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Contemporary Mathematics

Chapter 11

Contemporary MathematicsChapter 11
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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Your Turn

11.1
1.
Yes, Joe Biden won the majority.
11.2
1.
Tony Cambell won a plurality of votes in the Republican primary.
11.3
1.
Hearn and Lim tied. Yes, there must be a third election.
2.
Kelly must be removed in the runoff.
11.4
1.
Option B.
11.5
1.
4 + 6 + 4 + 7 = 21
2.
None
3.
4
11.6
1.
Using ranked-choice voting, Yoda is determined to be the winner.
11.7
1.
Blue received 78 points.
11.8
1.
Candidate B would be considered the winner using the ranked-choice voting method.
2.
Candidate C would be considered the winner using the Borda count method.
3.
Different candidates won. It appears that the vote counts were so close that a small shift in either direction could change the results of either method.
11.9
1.

A table shows the best director pairwise comparison matrix. The table has 6 rows and 7 columns. The row headers are opponent/runner, Scorsese (s), Phillips (P), Mendes (M), Tarantino (T), Bong (B), and points. Under opponent/runner it is labeled as Scorsese wins, Phillips wins, Mendes wins, Tarantino wins, and Bong wins. Under Scorsese (s) a symbol hyphen is labeled in a box, a box colored in blue and labeled as P S 3600, and the entire box is stricken out, a box labeled as M S 3100 is colored in yellow, a box colored in orange and labeled as B S 6600 and the entire box is stricken out. Under Phillips (P), a box colored in blue and labeled as S P 5100, a symbol hyphen is labeled in a box, a box labeled as M P 1900 is colored in blue, a box colored in yellow and labeled as T P 5100, a box colored in grey and labeled as B P 6300. Under Mendes (M), a box colored in yellow and labeled as S M 5600, a box labeled as P M 6800 is colored in blue, a symbol hyphen is labeled in a box, a box colored in blue and labeled as T M 8700, a box colored in yellow and labeled as B M 8700. Under Tarantino (T), a box colored in grey and labeled as S T 5600, a box labeled as P T 3600 is colored in yellow, a box colored in blue and labeled as M T 0, a hyphen is labeled in a box, a box colored in blue and labeled as B T 4700. Under Bong (B), a box colored in orange and labeled as S B 2100 and the entire box is stricken out, a box labeled as P B 2400 is colored in grey and the entire box is stricken out, a box colored in yellow and labeled as M B 0 and the entire box is stricken out, a box colored in blue and labeled as T B 4000 and the entire box is stricken out. Under points, it is labeled as 3, 1, 0, 2, 4.
2.
Bong Joon-ho won according to the pairwise comparison method.
3.
Yes, the winner is a Condorcet candidate.
11.10
1.
Candidate P wins. Candidate P is not a Condorcet candidate because P lost to Candidate Q.
11.11
1.
Candidate C won 25 points.
2.
The greatest number of points another candidate could win is 24 points.
3.
No, Candidate C cannot lose or tie.
11.12
1.
Dough Boys Pizza is the winner for dinner.
11.13
1.
There is a tie between Rainbow China and Dough Boys Pizza.
11.14
1.
Al Gore is the winner.
11.15
1.
Animal Kingdom
2.
Animal Kingdom
3.
Magic Kingdom
11.16
1.
Using the plurality voting method, there would be a tie between Option B and Option C.
2.
Using the ranked-choice voting method, Option B would be the winner.
3.
No, there was not a majority candidate in Round 1.
11.17
1.
Using the pairwise comparison voting method, Option B wins.
2.
Using the Borda count method, Option A and Option B tie.
3.
Yes. The majority criterion fails for Borda count.
11.18
1.
Yes, because Option A is a Condorcet candidate.
11.19
1.
Option A. Yes, the Condorcet criterion is satisfied.
2.
Option A. Yes, the Condorcet criterion is satisfied.
3.
Option B. No, the Condorcet criterion is not satisfied.
11.20
1.
Standard Poodle.
2.
Standard Poodle.
3.
This election does not violate the monotonicity criterion.
4.
Increasing the ranking for a winner of a Borda count election on a ballot will increase that candidate’s Borda score while decreasing another candidate’s Borda score, but leaving the remaining candidates’ Borda score unchanged. So, a Borda count election will never violate the monotonicity criterion.
11.21
1.
Labrador retriever wins the election.
2.
Golden retriever wins the election.
3.
This election violates the monotonicity criterion.
4.
It is possible that the monotonicity criterion would be met in other ranked-choice election scenarios, but overall, the ranked-choice voting method is said to fail the monotonicity criterion even if it failed in only one scenario.
11.22
1.
Chihuahua.
2.
Yorkshire Terrier.
3.
Yes, this election violates IIA.
11.23
1.
Jim wins.
2.
Pam wins.
3.
Yes, this is a violation of IIA.
11.24
1.
B 2 3 ; 0.67, C 2 3 ; 0.67, D 2 3 ; 0.67, and E 2 3 ; 0.67
2.
B 3 2 ; 1.5, C 3 2 ; 1.5, D 3 2 ; 1.5, and E 3 2 ; 1.5
3.
Yes, the constant ratio is 3 2 pencils per desk.
11.25
1.
42 pencils would be allocated.
2.
42 pencils would be allocated.
3.
34 desks
11.26
1.
IL 700,000; OH 700,000; MI 700,000; GA 800,000; NC 700,000
2.
IL 0.0000014; OH 0.0000014; MI 0.0000014; GA 0.0000013; NC 0.0000014
3.
IL 0.000001; OH 0.000001; MI 0.000001; GA 0.000001; NC 0.000001
4.
The ratio of State Population to Representative Seats seems to be either 700,000 or 800,000 to 1. There does appear to be a constant ratio of about 0.000001 to 1 of Representative Seats to State Population when rounding to six decimal places. This is the same as the top five states.
11.27
1.
60,449.4
11.28
1.
The states are the Hernandez family and the Higgins family. The seats are the pieces of candy. The house size is 313. The state populations are three in the Hernandez family and four in the Higgins family. The total population is 7.
2.
The standard divisor is the ratio of the number of children to the number of pieces of candy. 0.0224 children per piece of candy.
11.29
1.
0.98
11.30
1.
Family Family’s Standard Quota
Hernandez 175.6667 candies
Higgins 234.2222 candies
Ho 117.1111 candies
Total 527

The sum is very close to 527.

11.31
1.
6, 5, 6, 7, 5, 6
2.
35
3.
No.
11.32
1.
It is not possible to give a fractional part of a gift card. Also, traditional rounding to the nearest integer results in 4 gift cards for each student, which leaves one extra gift card.
11.33
1.
30, 18, 20
2.
68
3.
2
11.34
1.
The final Hamilton apportionment is Fictionville as follows: 1, Pretendstead 3, Illusionham 5, and Mythbury 26.
11.35
1.
Method V violates the quota rule because State F receives 4 seats instead of 5 or 6.
11.36
1.
53.33
2.
480
3.
96
4.
Answers may vary. With a modified divisor of 0.100, the modified quota is 120.
5.
The modified quota from part 3 was the smallest, because the divisor was the largest of the three. Dividing the same number by a larger value gives a smaller result.
11.37
1.

Each state would receive the following seats: Fictionville 1, Pretendstead 2, Illusionham 4, and Mythbury 28.

11.38
1.
Fictionville 2, Pretendstead 3, Illusionham 5, and Mythbury 25.
11.39
1.

The apportionment is Fictionville 2, Pretendstead 2, Illusionham 5, and Mythbury 26.

11.40
1.
The largest state is Mythbury. The citizens would likely favor the Jefferson method of apportionment most since they received the most seats by that method. They would likely favor the Adams and Webster methods of apportionment least because they received the least number of seats by those method.
2.
As a group the other three states received nine seats by either the Hamilton method, seven seats by the Jefferson Method, and ten seats by either the Adams method or the Webster method. They would likely favor the Adams method and Webster method most and favor the Jefferson methods least.
11.41
1.
State A loses 0, seats State B loses 0, and State C loses 3.
2.
State A loses 0 seats, State B loses 1, and State C loses 4.
3.
State C, the largest state, loses the most representatives each time the divisor is increased.
11.42
1.
The standard divisor is 214,079.1236 citizens per seat. The standard quota for Colorado is 2.5210 seats.
2.
The standard divisor is 213,479.4622 citizens per seat. The standard quota for Colorado is 2.5281 seats.
3.
The standard quota increased.
4.
It must have been the case that either the fractional part 0.5281 ranked lower amongst the other fractional parts of the state quotas than the fractional part 0.5210 did, or there were fewer remaining seats to be distributed, or both.
11.43
1.
The final apportionment is: A 183, B 124, C 14, and D 2, which sums to 323.
2.
The final apportionment is: A 184, B 125, C 13, and D 2, which sums to 324.
3.
Yes, this demonstrates the Alabama paradox because State C receives 14 seats if the house size is 323, but only 13 seats if the house size is 324.
11.44
1.
Hospital C lost a respirator while hospital A gained a seat, but hospital C has a higher growth rate than hospital A.
11.45
1.
The Hamilton reapportionment is: 19 for Mudston, 13 for WallaWalla, and 6 for Dilberta. This is an example of the population paradox because WallaWalla lost a seat to Dilberta, even though WallaWalla’s population grew by 2.24 percent while Dilberta’s only grew by 1.56 percent.
11.46
1.
90
2.
95
3.
The original state of Beaversdam lost a seat to the original state of Beruna when the new state of Chippingford was added.
11.47
1.
The reapportionment gives 38 seats to Neverwood, 46 seats to Mermaids Lagoon, and 9 seats to Marooners Rock. This is an example of the new-states paradox because the original state Mermaids Lagoon lost a seat to the original state Neverwood when the new state was added to the union.
11.48
1.

Original Standard Divisor ≈ 76 , 000 , 000 391 ≈ 194 , 400

New Population ≈ 76 , 000 , 000 + 300 , 000 = 76 , 300 , 000

New House Size ≈ 76,300,000 194 , 400 ≈ 392

There are 392 − 391 = 1 new seats to be apportioned to New Mexico.

11.49
1.
Only Hamilton’s method violates the population paradox.

Check Your Understanding

1.
Answers may vary. Example: Ranked-choice, Borda count, and pairwise comparison.
2.
True.
3.
True
4.
Borda count
5.
Pairwise comparison
6.
Hare
7.
In two-round voting, only the top two candidates from Round 1 move on to Round 2, and there are only two rounds. In ranked-choice voting, all candidates except those in last place move on to the next round, and there can be many rounds of voting.
8.
(Independence of) Irrelevant Alternatives Criterion
9.
Pairwise comparison
10.
Borda count method
11.
Pairwise comparison
12.
All of them: Plurality, ranked-choice, pairwise comparison, and Borda count
13.
True, because a majority candidate is always the Condorcet candidate.
14.
False, because the ranked-choice method violates the Condorcet criterion, but it doesn’t violate the majority criterion.
15.
No. Arrow’s Impossibility Theorem does not apply to approval voting because it is not a ranked voting system
16.
False. Arrow’s Impossibility Theorem says that no ranked voting system is perfect and that voter profiles may arise that will lead to a violation of one or more fairness criteria, but it does not guarantee that those voter profiles will occur or are even likely to occur.
17.
True. Candidates are rated as approved or not approved, and voters can give multiple candidates the same rating.
18.
Student 1 is correct.
19.
Both are correct.
20.
Neither are correct.
21.
Neither are correct.
22.
Student 2 is correct.
23.
Student 2 is correct.
24.
Both are correct.
25.

The Hamilton method.

26.

The Hamilton method.

27.

The Jefferson method.

28.

The Adams method.

29.

The Adams method.

30.

The Hamilton and Jefferson methods.

31.

Webster’s method.

32.

Larger

33.
The Alabama paradox
34.
Population paradox
35.
Quota rule violation
36.
None of these
37.
New-states paradox
38.
The population paradox could occur.
39.
The Alabama paradox could occur.
40.
The new-state paradox could occur.
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