Mathematics for Business E-Book

Gary Bronson

• Herausgeber: Packt Publishing
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 406

Veröffentlichungsjahr: 2024

Beschreibung

This course provides a comprehensive understanding of quantitative methods essential for economic forecasting, resource allocation, portfolio analysis, inventory management, data-mining, and addressing social and climate challenges. Starting with foundational topics like finite mathematics and the mathematics of finance, it progresses to differential calculus, optimization, and curve fitting. These concepts are vital for solving contemporary business problems.
Learners will explore algebra, finite math, finance mathematics, calculus, optimization techniques, and curve fitting, applying these methods to realistic business scenarios. Topics include cash flow, amortization, interest, loans, annuities, revenue/cost models, break-even analysis, inventory control, and econometrics. Each section includes extensive examples and exercises, reinforced by key terms and concepts, making the material accessible and practical.
The course begins with basic mathematical concepts and advances through increasingly complex topics. By the end, learners will have the tools to tackle various business problems using quantitative methods, making this course invaluable for anyone in the business field. This structured approach ensures both theoretical knowledge and practical application, preparing learners for real-world challenges.

Leseprobe

MATHEMATICSFORBUSINESS

Seventh Edition

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MATHEMATICSFORBUSINESS

Seventh Edition

Gary Bronson, Ph.D.

Fairleigh Dickinson University

Richard Bronson, Ph.D.

Fairleigh Dickinson University (Emeritus)

Maureen Kieff

Fairleigh Dickinson University

MERCURY LEARNING AND INFORMATIONDulles, VirginiaBoston, MassachusettsNew Delhi

Copyright ©2021 by MERCURY LEARNING AND INFORMATION LLC. All rights reserved.

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G. Bronson, R. Bronson, M. Kieff. Mathematics for Business, Seventh Edition.

ISBN: 978-1-68392-766-2

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To Oliver and Benjamin—Gary Bronson

To Casley, Sarah, Asher, and Max—Richard Bronson

To Richard, Kristen, Annie, and My Parents—Maureen Kieff

CONTENTS

Dedication

Preface

About the Authors

Chapter 1:The Basics

1.1Signed Numbers

1.2Exponents

1.3Basics of Solving Equations

1.4Sigma Notation

1.5Numerical Considerations

1.6Summary of Key Points

Chapter 2:Graphs and Linear Equations

2.1The Cartesian Coordinate System

2.2Line Graphs

2.3Graphing Linear Equations

2.4Properties of Straight Lines

2.5Break-Even Analysis

2.6Constructing Line Graphs using Excel

2.7Summary of Key Points

Chapter 3:Functions

3.1Concept of a Function

3.2Mathematical Functions

3.3Polynomial Functions

3.4Quadratic Functions

3.5Exponential Functions

3.6Summary of Key Points

3.7Chapter Appendix: The Quadratic Formula

Chapter 4:The Mathematics of Finance

4.1Simple Interest

4.2Compound Interest

4.3Lump-Sum Future and Present Values

4.4Cash Flow Net Present Values (NPVs)

4.5Ordinary Annuities

4.6Mortgages and Amortization

4.7Installment Loans and Interest Charges

4.8Annuities Due

4.9Annual Percentage Yield (APY) and Rate (APR)

4.10Summary of Key Points

Chapter 5:Rates of Change: The Derivative

5.1Average Rates of Change

5.2Instantaneous Rates of Change

5.3The Derivative

5.4Basic Derivatives

5.5Addition and Subtraction Rules

5.6Product and Quotient Rules

5.7Additional Rules

5.8Higher-Order Derivatives

5.9Summary of Key Points

Chapter 6:Optimization

6.1Optimization through Differentiation

6.2Modeling

6.3Maximizing Sales Profit

6.4Minimizing Inventory Costs

6.5Econometrics

6.6Summary of Key Points

Chapter 7:Curve Fitting and Trend Lines

7.1Constant Curve Fit

7.2Linear Least-Squares Trend Lines

7.3Creating Trend Lines using Excel

7.4Selecting an Appropriate Trend Line using R2

7.5Summary of Key Points

Appendix: Solutions to Selected Odd-Numbered Exercises

Index

PREFACE

Quantitative methods have become essential in economic forecasting, allocation of resources, portfolio analysis, inventory analysis, data-mining, and innovative solutions to myriad new social and climate challenges. The aim of this text is to introduce and provide a basic and necessary understanding of these quantitative methods in a student centered and affordable text.

We understand that the average student knows that mathematics is a powerful tool for solving problems, but may feel uneasy, sometimes fearful, about the subject matter. Our primary concern in writing this text is to present the material in a clear, understandable, and non-intimidating manner. By numerous examples each new mathematical concept is tied to a practical problem, so that the student never loses sight of the ultimate goal: to develop mathematical tools to solve business and social problems. We are not interested in mathematics as an end unto itself. Additionally, because many of the topics are now solved using spreadsheets, we show where, when, and how to apply these increasingly necessary tools.

The text assumes student familiarity with algebraic concepts but not facility in using them. As such, Chapter 1 is a review of elementary concepts typically covered at the secondary school level.

Chapters 2 and 3 deal with mathematical equations, and graphs. Material on polynomial equations, their solutions, and graphs is presented.

Interest rates, cash flows, and annuities are presented in Chapter 4. This serves as both motivation and introduction to applying mathematical equations to areas typically of interest to students. Here, mortgages, consumer loans, annuities and their relationship to pension plans and lottery winnings paid over time are presented. The subject matter is relatively easy to grasp and since most students find it interesting, it provides an opportunity for early success in using mathematical methods.

The next two chapters (5 and 6) bring the student to an appreciation and awareness of differential calculus as a powerful mathematical approach for analyzing and modeling commercial systems. The applications in these chapters were developed with this philosophy in mind. They provide a setting for introducing mathematical models and exposing students to realistic applications of average and instantaneous rates of change. Although we do not expect the reader of the book to become an expert in calculus, we do hope that these applications develop an understanding of how differential calculus is used in real-life problems.

The material in Chapter 7, on least-squares analysis, is included to answer the usual question asked by students as to the origin of the equations they have been using. An integral part of this chapter is the use of spreadsheets to create trend lines and their associated equations.

A number of people have been instrumental for helping to make this book a reality. First, our appreciation goes to our students, who used most of the material in this book in prepublication form. We sincerely hope this text provides a very low-cost alternative to the otherwise extremely expensive texts they have been required to purchase in the past, with the same quality that has characterized all of our published texts. We would also like to express our thanks to the hard-working and dedicated staff at Mercury Learning and Information. To David Pallai, Publisher, for expressing confidence in and publishing this book, and for tirelessly working with us closely on all aspects of the publication process, from beginning to end. It is rare that authors can receive the “personal touch” from such a dedicated and experienced publishing professional, who guided us through the various intricacies and challenges of the process. To Jennifer Blaney, for her professionalism, production expertise, and in doing a great job keeping the book on schedule. In particular, we would like to recognize the commitment to quality that has been shown by these individuals, in terms of the care shown during copyediting, resolving issues, requesting multiple rounds of page proofs, and providing us the opportunity to provide input into just about every aspect of our book and its production, marketing, and promotions. For this, we are grateful. Finally, and most importantly, we owe our deep appreciation and thanks to our spouses, Rochelle, Evelyn, and Richard.

Gary Bronson, Ph.D.

Richard Bronson, Ph.D.

Maureen Kieff

September 2021

ABOUT THE AUTHORS

Gary Bronson, Ph.D., is a professor of marketing, information systems, and decision sciences at the Silberman College of Business, Fairleigh Dickinson University, where he was twice voted Teacher of the Year of the college and received the Distinguished Research award of the university. He has worked as a senior engineer at Lockheed Electronics, an invited lecturer and consultant to Bell Laboratories, and a software consultant to a number of Wall Street financial firms. He is the author of the highly acclaimed A First Book of C and has authored several other successful programming textbooks on C++, Java, and Visual Basic. He is a co-author of Excel Basics and the Excel 2019 Project Book with Jeffrey Hsu. Additionally, he is the author of a number of journal articles in the fixed-income financial and programming areas. Dr. Bronson received his Ph.D. from Stevens Institute of Technology.

Richard Bronson, Ph.D., is Professor Emeritus in the Department of Mathematics at Fairleigh Dickinson University, where he has won the university’s Distinguished Teaching, Research, and Faculty Service awards, including the Distinguished College or University Teaching award by the New Jersey Section of the Mathematical Association of America. He has authored several successful texts in matrix methods, differential equations, linear algebra, finite mathematics, and operations research. He has also published a number of journal articles in mathematics and system simulation. His latest publication is a political thriller titled, Antispin.

Maureen Kieff, is a Clinical Assistant Professor in the Department of Information Systems and Decision Sciences at Fairleigh Dickinson. She has received 12 awards during her career for outstanding teaching and service to students, the latest being the 2014 College Teacher of The Year. She has co-authored ten research articles published in academic journals, primarily addressing mutual fund performance.

CHAPTER 1

THE BASICS

In this Chapter

1.1Signed Numbers

1.2Exponents

1.3Basics of Solving Equations

1.4Sigma Notation

1.5Numerical Considerations

1.6Summary of Key Points

This chapter is a review of the topics in algebra that are used throughout the text. Readers who already have a working knowledge of this material are advised to skim over the chapter and go directly to Chapter 2. Others are advised to spend as much time as is necessary to master this material before proceeding further.

1.1SIGNED NUMBERS

Multiplication Rules:

Division Rules:

Order of Operations:

Exercises 1.1

Evaluate the following expressions.

1.3 + (−6)

2.−4 + 7

3.19.7 + (−18.1)

4.−6.2 +) + (−8.1)

5.−9 + (−1/2)

6.−4.1 + 7

7.9(18)

8.9(−8)

9.(−9)(18)

10.(−9)(−18)

11.(2)(−1/3)

12.(−5)(−1/6)

13.(−6.1)(2.3)

14.(−8)(−1.4)

15.(−8)/(−2)

16.8/(−2)

17.−8/2

18.−2/8

19.4/(−5)

20.(−5)/(−4)

21.−22/4

22.8 – 4

23.4 – 8

24.−4 −8

25.−4 − (−8)

26.−8 −4

27.−8 − (−4)

28.2.1 − 5.6

29.−5.6 − 2.1

30.1/10 – 1/5

31.2[5 + (−3)]

32.−2[1 + (−6)]

33.−4(1 − 3) + 2(2 − 5)

34.6[2(−1 + 7) − 3]

35.(1.6)(1.9 − 2.1)

36.4[(−1)(2−9) + 7(3)]

37.

38.

39.

40.

41.

42.

1.2EXPONENTS

Exercises 1.2

In Exercises 1 through 28 simplify each of the given expressions into one exponent. Make sure that each solution is given with a positive exponent.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Using a calculator, determine the values of the quantities given in Exercises 17 through 28.

17.93/2

18.16−5/4

19.272/3

20.100−3/2

21.(31/2)(121/2)

22.(31/3)(91/3)

23.(5)−1/2(20)−1/2

24.(2−3/2)(32)−3/2

25.

26.

27.

28.

For Exercises 29 through 34 solve the following equations for the unknown quantity (note: factoring can be used whenever it is appropriate). No rounding.

29.

30.

31.

32.

33.

34.

1.3BASICS OF SOLVING EQUATIONS

The Fundamental Rule of Algebra

Solving Equations Having One Unknown

Exercises 1.3

1.4SIGMA NOTATION

In various parts of this text, we will need the sum of large numbers of terms. Sometimes we will need the actual total, and then we will have to sum the numbers physically. Other times, however, we will only need to indicate the appropriate sum. An example is a statement, “Yearly expenditures are the sum of weekly expenditures.” Here we are not explicitly calculating the total yearly expenditure but simply indicating that it is the sum of 52 numbers. In cases like this, there is a useful mathematical notation for indicating the appropriate sum.

Consider the case of a teacher who has a list of seven grades and wants their sum. If we denote the first grade as G1, the second grade as G2, and so on through the seventh grade, which we denote as G7, the final sum can be given as

Sum (Eq. 1.7)

Of course, if we had 52 items to add as opposed to only 7, writing an expression similar to Equation 1.7 would be tedious indeed. A more convenient way to represent the right side of Equation 1.7 symbolically as

The capital Greek letter sigma (Σ) denotes a sum. The term(s) after the sigma, in this case, the letter G, tells us what values are to be added, in this case, we are adding G terms.

The quantity i 1 at the bottom of the sigma indicates where the sum is to start from; in this case, the sum starts with G1, that is, Gi, with i replaced with the starting value of 1. The number at the top of the sigma indicates where the sum is to stop, which in this case is G7. Intermediate values in the sum are obtained by replacing the subscript i on individual G terms with consecutive integers between the starting and ending numbers at the bottom and top of the sigma, respectively.

For example, the notation

indicates a sum of S terms. The sum starts with S3, because the starting i value is given as 3 at the bottom of the sigma sign, and ends at S8, because the ending value of i is given as 8 above the sigma sign. Thus, the notation

is shorthand for

Additional examples are

and

In general,

(Eq. 1.8)

is the sigma notation for the sum of the quantities qi ranging from qm through qn successively. Obviously, the left side of Equation 1.8 is more compact than the expanded form given on the right side, and is the reason for using sigma notation. When actual values are given for the terms being summed, a final value for the sum can be obtained.

Example 1 Give the expanded form of

Solution The sigma notation indicated the sum of terms having the form (xi– i) beginning with i 4 and continuing successively through i 8. Therefore,

Example 2 Give the expanded form of

Solution

Example 3 Determine the sigma notation for the sum

Solution Each term in the sum is of the form where i is an integer starting at 3 and ending at 100, with all terms in between. The sum can be given as

Example 4 Weekly expenditures for a given year are denoted as W1 through W52 successively. Develop a formula for the yearly expenditure.

Solution Denote the yearly expenditure as Y. Because yearly expenditure is the sum of the individual weekly expenditures,

All the examples so far have used the subscript i. Any other letter would do equally well. Thus

and

Sometimes we are given data, and we would like to indicate that some portion of this data is to be summed. For example, if test scores for a particular student are 60, 70, 75, 80, 82, 83, 87, 90, and we only want to sum the 2nd through 7th scores for some reason, this is easily indicated by the notation . Of course, this notation is good only if we understand that G1 signifies the first score, G2 the second score, and G7 the next to last score. In general, we always assume that data are ordered as they appear.

Finally, when we write a sigma without any numbers below or above it, we mean that the sum is to include all possible terms. It should be clear from the context which terms are being considered. For example, certain data may include pairs of numbers, and we may wish to multiply the members of each pair and then sum over all the pairs. If the number of data points is not known in advance, we can still indicate the desired sum using the notation Σxiyi. For the data listed in Table 1.2, this sum is (because there are only four pairs of data points)

TABLE 1.2

Exercises 1.4

1.Write the expanded form of the following expressions:

2.Write the expanded form of the following expressions:

3.Write the following expressions in sigma notation.

a.3(2)2+ 3(3)2+ 3(4)2+ 3(5)2+ … + 3(29)2

b.2(3)2+ 3(3)2+ 4(3)2+ 5(3)2+ … + 29(3)2

c.2(3)2+ 2(3)3+ 2(3)4+ 2(3)5+ … + 2(3)29

d.3(2)2− 3(3)2+ 3(4)2− 3(5)2+ … + 3(28)2− 3(29)2

4.For the data given in Table 1.3, calculate the given sums.

TABLE 1.3

g.What can you conclude about the sums found in parts d and e?

5.For the data given in Table 1.4, calculate the indicated sums

TABLE 1.4

g.What can you conclude about the sums found in parts e and f ?

6.Write each of the following expressions in expanded form and verify that they are equal.

a.

b.

c.

7.Prove the following identities by converting each side to expanded form:

a.

b.

c.

8.Determine if the following statement is valid or not.

9.Derive a formula using sigma notation for the average of a set of grades

1.5NUMERICAL CONSIDERATIONS

Expressing numbers in decimal form is necessary for most commercial transactions. One reason involves money. All financial figures are given in decimal form; the numbers to the left of the decimal point represent dollars, and the numbers to the right of the decimal point represent cents. Quoting the cost of an item as $1.20 is clearer than quoting six-fifths of a dollar. A second reason for using decimals is mathematical. It is easier to add 0.2 and 1.5 than it is to 1/5 and 1½. In this section, we present two considerations when dealing with numerical values; rounding when dealing with dollar and cents commercial transactions and exponential notation that you will sometimes encounter when using a calculator.

Rounding

Exponential Notation

Calculators that have the ability to display more decimal values than can be accommodated on their displays typically display both very small and very large numbers using exponential notation. In this notation, the letter E stands for “exponent” and the number following the E indicates the number of places the decimal point should be moved to obtain the standard decimal value. The decimal point is moved to the right if the number after E is positive, or it is moved to the left if the number after the E is negative

For example, the E10 in a display such as 1.625 E10 means move the decimal place ten places to the right, so the number becomes 16250000000. The E-8 in a display such as 7.31 E-8 means move the decimal point eight places to the left, so the number becomes .0000000731. Table 1.5 provides a number of additional examples using exponential notation.

TABLE 1.5

Exponential NotationDecimal Notation2.689 E5268900.2.689 E-5.000026894.896723 E1048967230000.4.896723 E-10.0000000004896723

Exercises 1.5

In Exercises 1 through 5, use a calculator to find the equivalent decimal values for the given fraction, and then round (that is, use arithmetic rounding) the numbers to two decimal places.

6.Repeat Exercises 1 through 5, but round the numbers to three decimal places.

7.Round up the numbers given in Exercises 1 through 5 to three decimals.

8.Truncate the numbers given in Exercises 1 through 5 to two decimal places.

In Exercises 9 through 14, convert the numbers written in exponential notation to standard decimal numbers:

1.6SUMMARY OF KEY POINTS

Key Terms

•Exponential notation

•Exponents

•Fundamental Rule of Algebra

•Order of operations

•Sigma Notation

•Signed numbers

•Solution

Key Concepts