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Contemporary Mathematics

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Contemporary MathematicsChapter Test
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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

Chapter Test

For the following exercises, use the table below.
Number of Ballots 12 17 15 13
Option A 1 2 4 3
Option B 3 1 3 2
Option C 4 3 1 4
Option D 2 4 2 1
1.
Determine the winner of the election by plurality.
2.
Determine the Borda scores for each candidate to determine the winner by Borda count method.
3.
Create and analyze a pairwise comparison matrix based on the preference summary to determine the winner of the election by pairwise comparison.
4.
From the table below use ranked-choice voting to determine the winner of the election.
Number of Ballots 28 5 30 5 16 16
Option L 3 2 1 1 2 3
Option R 1 1 3 2 3 2
Option E 2 3 2 3 1 1
For the following exercises, identify which fairness criteria, if any, are violated by characteristics of the described voter profile. Explain your reasoning
5.
In a Borda count election, the candidates have the following Borda scores: A 1345, B 1260, C 685. Candidate B received 51% of the first-place rankings.
6.
In a plurality election, the candidates have the following percentages of first place votes: A 25, B 21, C 30, D 24. The pairwise matchup points for the same voter profiles would have been A 3, B 0, C 2, D 2.
For the following exercises, use the table below.
Number of Ballots 13 14 11 12
Option A 2 1 3 3
Option B 3 2 4 1
Option C 4 4 1 2
Option D 1 3 3 4
7.
Determine the winner by ranked-choice voting if two of the voters in the second column up-rank the original winner. Refer to Question 4. Which fairness criterion, if any, is violated?
8.
Determine the winner by ranked-choice voting if candidate R is removed from the election. Refer to Question 4. Which fairness criterion, if any, is violated?
For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations with representation proportionate to the number of residents.
9.
Identify the states, the seats, and the state population (the basis for apportionment) in the given scenario.
10.
Determine the standard divisor for the apportionment
11.
Determine each state's standard quota rounded to two decimal places.
For the following exercises, use this information: Air Force administration wanted to distribute 27 aircraft across 6 bases based on the number of qualified pilots stationed at those bases. The standard quota is 2.2963. The standard quotas for each base are listed in the table below.
Air Force Base (A) Alpha (B) Bravo (C) Charlie (D) Delta (E) Echo (F) Foxtrot
Pilots 13 12 5 16 7 9
Standard Quota 5.66 5.23 2.18 6.97 3.05 3.92
12.
Determine the states' lower quotas and the states' upper quotas.
13.
Use Adams's method to apportion the aircraft.
14.
Use Jefferson's method to apportion the aircraft.
15.
The apportionment of 616 schools to 5 Hawaiian counties by various methods is displayed in the table below.
County Hawaii Honolulu Kalawao Kauai Maui
Lower Quota 87 424 0 31 72
Upper Quota 88 425 1 32 73
Jefferson 87 425 1 31 72
Adams 88 422 1 32 73
Webster 87 424 1 31 73

Apportionment by which methods, if any, fail to satisfy the quota rule? Explain your reasoning.
For the following exercises, use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations. The Hamilton method was used to apportion the council seats. The Oaks has 10 seats on the council, while The Villages has 90 seats. The council votes to annex an unincorporated subdivision called The Lakes with a population of 630. They plan to increase the size of the council to maintain the ratio of seats to residents such that the new council will have 100 seats plus the number of seats given to The Lakes.
16.
What is the standard divisor from the original apportionment?
17.
What is the new house size?
18.
Use the Hamilton method to reapportion the seats.
19.
Is the reapportionment an example of the new-states paradox? If so, how?
For the following exercises, use this information: determine whether the reapportionment violates the Alabama paradox, the population paradox, or neither. Justify your answer.
20.
States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the population remains the same, but house size is increased, the reapportionment is A 20, B 26, C 27, and D 29.
21.
States A, B, C, and D received 21, 25, 26, and 28 seats respectively. When the house size remains the same, the population of state A increased, the population of state B decreased, and the populations of states C and D remained the same, the reapportionment is A 20, B 26, C 26, and D 28.
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