(i) (ii) • Head Office : B-32, Shivalik Main Road, Malviya Nagar, New Delhi-110017 • Sales Office : B-48, Shivalik Main Road, Malviya Nagar, New Delhi-110017 Tel. : 011-26691021 / 26691713 Price : ` 600 Typeset by Disha DTP Team DISHA PUBLICATION ALL RIGHTS RESERVED © Copyright Publisher No part of this publication may be reproduced in any form without prior permission of the publisher. The author and the publisher do not take any legal responsibility for any errors or misrepresentations that might have crept in. We have tried and made our best efforts to provide accurate up-to-date information in this book. For further information about the books from DISHA, Log on to www.dishapublication.com or email to [email protected] (iii) Preface Quantitative Aptitude forms a very important part of preparation of MBA aspirants. Not just the Quant section but it forms the backbone of the Data Interpretation, Data Sufficiency and Reasoning. Disha’s Quantitative Aptitude for CAT/ XAT/ IIFT/ CMAT/ MAT/ Bank PO/ SSC is a book focused on mastering techniques to crack these examinations. The book starts from a foundation level and moves to an expert level. Structure of the book: The book comprises of 5 Units (Numbers, Arithmetic, Algebra, Geometry and Counting Principles) which have been further divided into 22 chapters followed by 3 Mock Tests. Each chapter consists of • Theory with Illustrations • Foundation Level Exercise • Standard Level Exercise • Expert Level Exercise • Solutions to the 3 levels of exercises • Test Yourself • Solutions to Test Yourself Salient Features of the Book: • Each chapter includes detailed review of all the concepts involved with exhaustive number of well discussed Illustrations. • The theory is followed by 3 levels of exercises – Foundation Level, Standard Level and Expert Level. The detailed solution to each and every question has been provided immediately at the end of the 3 exercises. • Foundation Level : Here the focus is to expose the students to solve problems based on the concepts they have learned in theory part. The student develops a good foundation and is ready for the Standard level. • Standard Level : The Standard level is a collection of excellent quality problems which will test a student on the application of the concepts learned in various real-life situations. The problems provide a good platform to develop a very good problem solving aptitude so as to take up the expert level confidently. • Expert Level : This is the toughest part of the book and involves the trickiest questions on the concepts involved. Here most of the problems will pose good challenge to the students. • The book contains 22 Chapter-wise Tests – Test Yourself - on the basis of latest CAT pattern after the exercises in each chapter. The students must attempt these tests in specified time limits and conditions. • At the end of the book 3 Mock Tests are provided based on the exact pattern of latest CAT exams. The solutions to the test are provided at the end of the tests. • The book contains questions of past exams of CAT/ XAT/ IIFT/ SNAP/ NMAT/ ATMA/ FMS in the various exercises and Illustrations. We would like to thank the DTP team at Disha, especially Mr. Amit Kumar Jha, who have worked really hard to bring the book to the present shape. Although we have taken utmost care while preparing the book but errors might have crept in. We would like to request our readers to highlight these errors. Authors (iv) Contents Unit-I : Numbers ● Last Two Digits of a Number with Large Power ● Number of Zeroes in an Expression like 1. Fundamentals 1-28 a × b × c × ..., where a, b, c,... are Natural Numbers ● Powers of a Number Contained in a Factorial ● Introduction ● Base System ● Shortcuts for Addition and Subtraction ● Successive Division ● Shortcuts for Multiplication ● Factors and Multiples ● Rounding off and Its Uses ● Highest Common Factor (HCF) or ● ‘BODMAS’ Rule Greatest Common Divisor (GCD) ● Brackets ● Least Common Multiple (LCM) ● Factorial ● Greatest Integral Value ● Roman Numbers ● Practice Exercises : ● Important Conversion u Foundation Level u Standard Level u Expert Level ● Absolute Value or Modulus of a Number ● Test Yourself ● Hints & Solutions ● Properties of a Modulus ● Explanation of Test Yourself ● Powers or Exponents ● Algebraic Identities Unit-II : Arithmetic ● Squares ● Properties of Squares 3. Averages 67-86 ● Square Roots ● Average ● Cubes ● Position of the Average on the Number Line ● Practice Exercises : ● Weighted Average u Foundation Level u Standard Level u Expert Level ● Properties of Average (Arithmetic Mean) ● Test Yourself ● Practice Exercises : ● Hints & Solutions u Foundation Level u Standard Level u Expert Level ● Explanation of Test Yourself ● Test Yourself ● Hints & Solutions 2. Number System 29-66 ● Explanation of Test Yourself ● Introduction 4. Alligations 87-102 ● Tabular Classification of Numbers ● Alligation ● Concept of Number Line (or Number Line) ● Solving the Problems of Alligations Using Alligation Formula ● Conversion of Rational Number of the Form Non-terminating ● Graphical Representation of Alligation-Cross Method Recurring Decimal into the Rational Number of the form p/q ● The Straight Line Approach to Solve the Problems Related to ● Division Alligations ● Tests of Divisibility ● Recognition of Different Situations Where Alligation can be ● Prime Numbers Used ● Complex Numbers, Real Numbers and Imaginary Numbers ● A Typical Problem ● General or Expanded Form of 2 and 3 Digits Numbers ● Practice Exercises : ● Sum of Numbers Formed with given Different Digits u Foundation Level u Standard Level u Expert Level ● Factorisation ● Test Yourself ● Number of Factors of a Composite Number ● Hints & Solutions ● Number of ways of Expressing a Composite ● Explanation of Test Yourself Number as a Product of Two Factors ● Sum of Factors (or Divisors) of a composite Number 5. Percentages 103-130 ● Sum of Unit Digits ● Introduction ● The Last Digit From Left (i.e., unit digit) of ● Basic Definition of Percentage Any Power of a Number ● Percentage Increase, Percentage Decrease and Percentage ● Concept of Remainders Change ● To Find the Last Digits of the ● Percentage Point Change and Percentage Change Expression like a1 × a2 × a3 × ... × an ● Calculation of Percentage Value Through Addition (v) ● Effect of Percentage Change in the Numerator on the Value of 9. Time and Work 207-238 a Ratio ● Percentage Change Graphic ● Introduction ● Application of Percentage Change Graphic (PCG) ● Concept of Efficiency ● Concept of Negative Work ● Calculation of Multiplication by Numbers Like 2.14, 1.04, 0.35, 0.94 and so on Using Percentage ● Concept of Man-days ● Practice Exercises : ● Work Done u Foundation Level u Standard Level u Expert Level ● Work Done Equation ● Test Yourself ● Work in Terms of Volume (special case as building a wall) ● Hints & Solutions ● Extension of the Concept of Time and Work ● Explanation of Test Yourself ● Practice Exercises : u Foundation Level u Standard Level u Expert Level 6. Profit, Loss and Discount 131-158 ● Test Yourself ● Hints & Solutions ● Introduction ● Explanation of Test Yourself ● Total Cost Price (CP) ● Selling Price (SP) 10. Time, Speed and Distance 239-280 ● Profit (or Gain) and Loss ● Introduction ● Use of PCG (Percentage Change Graphic) in Profit and Loss ● Motion or Movement ● Marked Price, List Price, Discount and Successive Discounts ● Conversion of kmph (kilometer per hour) to m/s ● Contribution Margin (CM) (metre per second) and vice-versa ● Break-Even Point and Break-Even Sales ● Direct and Inverse Proportionality Between any Two of the Speed (S), ● Practice Exercises : Time (T) and Distance (D) When the Third One is Constant u Foundation Level u Standard Level u Expert Level ● Average Speed ● Test Yourself ● Relative Speed ● Hints & Solutions ● To and Fro Motion in a Straight Line Between Two Points A ● Explanation of Test Yourself and B ● Uniform Acceleration and Uniform Deceleration 7. Interest 159-180 ● Application of Alligation in the Problems Related to Time, Speed ● Introduction and Distance ● Interest ● Concept Related to Motion of Trains ● Simple Interest (S.I.) ● Boats and Streams ● Compound Interest (C.I.) ● Basic Terminology Related to Races ● Practice Exercises : ● Circular Motion u Foundation Level u Standard Level u Expert Level ● Clocks ● Test Yourself ● Practice Exercises : ● Hints & Solutions u Foundation Level u Standard Level u Expert Level ● Test Yourself ● Explanation of Test Yourself ● Hints & Solutions 8. Ratio, Proportion and Variation 181-206 ● Explanation of Test Yourself ● Introduction Unit-III : Algebra ● Ratio ● Decimal and Percentage Value of a Ratio 11. Progressions 281-306 ● Properties of Ratios ● Introduction ● Uses of Ratios ● Arithmetic Progressions (A.P.) ● Comparison of Ratios ● Calculation of Percentage Change in Ratio Using PCG ● nth Term of an A.P. (Percentage Change Graphic) ● Sum of First n Terms of an A.P. ● Proportion ● Special Cases of A.P.s in which Sum upto Different Terms are ● Properties of Proportion the Same ● Variations ● Arithmetic Mean of n Numbers ● Types of Variations ● Geometric Progression (G.P.) ● Compound Variations ● Considering the Terms in a G.P. ● Practice Exercises : ● Geometric Mean of n Numbers u Foundation Level u Standard Level u Expert Level ● Harmonic Progression (H.P.) ● Test Yourself ● Relations between Arithmetic Mean (A.M.), Geometric Mean ● Hints & Solutions (G.M.) and Harmonic Mean (H.M.) ● Explanation of Test Yourself ● Useful Results (vi) ● Practice Exercises : ● Solutions of Quadratic Inequalities u Foundation Level u Standard Level u Expert Level ● Solution of System of Inequalities in one Variable ● Test Yourself ● Inequalities Containing a Modulus ● Hints & Solutions ● Practice Exercises : ● Explanation of Test Yourself u Foundation Level u Standard Level u Expert Level ● Test Yourself 12. Linear Equations 307-326 ● Hints & Solutions ● Linear Equations ● Explanation of Test Yourself ● Steps to be Followed to Solve a Word 16. Logarithms 387-408 Problem Using Linear Equation(s) ● Practice Exercises : ● Introduction u Foundation Level u Standard Level u Expert Level ● Definition ● Test Yourself ● Laws of Logarithm ● Hints & Solutions ● Some Important Properties ● Explanation of Test Yourself ● Characteristics and Mantissa ● Very Useful Results 13. Functions 327-346 ● Practice Exercises : ● Introduction u Foundation Level u Standard Level u Expert Level ● Test Yourself ● Function ● Hints & Solutions ● Rules for Finding the Domain of a Function ● Methods of Representation of Functions ● Explanation of Test Yourself ● Some Special Functions 17. Set Theory 409-430 ● Shifting of Graphs ● Combination of Shifting of a Graph ● Introduction ● Practice Exercises : ● Sets u Foundation Level u Standard Level u Expert Level ● Representations of Sets ● Test Yourself ● Standard Symbols of Some Special Sets ● Hints & Solutions ● Types of Sets ● Explanation of Test Yourself ● Subsets ● Intervals as Subsets of a Set of Real Numbers (R) 14. Quadratic & Cubic Equations 347-368 ● Power Set of a Set ● Introduction ● Universal Set ● Quadratic Polynomials ● Venn Diagrams ● Quadratic Equations ● Operation on Sets ● Graph of a Quadratic Expression ● Disjoint Sets ● Geometrical Meaning of Roots or Solutions of a ● Cardinal Number Quadratic Equation ● Situation Based Venn Diagrams ● Sign of a Quadratic Expression ● Practice Exercises : ● Sum and Product of Roots u Foundation Level u Standard Level u Expert Level ● Formation of an Equation with Given Roots ● Test Yourself ● Greatest and Least Value of a Quadratic Expression ● Hints & Solutions ● Cubic Equations ● Explanation of Test Yourself ● Bi-quadratic equation ● Practice Exercises : Unit-IV : Geometry u Foundation Level u Standard Level u Expert Level ● Test Yourself 18. Geometry 431-492 ● Hints & Solutions ● Introduction ● Explanation of Test Yourself ● Points, Lines, Line Segment, Ray and Plane 15. Inequalities 369-386 ● Lines and Angles ● Polygons ● Introduction ● Triangles ● Inequality ● Basic Properties and Some Important Theorems of Triangles ● Types of Inequalities ● Important Terms Related to a Triangle ● Some Properties of Inequality ● Congruency of Two Triangles ● Important Results ● Solution of an Inequality ● Similarity of Two Triangles ● Equivalent Inequalities ● Quadrilaterals ● Notation and Ranges ● Circles ● Solutions of Linear Inequalities in one Unknown ● Basic Pythagorean Triplets (vii) ● Determination of Nature of Triangle ● Practice Exercises : ● Important Points u Foundation Level u Standard Level u Expert Level ● Locus ● Test Yourself ● Sine and Cosine Rule ● Hints & Solutions ● Practice Exercises : ● Explanation of Test Yourself u Foundation Level u Standard Level u Expert Level Unit-V : Counting Principles ● Test Yourself ● Hints & Solutions 21. Permutations and Combinations 575-608 ● Explanation of Test Yourself ● Introduction 19. Mensuration 493-548 ● Fundamental Principle of Counting ● Introduction ● Factorials ● Basic Conversion of Units ● Meaning of Permutation and Combination ● Plane Figures ● Counting Formula for Linear Permutations ● Area of a Triangle ● Number of Linear Permutations Under Certain Conditions ● Area of a Quadrilateral ● Circular Permutations ● Area of a Regular Hexagon ● Counting Formula for Combination ● Area of Irregular Plane Figures ● Division and Distribution of Objects ● Paths ● Dearrangement ● Area Related to a Circle ● Important Results about Points ● Surface area and Volume of Solids ● Finding the Rank of a Word ● Euler’s Rule ● Practice Exercises : ● Circle Packing in a Square u Foundation Level u Standard Level u Expert Level ● Circles Packing in a Circle ● Test Yourself ● Some Other Important Concepts ● Hints & Solutions ● Practice Exercises : ● Explanation of Test Yourself u Foundation Level u Standard Level u Expert Level ● Test Yourself 22. Probability 609-652 ● Hints & Solutions ● Introduction ● Explanation of Test Yourself ● Concept of probability 20. Coordinate Geometry 549-574 ● Basic Terms ● Mathematical Definition of Probability ● Introduction ● Odds Against and Odds in Favour of an Event ● Rectangular Coordinate Axes ● Addition Theorem ● Sign Conventions in the xy-Plane ● Independent Events ● Quadrants of xy-Plane and Sign of x and y-Coordinate of a ● Conditional Probability Point in Different Quadrants ● Geometrical Applications ● Plotting a Point Whose Coordinates are Known ● Probability Regarding n Letters and Their Envelopes ● Distance Formula ● Expectation ● Applications of Distance Formula ● Practice Exercises : ● Section Formula u Foundation Level u Standard Level u Expert Level` ● Coordinates of Some Particular Points ● Test Yourself ● Area of Triangle and Quadrilateral ● Hints & Solutions ● Transformation of Axes ● Explanation of Test Yourself ● Image of a Point ● Equation of Straight Line Parallel to An Axis Mock Tests ● Inclination of a Straight Line ● Slope (or Gradient) of a Straight Line Mock Test - 1 653-654 ● Equation of Straight Lines ● Different Forms of the Equation of a Straight Line Mock Test - 2 655-656 ● Point of Intersection of Two Lines ● Position of a Point Relative to a Line Mock Test - 3 657 ● Angle Between Two Straight Lines ● Equation of Parallel and Perpendicular Lines Hints & Solutions (Mock Test - 1 to 3) 658-666 ● Distance of a Line from a Point ● Distance Between Two Parallel Lines s r e I - Chapter 1 Fundamentals T b Chapter 2 Number System I m N U u N 1 FUNDAMENTALS l Introduction l Absolute Value or Modulus of a Number l Shortcuts For Addition and Subtraction l Properties of a Modulus l Shortcuts for Multiplication l Powers or Exponents l Rounding off and Its Uses l Algebraic Identities l ‘BODMAS’ Rule l Squares l Brackets l Properties of Squares l Factorial l Square Roots l Roman Numbers l Cubes l Important Conversion INTRODUCTION First add the thousands like 64 + 43 (= 107) + 94 (= 201) + 8 (= 209) + 7 = 216 In the CAT and the likes of competitions, 25% to 35% questions At this stage you know the answer would be 216000 + are based on numeracy. So, to save the time for other questions in (a maximum of 5000), as there are five numbers whose last competitions, it is essential to command over shortcuts of addition, 3 digits numbers are not added. If the range from 216000 to subtraction and multiplications given in this chapter. 221000 is sufficient to choose the correct option, then no need SHORTCUTS FOR ADDITION AND to add further otherwise add the hundredth digits of given numbers 1 + 3 + 9 = 13. SUBTRACTION At this stage you know the answer would be Addition is the mother of all calculations, which gives you an 217300 + (a maximum of 500) extra edge that makes your calculations faster. Subtraction is the If the range from 217300 to 217800 is sufficient to choose extension of addition. the correct option, then no need to add further otherwise add I. Addition of smaller number to larger number is easier than the last two digits of numbers addition of larger number to smaller number. For example 83 + 2 + 20 + 15 + 42 = 162. addition in the order 5817 + 809 + 67 + 8 is easier than the Hence the correct sum will be 217300 + 162 = 217462. addition in the order 8 + 67 + 809 + 5817. Hence to add the There are two advantages of process of addition (i) No need numbers, it is better to first arrange them in decreasing order to get final sum as in this process of addition, you could and then add them. choose the correct option at earlier stage also. (ii) In the entire II. To find the sum like 6345 + 2476 + 802, first add the calculation, you have not gone above two digits additions. thousands and then hundreds, tens and once in order. Thus IV. Sometimes you have to add so many large numbers. In that case 6345 + 2476 + 802 = 6000 + 2000 ( = 8000) + 300 (= 8300) you can find the required sum using the following methods. + 400 (= 8700) + 800 (= 9500) + 40 (= 9540) + 70 (= 9610) (A) Column Form + 5 (= 9615) + 6 (9621) + 2 Write the given numbers one below the other with right align if = 9623 the given numbers are whole numbers and with decimal point III. To find the sum of large numbers like align if the given numbers are decimal numbers as we write in 64083 + 43102 + 94320 + 8915 + 7042 conventional method of addition.