WORKING WITH ARITHMETIC Working With ARITHMETIC by D. M. NEAL, B.sc. Lecturer in Mathematics, Mid-Essex Technical College Formerly Assistant Master, Owen's School and J. D. NEW, B.Sc. (Eng.), A.C.G.I. Formerly Assistant Master, Owen's School Macmillan Education MACMILLAN AND CLEAVER LTD Little Essex Street, London W C 2 ISBN 978-1-349-00045-6 ISBN 978-1-349-00043-2 (eBook) DOI 10.1007/978-1-349-00043-2 © 1967 D. M. NEAL D . .J. NEW + Crown Bvo x 401 pages 82 line illustrations Preface It is the aim of this book to be a concise and yet complete course in basic arithmetic, meeting the requirements of stu dents studying for the General Certificate of Education and Certificate of Secondary Education and similar ' ordinary level ' examinations. It can be divided into three parts, concerning respectively: (i) the fundamental operations (ii) the use of mathematical tables (iii) applications. The first eleven chapters give the student an introduction to the fundamental operations of arithmetic. The authors have attempted, by introducing such ideas as, ' The origin of number ', ' What is a fraction?', ' Why decimals?', ' What do we mean by measurement?', 'What is area?', 'What is volume?', to give the student a sound understanding of these basic ideas. At the same time, being guided by their class room experiences, they recognise the importance of securing accuracy. Thus it will be found that each chapter also in cludes many worked examples and sufficient exercises for the student, so that he may become thoroughly proficient in the use of, and manipulation of, such basic principles as fractions, decimals, ratio, proportion, and percentage. In Chapter II the floating point representation is intro duced. This is used in both Chapters II and 12 to give the student a simple introduction to the working of logarithm tables and all other tables likely to be needed in a course in basic arithmetic. By introducing logarithms as the ' area under the curve ', their properties are easily deduced with the vi PREFACE aid of the trapezoidal rule. Both chapters contain many exercises designed to give a complete working knowledge of mathematical tables. The final chapters are devoted to applications of arithmetic in everyday life. The numerous worked examples, and pro blems set in each chapter, are of the type that the student is likely to encounter in G.C.E., C.S.E., or other 'ordinary level ' examinations. The authors wish to express their gratitude to the Univer sity of London Examination Board and the Associated Exam ining Board for their permission to make free use of past examination questions. Spring 1967 D. M. Neal D.J. New Contents Preface v Chapter 1. Integral Numbers Number Patterns. Factors. Prime Numbers and Prime Factors. Multiples. Highest Common Factor. Lowest Common Multiple. Use ofFactors. Tests for Divisibility. Factorisation of Large Numbers. To Find the H.C.F. Application of Factors to L.C.M. Powers of Numbers. Index Notation. Roots of Numbers. Roots by the Use of Prime Factors. Chapter 2. Fractions 16 Definition of Fractions. Mixed Numbers. Reduction of Fractions. Magnitude of Fractions. Addition and Subtraction. Multiplication. Division. Combined Operations. Problems in Fractions. Chapter 3. Decimals 35 Ancient Numerals. Position-Value System of Count- ing. Working with Decimals. Addition and Subtrac- tion. Multiplication and Division by powers of ten. Multiplication and Division by other numbers. Con- version of Fractions to Decimals. Recurring Decimals. Degrees of Accuracy. Corrected Approximations. Decimalisation of Compound Quantities. Square Roots of Decimals by Factors. Chapter 4. Averages and Unitary Methods 59 Averages. Direct Variation. Inverse Variation. Chapter 5. The Common Systems of Measurement 70 First Ideas of Measurement. British System. Metric System. Table of Standard Units. Decimals and the Metric System. Calculations in the British and Metric Systems. Conversion of Units of Measurement. viii CONTENTS Chapter 6. Measurement of Area 89 Area by Weighing. Unit of Measurement. Fitting Squares into Rectangles. Land Measure. Calcula~ tion of Areas by Formulae: Square, Rectangle, Tri~ angle, Parallelogram, Trapezium. Perimeter of Recti~ linear Plane Figure. Problems involving use of Area. Pythagoras Theorem. Chapter 7. Manipulation of Compound Quantities. . 124 Working with Mixed Units. Fractional Method. Simple Practice. Application of Simple Practice. Sub~ rraction Method. Compound Practice. Chapter B. Volumes 131 Volume and Capacity. Volume of Rectangular Ob~ ect. Volume of Prisms. Chapter 9. Ratio and Proportion 142 Comparison. Ratio in the Form 1 : m and n : I. Ratio Expressed in Whole Numbers. Direct Proportion. Inverse Proportion. Ratio of more than Two Quan~ tities. Proportional Parts. Applications of Ratio: ( 1) Scale Drawings (2) Income Tax (3) Rates and Rateable Value (4) Bankruptcy. Chapter 10. Percentage 164 Meaning of Percentage. Fractions as Percentages. Percentage in Daily Life: (I) Hire Purchase (2) In~ surance (3) Discount. Percentage Change. Profit and Loss. Simple Interest. Use of Formula. Compound Interest. Chapter 11. Graphs 191 Plotting. Travel Graphs. Intersecting Graphs. Other Straight Line Graphs. Trapezoidal Rule. Revision Exercises Chapters 1 to 11 225 CONTENTS IX Chapter 12. Use of Tables 234 Squares of Numbers from 1 to 10. Square Roots of Numbers from 1 to 100. Square Root Tables. Re ciprocals of Numbers from l to 10. Floating Point Numbers. Operations with Floating Point Numbers: Multiplication, Division, Addition, Subtraction. Squares of Other Numbers. Square Roots of Other Numbers. Reciprocals of Other Numbers. Exam ples on Combined Tables. Chapter 13. Logarithms 256 Purpose. Evaluation of Logarithms from a Graph. Fundamental Properties of Logarithms. Logarithm of any Number. The Bar Notation. Basic Rules for Multiplication, Division, Powers and Roots. Use of Four-Figure Tables. To Find the Logarithm of a Number. To Find a Number given the Logarithm. Multiplication. Division. Powers. Roots. Opera- tions involving Negative Characteristics. Examples on the use of Logarithms. Chapter 14. The Circle 278 Diameter and Circumference. Experimental Deter mination of Ratio between Diameter ami. Circum ference. The Number Length of an Arc. Area of 7T. the Circle. Area of the Annulus. Area of the Sector. Surface Area of Cylinder. Surface Area of Cone. Chapter 15. Rates and Percentages 298 Problems involving the Principles of Unitary Method, Rate, Proportion, Compound Units, Mixtures, and Percentage. Chapter 16. Further Volumes 307 Cylinder. Sphere. Cone. Specific Gravity and Archi medes' Principle. Chapter 17 . Shares and Stocks 317 Formation of a Company. Dividend. Buying and Selling of Shares. Stocks. Buying and Selling of Stocks and Shares. _ Other Types of Shares. X CONTENTS Chapter 18. Miscellaneous Examples and Exercises . . 333 Answers 347 Tables 381 Square Roots 382 Reciprocals 390 Squares 396 Logarithms 398 Antilogarithms 400