Mathematics 209: Finite Mathematics

Study Guide

Unit 3: Functions and Graphs

In this unit, you will learn to solve problems involving compound interest, annuities and amortization.

Objectives

When you have completed this unit, you should be able to

1. determine the interest rate earned on an investment.
2. evaluate the commissions due on a transaction involving the purchase or the sale of stock.
3. determine the growth time for an investment.
4. compare interest for various compounding periods.
5. determine the present value of an investment needed to achieve a desired amount over a given compounding period and at a given rate of interest.
6. determine the future value of an annuity.
7. determine the monthly payments and total interest on an amortized debt.
8. construct an amortization schedule.

Simple Interest

Indications

1. Read Section 3-1, Simple Interest, on pages 130–134 of the textbook.
2. Do odd-numbered exercises 5–21 on page 134 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 33–51 on pages 135–136 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

1. In the formula $A=P\left(1+rt\right)\text{,}$ what are $A\text{,}$ $P$, and $r\text{?}$
2. When do we use the formula ${\displaystyle P=\frac{A}{1+rt}\text{?}}$
3. When do we use the formula ${\displaystyle r=\frac{A-P}{Pt}\text{?}}$
4. When do we use the formula ${\displaystyle t=\frac{A-P}{Pr}\text{?}}$

Compound and Continuous Compound Interest

Indications

1. Read Section 3-2, Compound and Continuous Compound Interest, on pages 136–147 of the textbook.
2. Do odd-numbered exercises 3–29 and 35–45 on pages 148–149 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 61–95 on pages 149–150 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

5. What is the difference between simple interest and compound interest?
6. What is the difference between compound interest and continuous compound interest?
7. How long does it take for an investment to double in value if it is continuously compounded at a rate $r\text{?}$
8. What would the rate be for an investment that doubles in value in $t$ years?
9. What is the annual percentage yield?

Future Value of an Annuity; Sinking Funds

Indications

1. Read Section 3-3, Future Value of an Annuity; Sinking Funds, on pages 151–158 of the textbook.
2. Do odd-numbered exercises 3–17 on page 158 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 21–43 on pages 158–159 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

10. What is an annuity?
11. Formula (6) on page 153 of the textbook gives the future value of an ordinary annuity when the interest is compounded annually. What is the value of an ordinary annuity after the second period, if the interest is compounded $m$ times per year?
12. What is a sinking fund?

Present Value of an Annuity; Amortization

Indications

1. Read Section 3-4, Present Value of an Annuity; Amortization, on pages 160–168 of the textbook.
2. Do odd-numbered exercises 5–15 on page 169 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 21–45 and 49–57 on pages 169–171 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

13. What is an amortization?
14. What is the difference between the present value of an ordinary annuity and an amortization payment?
15. Which formula do you use to find the monthly payments of a loan at an annual rate of $r\text{,}$ to be paid in $m$ years?

Finishing This Unit

1. Review the objectives of this unit and make sure you are able to meet each of them.
2. Study the section of the Chapter 3 Review titled Important Terms, Symbols and Concepts, on pages 172–173 of the textbook.
3. If there is a concept, definition, example or exercise that is not yet clear to you, go back and re-read it. Contact your tutor if you need help.
4. You might want to do exercises 1–13 and 15–55 from the Review Exercise section on pages 173–176 of the textbook. The questions on the practice examination are taken from this exercise. If you have difficulty, consult your tutor to discuss the problem.
5. Complete the practice examination provided for this unit. Evaluate yourself, first checking your answers against those provided in the Answers section at the end of the textbook, and then comparing your solutions with those provided on pages S-116 to S-133 of the Student Solutions Manual portion of the textbook. The number of points in a question may indicate the number of steps in the solution. Give yourself full credit if your answer is correct and you give a complete solution, even if your solution differs from that shown in the Student Solutions Manual.

Practice Examination

Time: 2 hours
Total points: 48
Passing grade: 50%

Do the following exercises from the Chapter 3 Review Exercise on pages 171–174 of the textbook.

To obtain full credit you must justify all your answers and show your work.

1. Exercise 20  (Marks: 6 pts.)
2. Exercise 30  (Marks: 6 pts.)
3. Exercise 36  (Marks: 12 pts.)
4. Exercise 40  (Marks: 8 pts.)
5. Exercise 46  (Marks: 8 pts.)
6. Exercise 56  (Marks: 8 pts.)