FOUNDATION : PAPER - 4 US - 2016 B A LL SY 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 : August 2016 Reprint : April 2017 Reprint : January 2018 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 : Mega Calibre Enterprises Pvt. Ltd. 06/315 Action Area 3, New Town, Rajarhat, Kolkata 700 160. 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 - 2016 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 1. Arithmetic 20% 2. Algebra 20% Section B: Fundamentals of Business Statistics 3. Statistical representation of Data 10% 4. Measures of Central Tendency and Dispersion 30% 5. Correlation and Regression 10% 6. Probability 10% SECTION A: FUNDAMENTALS OF BUSINESS MATHEMATICS [40 MARKS] 1. Arithmetic (a) Ratios, Variations and Proportions (b) Simple and Compound interest (c) Arithmetic Progression and Geometric Progression 2. Algebra (a) Set Theory (b) Indices and Logarithms (basic concepts) (c) Permutation and Combinations (basic concepts) (d) Quadratic Equations (basic concepts) SECTION B: FUNDAMENTALS OF BUSINESS STATISTICS [60 MARKS] 3. Statistical Representation of Data (a) Diagrammatic representation of data (b) Frequency distribution (c) Graphical representation of Frequency Distribution – Histogram, Frequency Polygon Curve, Ogive, Pie-chart 4. Measures of Central Tendency and Dispersion (a) Mean, Median, Mode, Mean Deviation (b) Range, Quartiles and Quartile Deviation (c) Standard Deviation (d) Co-efficient of Variation (e) Karl Pearson and Bowley’s Coefficient of Skewness 5. Correlation and Regression (a) Scatter diagram (b) Karl Pearson’s Coefficient of Correlation (c) Regression lines, Regression equations, Regression coefficients 6. Probability (a) Independent and dependent events; Mutually exclusive events (b) Total and Compound Probability; Baye’s theorem; Mathematical Expectation 4 I FUNDAMENTALS OF LAWS AND ETHICS Contents SECTION - A BUSINESS MATHEMATICS Study Note 1 : Arithmetic 1.1 Ratios, Variations and Proportions 1 1.2 Simple and Compound Interest 13 1.3 Arithmetic Progression and Geometric Progression 22 Study Note 2 : Algebra 2.1 Set Theory 37 2.2 Indices and Logarithms 45 2.3 Permutations and Combinations 57 2.4 Quadratic Equations 70 SECTION - B STATISTICS Study Note 3 : Statistical Representation of Data 3.1 Introduction to Statistics 77 3.2 Diagramatic Representation of Data 86 3.3 Frequency Distribution 89 3.4 Graphical Representation of Frequency Distribution 93 Study Note 4 : Measures of Central Tendency and Dispersion 4.1 Measures of Central Tendency 99 4.2 Measures of Dispersion 135 4.3 Measures of Skewness 150 Study Note 5 : Correlation and Regression 5.1 Correlation 175 5.2 Regression 194 Study Note 6 : Probability 6.1 General Concept 213 6.2 Some Useful Terms 214 6.3 Measurement of Probability 216 6.4 Theorem of Probability 224 6.5 Bayes’ Theorem 227 6.6 ODDS 231 6.7 Some Important Terms and Concept 239 Section A Fundamentals of Business Mathematics (Syllabus - 2016) Study Note - 1 ARITHMETIC This Study Note includes 1.1 Ratios, Variations and Proportions 1.2 Simple and Compound Interest 1.3 Arithmetic Progression and Geometric Progression 1.1 RATIOS, VARIATIONS AND PROPORTIONS Ratios: Ratio is the comparative relation between two quantities of same kind expressed in the same units. Example: In a class test A secured 80 marks and B secured 40 marks out of 100 then we can compare that A secures double that of B. i.e. Ratio is 80/40 = 2, is a pure natural number, and no unit is associated with it. Note: 1. In the ratio a:b and a and b are called the terms of the ratio. Here a is called antecedent and B is called consequent. Properties of Ratio: The value of ratio remains unchanged when the terms of the ratio are multiplied and divided by the same number. Ex: 2:4, Multiplied by 2, 4:8 9:27, divided by 9, 1:3 2. Two or more ratios can be compared by reducing them to the same denominator. Ex: In the ratios 3:4 and 8:12 which is greater 3 8 , 4 12 98 > 12 12 Ratio of equality and in-equality: 1. If a = b then the ratio a:b is called equal ratio. Ex: 3:3, 4:4 etc are equal ratios. 2. If a>b then the ratio a:b is called greater inequality. Ex: 4:3, 9:7 etc are greater inequalities. 3. If a < b then the ratio a:b is called lesser inequality. Ex: 3:4, 7:9 etc FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS 1 Arithmetic Inverse ratio or reciprocal ratio: The inverse ratio of a:b is b:a. Different kinds of ratios: 1. Compound ratio: If two or more ratios are multiplied together then the ratio is called compounded ratio. Ex: For the ratios a:b, c:d, e:f the compounded ratio is ace : bdf Note: The compounded ratio of two reciprocal ratios is unity. i.e a:b is the reciprocal ratio of b:a then the compounded ratio is ab:ba.. ab →=1 ba 2. Duplicate ratio: If two equal ratios are compounded together then the resulting ratio is called duplicate ratio i.e the duplicate ratio of two equal ratios a:b and a:b is a2:b2. 3. Triplicate ratio: If three equal ratios are multiplied together then the resulting ratio is called triplicate ratio. i.e. the triplicate ratio of a:b,a:b,a:b is a3:b3. 4. Sub duplicate ratio and sub triplicate ratios: a: b is the sub duplicate ratio of a:b and 33a: b is the sub triplicate ratio of a:b. Ex: The sub-duplicate ratio of 4:9 is 4: 9is 2:3. The sub triplicate ratio of 8:64 is 338 : 64 is 2:4= 1:2 5. Continued ratio: The continued ratio is the relation between the magnitudes of two or more ratios and is denoted by a:b:c. Ex: The continued ratio of 2:3 and 4:10. 8:12:30, 4:6:15 Points to be remember:- 1. Reduce the quantities to same units. Ex: if A= ` 2 and B = 50 p Then a:b = 200 : 50 = 4 : 1 = 4 2. When the quantity is increased by given ratio multiply the quantity by greater ratio. 3. When the quantity is decreased by given ratio multiply the quantity by lesser ratio. 4. When both increasing and decreasing of quantities are present in a problem multiply the quantity by greater ratio in increase and multiply the result by lesser ratio to obtain the final result. Proportions (OR) proportional: If two rations are equal then we say that the two ratios are in proportion. In other words the four quantities a,b,c and d are said to be in proportion if a:b = c:d ac ⇒= bd 2 FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS Here the first and last quantities i.e a and d are extremes and the two middle terms b and c are called means. Property: ac The four quantities a, b, c, d are in proportion of ⇔=⇔ad = bc converse is also true. bd Continued proportion: If 3 quantities a,b, and c such that a:b = b:c. then we say that those 3 qualities are in continued proportion. If 3 quantities are in continued proportion then we get b2=ac. Ex: 3,6,12 are in continued proportion. = 62 = 3×12 = 36 Ex: 2, -4, 8 are in continued proportion ⇔ (-4)2 = 8×2 ⇔ 16 = 16 Basic rules of proportions: 1. Invertendo: If a:b = c:d which implies b:a = d:c then we say that the proportion is invertendo. 2. Alternendo: If a:b = c:d ⇒ a:c = b:d, then we say that the proportion in alternendo. 3. Componendo: If a:b = c:d ⇒ a + b:b = c + d:d then we say that proportion is componendo. 4. Dividendo: If a:b = c:d ⇒ a – b:b = c – d:d then we say that the proportion is in dividendo 5. Componendo and dividendo: If a:b = c:d ab++ cd ⇒= then we say that the proportion is in componendo and dividendo. ab−− cd 6. Important theorem: 1/ n ace pan+ qc nn ++ re If === ... ... ... then each ratio = n nn bdf pb+ qd ++ rf Where p, q, r…… are quantities Proof: Let ace ==... ... ... k (say) bdf a ∴ =⇒k a = bk. b c =⇒k c = dk. d e = k = e = fk. f FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS 3 Arithmetic R.H.s: 1/ n pan+ qc nn ++ re ... ... = n nn pb+ qd ++ rf ... ... 1/ n pbnn k+ qc nn k ++ rf nn k ... ... = n nn pb+ qd ++ rf ..