Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Contemporary Mathematics

6.4 Compound Interest

Contemporary Mathematics6.4 Compound Interest

Menu
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
Coins are placed in stacks ordered from smallest to largest.
Figure 6.6 The impact of compound interest (credit: "English Money" by Images Money/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Compute compound interest.
  2. Determine the difference in interest between simple and compound calculations.
  3. Understand and compute future value.
  4. Compute present value.
  5. Compute and interpret effective annual yield.

For a very long time in certain parts of the world, interest was not charged due to religious dictates. Once this restriction was relaxed, loans that earned interest became possible. Initially, such loans had short terms, so only simple interest was applied to the loan. However, when loans began to stretch out for years, it was natural to add the interest at the end of each year, and add the interest to the principal of the loan. After another year, the interest was calculated on the initial principal plus the interest from year 1, or, the interest earned interest. Each year, more interest was added to the money owed, and that interest continued to earn interest.

Since the amount in the account grows each year, more money earns interest, increasing the account faster. This growth follows a geometric series (Geometric Sequences). It is this feature that gives compound interest its power. This module covers the mathematics of compound interest.

Understand and Compute Compound Interest

As we saw in Simple Interest, an account that pays simple interest only pays based on the original principal and the term of the loan. Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added previously.

After each period, the interest on the account is computed, then added to the account. Then, after the next period, when interest is computed, it is computed based on the original principal AND the interest that was added in the previous periods.

The following example illustrates how compounded interest works.

Example 6.39

Interest Compounded Annually

Abena invests $1,000 in a CD (certificate of deposit) earning 4% compounded annually. How much will Abena’s CD be worth after 3 years?

Your Turn 6.39

1.
Oksana deposits $5,000 in a CD that earns 3% compounded annually. How much is the CD worth after 4 years?

Determine the Difference in Interest Between Simple and Compound Calculations

It is natural to ask, does compound interest make much of a difference? To find out, we revisit Abena’s CD.

Example 6.40

Comparing Simple to Compound Interest on a 3-Year CD

Abena invested $1,000 in a CD that earned 4% compounded annually, and the CD was worth $1,124.86 after 3 years. Had Abena invested in a CD with simple interest, how much would the CD have been worth after 3 years? How much more did Abena earn using compound interest?

Your Turn 6.40

1.
Oksana deposits $5,000 in a CD that earned 3% compounded annually and was worth $5,627.54 after 4 years. Had Oksana invested in a CD with simple interest, how much would the CD have been worth after 4 years? How much more did Oksana earn using compound interest?

Understand and Compute Future Value

Imagine investing for 30 years and compounding the interest every month. Using the method above, there would be 360 periods to calculate interest for. This is not a reasonable approach. Fortunately, there is a formula for finding the future value of an investment that earns compound interest.

FORMULA

The future value of an investment, AA, when the principal PP is invested at an annual interest rate of rr (in decimal form), compounded nn times per year, for tt years, is found using the formula A=P(1+rn)ntA=P(1+rn)nt. This is also referred to as the future value of the investment.

Checkpoint

Note, sometimes the formula is presented with the total number of periods, nn, and the interest rate per period, rr. In that case the formula becomes A=P(1+r)nA=P(1+r)n.

Example 6.41

Computing Future Value for Compound Interest

In the following, compute the future value of the investment with the given conditions.

  1. Principal is $5,000, annual interest rate is 3.8%, compounded monthly, for 5 years.
  2. Principal is $18,500, annual interest rate is 6.25%, compounded quarterly, for 17 years.

Your Turn 6.41

In the following, compute the future value of the investment with the given conditions.
1.
Principal is $7,600, annual interest rate is 4.1%, compounded monthly, for 10 years.
2.
Principal is $13,250, annual interest rate is 2.79%, compounded quarterly, for 25 years.

Example 6.42

Interest Compounded Quarterly

Cody invests $7,500 in an account that earns 4.5% interest compounded quarterly (4 times per year). Determine the value of Cody’s investment after 10 years.

Your Turn 6.42

1.
Maggie invests $3,000 in an account that earns 5.1% interest compounded monthly. How much is the account worth after 13 years?

Example 6.43

Interest Compounded Daily

Kathy invests $10,000 in an account that yields 5.6% compounded daily. How much money will be in her account after 20 years?

Your Turn 6.43

1.
Jacob invests $3,000 in a CD that yields 3.4% compounded daily for 5 years. How much is his CD worth after 5 years?

WORK IT OUT

To truly grasp how compound interest works over a long period of time, create a table comparing simple interest to compound interest, with different numbers of periods per year, for many years would be useful. In this situation, the principal is $10,000, and the annual interest rate is 6%.

  1. Create a table with five columns. Label the first column YEARS, the second column SIMPLE INTEREST, the third column COMPOUND ANNUALLY, the fourth column COMPOUND MONTHLY and the last column COMPOUND DAILY, as shown below.
    YEARS SIMPLE INTEREST COMPOUND ANNUALLY COMPOUND MONTHLY COMPOUND DAILY
  2. In the years column, enter 1, 2, 3, 5, 10, 20, and 30 for the rows.
  3. Calculate the account value for each column and each year.
  4. Compare the results from each of the values you find. How do the number of periods per year (compoundings per year) impact the account value? How does the number of years impact the account value?
  5. Redo the chart, with an interest rate you choose and a principal you choose. Are the patterns identified earlier still present?

Understand and Compute Present Value

When investing, there is often a goal to reach, such as “after 20 years, I’d like the account to be worth $100,000.” The question to be answered in this case is “How much money must be invested now to reach the goal?” As with simple interest, this is referred to as the present value.

FORMULA

The money invested in an account bearing an annual interest rate of rr (in decimal form), compounded nn times per year for tt years, is called the present value, PVPV, of the account (or of the money) and found using the formula PV=A(1+rn)n×tPV=A(1+rn)n×t, where AA is the value of the account at the investment’s end. Always round this value up to the nearest penny.

Example 6.44

Computing Present Value

Find the present value of the accounts under the following conditions.

  1. AA = $250,000, invested at 6.75 interest, compounded monthly, for 30 years.
  2. AA = $500,000, invested at 7.1% interest, compounded quarterly, for 40 years.

Your Turn 6.44

Find the present value of the accounts under the following conditions.
1.
A = $1,000,000, invested at 5.75% interest, compounded monthly, for 40 years.
2.
A = $175,000, invested at 3.8% interest, compounded quarterly, for 20 years.

Example 6.45

Investment Goal with Compound Interest

Pilar plans early for retirement, believing she will need $1,500,000 to live comfortably after the age of 67. How much will she need to deposit at age 23 in an account bearing 6.35% annual interest compounded monthly?

Your Turn 6.45

1.
Hajun turns 30 this year and begins to think about retirement. He calculates that he will need $1,200,000 to retire comfortably. He finds a fund to invest in that yields 7.23% and is compounded monthly. How much will Hajun need to invest in the fund when he turns 30 so that he can reach his goal when he retires at age 65?

Compute and Interpret Effective Annual Yield

As we’ve seen, quarterly compounding pays interest 4 times a year or every 3 months; monthly compounding pays 12 times a year; daily compounding pays interest every day, and so on. Effective annual yield allows direct comparisons between simple interest and compound interest by converting compound interest to its equivalent simple interest rate. We can even directly compare different compound interest situations. This gives information that can be used to identify the best investment from a yield perspective.

Using a formula, we can interpret compound interest as simple interest. The effective annual yield formula stems from the compound interest formula and is based on an investment of $1 for 1 year.

Effective annual yield is Y=(1+rn)n-1Y=(1+rn)n-1 where YY = effective annual yield, rr = interest rate in decimal form, and nn = number of times the interest is compounded in a year. YY is interpreted as the equivalent annual simple interest rate.

Example 6.46

Determine and Interpret Effective Annual Yield for 6% Compounded Quarterly

Suppose you have an investment paying a rate of 6% compounded quarterly. Determine and interpret that effective annual yield of the investment.

Your Turn 6.46

1.
Calculate and interpret the effective annual yield for an investment that pays at a 7% interest compounded quarterly.

Example 6.47

Determine and Interpret Effective Annual Yield for 5% Compounded Daily

Calculate and interpret the effective annual yield on a deposit earning interest at a rate of 5% compounded daily.

Your Turn 6.47

1.
Calculate and interpret the effective annual yield on a deposit earning 2.5% compounded daily.

Example 6.48

Choosing a Bank

Minh has a choice of banks in which he will open a savings account. He will deposit $3,200 and he wants to get the best interest he can. The banks advertise as follows:

Bank Interest Rate
ABC Bank 2.08% compounded monthly
123 Bank 2.09% compounded annually
XYZ Bank 2.05% compounded daily

Which bank offers the best interest?

Your Turn 6.48

1.
Isabella decides to deposit $5,500 in a CD but needs to choose between banks that offer CDs. She identifies four banks and finds out the terms of their CDs. Her findings are in the table below.
Bank Interest Rate
Smith Bank 3.08% compounded quarterly
Park Bank 3.11% compounded annually
Town Bank 3.09% compounded daily
Community Bank 3.10% compounded monthly
Which bank has the best yield?

Check Your Understanding

19.
What is compound interest?
20.
Which yields more money, simple interest or compound interest?
21.
Find the future value after 15 years of $8,560.00 deposited in an account bearing 4.05% interest compounded monthly.
22.
$10,000 is deposited in an account bearing 5.6% interest for 5 years. Find the difference between the future value when the interest is simple interest and when the interest is compounded quarterly.
23.
Find the present value of $75,000 after 28 years if money is invested in an account bearing 3.25% interest compounded monthly.
24.
What can be done to compare accounts if the rates and number of compound periods per year are different?
25.
Find the effective annual yield of an account with 4.89% interest compounded quarterly.

Section 6.4 Exercises

1.
What is the difference between simple interest and compound interest?
2.
What is a direct way to compare accounts with different interest rates and number of compounding periods?
3.
Which type of account grows in value faster, one with simple interest or one with compound interest?
How many periods are there if interest is compounded?
4.
Daily
5.
Weekly
6.
Monthly
7.
Quarterly
8.
Semi-annually
In the following exercises, compute the future value of the investment with the given conditions.
9.
Principal = $15,000, annual interest rate = 4.25%, compounded annually, for 5 years
10.
Principal = $27,500, annual interest rate = 3.75%, compounded annually, for 10 years
11.
Principal = $13,800, annual interest rate = 2.55%, compounded quarterly, for 18 years
12.
Principal = $150,000, annual interest rate = 2.95%, compounded quarterly, for 30 years
13.
Principal = $3,500, annual interest rate = 2.9%, compounded monthly, for 7 years
14.
Principal = $1,500, annual interest rate = 3.23%, compounded monthly, for 30 years
15.
Principal = $16,000, annual interest rate = 3.64%, compounded daily, for 13 years
16.
Principal = $9,450, annual interest rate = 3.99%, compounded daily, for 25 years
In the following exercises, compute the present value of the accounts with the given conditions.
17.
Future value = $250,000, annual interest rate = 3.45%, compounded annually, for 25 years
18.
Future value = $300,000, annual interest rate = 3.99%, compounded annually, for 15 years
19.
Future value = $1,500,000, annual interest rate = 4.81%, compounded quarterly, for 35 years
20.
Future value = $750,000, annual interest rate = 3.95%, compounded quarterly, for 10 years
21.
Future value = $600,000, annual interest rate = 3.79%, compounded monthly, for 17 years
22.
Future value = $800,000, annual interest rate = 4.23%, compounded monthly, for 35 years
23.
Future value = $890,000, annual interest rate = 2.77%, compounded daily, for 25 years
24.
Future value = $345,000, annual interest rate = 2.99%, compounded daily, for 19 years
In the following exercises, compute the effective annual yield for accounts with the given interest rate and number of compounding periods. Round to three decimal places.
25.
Annual interest rate = 2.75%, compounded monthly
26.
Annual interest rate = 3.44%, compounded monthly
27.
Annual interest rate = 5.18%, compounded quarterly
28.
Annual interest rate = 2.56%, compounded quarterly
29.
Annual interest rate = 4.11%, compounded daily
30.
Annual interest rate = 6.5%, compounded daily
The following exercises explore what happens when a person deposits money in an account earning compound interest.
31.
Find the present value of $500,000 in an account that earns 3.85% compounded quarterly for the indicated number of years.
  1. 40 years
  2. 35 years
  3. 30 years
  4. 25 years
  5. 20 years
  6. 15 years
32.
Find the present value of $1,000,000 in an account that earns 6.15% compounded monthly for the indicated number of years.
  1. 40 years
  2. 35 years
  3. 30 years
  4. 25 years
  5. 20 years
  6. 15 years
33.
In the following exercises, the number of years can reflect delaying depositing money. 40 years would be depositing money at the start of a 40-year career. 35 years would be waiting 5 years before depositing the money. Thirty years would be waiting 10 years before depositing the money, and so on. What do you notice happens if you delay depositing money?
34.
For each 5-year gap for exercise 32, compute the difference between the present values. Do these differences remain the same for each of the 5-year gaps, or do they differ? How do they differ? What conclusion can you draw?
35.
Daria invests $2,500 in a CD that yields 3.5% compounded quarterly for 5 years. How much is the CD worth after those 5 years?
36.
Maurice deposits $4,200 in a CD that yields 3.8% compounded annually for 3 years. How much is the CD worth after those 3 years?
37.
Georgita is shopping for an account to invest her money in. She wants the account to grow to $400,000 in 30 years. She finds an account that earns 4.75% compounded monthly. How much does she need to deposit to reach her goal?
38.
Zak wants to create a nest egg for himself. He wants the account to be valued at $600,000 in 25 years. He finds an account that earns 4.05% interest compounded quarterly. How much does Zak need to deposit in the account to reach his goal of $600,000?
39.
Eli wants to compare two accounts for their money. They find one account that earns 4.26% interest compounded monthly. They find another account that earns 4.31% interest compounded quarterly. Which account will grow to Eli’s goal the fastest?
40.
Heath is planning to retire in 40 years. He’d like his account to be worth $250,000 when he does retire. He wants to deposit money now. How much does he need to deposit in an account yielding 5.71% interest compounded semi-annually to reach his goal?
41.
Jo and Kim want to set aside some money for a down payment on a new car. They have 6 years to let the money grow. If they want to make a $15,000 down payment on the car, how much should they deposit now in an account that earns 4.36% interest compounded monthly?
42.
A newspaper’s business section runs an article about savings at various banks in the city. They find six that offer accounts that offer compound interest.
Bank A offers 3.76% compounded daily.
Bank B offers 3.85% compounded annually.
Bank C offers 3.77% compounded weekly.
Bank D offers 3.74% compounded daily.
Bank E offers 3.81% compounded semi-annually.
To earn the most interest on a deposit, which bank should a person choose?
43.
Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the second highest effective annual yield?
44.
Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the lowest effective annual yield?
45.
Jesse and Lila need to decide if they want to deposit money this year. If they do, they can deposit $17,400 and allow the money to grow for 35 years. However, they could wait 12 years before making the deposit. At that time, they’d be able to collect $31,700 but the money would only grow for 23 years. Their account earns 4.63% interest compounded monthly. Which plan will result in the most money, depositing $17,400 now or depositing $31,700 in 12 years?
46.
Veronica and Jose are debating if they should deposit $15,000 now in an account or if they should wait 10 years and deposit $25,000. If they deposit money now, the money will grow for 35 years. If they wait 10 years, it will grow for 25 years. Their account earns 5.25% interest compounded weekly. Which plan will result in the most money, depositing $15,000 now or depositing $25,000 in 10 years?
Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction
Citation information

© Apr 17, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.