FOUNDATION : PAPER - 4 FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS FOUNDATION STUDY NOTES The Institute of Cost Accountants of India CMA Bhawan, 12, Sudder Street, Kolkata - 700 016 First Edition : January 2013 Second Edition : September 2014 Published by : Directorate of Studies The Institute of Cost Accountants of India (ICAI) CMA Bhawan, 12, Sudder Street, Kolkata - 700 016 www.icmai.in Printed at : Repro India Limited Plot No. 02, T.T.C. MIDC Industrial Area, Mahape, Navi Mumbai 400 709, India. Website : www.reproindialtd.com Copyright of these Study Notes is reserved by the Institute of Cost Accountants of India and prior permission from the Institute is necessary for reproduction of the whole or any part thereof. Syllabus PAPER 4: FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS (FBMS) Syllabus Structure A Fundamentals of Business Mathematics 40% B Fundamentals of Business Statistics 60% A 40% B 60% ASSESSMENT STRATEGY There will be written examination paper of three hours. OBJECTIVES To gain understanding on the fundamental concepts of mathematics and statistics and its application in business decision- making Learning Aims The syllabus aims to test the student’s ability to: Understand the basic concepts of basic mathematics and statistics Identify reasonableness in the calculation Apply the basic concepts as an effective quantitative tool Explain and apply mathematical techniques Demonstrate to explain the relevance and use of statistical tools for analysis and forecasting Skill sets required Level A: Requiring the skill levels of knowledge and comprehension CONTENTS Section A: Fundamentals of Business Mathematics 40% 1. Arithmetic 2. Algebra 3. Calculus Section B: Fundamentals of Business Statistics 60% 4. Statistical representation of Data 5. Measures of Central Tendency and Dispersion 6. Correlation and Regression 7. Index Numbers 8. Time Series Analysis- basic applications including Moving Average 9. Probability 10. Theoretical Distribution SECTION A: FUNDAMENTALS OF BUSINESS MATHEMATICS [40 MARKS] 1. Arithmetic (a) Ratios and Proportions (b) Simple and Compound interest including application of Annuity (c) Bill Discounting and Average Due Date (d) Mathematical reasoning – basic application 2. Algebra (a) Set Theory and simple application of Venn Diagram (b) Variation, Indices, Logarithms (c) Permutation and Combinations – basic concepts (d) Linear Simultaneous Equations ( 3 variables only) (e) Quadratic Equations (f) Solution of Linear inequalities (by geometric method only) (g) Determinants and Matrices 3. Calculus (a) Constant and variables, Functions, Limit & Continuity (b) Differentiability & Differentiation, Partial Differentiation (c) Derivatives – First order and Second order Derivatives (d) Maxima & Minima – without constraints and with constraints using Lagrange transform (e) Indefinite Integrals: as primitives, integration by substitution, integration by part (f) Definite Integrals: evaluation of standard integrals, area under curve SECTION B: FUNDAMENTALS OF BUSINESS STATISTICS [60 MARKS] 4. Statistical Representation of Data (a) Diagrammatic representation of data (b) Frequency distribution (c) Graphical representation of Frequency Distribution – Histogram, Frequency Polygon, Ogive, Pie-chart 5. Measures of Central Tendency and Dispersion (a) Mean, Median, Mode, Mean Deviation (b) Quartiles and Quartile Deviation (c) Standard Deviation (d) Co-efficient of Variation, Coefficient of Quartile Deviation 6. Correlation and Regression (a) Scatter diagram (b) Karl Pearson’s Coefficient of Correlation (c) Rank Correlation (d) Regression lines, Regression equations, Regression coefficients 7. Index Numbers (a) Uses of Index Numbers (b) Problems involved in construction of Index Numbers (c) Methods of construction of Index Numbers 8. Time Series Analysis – basic application including Moving Average (a) Moving Average Method (b) Method of Least Squares 9. Probability (a) Independent and dependent events; Mutually exclusive events (b) Total and Compound Probability; Baye’s theorem; Mathematical Expectation 10. Theoretical Distribution (a) Binomial Distribution, Poisson Distribution – basic application (b) Normal Distribution – basic application 4 I FUNDAMENTALS OF LAWS AND ETHICS Content SECTION - A BUSINESS MATHEMATICS Study Note 1 : Arithmetic 1.1 Ratio & Proportion 1.1 1.2 Simple & Compound Interest (Including Application of Annuity) 1.6 1.3 Discounting of Bills and Average Due Date 1.19 1.4 Mathematical Reasoning - Basic Application 1.37 Study Note 2 : Algebra 2.1 Set Theory 2.1 2.2 Inequations 2.15 2.3 Variation 2.18 2.4 Logarithm 2.23 2.5 Laws of Indices 2.33 2.6 Permutation & Combination 2.39 2.7 Simultaneous Linear Equations 2.51 2.8 Matrices & Determinants 2.56 Study Note 3 : Calculus 3.1 Function 3.1 3.2 Limit 3.6 3.3 Continuity 3.17 3.4 Derivative 3.23 3.5 Integration 3.57 SECTION - B STATISTICS Study Note 4 : Statistical Representation of Data 4.1 Diagramatic Representation of Data 4.1 4.2 Frequency Distribution 4.5 4.3 Graphical Representation of Frequency Distribution 4.10 Study Note 5 : Measures Of Central Tendency and Measures of Dispersion 5.1 Measures of Central Tendency or Average 5.1 5.2 Quartile Deviation 5.42 5.3 Measures of Dispersion 5.44 5.4 Coefficient Quartile & Coefficient variation 5.61 Study Note 6 : Correlation and Regression 6.1 Correlation & Co-efficient 6.1 6.2 Regression Analysis 6.20 Study Note 7 : Index Numbers 7.1 Uses of Index Numbers 7.1 7.2 Problems involved in construction of Index Numbers 7.2 7.3 Methods of construction of Index Numbers 7.2 7.4 Quantity Index Numbers 7.12 7.5 Value Index Number 7.13 7.6 Consumber Price Index 7.13 7.7 Aggregate Expenditure Method 7.14 7.8 Test of Adequacy of the Index Number Formulae 7.15 7.9 Chain Index Numbers 7.19 7.10 Steps in Construction of Chain Index 7.20 Study Note 8 : Time Series Analysis 8.1 Definition 8.1 8.2 Components of Time Series 8.1 8.3 Models of Time Series Analysis 8.2 8.4 Measurement of Secular Trend 8.3 8.5 Method of Semi Averages 8.3 8.6 Moving Average Method 8.3 8.7 Method of Least Squares 8.6 Study Note 9 : Probability 9.1 General Concept 9.1 9.2 Some Useful Terms 9.1 9.3 Measurement of Probability 9.3 9.4 Theorems of Probability 9.8 9.5 Bayes’ Theorem 9.11 9.6 Odds 9.15 Study Note 10 : Theoretical Distribution 10.1 Theoretical Distribution 10.1 10.2 Binomial Distribution 10.1 10.3 Poisson Distribution 10.13 10.4 Normal Distribution 10.19 Section - A BUSINESS MATHEMATICS Study Note - 1 ARITHMETIC This Study Note includes 1.1 Ratio & Proportion 1.2 Simple & Compound Interest (Including Application of Annuity) 1.3 Discounting of Bills & Average Due Date 1.4 Mathematical Reasoning - Basic Application 1.1 RATIO AND PROPORTION 1.1.1 Ratio The ratio between quantities a and b of same kind is obtained by dividing a by b and is denoted by a : b. Inverse Ratio: For the ratio a : b inverse ratio is b : a. A ratio remains unaltered if its terms are multiplied or divided by the same number. a : b = am : bm (multiplied by m) a b a : b = : (divided by m ≠ 0) m m 4 6 Thus 2 : 3 = 2 x 2 : 3 x 2 = 4 : 6 = : = 2 : 3 2 2 If a = b, the ratio a : b is known as ratio of equality. If a > b, then ratio a : b is known as ratio of greater inequality i.e. 7 : 4 And for a < b, ratio a : b will be the ratio of Lesser inequality i.e. 4 : 7. Solved Examples: Example 1 : Reduce the two quantities in same unit. If a = 2kg., b = 400gm, then a : b = 2000 : 400 = 20 : 4 = 5 : 1 (here kg is changed to gm) Example 2 : If a quantity increases by a given ratio, multiply the quantity by the greater ratio. 5 If price of crude oil increased by 4 : 5, which was ` 20 per unit of then present price = 20 × 4 = ` 25 per unit. Example 3 : If again a quantity decreases by a given ratio, then multiply the quantity by the lesser ratio. 3 In the above example of the price of oil is decreased by 4 : 3, the present price = 20 × 4 = ` 15 per unit. If both increase and decrease of a quantity are present is a problem, then multiply the quantity by greater ratio for increase and lesser ratio for decrease, to obtain the final result. 1.1.2 Proportion a c An equation that equates two ratios is a proportion. For instance, if the ratio is equal to the ratio , b d then the following proportion can be written: Means Extremes a c = b d FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 1.1 Arithmetic The numbers a and d are the extremes of the proportion. The numbers b and c are the means of the proportion. Properties of proportions 1. Cross product property: The product of the extremes equals the product of the means. a c If = , then ad = bc. b d 2. Reciprocal property: If two ratios are equal, then their reciprocals are also equal. a c b d If = , then = . b d a c Few Terms : 1. Continued proportions : The quantities a, b, c, d, e…. are said to be in continued proportion of a : b = b : c = c …. Thus 1, 3, 9, 27, 81, ….. are in continued proportion as 1 : 3 = 3 : 9 = 9 : 27 = 27 : 81 = …. Say for example : If 2, x and 18 are in continued proportion, find x. Now 2 : x = x : 18 or, 2 x =or,x2 = 36or,x = ± 6 x 18 Observation: If a, b, c are in continued proportion, the b2 = ac, b = ± ac . 2. Compound Proportion : If two or more ratios are multiplied together then they are known as compounded. Thus a1 a2 a3 : b1 b2 b3 is a compounded ratios of the ratios a1 : b1 ; a2 : b2 and a3 : b3.