COM M ON PROFICIENCY TEST QUANTITATIVE APTITUDE The Institute of Chartered Accountants of India
The objective of the study material is to provide teaching material to the students to enable them to obtain knowledge and skills in the subject. In case students need any clarifications or have any suggestions to make for further improvement of the material contained herein they may write to the Director of Studies. All care has been taken to provide interpretations and discussions in a manner useful for the students. However, the study material has not been specifically discussed by the Council of the Institute or any of its Committees and the views expressed herein may not be taken to necessarily represent the views of the Council or any of its Committees. Permission of the Institute is essential for reproduction of any portion of this material. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission, in writing, from the publisher. Price : ` 300.00 Revised Edition November 2014 Website : www.icai.org E-mail : bosnoida@icai.in ISBN : 978-81-8441-036-5 Published by The Publication Department on behalf of The Institute of Chartered Accountants of India, ICAI Bhawan, Post Box No. 7100, Indraprastha Marg, New Delhi - 110 002, India. Designed & Printed at Repro India Limited, 50/2, TTC MIDC Industrial Area, Mahape, Navi Mumbai - 400 710, India. November / 2014 / P1659 (Revised)
PREFACE Developing Quantitative Aptitude is important for the students of Chartered Accountancy Course as professional work in future will demand analytical and quantitative skills. Through this section of CPT, it is intended to develop analytical ability of the students using basic mathematical and statistical techniques. By this, students will be equipped with the knowledge to absorb various concepts of other subjects of the chartered accountancy course like accounting, auditing and assurance, financial management, cost accounting, management accounting, etc. The first part of the study material (Chapters 1-9) covers basic mathematical techniques like ratio, proportion, indices, logarithms, equations and inequalities, simple and compound interests, permutations and combinations, sequence and series, sets, relations and basics of differential integral calculus. The second part of the study material (Chapters 10 16) covers basic principles of statistical techniques and measurement thereof. The entire study material has been written in a simple and easy to understand language. A number of illustrations have been incorporated in each chapter to explain various concepts and related computational techniques dealt within each chapter. A reasonably good question bank has been included in the study material which will help the students to prepare for the CPT examination. This study material has been prepared by a team of experts comprising of Dr. Bishwapati Chaudhuri, Prof. Swapan Banerjee, Dean of Commerce St. Xavier College, Kolkata, Dr. Sampa Bose, Dr. Shaligram Shukla, CA. Anjan Bhattacharyya, Shri Indrajit Das, Dr. S.K.Chatterjee, Former Additional Director (SG) and Shri A.K. Aggarwal, Former Additional Director of ICAI. The entire work was co-ordinated by Shri S. Bardhan, Assistant Director, EIRC of the ICAI.
SYLLABUS Quantitative Aptitude (50 Marks) Objective : To test the grasp of elementary concepts in Mathematics and Statistics and application of the same as useful quantitative tools. Contents 1. Ratio and proportion, Indices, Logarithms 2. Equations Linear simultaneous linear equations up to three variables, quadratic and cubic equations in one variable, equations of a straight line, intersection of straight lines, graphical solution to linear equations. 3. Inequalities Graphs of inequalities in two variables common region. 4. Simple and Compound Interest including annuity Applications 5. Basic concepts of Permutations and Combinations 6. Sequence and Series Arithmetic and geometric progressions 7. Sets, Functions and Relations 8. Limits and Continuity Intuitive Approach 9. Basic concepts of Differential and Integral Calculus (excluding trigonometric functions) 10. Statistical description of data (a) (b) (c) Textual, Tabular & Diagrammatic representation of data. Frequency Distribution. Graphical representation of frequency distribution Histogram, Frequency Polygon, Ogive 11. Measures of Central Tendency and Dispersion Arithmetic Mean, Median Partition Values, Mode, Geometric Mean and Harmonic, Mean, Standard deviation, Quartile deviation 12. Correlation and Regression 13. Probability and Expected Value by Mathematical Expectation 14. Theoretical Distributions Binomial, Poisson and Normal. 15. Sampling Theory Basic Principles of sampling theory, Comparison between sample survey and complete enumeration, Errors in sample survey, Some important terms associated with sampling, Types of sampling, Theory of estimation, Determination of sample size. 16. Index Numbers
CONTENTS MATHEMATICS Chapter 1 - Ratio and Proportion, Indices, Logarithms 1.1 Ratio 1.2 1.2 Proportion 1.7 1.3 Indices 1.14 1.4 Logarithm 1.22 Additional Question Bank 1.34 Chapter 2 - Equations 2.1 Introduction 2.2 2.2 Simple Equation 2.2 2.3 Simultaneous Linear Equations in two unknowns 2.6 2.4 Method of Solution 2.6 2.5 Method of Solving Simultaneous Linear Equation with three variables 2.8 2.6 Problems Leading to Simultaneous Equations 2.13 2.7 Quadratic Equation 2.15 2.8 How to Construct a Quadratic Equation 2.16 2.9 Nature of the Roots 2.16 2.10 Problems on Quadratic Equation 2.23 2.11 Solution of Cubic Equation 2.26 2.12 Application of Equations in Co-ordinate Geometry 2.28 2.13 Equation of a Straight Line 2.29 2.14 Graphical Solution to Linear Equations 2.35 Additional Question Bank 2.39 Chapter 3 - Inequalities 3.1 Inequalities 3.2 3.2 Linear Inequalities in one variable and the Solution space 3.2 Additional Question Bank 3.19
CONTENTS Chapter 4 - Simple and Compound Interest Including Annuity - Applications 4.1 Introduction 4.2 4.2 Why is Interest Paid? 4.2 4.3 Definition of Interest and some other Related Terms 4.3 4.4 Simple Interest and Compound Interest 4.3 4.5 Effective Rate of Interest 4.17 4.6 Annuity 4.21 4.7 Future Value 4.23 4.8 Present Value 4.27 4.9 Sinking Fund 4.33 4.10 Applications 4.34 Additional Question Bank 4.39 Chapter 5 - Basic Concepts of Permutations and Combinations 5.1 Introduction 5.2 5.2 The Factorial 5.3 5.3 Permutations 5.3 5.4 Results 5.5 5.5 Circular Permutations 5.9 5.6 Permutation with Restrictions 5.10 5.7 Combinations 5.15 5.8 Standard Results 5.22 Additional Question Bank 5.31 Chapter 6 - Sequence and Series - Arithmetic and Geometric Progressions 6.1 Sequence 6.2 6.2 Series 6.3 6.3 Arithmetic Progression (A.P.) 6.3 6.4 Geometric Progression (G.P.) 6.9 6.5 Geometric Mean 6.11 Additional Question Bank 6.21
CONTENTS Chapter 7 - Sets, Functions and Relations 7.1 Sets 7.2 7.2 Venn Diagrams 7.5 7.3 Product Sets 7.9 7.4 Relations and Functions 7.10 7.5 Domain & Range of a Function 7.10 7.6 Various Types of Function 7.10 Additional Question Bank 7.22 Chapter 8 - Limits and Continuity - Intuitive Approach 8.1 Introduction 8.2 8.2 Types of Functions 8.3 8.3 Concept of Limit 8.5 8.4 Useful Rules of Theorems on Limits 8.7 8.5 Some Important Limits 8.8 8.6 Continuity 8.16 Additional Question Bank 8.24 Chapter 9 - Basic Concepts of Differential and Integral Calculus (A) Differential Calculus 9.A.1 Introduction 9.2 9.A.2 Derivative or Differential Coefficient 9.2 9.A.3 Some Standard Results (Formulas) 9.5 9.A.4 Derivative of a Function of Function 9.8 9.A.5 Implicit Functions 9.8 9.A.6 Parametric Equation 9.9 9.A.7 Logarithmic Differentiation 9.9 9.A.8 Some More Examples 9.10 9.A.9 Basic Idea about Higher Order Differentiation 9.12 9.A.10 Geometric Interpretation of the Derivative 9.13 (B) Integral Calculus 9.B.1 Integration 9.18 9.B.2 Basic Formulas 9.19
CONTENTS 9.B.3 Method of Substitution (change of variable) 9.21 9.B.4 Integration By Parts 9.23 9.B.5 Method of Partial Fraction 9.24 9.B.6 Definite Integration 9.27 9.B.7 Important Properties 9.28 Additional Question Bank 9.37 STATISTICS Chapter 10 - Statistical Description of Data 10.1 Introduction of Statistics 10.2 10.2 Collection of Data 10.4 10.3 Presentation of Data 10.6 10.4 Frequency Distribution 10.14 10.5 Graphical representation of Frequency Distribution 10.19 Additional Question Bank 10.37 Chapter 11 - Measures of Central Tendency and Dispersion 11.1 Definition of Central Tendency 11.2 11.2 Criteria for an ideal measure of Central Tendency 11.2 11.3 Arithmetic Mean 11.3 11.4 Median Partition Values 11.8 11.5 Mode 11.14 11.6 Geometric Mean and Harmonic Mean 11.15 11.7 Exercise 11.23 11.8 Definition of Dispersion 11.30 11.9 Range 11.31 11.10 Mean Deviation 11.32 11.11 Standard Deviation 11.38 11.12 Quartile Deviation 11.47 11.13 Exercise 11.54 Additional Question Bank 11.61
CONTENTS Chapter 12 - Correlation and Regression 12.1 Introduction 12.2 12.2 Bivariate Data 12.2 12.3 Correlation Analysis 12.5 12.4 Measures of Correlation 12.6 12.5 Regression Analysis 12.25 12.6 Properties of Regression Lines 12.34 12.7 Review of Correlation and Regression Analysis 12.37 Additional Question Bank 12.51 Chapter 13 - Probability and Expected Value by Mathematical Expectation 13.1 Introduction 13.2 13.2 Random Experiment 13.2 13.3 Classical Definition of Probability 13.3 13.4 Statistical Definition of Probability 13.8 13.5 Operations on Events: Set Theoretic Approach to Probability 13.10 13.6 Axiomatic or Modern Definition of Probability 13.13 13.7 Addition Theorems 13.14 13.8 Conditional Probability and Compound Theorem of Probability 13.17 13.9 Random Variable-Its Probability Distribution 13.26 13.10 Expected Value of a Random Variable 13.28 Additional Question Bank 13.49 Chapter 14 - Theoretical Distributions 14.1 Introduction 14.2 14.2 Binomial Distribution 14.3 14.3 Poisson Distribution 14.10 14.4 Normal Distribution or Gaussian Distribution 14.19 14.5 Chi-square Distribution, t-distribution and F-Distribution 14.33 Additional Question Bank 14.49
CONTENTS Chapter 15 - Sampling Theory 15.1 Introduction 15.2 15.2 Basic Principles of Sample Survey 15.3 15.3 Comparison between Sample Survey and Complete Enumeration 15.4 15.4 Errors in Sample Survey 15.4 15.5 Some important terms associated with Sampling 15.5 15.6 Types of Sampling 15.11 15.7 Theory of Estimation 15.14 15.8 Determination of sample size for a Specific Precision 15.23 Additional Question Bank 15.31 Chapter 16 - Index Numbers 16.1 Introduction 16.2 16.2 Issues Involved 16.3 16.3 Construction of Index Number 16.3 16.4 Usefulness of Index Numbers 16.10 16.5 Deflating Time Series using Index Numbers 16.10 16.6 Shifting and Splicing of Index Numbers 16.11 16.7 Test of Adequacy 16.12 Additional Question Bank 16.20 Appendices