Business Mathematics

Cover
Copyright
Brief Contents
Contents
Foreword
Preface
Acknowledgement
About the Author
Chapter 1: Introduction to Business Mathematics
	1.1 Introduction to Business Mathematics
	1.2 Scope and Importance
		1.2.1 Steps in Quantitative Analysis Approach
	1.3 Number System
		1.3.1 Natural Numbers
		1.3.2 Whole Numbers
	1.4 Positive and Negative Integers
		1.4.1 Positive Integers
		1.4.2 Negative Integers
	1.5 Fractions
		1.5.1 Proper Fraction
		1.5.2 Improper Fraction
	1.6 Rational and Irrational Numbers
		1.6.1 Rational Numbers
		1.6.2 Irrational Numbers
	1.7 Real Numbers
	1.8 Sum of First n Natural Numbers
	1.9 Sum of Square and Sum of Cube of First n Natural Numbers
Chapter 2: Algebra
	2.1 Algebra
	2.2 Algebraic Identities
	2.3 Simple Factorizations
	2.4 Equations
	2.5 Linear and Quadratic Equations
		2.5.1 Linear Equations
		2.5.2 Simultaneous Linear Equations
		2.5.3 Quadratic Equation
	2.6 Indices and Surds
		2.6.1 Indices
		2.6.2 Surds
	2.7 Logarithms and Their Properties
		2.7.1 Definition
		2.7.2 Properties
		2.7.3 Common Logarithms
	2.8 Problems Based on Logarithms
	2.9 Inequality in Graphs
	2.10 Remainder Theorem and Factor Theorem
		2.10.1 Remainder Theorem
		2.10.2 Factor Theorem
	Exercise
Chapter 3: Theory of Sets and Its Business Applications
	3.1 Set Theory
	3.2 Representation of Sets
		3.2.1 Tabulation Method
		3.2.2 Set Builder Form
		3.2.3 Cardinal Number
		3.2.4 Element of a Set
		3.2.5 Order of a Set
		3.2.6 Finite Set
		3.2.7 Infinite Set
		3.2.8 Singleton Set
	3.3 Union of Sets
	3.4 Intersection of Two Sets
	3.5 Complement of a Set
	3.6 Subset
		3.6.1 Identical/Equal Set
		3.6.2 Equivalent Set
		3.6.3 Disjoint Set
		3.6.4 Universal Set
	3.7 Null Set/Empty Set/Void Set
	3.8 Difference of a Set
	3.9 De Morgan's Law
	3.10 Universal Set
	3.11 Basic Operations on Set
		3.11.1 Union of Sets
		3.11.2 Intersection of Two Sets
		3.11.3 Difference of a Set
		3.11.4 Complement of a Set
	3.12 Cartesian Products
	3.13 Laws of Set Algebra
	3.14 Cardinal Numbers
	3.15 Venn Diagram
	3.16 Business Applications
	Exercise
Chapter 4: Ratio
	4.1 Ratio
	4.2 Proportion
	4.3 Types of Proportion
		4.3.1 Direct Proportion
		4.3.2 Inverse Proportion
		4.3.3 Indirect Proportion
	4.4 Ratio, Proportion and its Type of Comparison
		4.4.1 Addition and Subtraction of Ratios
	4.5 Variations
		4.5.1 Direct Variation
		4.5.2 Inverse Variation
		4.5.3 Joint Variation
	4.6 Percentage
		4.6.1 Uses of Percentages
		4.6.2 Converting Fractions into Percentage
		4.6.3 Problem Based on Quantity Application
		4.6.4 Problem Based on Election Application
		4.6.5 Problem Based on Examination Application
		4.6.6 Price Mixture
		4.6.7 Decimal Fraction
	4.7 Annexing Zeros and Removing Decimal Signs
		4.7.1 Annexing Zeros
	4.8 Average (Arithmetic Mean)
		4.8.1 Discrete Data
		4.8.2 Discrete Data with Frequency
	4.9 Computation of Mean for Grouped Series
	4.10 Mean of Composite Group
	Exercises
Chapter 5: Functions and Graphs
	5.1 Introduction
	5.2 Function
		5.2.1 Constant
		5.2.2 Variable
	5.3 Domain and Range of a Function
	5.4 Continuous and Discrete Functions
		5.4.1 Continuous Function
		5.4.2 Discrete Function
	5.5 Composition and Inverse Functions
		5.5.1 Composition Functions
		5.5.2 Order of Composition
		5.5.3 Decomposition of a Function
		5.5.4 Inverse Function
	5.6 Classification of Functions
	5.7 Exponential and Logarithmic Functions
		5.7.1 Exponential Functions
		5.7.2 Logarithmic Functions
	5.8 Graphs
		5.8.1 The General Idea of Curve Sketching and Graphs of Straight Lines
	5.9 Modulus Function
	Exercise
Chapter 6: Quadratic Functions and Theory of Quadratic Equations
	6.1 Quadratic Functions
		6.1.1 Quadratic Modeling
	6.2 Evaluating the Solutions of the Quadratic Equation ax2 + bx + c = 0
	6.3 Nature of the Roots of Quadratic Equation
	6.4 Sum and Product of the Roots
	6.5 Relations Between the Roots
	6.6 Condition for the Existence of a Common Root
	6.7 To form the Equation Whose Roots are r1 and r2
	6.8 Graph of the Function f(x) = ax2 + bx + c; a ≠ 0
	6.9 Sign of the Expression f(x) = ax2 + bx + c
	6.10 Maximum/Minimum Values of a Quadratic Function
	6.11 Solution of Quadratic Inequalities
	6.12 Additional Problems
	Exercise
Chapter 7: Permutation and Combination
	7.1 Fundamental Counting Principle
	7.2 Factorial
	7.3 Permutation
		7.3.1 Permutation of n Different Things
		7.3.2 Permutation of n Things not all Different (Taken all Together)
	7.4 Restricted Permutation
	7.5 Circular Permutations
	7.6 Combination
		7.6.1 To Find the Number of Ways of Selecting r Things Out of n Things
		7.6.2 Complementary Combination
		7.6.3 Different Formulas on Combination
	7.7 Restricted Combinations
	7.8 Division into Groups
	7.9 Mixed Problems on Permutation and Combination
	Exercise
Chapter 8: Mathematical Induction and Binomial Theorem
	8.1 Principle of Mathematical Induction
	8.2 Binomial Theorem for a Positive Integral Index
	8.3 Properties of the Binomial Coefficients
	8.4 General Term in Binomial Expansion
	8.5 Middle Term in the Binomial Expansion
	8.6 Greatest Term in the Binomial Expansion
	8.7 Binomial Theorem for Any Index
		8.7.1 Binomial Theorem for a Rational Index
	Exercise
Chapter 9: Sequence and Series—Its Application to Business
	9.1 General Idea and Different Types of Sequences
	9.2 Different Types of Sequence
		9.2.1 Finite Sequence
		9.2.2 Infinite Sequence
		9.2.3 Fibonacci Sequence
	9.3 Kinds of sequence
		9.3.1 Arithmetic Sequence
		9.3.2 Geometric Sequence
	9.4 Arithmetic and Geometric Means
	9.5 Arithmetic Progression (AP)
	9.6 Geometric Progression (GP)
		9.6.1 Sum to Infinity of a Geometric Series
		9.6.2 Recurring Decimals as Geometric Progression
	9.7 Harmonic Progression (HP)
		9.7.1 Harmonic Mean
		9.7.2 Special Case
		9.7.3 Method of Differences
	Exercise
Chapter 10: Compound Interest and Annuities
	10.1 Interest
		10.1.1 Simple Interest
		10.1.2 Compound Interest/Continuous Compounding
	10.2 Present Value and Amount of a Sum
	10.3 Annuity and Its Types
	10.4 Present Value of an Annuity (PVA)
	10.5 Debenture
	Exercise
Chapter 11: Coordinate Geometry
	11.1 Fundamentals
	11.2 Rectangular Cartesian Coordinates/Cartesian Coordinates System
	11.3 Polar Coordinates
	11.4 Distance Formula
	11.5 Section Formula
		11.5.1 To Find the Mid-point of the Line Joining (x1, y1) and (x2, y2)
	11.6 Centroid of a Triangle
	11.7 Area of a Triangle
	Exercise
Chapter 12: Locus and Equation to a Locus
	12.1 Locus
	12.2 Equation to a Locus
	Exercise
Chapter 13: Straight Line
	13.1 Introduction
	13.2 Definitions of the Terms
		13.2.1 To Find the Equation of a Line Parallel to the x-axis
	13.3 Different Forms of the Equation of a Straight Line
		13.3.1 Gradient/Slope-intercept Form
		13.3.2 Point Slope Form
		13.3.3 Intercept Form of the Equation of a Straight Line
		13.3.4 Normal Form of the Equation of a Straight Line
		13.3.5 Distance Form of the Equation of a Straight Line
		13.3.6 General Form ax + by + c = 0
	13.4 Distance of a Point from a Straight Line
		13.4.1 To Find the Length of the Perpendicular from the Origin on the Line ax + by + c = 0
	13.5 Angle Between Two Lines and Condition of Parallelism and Perpendicularity of Lines
		13.5.1 Angle Between Two Lines
		13.5.2 Condition for Parallelism of Two Lines
		13.5.3 Condition for Perpendicularity of Two Lines
		13.5.4 Equations of Straight Lines Parallel or Perpendicular to the Given Line
	13.6 Point of Intersection of Two Lines
		13.6.1 General Form of the Equation of a line
	13.7 Condition for Concurrency of Three Given Straight Lines
	13.8 Position of a Point with Respect to a Given Line
	Exercise
Chapter 14: Circle
	14.1 Introduction
	14.2 Standard Form/General Form/Center Radius Form of the Equation of a Circle
	14.3 Finding the Centre and Radius of a Circle
	14.4 Position of a Point with Respect to a Given Circle
	14.5 Condition for a Line to be a Tangent
		14.5.1 Condition for a Line to Touch the Circle with Centre O(0, 0)
		14.5.2 Condition for a Line to Be Tangent to the Circle with Centre C(g, f)
	14.6 Diameter Form of the Equation of a Circle
	14.7 Tangents
		14.7.1 Equations of Tangents to the Circle of the Form x2 + y2 = a2
		14.7.2 Equations of Tangents to the Circle of the Form x2 + 2gx + y2 + 2fy + c = 0
	14.8 Points of Intersection of a Circle and a Straight Line
	14.9 To Find the Length of the Tangent from P(x1, y1) to the Circle
	14.10 Condition for Two Circles to Touch Each Other
	14.11 Normal
	14.12 Equations of the Circles in Some Special Cases
		14.12.1 The Equation of a Circle of Radius r, Passing Through the Origin and Having Its Center on the (i) x-axis and (ii) y-axis
		14.12.2 Equation of a Circle of Radius r and Touching Both the Axes
		14.12.3 Equation of a Circle on the Join of A (x1, y1) and B (x2, y2) as a Diameter
		14.12.4 Equation of a Circle Passing Through the Origin and Making Intercepts a and b on the Axis
	14.13 Angle of Intersection of Two Circles
		14.13.1 Orthogonal Circles
	14.14 Equation of the Common Chord (Radical Axis) of the Two Circles
	14.15 Family of Circles
	Exercise
Chapter 15: Matrix and Its Applications to Business
	15.1 Introduction
	15.2 Definitions and Notations
		15.2.1 Matrix
		15.2.2 General Form of the Matrix
		15.2.3 Square Matrix
		15.2.4 Rectangular Matrix
		15.2.5 Row Matrix
		15.2.6 Column Matrix
		15.2.7 Transpose of a Matrix
		15.2.8 Diagonal Matrix
		15.2.9 Scalar Matrix
		15.2.10 Identity Matrix
		15.2.11 Null Matrix
		15.2.12 Upper Triangular Matrix
		15.2.13 Lower Triangular Matrix
		15.2.14 Triangular Matrix
		15.2.15 Matrix Representation of Multivariable Data
	15.3 Operations on Matrices
		15.3.1 Rules for Matrices Operation
		15.3.2 Properties of Matrix Addition
		15.3.3 Properties of Matrix Multiplication
		15.3.4 Scalar Multiplication of Matrix
	15.4 Conversion of a Business Problem into a Linear System of Equations (LSE) (Figure 15.1)
		15.4.1 Equation
		15.4.2 Linear Equation
	15.5 Determinant of a Square Matrix
		15.5.1 Non-singular Matrix and Singular Matrix
		15.5.2 Sarrus Diagram for Expansion of Determinant of Order 3
		15.5.3 Properties of Determinants
		15.5.4 Rank of a Matrix
		15.5.5 Application of Rank Concept
		15.5.6 Minor and Cofactor
		15.5.7 Adjoint of a Square Matrix
		15.5.8 Inverse of a Square Matrix (Figure 15.2)
		15.5.9 Finding Inverse Using Row Transformation Technique (Figure 15.3)
		15.5.10 The Echelon Matrix: Row Echelon Forms (REF)
		15.5.11 Depreciation Lapse Schedule
	15.6 Matrix Equation
		15.6.1 Conversion
	15.7 Methods to Solve Linear System of Equations
		15.7.1 Solution to the Linear System of Equations
		15.7.2 Types of Solutions
		15.7.3 Cramer's Rule (Figure 15.4)
		15.7.4 Matrix Inversion Method (Figure 15.5)
		15.7.5 Gaussian Elimination Method (Figure 15.6)
		15.7.6 Gauss–Jordan Method (Figure 15.7)
		15.7.7 Variance Analysis
	Exercise
Chapter 16: Application of Matrix Concept
	16.1 Input-Output Analysis
		16.1.1 Assumptions of a Model
	16.2 Markov Analysis
		16.2.1 Stochastic Process
		16.2.2 Markov Process
		16.2.3 One Stage Transition Probability
		16.2.4 Markov Chain
		16.2.5 State Transition Matrix
		16.2.6 Steady State
		16.2.7 Existence of the Steady State
	Exercise
Chapter 17: Differential Calculus and Its Business Applications
	17.1 Introduction
	17.2 Limit of a Function and Rules for Evaluating the Limit of a Function
		17.2.1 Theorems on Limit
		17.2.2 List of Important Results
	17.3 Continuous Function
	17.4 Slope and Rate of Change
		17.4.1 Derivative
	17.5 Method for Evaluating the Differential Coefficient Using the First Principle and Standard Results
		17.5.1 Derivative of a Sum/Difference of Two/More Functions
		17.5.2 Derivative of a Product of Two Functions
		17.5.3 Derivative of a Quotient of Two Functions
		17.5.4 Derivative of the nth Power of a Function
		17.5.5 Derivative of a Logarithmic Function
		17.5.6 Derivative of an Exponential Function
	17.6 Differentiation of an Implicit Function
	17.7 Parametric Differentiation
	17.8 Logarithmic Differentiation
		17.8.1 Properties of Logarithms
		17.8.2 Differentiation of ax w.r.t. x
	17.9 Applications of Differential Calculus
		17.9.1 Average Concept
		17.9.2 Marginal Concept
		17.9.3 Average Cost
		17.9.4 Marginal Cost
		17.9.5 Elasticity
		17.9.6 Price Elasticity of Supply
		17.9.7 Price Elasticity of Demand
		17.9.8 Marginal Revenue and Elasticity of Demand
	17.10 Successive Differentiation
		17.10.1 Definition and Notations
	17.11 Concept of Maxima and Minima
	17.12 Lagrangian Method for Equality Constraints
	17.13 Constrained Optimization with Inequality Constraints: Kuhn-Tucker Conditions
	17.14 Differential Equation
		17.14.1 Formation of a Differential Equation
		17.14.2 Solution to a Given Linear Differential Equation
		17.14.3 Method for Solving the Equation of the Type dy/dx = f(x)/f(y)
		17.14.4 Method for Solving the Differential Equation of the Type dy/dx = f1(x, y)/f2(x, y)
	Exercise
Chapter 18: Applications of Integral Calculus to Business
	18.1 Introduction
	18.2 Constant of Integration
	18.3 List of Standard Integrals
	18.4 Standard Results
	18.5 Definite Integral
		18.5.1 Properties of Definite Integral
	18.6 Integration by Substitution
		18.6.1 Some Special Types
	18.7 Method of Partial Fractions
	18.8 Method of Integration by Parts
	18.9 Application of Integral Calculus to Business
		18.9.1 Cost Function
		18.9.2 Revenue Function
		18.9.3 Consumer's Surplus and Producer's Surplus
	18.10 Learning Curves
		18.10.1 Arithmetic Approach
		18.10.2 Logarithmic Approach
	Exercise
Chapter 19: Functions and Their Applications to Business
	19.1 Introduction
	19.2 Function
		19.2.1 Constant
		19.2.2 Variable
		19.2.3 Single Valued Function
		19.2.4 Many Valued Function
		19.2.5 Explicit and Implicit Functions
		19.2.6 Even and Odd Functions
		19.2.7 Algebraic Function
		19.2.8 Transcendental Function
		19.2.9 Inverse Function
	19.3 Functions Related to Commerce and Economic Theory
		19.3.1 Demand Function
		19.3.2 Cost Function
		19.3.3 Total Cost Function
		19.3.4 Average Cost Function
		19.3.5 Marginal Cost Function
		19.3.6 Revenue Function
		19.3.7 Profit Function
	Exercise
Chapter 20: Linear Programming and Its Application to Business
	20.1 Introduction
	20.2 General Model of the Linear Programming Problem
	20.3 Characteristics of a Linear Programming Problem
	20.4 Assumptions of Linear Programming
		20.4.1 Limitations of Linear Programming Program
	20.5 Formulation of Linear Programming Program
	20.6 Standard form of an LPP
		20.6.1 Conversion of an LPP into Standard Form with Maximization Objective (Figure: 20.1)
	20.7 Solution to an LPP
		20.7.1 Types of Possible Solutions to an LPP
	20.8 Convex Set and Extreme Point
	20.9 Graphical Solution to an LPP
	20.10 Simplex Methods
		20.10.1 Simplex Method-I/Ordinary Simplex Method
		20.10.2 Penalty Method/Big-M Method/Charnes Method (Figure 20.5)
		20.10.3 Two-Phase Method (Figure 20.6)
		20.10.4 Duality Concept in a Linear Programming
		20.10.5 Dual Simplex Method (DSM) (Figure 20.7)
	Exercise
Chapter 21: Transportation Problem
	21.1 Introduction
	21.2 Conversion of a TP into an Equivalent LPP Form
	21.3 Formulation of a Transportation Problem
	21.4 Concept of Feasibility, Basicness and Degeneracy
	21.5 Methods Used to Find the Solution to a Transportation Problem
	21.6 Description of Various Methods to Find the Initial Basic Feasible Solution
		21.6.1 North–West Corner Rule
		21.6.2 Row Minima Method
		21.6.3 Column Minima Method
		21.6.4 Least Cost Method/Matrix Minima Method
		21.6.5 Vogel's Approximation Method
		21.6.6 Effectiveness of Various Methods
	21.7 Stepping Stone Method/Modified Distributive Method
	21.8 Transshipment Problems
	21.9 Sensitivity Analysis for Transportation Problem
		21.9.1 Change in the Objective Function Coefficient by a Non-basic Variable
		21.9.2 Change in the Objective Function Coefficient of a Basic Variable
		21.9.3 Increasing Both Supply Si and Demand dj by Δ
	21.10 Introduction to Assignment Programming Problem
		21.10.1 General Model of the Assignment Problem
		21.10.2 Conversion into an Equivalent LPP
		21.10.3 Solution to the Assignment Problem
		21.10.4 Travelling Salesman Problem
	21.11 Introduction to Game Theory
		21.11.1 Game
		21.11.2 Algebraic Method to Solve a 2 × 2 Game
		21.11.3 Algebraic Method to Solve a 3 × 3 Game
		21.11.4 Graphical Method to Solve a 2 × n or n × 2 Game
		21.11.5 Dominance Property
		21.11.6 Linear Programming Application to Solve a Game
	Exercise
Chapter 22: Probability Theory
	22.1 Introduction
	22.2 Definition for Certain Key Terms
	22.3 Meaning of Probability
		22.3.1 Addition Rules for Probability
		22.3.2 Multiplication Rule on Probability When Events are Independent
		22.3.3 Compound Probability or Conditional Probability
	22.4 Baye's Theorem
	22.5 Random Variables and Expectation
		22.5.1 Introduction
		22.5.2 Random Variable
		22.5.3 Discrete Random Variable
		22.5.4 Continuous Random Variable
	22.6 Probability Distribution
		22.6.1 Discrete Probability Distribution
		22.6.2 Characteristics of a Discrete Distribution
		22.6.3 Probability Function
	22.7 Mathematical Expectation
	22.8 Discrete Probability Distribution (Binomial and Poisson Distributions)
		22.8.1 Introduction
		22.8.2 Binomial Distribution
		22.8.3 Characteristics of a Bernoulli Process
		22.8.4 Definition of Binomial Distribution
		22.8.5 Conditions of Binomial Distribution
		22.8.6 Properties of Binomial Distribution
		22.8.7 Mean of Binomial Distribution
		22.8.8 Variance of Binomial Distribution
	22.9 Poisson Distribution
		22.9.1 Definition of Poisson Distribution
		22.9.2 Properties of Poisson Distribution
		22.9.3 Mean of Poisson Distribution
		22.9.4 Variance of Poisson Distribution
	22.10 Continuous Probability Distribution (Normal Distribution)
		22.10.1 Introduction
		22.10.2 Definition of Normal Distribution
		22.10.3 Standard Normal Distribution
		22.10.4 Properties of Normal Distribution
	Exercise
Chapter 23: Correlation and Regression Analysis
	23.1 Introduction
	23.2 Correlation
		23.2.1 Simple Correlation/Correlation
		23.2.2 Rank Correlation
		23.2.3 Group Correlation
		23.2.4 Assumptions for Karl Pearson's Coefficient of Correlation
		23.2.5 Limitations
		23.2.6 Properties
		23.2.7 Scatter Diagram
	23.3 Karl Pearson Coefficient of Correlation
	23.4 Coefficient of Correlation for a Grouped Data
	23.5 Probable Error of the Coefficient of Correlation
	23.6 Rank Correlation
	23.7 Regression Equations
	Exercise
Chapter 24: Concept of Shares
	24.1 Shares
	24.2 Stock Exchange
		24.2.1 Role of Stock Exchanges in the Economy
	24.3 Nature of Shares
		24.3.1 Face Value
		24.3.2 Book Value
		24.3.3 Market Value
		24.3.4 Difference Between Face Value and Market Value
		24.3.5 Brokerage
		24.3.6 Dividend
	24.4 Different Types of Shares
		24.4.1 Equity Shares
		24.4.2 Preferential Shares
	24.5 Bonus Share
Index