = 2500

(1 + .032

12

)12(4)=$2840.90 (c)

A = 2500e.032(4) =$2841.38

8. $2331 = $2300 (1 + r × 8/12) =⇒ r = 0.02 =⇒ 2%.

9. $3000 = $1500 (1 + .0333t) =⇒ t = 30 years.

10. At First Bank the effective annual yield is Y = (1 + 0.05/365)365−1 ≈ 0.0512.At National Bank the effective annual yield is Y = (1 + 0.0525/12)12 − 1 ≈0.0538. Since 0.0538 > 0.0512 we should pick National Bank.

11. $125, 000 = P (1 + 0.06/52)52(15) =⇒ P = $50, 847.58.

12. (a) $1199−$550 = $649 (b) $550+$35(22) = $1320 (c) $1320−$1199 = $121.

13.$2180 (1 + .115× 30/12)

30= $93.56

14. 20 × $38.72 = $774.40; $774.40 = $650 (1 + r × 20/12) =⇒ r = .1148 =⇒11.48%.


15. (a) $155, 000− 0.20× $155, 000 = $124, 000

(b) $7.16× $124, 000/$1000 + $110 + $135 = $1132.84

(c) 0.28 ($3500− $220− $95) = $891.80

(d) No.

16. (a) $144, 500× 0.15 = $21, 675

(b) $144, 500− $21, 675 = $122, 825

(c) $86, 500/12− $220− $170− $210 = $6608.33; 0.28× $6608.33 = $1850.33

(d) $6.99× $122, 825/$1000 = $858.55

(e) $858.55 + $3200/12 + $450/12 = $1162.71

(f) Since total monthly payment = $1162.71 < $1850.33 = 28% of adjustedmonthly income they qualify for the mortgage.

concordia college mathematics department spring 2012faculty.cord.edu/andersod/textspring2012.pdf ·...

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