Essential Mathematics for Economics and Business / Edition 4 available in Paperback
Essential Mathematics for Economics and Business isestablished as one of the leading introductory textbooks onmathematics for students of business and economics. Combining auser–friendly approach to mathematics with practicalapplications to the subjects, the text provides students with aclear and comprehensible guide to mathematics. The fundamentalmathematical concepts are explained in a simple and accessiblestyle, using a wide selection of worked examples, progressexercises and real–world applications.
New to this Edition
- Fully updated text with revised worked examples and updatedmaterial on Excel and Powerpoint
- New exercises in mathematics and its applications to givefurther clarity and practice opportunities
- Fully updated online material including animations and a newtest bank
- The fourth edition is supported by a companion website atwww.wiley.com/college/bradley, which contains: Animations ofselected worked examples providing students with a new way ofunderstanding the problems Access to the Maple T.A. test bank,which features over 500 algorithmic questions Further learningmaterial, applications, exercises and solutions.
- Problems in context studies, which present the mathematics in abusiness or economics framework.
- Updated PowerPoint slides, Excel problems and solutions.
"The text is aimed at providing an introductory-level expositionof mathematical methods for economics and business students. Interms of level, pace, complexity of examples and user-friendlystyle the text is excellent - it genuinely recognises and meets theneeds of students with minimal maths background."—Colin Glass, Emeritus Professor, University ofUlster
"One of the major strengths of this book is the range ofexercises in both drill and applications. Also the 'workedexamples' are excellent; they provide examples of the use ofmathematics to realistic problems and are easy to follow."—Donal Hurley, formerly of University College Cork
"The most comprehensive reader in this topic yet, this book isan essential aid to the avid economist who loathesmathematics!"—Amazon.co.uk
|Edition description:||New Edition|
|Product dimensions:||7.40(w) x 9.60(h) x 1.20(d)|
About the Author
Until 2012, Teresa Bradley lectured in mathematics andstatistics at Limerick Institute of Technology, and has beeninvolved for many years with the University of London on theexternal Diploma in Economics as well as the BSc in Economics,Business and Management.
Teresa Bradley is also author of Essential Statistics forEconomics, Business and Management, published by John Wiley& Sons, Ltd.
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Read an Excerpt
Essential Mathematics for Economics and Business
By Teresa Bradley Paul Patton
John Wiley & SonsISBN: 0-470-84466-3
Chapter OneMathematical Preliminaries
At the end of this chapter you should be able to:
Perform basic arithmetic operations and simplify algebraic expressions
Perform basic arithmetic operations with fractions
Solve equations in one unknown, including equations involving fractions
Understand the meaning of no solution and infinitely many solutions
Solve simple inequalities
In addition, you will be introduced to the calculator and a spreadsheet.
Some mathematical preliminaries
Brackets in mathematics are used for grouping and clarity. Brackets may also be used to indicate multiplication. Brackets are used in functions to declare the independent variable (see later). Powers: positive whole numbers such as [2.sup.3], which means 2 x 2 x 2 = 8:
[(anything).sup.3] = (anything) x (anything) x (anything)
[(x).sup.3] = x x x x x
[(x + 4).sup.5] = (x + 4)(x + 4)(x + 4) (x + 4) (x + 4)
Brackets: (A)(B) or A x B or AB all indicate A multiplied by B.
Variables and letters: When we don't know the value of a quantity, we give that quantity a symbol, such as x. We may then make general statements about the unknownquantity, x, for example 'For the next 15 weeks, if I save £x per week I shall have £4000 to spend on a holiday'. This statement may be expressed as a mathematical equation:
15 x weekly savings = 4000 15 x x = 4000 or 15x = 4000
Now that the statement has been reduced to a mathematical equation, we may solve the equation for the unknown, x:
15x = 4000 15x/15 = 4000/15 divide both sides of the equation by 15 x = 266.67
Square roots: the square root of a number is the reverse of squaring:
[(2).sup.2] = 4 [right arrow] [square root of (4)] = 2
[(2.5).sup.2] = 6:25 [right arrow] [square root of (6.25)] = 2:5
1.1 Arithmetic Operations
Addition and subtraction
Adding: If all the signs are the same, simply add all the terms and give the answer with the common overall sign.
Subtracting: When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term.
If terms are identical, for example all x-terms, all xy-terms, all [x.sup.2]-terms, then they may be added or subtracted as shown in the following examples:
Add/subtract with numbers, mostly Add/subtract with variable terms
5 + 8 + 3 = 16 similarity [right arrow] 5x + 8x + 3x = 16x
5 + 8 + 3 + y = 16 + y similarity [right arrow] (i) 5x + 8x + 3x + y = 16x + y
The y-term is different, so it cannot be (ii) 5xy + 8xy + 3xy + added to the others y = 16xy + y
The y-term is different, so it cannot be added to the others
7 - 10 = -3 similarity [right arrow] (i) 7x - 10x = -3x
(ii) 7[x.sup.2] - 10[x.sup.2] = -3[x.sup.2]
7 - 10 - 10x = -3 - 10x similarity [right arrow] 7[x.sup.2] - 10[x.sup.2] - 10x = -3[x.sup.2] - 10x
The x-term is different, so it cannot be The x-term is different, so it cannot be subtracted from the others subtracted from the others
Worked example 1.1 Addition and subtraction
For each of the following, illustrate the rules for addition and subtraction:
(a) 2 + 3 + 2·5 = (2 + 3 + 2.5) = 7.5
(b) 2x + 3x + 2.5x = (2 + 3 + 2.5)x = 7.5x
(c) -3xy - 2.2xy - 6xy = (-3 -2.2 -6)xy = -11.2 xy
(d) 8x + 6xy -12x + 6 + 2xy = 8x 12x + 6xy + 2xy)6 = -4x + 8xy + 6
(e) 3[x.sup.2] + 4x + 7 - 2[x.sup.2] - 8x + 2 = 3[x.sup.2] - 2[x.sup.2] + 4x - 8[x + 7 + 2 = [x.sup.2] - 4x + 9
Multiplying and dividing
Multiplying or dividing two quantities with like signs gives an answer with a positive sign. Multiplying or dividing two quantities with different signs gives an answer with a negative sign.
Worked example 1.2 Multiplication and division
Each of the following examples illustrate the rules for multiplication.
(a) 5 x 7 = 35 (b) -5 x -7 = 35 (c) 5 x -7 = -35 (d) -5 x 7 = -35 (e) 7/5 = 1.4 (f) (-7)/(-5) = 1.4 (g) (-7)/5= -1.4 (h) 7/(-5) = -1.4 (i) 5(7) = 35 (j) (-5)(-7) = 35 (k) (-5)y = -5y (l) (-x)(-y) = xy (m) 2(x + 2) = 2x + 4 (n) (x + 4)(x + 2) = x(x + 2) + 4(x + 2) = [x.sup.2] + 2x + 4x + 8 = [x.sup.2] + 6x + 8
(o) [(x + y).sup.2]
= (x + y)(x + y) = x(x + y) + y(x + y) = xx + xy + yx + yy = [x.sup.2] + 2xy + [y.sup.2]
It is very useful to remember that a minus sign is a -1, so -5 is the same as -1 x 5
0 x (any real number) = 0 0 / (any real number) = 0 But you cannot divide by 0
multiply each term inside the bracket by the term outside the bracket multiply the second bracket by x, then multiply the second bracket by (+4) and add,
multiply each bracket by the term outside it add or subtract similar terms, such as 2x + 4x = 6x
multiply the second bracket by x and then by y add the similar terms: xy + yx = 2xy
The following identities are important:
1. (x + y).sup.2] = [x.sup.2] + 2xy + [y.sup.2] 2. (x - y).sup.2] = [x.sup.2] - 2xy + [y.sup.2] 3. (x + y) = (x - y) = [x.sup.2] -[y.sup.2]
* Remember: Brackets are used for grouping terms together in maths for:
(i) Clarity (ii) Indicating the order in which a series of operations should be carried out
fraction = numerator/denominator
3 is called the numerator
7 is called the denominator
1.2.1 Add/subtract fractions: method
The method for adding or subtracting fractions is:
Step 1: Take a common denominator, that is, a number or term which is divisible by the denominator of each fraction to be added or subtracted. A safe bet is to use the product of all the individual denominators as the common denominator.
Step 2: For each fraction, divide each denominator into the common denominator, then multiply the answer by the numerator.
Step 3: Simplify your answer if possible.
Worked example 1.3 Add and subtract fractions
Each of the following illustrates the rules for addition and subtraction of fractions.
1/7 + 2/3 - 4/5
Step 1: The common denominator is (7)(3)(5)
Step 2: 1/7 + 2/3 - 4/5
Step 3: = 15 + 70 - 84/105 =1/105
1/7 + 2/3
Step 1: The common denominator is (7)(3)
Step 2: 1/7 + 2/3 = 1(3) + 2(7)/(7)(3)
Step 3: = 3 + 14/21 = 17/21
Same example, but with variables
x/7 + 2x/3-4x/5
= x(3)(5) + 2x(7)(5)-4x(7)(3) (7)(3)(5)
= 15x + 70x - 84x/105
1/x + 4 + 2/x = 1(x) + 2(x + 4)/(x + 4) (x)
= x + 2x + 8/[x.sup.2]+ 4x
= 3x + 8/[x.sup.2]+ 4x
1.2.2 Multiplying fractions
In multiplication, write out the fractions, multiply the numbers across the top lines and multiply the numbers across the bottom lines.
Note: Write whole numbers as fractions by putting them over 1.
Terminology: RHS means right-hand side and LHS means left-hand side.
Worked example 1.4
(a) (2/3) (5/7) = (2)(5)/(3)(7) = 10/21
(b) (-2/3) (7/5) = (-2)(7)/(3)(5) = -14/15
(c) 3 x 2/5 (3/1)(2/5)=(3)(2)/(1)(5) = 6/5 = 1 1/5
The same rules apply for fractions involving variables, x, y, etc.
(d) (3/x) (x + 3)/(x - 5) = 3(x + 3)/x(x - 5) = 3x + 9/[x.sup.2]- 5x]
1.2.3 Dividing fractions
Dividing by a fraction is the same as multiplying by the fraction inverted
Worked example 1.5
Division with fractions
The following examples illustrate how division with fractions operates.
(a) (2/3)/(5/11) = (2/3)(11/5) = 22/15
(b) 5/(3/4) = 5 x 4/3 = 5/1 x 4/3 = 20/3 = 6 2/3
(c) (7/3)/8 = (7/3)/8/1 = 7/3 x 1/8 = 7/24
(d) 2x/x + y/3x/2(x - y) = 2x/x + y 2(x - y)/ 3x
= 4x(x - y)/3x(x + y) = 4(x - y)/3(x + y)
Note: The same rules apply to all fractions, whether the fractions consist of numbers or variables.
Progress Exercises 1.1 Revision on Basics
Show, step by step, how the expression on the left-hand side simplifies to that on the right.
1. 2x + 3x + 5(2x - 3) = 15(x - 1)
2. 4[x.sup.2] + 7x + 2x (4x - 5) = 3x(4x - 1)
3. 2x (y + 2) - 2y(x + 2) = 4(x - y)
4. (x + 2)(x - 4) - 2(x - 4) = x(x - 4)
5. (x + 2)(y - 2) + (x - 3)(y + 2)
= 2xy - y - 10
6. [(x + 2).sup.2] + [(x - 2).sup.2] = 2[(x.sup.2] + 4)
7. [(x + 2).sup.2] - [(x - 2).sup.2] = 8x
8. (x + 2)2 - x(x + 2) = 2(x + 2)
9. 1/3 + 3/5 + 5/7 = 1 68/105
10. x/2 - x/3 = x/6
11. (2/3)/(1/5) = 10/3
12. (2/7)/3 = 2/21
13. 2(2/x - x/2) = 4-[x.sup.2]/x = 4/x - x
14. -12/P (3P/2 + P/2) = -24
15. (3/x)/x + 3 = 3/x(x +3)
16. (5Q/P + 2)/(1/(P + 2)) = 5Q
1.3 Solving Equations
The solution of an equation is simply the value or values of the unknown(s) for which the left-hand side (LHS) of the equation is equal to the right-hand side (RHS).
For example, the equation, x + 4 = 10, has the solution x = 6. We say x = 6 'satisfies' the equation. We say this equation has a unique solution.
Not all equations have solutions. In fact, equations may have no solutions at all or may have infinitely many solutions. Each of these situations is demonstrated in the following examples.
Case 1: Unique solutions An example of this is given above: x + 4 = 10 etc.
Case 2: Infinitely many solutions The equation, x + y = 10 has solutions (x = 5, y = 5); (x = 4, y = 6); (x = 3, y = 7), etc. In fact, this equation has infinitely many solutions or pairs of values (x, y) which satisfy the formula, x + y = 10.
Case 3: No solution The equation, 0(x) = 5 has no solution. There is simply no value of x which can be multiplied by 0 to give 5.
* Methods for solving equations
Solving equations can involve a variety of techniques, many of which will be covered later.
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Table of Contents
CHAPTER 1 Mathematical Preliminaries
1.1 Some Mathematical Preliminaries
1.2 Arithmetic Operations
1.4 Solving Equations
1.5 Currency Conversions
1.6 Simple Inequalities
1.7 Calculating Percentages
1.8 The Calculator. Evaluation and Transposition of Formulae
1.9 Introducing Excel
CHAPTER 2 The Straight Line and Applications
2.1 The Straight Line
2.2 Mathematical Modelling
2.3 Applications: Demand, Supply, Cost, Revenue
2.4 More Mathematics on the Straight Line
2.5 Translations of Linear Functions
2.6 Elasticity of Demand, Supply and Income
2.7 Budget and Cost Constraints
2.8 Excel for Linear Functions
CHAPTER 3 Simultaneous Equations
3.1 Solving Simultaneous Linear Equations
3.2 Equilibrium and Break-even
3.3 Consumer and Producer Surplus
3.4 The National Income Model and the IS-LM Model
3.5 Excel for Simultaneous Linear Equations
CHAPTER 4 Non-linear Functions and Applications
4.1 Quadratic, Cubic and Other Polynomial Functions
4.2 Exponential Functions
4.3 Logarithmic Functions
4.4 Hyperbolic (Rational) Functions of the Form a/(bx + c)
4.5 Excel for Non-linear Functions
CHAPTER 5 Financial Mathematics
5.1 Arithmetic and Geometric Sequences and Series
5.2 Simple Interest, Compound Interest and Annual Percentage Rates
5.4 Net Present Value and Internal Rate of Return
5.5 Annuities, Debt Repayments, Sinking Funds
5.6 The Relationship between Interest Rates and the Price of Bonds
5.7 Excel for Financial Mathematics
CHAPTER 6 Differentiation and Applications
6.1 Slope of a Curve and Differentiation
6.2 Applications of Differentiation, Marginal Functions, Average Functions
6.3 Optimisation for Functions of One Variable
6.4 Economic Applications of Maximum and Minimum Points
6.5 Curvature and Other Applications
6.6 Further Differentiation and Applications
6.7 Elasticity and the Derivative
CHAPTER 7 Functions of Several Variables
7.1 Partial Differentiation
7.2 Applications of Partial Differentiation
7.3 Unconstrained Optimisation
7.4 Constrained Optimisation and Lagrange Multipliers
CHAPTER 8 Integration and Applications
8.1 Integration as the Reverse of Differentiation
8.2 The Power Rule for Integration
8.3 Integration of the Natural Exponential Function
8.4 Integration by Algebraic Substitution
8.5 The Definite Integral and the Area under a Curve
8.6 Consumer and Producer Surplus
8.7 First-order Differential Equations and Applications
8.8 Differential Equations for Limited and Unlimited Growth
8.9 Integration by Substitution and Integration by Parts (website only)
CHAPTER 9 Linear Algebra and Applications
9.1 Linear Programming
9.3 Solution of Equations: Elimination Methods
9.5 The Inverse Matrix and Input/Output Analysis
9.6 Excel for Linear Algebra
CHAPTER 10 Difference Equations
10.1 Introduction to Difference Equations
10.2 Solution of Difference Equations (First-order)
10.3 Applications of Difference Equations (First-order)
Solutions to Progress Exercises