The Free High School Science Texts: Textbooks for High School Students Mathematics

The Free High School Science Texts: Textbooks for High School Students Mathematics
FHSST Authors
The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Mathematics
Grades 10 - 12
Version 0
September 17, 2008
ii
iii
Copyright 2007 “Free High School Science Texts”
Permission is granted to copy, distribute and/or modify this document under the
terms of the GNU Free Documentation License, Version 1.2 or any later version
published by the Free Software Foundation; with no Invariant Sections, no FrontCover Texts, and no Back-Cover Texts. A copy of the license is included in the
section entitled “GNU Free Documentation License”.
STOP!!!!
Did you notice the FREEDOMS we’ve granted you?
Our copyright license is different! It grants freedoms
rather than just imposing restrictions like all those other
textbooks you probably own or use.
• We know people copy textbooks illegally but we would LOVE it if you copied
our’s - go ahead copy to your hearts content, legally!
• Publishers revenue is generated by controlling the market, we don’t want any
money, go ahead, distribute our books far and wide - we DARE you!
• Ever wanted to change your textbook? Of course you have! Go ahead change
ours, make your own version, get your friends together, rip it apart and put
it back together the way you like it. That’s what we really want!
• Copy, modify, adapt, enhance, share, critique, adore, and contextualise. Do it
all, do it with your colleagues, your friends or alone but get involved! Together
we can overcome the challenges our complex and diverse country presents.
• So what is the catch? The only thing you can’t do is take this book, make
a few changes and then tell others that they can’t do the same with your
changes. It’s share and share-alike and we know you’ll agree that is only fair.
• These books were written by volunteers who want to help support education,
who want the facts to be freely available for teachers to copy, adapt and
re-use. Thousands of hours went into making them and they are a gift to
everyone in the education community.
iv
FHSST Core Team
Mark Horner ; Samuel Halliday ; Sarah Blyth ; Rory Adams ; Spencer Wheaton
FHSST Editors
Jaynie Padayachee ; Joanne Boulle ; Diana Mulcahy ; Annette Nell ; René Toerien ; Donovan
Whitfield
FHSST Contributors
Rory Adams ; Prashant Arora ; Richard Baxter ; Dr. Sarah Blyth ; Sebastian Bodenstein ;
Graeme Broster ; Richard Case ; Brett Cocks ; Tim Crombie ; Dr. Anne Dabrowski ; Laura
Daniels ; Sean Dobbs ; Fernando Durrell ; Dr. Dan Dwyer ; Frans van Eeden ; Giovanni
Franzoni ; Ingrid von Glehn ; Tamara von Glehn ; Lindsay Glesener ; Dr. Vanessa Godfrey ; Dr.
Johan Gonzalez ; Hemant Gopal ; Umeshree Govender ; Heather Gray ; Lynn Greeff ; Dr. Tom
Gutierrez ; Brooke Haag ; Kate Hadley ; Dr. Sam Halliday ; Asheena Hanuman ; Neil Hart ;
Nicholas Hatcher ; Dr. Mark Horner ; Mfandaidza Hove ; Robert Hovden ; Jennifer Hsieh ;
Clare Johnson ; Luke Jordan ; Tana Joseph ; Dr. Jennifer Klay ; Lara Kruger ; Sihle Kubheka ;
Andrew Kubik ; Dr. Marco van Leeuwen ; Dr. Anton Machacek ; Dr. Komal Maheshwari ;
Kosma von Maltitz ; Nicole Masureik ; John Mathew ; JoEllen McBride ; Nikolai Meures ;
Riana Meyer ; Jenny Miller ; Abdul Mirza ; Asogan Moodaly ; Jothi Moodley ; Nolene Naidu ;
Tyrone Negus ; Thomas O’Donnell ; Dr. Markus Oldenburg ; Dr. Jaynie Padayachee ;
Nicolette Pekeur ; Sirika Pillay ; Jacques Plaut ; Andrea Prinsloo ; Joseph Raimondo ; Sanya
Rajani ; Prof. Sergey Rakityansky ; Alastair Ramlakan ; Razvan Remsing ; Max Richter ; Sean
Riddle ; Evan Robinson ; Dr. Andrew Rose ; Bianca Ruddy ; Katie Russell ; Duncan Scott ;
Helen Seals ; Ian Sherratt ; Roger Sieloff ; Bradley Smith ; Greg Solomon ; Mike Stringer ;
Shen Tian ; Robert Torregrosa ; Jimmy Tseng ; Helen Waugh ; Dr. Dawn Webber ; Michelle
Wen ; Dr. Alexander Wetzler ; Dr. Spencer Wheaton ; Vivian White ; Dr. Gerald Wigger ;
Harry Wiggins ; Wendy Williams ; Julie Wilson ; Andrew Wood ; Emma Wormauld ; Sahal
Yacoob ; Jean Youssef
Contributors and editors have made a sincere effort to produce an accurate and useful resource.
Should you have suggestions, find mistakes or be prepared to donate material for inclusion,
please don’t hesitate to contact us. We intend to work with all who are willing to help make
this a continuously evolving resource!
www.fhsst.org
v
vi
Contents
I
Basics
1
1 Introduction to Book
1.1
II
3
The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grade 10
3
5
2 Review of Past Work
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
What is a number? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Letters and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.5
Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.6
Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.7
Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.8
Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9
2.8.1
What is a negative number? . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8.2
Working with Negative Numbers . . . . . . . . . . . . . . . . . . . . . . 11
2.8.3
Living Without the Number Line . . . . . . . . . . . . . . . . . . . . . . 12
Rearranging Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.10 Fractions and Decimal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.11 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.14 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.15 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Rational Numbers - Grade 10
23
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
The Big Picture of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vii
CONTENTS
CONTENTS
3.4
Forms of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5
Converting Terminating Decimals into Rational Numbers . . . . . . . . . . . . . 25
3.6
Converting Repeating Decimals into Rational Numbers . . . . . . . . . . . . . . 25
3.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Exponentials - Grade 10
29
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3
Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1
Exponential Law 1: a0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2
Exponential Law 2: am × an = am+n . . . . . . . . . . . . . . . . . . . 30
4.3.3
Exponential Law 3: a−n =
4.3.4
4.4
m
1
an , a
n
6= 0 . . . . . . . . . . . . . . . . . . . . 31
Exponential Law 4: a ÷ a = am−n . . . . . . . . . . . . . . . . . . . 32
4.3.5
Exponential Law 5: (ab)n = an bn . . . . . . . . . . . . . . . . . . . . . 32
4.3.6
Exponential Law 6: (am )n = amn . . . . . . . . . . . . . . . . . . . . . 33
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Estimating Surds - Grade 10
37
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2
Drawing Surds on the Number Line (Optional) . . . . . . . . . . . . . . . . . . 38
5.3
End of Chapter Excercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Irrational Numbers and Rounding Off - Grade 10
41
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2
Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3
Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.4
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Number Patterns - Grade 10
7.1
45
Common Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.1.1
Special Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2
Make your own Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3.1
7.4
Patterns and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Finance - Grade 10
53
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2
Foreign Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3
8.2.1
How much is R1 really worth? . . . . . . . . . . . . . . . . . . . . . . . 53
8.2.2
Cross Currency Exchange Rates
8.2.3
Enrichment: Fluctuating exchange rates . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . 56
Being Interested in Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
viii
CONTENTS
8.4
Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.4.1
8.5
8.6
8.7
CONTENTS
Other Applications of the Simple Interest Formula . . . . . . . . . . . . . 61
Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.5.1
Fractions add up to the Whole . . . . . . . . . . . . . . . . . . . . . . . 65
8.5.2
The Power of Compound Interest . . . . . . . . . . . . . . . . . . . . . . 65
8.5.3
Other Applications of Compound Growth . . . . . . . . . . . . . . . . . 67
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.2
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Products and Factors - Grade 10
71
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2
Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.1
Parts of an Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.2
Product of Two Binomials . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.3
Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.3
More Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4
Factorising a Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.5
Factorisation by Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.6
Simplification of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.7
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10 Equations and Inequalities - Grade 10
83
10.1 Strategy for Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.4 Exponential Equations of the form ka(x+p) = m . . . . . . . . . . . . . . . . . . 93
10.4.1 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.5 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.6 Linear Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.1 Finding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.2 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.3 Solution by Substitution
. . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.7 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.7.3 Application of Mathematical Modelling
. . . . . . . . . . . . . . . . . . 104
10.7.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.8 Introduction to Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . 107
10.9 Functions and Graphs in the Real-World . . . . . . . . . . . . . . . . . . . . . . 107
10.10Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
CONTENTS
CONTENTS
10.10.1 Variables and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.10.2 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.3 The Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.4 Drawing Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.10.5 Notation used for Functions
. . . . . . . . . . . . . . . . . . . . . . . . 110
10.11Characteristics of Functions - All Grades . . . . . . . . . . . . . . . . . . . . . . 112
10.11.1 Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . 112
10.11.2 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.3 Intercepts with the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.4 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.5 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.6 Lines of Symmetry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.7 Intervals on which the Function Increases/Decreases . . . . . . . . . . . 114
10.11.8 Discrete or Continuous Nature of the Graph . . . . . . . . . . . . . . . . 114
10.12Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.12.1 Functions of the form y = ax + q . . . . . . . . . . . . . . . . . . . . . 116
10.12.2 Functions of the Form y = ax2 + q . . . . . . . . . . . . . . . . . . . . . 120
10.12.3 Functions of the Form y =
a
x
+ q . . . . . . . . . . . . . . . . . . . . . . 125
10.12.4 Functions of the Form y = ab(x) + q . . . . . . . . . . . . . . . . . . . . 129
10.13End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11 Average Gradient - Grade 10 Extension
135
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.2 Straight-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.3 Parabolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
12 Geometry Basics
139
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.2 Points and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
12.3.1 Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.2 Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.3 Special Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.3.4 Parallel Lines intersected by Transversal Lines . . . . . . . . . . . . . . . 143
12.4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12.4.3 Other polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.4.4 Extra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.5.1 Challenge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
x
CONTENTS
13 Geometry - Grade 10
CONTENTS
161
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2 Right Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2.1 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
13.2.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
13.3 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.3.1 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.4 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.2 Distance between Two Points . . . . . . . . . . . . . . . . . . . . . . . . 172
13.4.3 Calculation of the Gradient of a Line . . . . . . . . . . . . . . . . . . . . 173
13.4.4 Midpoint of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.1 Translation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.2 Reflection of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14 Trigonometry - Grade 10
189
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
14.2 Where Trigonometry is Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.3 Similarity of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.4 Definition of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 191
14.5 Simple Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . 195
14.5.1 Height and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14.5.2 Maps and Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.6 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.1 Graph of sin θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.2 Functions of the form y = a sin(x) + q . . . . . . . . . . . . . . . . . . . 200
14.6.3 Graph of cos θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.6.4 Functions of the form y = a cos(x) + q
. . . . . . . . . . . . . . . . . . 202
14.6.5 Comparison of Graphs of sin θ and cos θ . . . . . . . . . . . . . . . . . . 204
14.6.6 Graph of tan θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
14.6.7 Functions of the form y = a tan(x) + q . . . . . . . . . . . . . . . . . . 205
14.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
15 Statistics - Grade 10
211
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2 Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.1 Data and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.2 Methods of Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 212
15.2.3 Samples and Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3 Example Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
xi
CONTENTS
CONTENTS
15.3.1 Data Set 1: Tossing a Coin . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.2 Data Set 2: Casting a die . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.3 Data Set 3: Mass of a Loaf of Bread . . . . . . . . . . . . . . . . . . . . 214
15.3.4 Data Set 4: Global Temperature . . . . . . . . . . . . . . . . . . . . . . 214
15.3.5 Data Set 5: Price of Petrol . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4 Grouping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4.1 Exercises - Grouping Data
. . . . . . . . . . . . . . . . . . . . . . . . . 216
15.5 Graphical Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.1 Bar and Compound Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.2 Histograms and Frequency Polygons . . . . . . . . . . . . . . . . . . . . 217
15.5.3 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.5.4 Line and Broken Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . 220
15.5.5 Exercises - Graphical Representation of Data
. . . . . . . . . . . . . . . 221
15.6 Summarising Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.1 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15.6.3 Exercises - Summarising Data
. . . . . . . . . . . . . . . . . . . . . . . 228
15.7 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
15.7.1 Exercises - Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . 230
15.8 Summary of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
16 Probability - Grade 10
235
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2 Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2.1 Sample Space of a Random Experiment . . . . . . . . . . . . . . . . . . 235
16.3 Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
16.3.1 Classical Theory of Probability . . . . . . . . . . . . . . . . . . . . . . . 239
16.4 Relative Frequency vs. Probability . . . . . . . . . . . . . . . . . . . . . . . . . 240
16.5 Project Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.6 Probability Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.7 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
16.8 Complementary Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16.9 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
III
Grade 11
17 Exponents - Grade 11
249
251
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
17.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
√
m
17.2.1 Exponential Law 7: a n = n am . . . . . . . . . . . . . . . . . . . . . . 251
17.3 Exponentials in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
17.4 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
xii
CONTENTS
CONTENTS
18 Surds - Grade 11
18.1 Surd Calculations . . . . . . . . . .
√
√ √
18.1.1 Surd Law 1: n a n b = n ab
√
p
n
a
18.1.2 Surd Law 2: n ab = √
. .
n
b
√
m
18.1.3 Surd Law 3: n am = a n . .
255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.4 Like and Unlike Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.5 Simplest Surd form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
18.1.6 Rationalising Denominators . . . . . . . . . . . . . . . . . . . . . . . . . 258
18.2 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
19 Error Margins - Grade 11
261
20 Quadratic Sequences - Grade 11
265
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.2 What is a quadratic sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.3 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
21 Finance - Grade 11
271
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.2 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.3 Simple Depreciation (it really is simple!) . . . . . . . . . . . . . . . . . . . . . . 271
21.4 Compound Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
21.5 Present Values or Future Values of an Investment or Loan . . . . . . . . . . . . 276
21.5.1 Now or Later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
21.6 Finding i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
21.7 Finding n - Trial and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
21.8 Nominal and Effective Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 280
21.8.1 The General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
21.8.2 De-coding the Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 282
21.9 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
21.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
22 Solving Quadratic Equations - Grade 11
287
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.2 Solution by Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.3 Solution by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 290
22.4 Solution by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 293
22.5 Finding an equation when you know its roots . . . . . . . . . . . . . . . . . . . 296
22.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
xiii
CONTENTS
CONTENTS
23 Solving Quadratic Inequalities - Grade 11
301
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.2 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
24 Solving Simultaneous Equations - Grade 11
307
24.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
24.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
25 Mathematical Models - Grade 11
313
25.1 Real-World Applications: Mathematical Models . . . . . . . . . . . . . . . . . . 313
25.2 End of Chatpter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
26 Quadratic Functions and Graphs - Grade 11
321
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
26.2 Functions of the Form y = a(x + p)2 + q
. . . . . . . . . . . . . . . . . . . . . 321
26.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
26.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
26.2.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
26.2.4 Axes of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
26.2.5 Sketching Graphs of the Form f (x) = a(x + p)2 + q . . . . . . . . . . . 325
26.2.6 Writing an equation of a shifted parabola . . . . . . . . . . . . . . . . . 327
26.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
27 Hyperbolic Functions and Graphs - Grade 11
329
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
27.2 Functions of the Form y =
a
x+p
+q
. . . . . . . . . . . . . . . . . . . . . . . . 329
27.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
27.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
27.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
27.2.4 Sketching Graphs of the Form f (x) =
a
x+p
+ q . . . . . . . . . . . . . . 333
27.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
28 Exponential Functions and Graphs - Grade 11
335
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
28.2 Functions of the Form y = ab(x+p) + q . . . . . . . . . . . . . . . . . . . . . . . 335
28.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
28.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
28.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
28.2.4 Sketching Graphs of the Form f (x) = ab(x+p) + q . . . . . . . . . . . . . 338
28.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
29 Gradient at a Point - Grade 11
341
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.2 Average Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
xiv
CONTENTS
30 Linear Programming - Grade 11
CONTENTS
345
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.1 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.4 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.5 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.3 Example of a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.4 Method of Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5 Skills you will need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.1 Writing Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.2 Writing the Objective Function . . . . . . . . . . . . . . . . . . . . . . . 348
30.5.3 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
30.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
31 Geometry - Grade 11
357
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
31.2 Right Pyramids, Right Cones and Spheres . . . . . . . . . . . . . . . . . . . . . 357
31.3 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
31.4 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.4.1 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.5 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
31.5.1 Equation of a Line between Two Points . . . . . . . . . . . . . . . . . . 368
31.5.2 Equation of a Line through One Point and Parallel or Perpendicular to
Another Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.5.3 Inclination of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.1 Rotation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.2 Enlargement of a Polygon 1 . . . . . . . . . . . . . . . . . . . . . . . . . 376
32 Trigonometry - Grade 11
381
32.1 History of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2.1 Functions of the form y = sin(kθ) . . . . . . . . . . . . . . . . . . . . . 381
32.2.2 Functions of the form y = cos(kθ) . . . . . . . . . . . . . . . . . . . . . 383
32.2.3 Functions of the form y = tan(kθ) . . . . . . . . . . . . . . . . . . . . . 384
32.2.4 Functions of the form y = sin(θ + p) . . . . . . . . . . . . . . . . . . . . 385
32.2.5 Functions of the form y = cos(θ + p) . . . . . . . . . . . . . . . . . . . 386
32.2.6 Functions of the form y = tan(θ + p) . . . . . . . . . . . . . . . . . . . 387
32.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
32.3.1 Deriving Values of Trigonometric Functions for 30◦ , 45◦ and 60◦ . . . . . 389
32.3.2 Alternate Definition for tan θ . . . . . . . . . . . . . . . . . . . . . . . . 391
xv
CONTENTS
CONTENTS
32.3.3 A Trigonometric Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 392
32.3.4 Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
32.4 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
32.4.3 Solution using CAST diagrams . . . . . . . . . . . . . . . . . . . . . . . 403
32.4.4 General Solution Using Periodicity . . . . . . . . . . . . . . . . . . . . . 405
32.4.5 Linear Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 406
32.4.6 Quadratic and Higher Order Trigonometric Equations . . . . . . . . . . . 406
32.4.7 More Complex Trigonometric Equations . . . . . . . . . . . . . . . . . . 407
32.5 Sine and Cosine Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.1 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.2 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
32.5.3 The Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
32.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
33 Statistics - Grade 11
419
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2 Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.2 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
33.2.3 Interpretation and Application . . . . . . . . . . . . . . . . . . . . . . . 423
33.2.4 Relationship between Standard Deviation and the Mean . . . . . . . . . . 424
33.3 Graphical Representation of Measures of Central Tendency and Dispersion . . . . 424
33.3.1 Five Number Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
33.3.2 Box and Whisker Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 425
33.3.3 Cumulative Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
33.4 Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.1 Symmetric and Skewed Data . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.2 Relationship of the Mean, Median, and Mode . . . . . . . . . . . . . . . 428
33.5 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
33.6 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
33.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
34 Independent and Dependent Events - Grade 11
437
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2.1 Identification of Independent and Dependent Events
. . . . . . . . . . . 438
34.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
IV
Grade 12
35 Logarithms - Grade 12
443
445
35.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
xvi
CONTENTS
CONTENTS
35.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
35.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.4 Logarithm Law 1: loga 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.5 Logarithm Law 2: loga (a) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
35.6 Logarithm Law 3: loga (x · y) = loga (x) + loga (y) . . . . . . . . . . . . . . . . . 448
35.7 Logarithm Law 4: loga xy = loga (x) − loga (y) . . . . . . . . . . . . . . . . . 449
35.8 Logarithm Law 5: loga (xb ) = b loga (x) . . . . . . . . . . . . . . . . . . . . . . . 450
√
35.9 Logarithm Law 6: loga ( b x) = logab(x) . . . . . . . . . . . . . . . . . . . . . . . 450
35.10Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
35.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
35.11Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 454
35.11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
35.12End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
36 Sequences and Series - Grade 12
457
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2.1 General Equation for the nth -term of an Arithmetic Sequence . . . . . . 458
36.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.2 General Equation for the nth -term of a Geometric Sequence . . . . . . . 461
36.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
36.4 Recursive Formulae for Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 462
36.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
36.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 466
36.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
36.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
36.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
36.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
36.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
36.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
37 Finance - Grade 12
477
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
37.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 477
37.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
37.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
xvii
CONTENTS
CONTENTS
37.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 479
37.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 484
37.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 485
37.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.2 Exercises - Investments and Loans . . . . . . . . . . . . . . . . . . . . . 489
37.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 489
37.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
37.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
38 Factorising Cubic Polynomials - Grade 12
493
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.2 The Factor Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.3 Factorisation of Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 494
38.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 498
38.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
39 Functions and Graphs - Grade 12
501
39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 503
39.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.3 Inverse Function of y = ax2
. . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.5 Inverse Function of y = ax . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
40 Differential Calculus - Grade 12
509
40.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 509
40.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
40.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 510
40.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 511
40.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
40.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 516
40.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 519
xviii
CONTENTS
CONTENTS
40.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
40.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 522
40.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 523
40.5.1 Finding Equations of Tangents to Curves
. . . . . . . . . . . . . . . . . 523
40.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
40.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 529
40.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 530
40.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 534
40.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
41 Linear Programming - Grade 12
539
41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.3 Linear Programming and the Feasible Region . . . . . . . . . . . . . . . . . . . 540
41.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
42 Geometry - Grade 12
549
42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
42.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 550
42.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 569
42.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
42.4.1 Rotation of a Point about an angle θ . . . . . . . . . . . . . . . . . . . . 571
42.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.4.3 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
43 Trigonometry - Grade 12
577
43.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.1 Derivation of sin(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.2 Derivation of sin(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.3 Derivation of cos(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.4 Derivation of cos(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.5 Derivation of sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.6 Derivation of cos 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 580
43.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 582
43.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 582
xix
CONTENTS
CONTENTS
43.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 584
43.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.3 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
43.3.4 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
43.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
44 Statistics - Grade 12
591
44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
44.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 594
44.4.1 The Method of Least Squares
. . . . . . . . . . . . . . . . . . . . . . . 596
44.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
44.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
44.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
45 Combinations and Permutations - Grade 12
603
45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 604
45.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 606
45.6 Permutations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
45.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
45.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
45.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
V
Exercises
613
46 General Exercises
615
47 Exercises - Not covered in Syllabus
617
A GNU Free Documentation License
619
xx
Chapter 8
Finance - Grade 10
8.1
Introduction
Should you ever find yourself stuck with a mathematics question on a television quiz show, you
will probably wish you had remembered the how many even prime numbers there are between 1
and 100 for the sake of R1 000 000. And who does not want to be a millionaire, right?
Welcome to the Grade 10 Finance Chapter, where we apply maths skills to everyday financial
situations that you are likely to face both now and along your journey to purchasing your first
private jet.
If you master the techniques in this chapter, you will grasp the concept of compound interest,
and how it can ruin your fortunes if you have credit card debt, or make you millions if you
successfully invest your hard-earned money. You will also understand the effects of fluctuating
exchange rates, and its impact on your spending power during your overseas holidays!
8.2
Foreign Exchange Rates
Is $500 (”500 US dollars”) per person per night a good deal on a hotel in New York City? The
first question you will ask is “How much is that worth in Rands?”. A quick call to the local bank
or a search on the Internet (for example on http://www.x-rates.com/) for the Dollar/Rand
exchange rate will give you a basis for assessing the price.
A foreign exchange rate is nothing more than the price of one currency in terms of another.
For example, the exchange rate of 6,18 Rands/US Dollars means that $1 costs R6,18. In other
words, if you have $1 you could sell it for R6,18 - or if you wanted $1 you would have to pay
R6,18 for it.
But what drives exchange rates, and what causes exchange rates to change? And how does this
affect you anyway? This section looks at answering these questions.
8.2.1
How much is R1 really worth?
We can quote the price of a currency in terms of any other currency, but the US Dollar, British
Pounds Sterling or even the Euro are often used as a market standard. You will notice that the
financial news will report the South African Rand exchange rate in terms of these three major
currencies.
So the South African Rand could be quoted on a certain date as 6,7040 ZAR per USD (i.e.
$1,00 costs R6,7040), or 12,2374 ZAR per GBP. So if I wanted to spend $1 000 on a holiday
in the United States of America, this would cost me R6 704,00; and if I wanted £1 000 for a
weekend in London it would cost me R12 237,40.
This seems obvious, but let us see how we calculated that: The rate is given as ZAR per USD,
or ZAR/USD such that $1,00 buys R6,7040. Therefore, we need to multiply by 1 000 to get the
53
8.2
CHAPTER 8. FINANCE - GRADE 10
Table 8.1: Abbreviations and symbols for some common currencies.
Currency
Abbreviation Symbol
South African Rand
ZAR
R
United States Dollar
USD
$
British Pounds Sterling
GBP
£
Euro
EUR
e
number of Rands per $1 000.
Mathematically,
$1,00 =
∴
1 000 × $1,00 =
=
R6,0740
1 000 × R6,0740
R6 074,00
as expected.
What if you have saved R10 000 for spending money for the same trip and you wanted to use
this to buy USD? How much USD could you get for this? Our rate is in ZAR/USD but we want
to know how many USD we can get for our ZAR. This is easy. We know how much $1,00 costs
in terms of Rands.
$1,00 =
$1,00
=
∴
6,0740
1,00
$
=
6,0740
R1,00 =
=
R6,0740
R6,0740
6,0740
R1,00
1,00
6,0740
$0,164636
$
As we can see, the final answer is simply the reciprocal of the ZAR/USD rate. Therefore, R10 000
will get:
R1,00 =
∴
10 000 × R1,00 =
=
$
1,00
6,0740
10 000 × $
1,00
6,0740
$1 646,36
We can check the answer as follows:
∴
$1,00 =
1 646,36 × $1,00 =
=
R6,0740
1 646,36 × R6,0740
R10 000,00
Six of one and half a dozen of the other
So we have two different ways of expressing the same exchange rate: Rands per Dollar (ZAR/USD)
and Dollar per Rands (USD/ZAR). Both exchange rates mean the same thing and express the
value of one currency in terms of another. You can easily work out one from the other - they
are just the reciprocals of the other.
If the South African Rand is our Domestic (or home) Currency, we call the ZAR/USD rate a
“direct” rate, and we call a USD/ZAR rate an “indirect” rate.
In general, a direct rate is an exchange rate that is expressed as units of Home Currency per
54
CHAPTER 8. FINANCE - GRADE 10
8.2
units of Foreign Currency, i.e., Domestic Currency / Foreign Currency.
The Rand exchange rates that we see on the news are usually expressed as Direct Rates, for
example you might see:
Table 8.2: Examples of
Currency Abbreviation
1 USD
1 GBP
1 EUR
exchange rates
Exchange Rates
R6,9556
R13,6628
R9,1954
The exchange rate is just the price of each of the Foreign Currencies (USD, GBP and EUR) in
terms of our Domestic Currency, Rands.
An indirect rate is an exchange rate expressed as units of Foreign Currency per units of Home
Currency, i.e. Foreign Currency / Domestic Currency
Defining exchange rates as direct or indirect depends on which currency is defined as the Domestic
Currency. The Domestic Currency for an American investor would be USD which is the South
African investor’s Foreign Currency. So direct rates from the perspective of the American investor
(USD/ZAR) would be the same as the indirect rate from the perspective of the South Africa
investor.
Terminology
Since exchange rates are simple prices of currencies, movements in exchange rates means that
the price or value of the currency changed. The price of petrol changes all the time, so does the
price of gold, and currency prices also move up and down all the time.
If the Rand exchange rate moved from say R6,71 per USD to R6,50 per USD, what does this
mean? Well, it means that $1 would now cost only R6,50 instead of R6,71. The Dollar is now
cheaper to buy, and we say that the Dollar has depreciated (or weakened) against the Rand.
Alternatively we could say that the Rand has appreciated (or strengthened) against the Dollar.
What if we were looking at indirect exchange rates, and the exchange rate moved from $0,149
1
1
per ZAR (= 6,71
) to $0,1538 per ZAR (= 6,50
).
Well now we can see that the R1,00 cost $0,149 at the start, and then cost $0,1538 at the end.
The Rand has become more expensive (in terms of Dollars), and again we can say that the Rand
has appreciated.
Regardless of which exchange rate is used, we still come to the same conclusions.
In general,
• for direct exchange rates, the home currency will appreciate (depreciate) if the exchange
rate falls (rises)
• For indirect exchange rates, the home currency will appreciate (depreciate) if the exchange
rate rises (falls)
As with just about everything in this chapter, do not get caught up in memorising these formulae
- that is only going to get confusing. Think about what you have and what you want - and it
should be quite clear how to get the correct answer.
Activity :: Discussion : Foreign Exchange Rates
In groups of 5, discuss:
1. Why might we need to know exchange rates?
2. What happens if one countries currency falls drastically vs another countries
currency?
55
8.2
CHAPTER 8. FINANCE - GRADE 10
3. When might you use exchange rates?
8.2.2
Cross Currency Exchange Rates
We know that the exchange rates are the value of one currency expressed in terms of another
currency, and we can quote exchange rates against any other currency. The Rand exchange rates
we see on the news are usually expressed against the major currency, USD, GBP and EUR.
So if for example, the Rand exchange rates were given as 6,71 ZAR/USD and 12,71 ZAR/GBP,
does this tell us anything about the exchange rate between USD and GBP?
Well I know that if $1 will buy me R6,71, and if £1.00 will buy me R12,71, then surely the GBP
is stronger than the USD because you will get more Rands for one unit of the currency, and we
can work out the USD/GBP exchange rate as follows:
Before we plug in any numbers, how can we get a USD/GBP exchange rate from the ZAR/USD
and ZAR/GBP exchange rates?
Well,
USD/GBP = USD/ZAR × ZAR/GBP.
Note that the ZAR in the numerator will cancel out with the ZAR in the denominator, and we
are left with the USD/GBP exchange rate.
Although we do not have the USD/ZAR exchange rate, we know that this is just the reciprocal
of the ZAR/USD exchange rate.
USD/ZAR =
1
ZAR/USD
Now plugging in the numbers, we get:
USD/GBP =
=
=
=
USD/ZAR × ZAR/GBP
1
× ZAR/GBP
ZAR/USD
1
× 12,71
6,71
1,894
Important: Sometimes you will see exchange rates in the real world that do not appear to
work exactly like this. This is usually because some financial institutions add other costs
to the exchange rates, which alter the results. However, if you could remove the effect of
those extra costs, the numbers would balance again.
Worked Example 8: Cross Exchange Rates
Question: If $1 = R 6,40, and £1 = R11,58 what is the $/£ exchange rate (i.e.
the number of US$ per £)?
Answer
Step 1 : Determine what is given and what is required
The following are given:
• ZAR/USD rate = R6,40
• ZAR/GBP rate = R11,58
56
CHAPTER 8. FINANCE - GRADE 10
8.2
The following is required:
• USD/GBP rate
Step 2 : Determine how to approach the problem
We know that:
USD/GBP = USD/ZAR × ZAR/GBP.
Step 3 : Solve the problem
USD/GBP =
=
=
=
USD/ZAR × ZAR/GBP
1
× ZAR/GBP
ZAR/USD
1
× 11,58
6,40
1,8094
Step 4 : Write the final answer
$1,8094 can be bought for £1.
Activity :: Investigation : Cross Exchange Rates - Alternate Method
If $1 = R 6,40, and £1 = R11,58 what is the $/£ exchange rate (i.e. the number
of US$ per £)?
Overview of problem
You need the $/£ exchange rate, in other words how many dollars must you pay
for a pound. So you need £1. From the given information we know that it would
cost you R11,58 to buy £1 and that $ 1 = R6,40.
Use this information to:
1. calculate how much R1 is worth in $.
2. calculate how much R11,58 is worth in $.
Do you get the same answer as in the worked example?
8.2.3
Enrichment: Fluctuating exchange rates
If everyone wants to buy houses in a certain suburb, then house prices are going to go up - because
the buyers will be competing to buy those houses. If there is a suburb where all residents want
to move out, then there are lots of sellers and this will cause house prices in the area to fall because the buyers would not have to struggle as much to find an eager seller.
This is all about supply and demand, which is a very important section in the study of Economics.
You can think about this is many different contexts, like stamp-collecting for example. If there
is a stamp that lots of people want (high demand) and few people own (low supply) then that
stamp is going to be expensive.
And if you are starting to wonder why this is relevant - think about currencies. If you are going
to visit London, then you have Rands but you need to “buy” Pounds. The exchange rate is the
price you have to pay to buy those Pounds.
Think about a time where lots of South Africans are visiting the United Kingdom, and other
South Africans are importing goods from the United Kingdom. That means there are lots of
Rands (high supply) trying to buy Pounds. Pounds will start to become more expensive (compare
this to the house price example at the start of this section if you are not convinced), and the
57
8.3
CHAPTER 8. FINANCE - GRADE 10
exchange rate will change. In other words, for R1 000 you will get fewer Pounds than you would
have before the exchange rate moved.
Another context which might be useful for you to understand this: consider what would happen
if people in other countries felt that South Africa was becoming a great place to live, and that
more people were wanting to invest in South Africa - whether in properties, businesses - or just
buying more goods from South Africa. There would be a greater demand for Rands - and the
“price of the Rand” would go up. In other words, people would need to use more Dollars, or
Pounds, or Euros ... to buy the same amount of Rands. This is seen as a movement in exchange
rates.
Although it really does come down to supply and demand, it is interesting to think about what
factors might affect the supply (people wanting to “sell” a particular currency) and the demand
(people trying to “buy” another currency). This is covered in detail in the study of Economics,
but let us look at some of the basic issues here.
There are various factors affect exchange rates, some of which have more economic rationale
than others:
• economic factors (such as inflation figures, interest rates, trade deficit information, monetary policy and fiscal policy)
• political factors (such as uncertain political environment, or political unrest)
• market sentiments and market behaviour (for example if foreign exchange markets perceived a currency to be overvalued and starting selling the currency, this would cause the
currency to fall in value - a self fulfilling expectation).
Exercise: Foreign Exchange
1. I want to buy an IPOD that costs £100, with the exchange rate currently at
£1 = R14. I believe the exchange rate will reach R12 in a month.
(a) How much will the MP3 player cost in Rands, if I buy it now?
(b) How much will I save if the exchange rate drops to R12?
(c) How much will I lose if the exchange rate moves to R15?
2. Study the following exchange rate table:
Country
Currency
United Kingdom (UK) Pounds(£)
United States (USA)
Dollars ($)
Exchange Rate
R14,13
R7,04
(a) In South Africa the cost of a new Honda Civic is R173 400. In England the
same vehicle costs £12 200 and in the USA $ 21 900. In which country is
the car the cheapest if you compare it to the South African Rand ?
(b) Sollie and Arinda are waiters in a South African Restaurant attracting many
tourists from abroad. Sollie gets a £6 tip from a tourist and Arinda gets
$ 12. How many South African Rand did each one get ?
8.3
Being Interested in Interest
If you had R1 000, you could either keep it in your wallet, or deposit it in a bank account. If it
stayed in your wallet, you could spend it any time you wanted. If the bank looked after it for
you, then they could spend it, with the plan of making profit off it. The bank usually “pays” you
to deposit it into an account, as a way of encouraging you to bank it with them, This payment
is like a reward, which provides you with a reason to leave it with the bank for a while, rather
than keeping the money in your wallet.
58
CHAPTER 8. FINANCE - GRADE 10
8.4
We call this reward ”interest”.
If you deposit money into a bank account, you are effectively lending money to the bank - and
you can expect to receive interest in return. Similarly, if you borrow money from a bank (or from
a department store, or a car dealership, for example) then you can expect to have to pay interest
on the loan. That is the price of borrowing money.
The concept is simple, yet it is core to the world of finance. Accountants, actuaries and bankers,
for example, could spend their entire working career dealing with the effects of interest on
financial matters.
In this chapter you will be introduced to the concept of financial mathematics - and given the
tools to cope with even advanced concepts and problems.
Important: Interest
The concepts in this chapter are simple - we are just looking at the same idea, but from many
different angles. The best way to learn from this chapter is to do the examples yourself, as you
work your way through. Do not just take our word for it!
8.4
Simple Interest
Definition: Simple Interest
Simple interest is where you earn interest on the initial amount that you invested, but not
interest on interest.
As an easy example of simple interest, consider how much you will get by investing R1 000 for
1 year with a bank that pays you 5% simple interest. At the end of the year, you will get an
interest of:
Interest = R1 000 × 5%
5
= R1 000 ×
100
= R1 000 × 0,05
= R50
So, with an “opening balance” of R1 000 at the start of the year, your “closing balance” at the
end of the year will therefore be:
Closing Balance =
=
=
Opening Balance + Interest
R1 000 + R50
R1 050
We sometimes call the opening balance in financial calculations Principal, which is abbreviated
as P (R1 000 in the example). The interest rate is usually labelled i (5% in the example), and
the interest amount (in Rand terms) is labelled I (R50 in the example).
So we can see that:
I =P ×i
and
Closing Balance =
=
=
=
Opening Balance + Interest
P +I
P + (P × i)
P (1 + i)
59
(8.1)
8.4
CHAPTER 8. FINANCE - GRADE 10
This is how you calculate simple interest. It is not a complicated formula, which is just as well
because you are going to see a lot of it!
Not Just One
You might be wondering to yourself:
1. how much interest will you be paid if you only leave the money in the account for 3 months,
or
2. what if you leave it there for 3 years?
It is actually quite simple - which is why they call it Simple Interest.
1. Three months is 1/4 of a year, so you would only get 1/4 of a full year’s interest, which
is: 1/4 × (P × i). The closing balance would therefore be:
Closing Balance = P + 1/4 × (P × i)
= P (1 + (1/4)i)
2. For 3 years, you would get three years’ worth of interest, being: 3 × (P × i). The closing
balance at the end of the three year period would be:
Closing Balance =
=
P + 3 × (P × i)
P × (1 + (3)i)
If you look carefully at the similarities between the two answers above, we can generalise the
result. In other words, if you invest your money (P ) in an account which pays a rate of interest
(i) for a period of time (n years), then, using the symbol (A) for the Closing Balance:
Closing Balance,(A) = P (1 + i · n)
(8.2)
As we have seen, this works when n is a fraction of a year and also when n covers several years.
Important: Interest Calculation
Annual Rates means Yearly rates. and p.a.(per annum) = per year
Worked Example 9: Simple Interest
Question: If I deposit R1 000 into a special bank account which pays a Simple
Interest of 7% for 3 years, how much will I get back at the end?
Answer
Step 1 : Determine what is given and what is required
• opening balance, P = R1 000
• interest rate, i = 7%
• period of time, n = 3 years
We are required to find the closing balance (A).
Step 2 : Determine how to approach the problem
We know from (8.2) that:
Closing Balance,(A) = P (1 + i · n)
60
CHAPTER 8. FINANCE - GRADE 10
8.4
Step 3 : Solve the problem
A =
=
=
P (1 + i · n)
R1 000(1 + 3 × 7%)
R1 210
Step 4 : Write the final answer
The closing balance after 3 years of saving R1 000 at an interest rate of 7% is R1 210.
Worked Example 10: Calculating n
Question: If I deposit R30 000 into a special bank account which pays a Simple
Interest of 7.5% ,for how many years must I invest this amount to generate R45 000
Answer
Step 1 : Determine what is given and what is required
• opening balance, P = R30 000
• interest rate, i = 7,5%
• closing balance, A = R45 000
We are required to find the number of years.
Step 2 : Determine how to approach the problem
We know from (8.2) that:
Closing Balance (A) = P (1 + i · n)
Step 3 : Solve the problem
Closing Balance (A) =
R45 000 =
(1 + 0,075 × n) =
0,075 × n
=
n
=
n
=
P (1 + i · n)
R30 000(1 + n × 7,5%)
45000
30000
1,5 − 1
0,5
0,075
6,6666667
Step 4 : Write the final answer
n has to be a whole number, therefore n = 7.
The period is 7 years for R30 000 to generate R45 000 at a simple interest rate of
7,5%.
8.4.1
Other Applications of the Simple Interest Formula
61
8.4
CHAPTER 8. FINANCE - GRADE 10
Worked Example 11: Hire-Purchase
Question: Troy is keen to buy an addisional hard drive for his laptop advertised for
R 2 500 on the internet. There is an option of paying a 10% deposit then making
24 monthly payments using a hire-purchase agreement where interest is calculated
at 7,5% p.a. simple interest. Calculate what Troy’s monthly payments will be.
Answer
Step 1 : Determine what is given and what is required
A new opening balance is required, as the 10% deposit is paid in cash.
•
•
•
•
10% of R 2 500 = R250
new opening balance, P = R2 500 − R250 = R2 250
interest rate, i = 7,5% = 0,075pa
period of time, n = 2 years
We are required to find the closing balance (A) and then the montly payments.
Step 2 : Determine how to approach the problem
We know from (8.2) that:
Closing Balance,(A) = P (1 + i · n)
Step 3 : Solve the problem
A =
=
=
Monthly payment =
=
P (1 + i · n)
R2 250(1 + 2 × 7,5%)
R2 587,50
2587,50 ÷ 24
R107,81
Step 4 : Write the final answer
Troy’s monthly payments = R 107,81
Worked Example 12: Depreciation
Question: Seven years ago, Tjad’s drum kit cost him R12 500. It has now been
valued at R2 300. What rate of simple depreciation does this represent ?
Answer
Step 1 : Determine what is given and what is required
• opening balance, P = R12 500
• period of time, n = 7 years
• closing balance, A = R2 300
We are required to find the rate(i).
Step 2 : Determine how to approach the problem
We know from (8.2) that:
Closing Balance,(A) = P (1 + i · n)
Therefore, for depreciation the formula will change to:
Closing Balance,(A) = P (1 − i · n)
62
CHAPTER 8. FINANCE - GRADE 10
8.5
Step 3 : Solve the problem
A = P (1 − i · n)
R2 300 = R12 500(1 − 7 × i)
i
= 0,11657...
Step 4 : Write the final answer
Therefore the rate of depreciation is 11,66%
Exercise: Simple Interest
1. An amount of R3 500 is invested in a savings account which pays simple interest
at a rate of 7,5% per annum. Calculate the balance accumulated by the end
of 2 years.
2. Calculate the simple interest for the following problems.
(a) A loan of R300 at a rate of 8% for l year.
(b) An investment of R225 at a rate of 12,5% for 6 years.
3. I made a deposit of R5 000 in the bank for my 5 year old son’s 21st birthday.
I have given him the amount of R 18 000 on his birtday. At what rate was the
money invested, if simple interest was calculated ?
4. Bongani buys a dining room table costing R 8 500 on Hire Purchase. He is
charged simple interest at 17,5% per annum over 3 years.
(a) How much will Bongani pay in total ?
(b) How much interest does he pay ?
(c) What is his montly installment ?
8.5
Compound Interest
To explain the concept of compound interest, the following example is discussed:
Worked Example 13:
Using Simple Interest to lead to the concept Com-
pound Interest
Question: If I deposit R1 000 into a special bank account which pays a Simple
Interest of 7%. What if I empty the bank account after a year, and then take the
principal and the interest and invest it back into the same account again. Then I
take it all out at the end of the second year, and then put it all back in again? And
then I take it all out at the end of 3 years?
Answer
Step 1 : Determine what is given and what is required
• opening balance, P = R1 000
• interest rate, i = 7%
63
8.5
CHAPTER 8. FINANCE - GRADE 10
• period of time, 1 year at a time, for 3 years
We are required to find the closing balance at the end of three years.
Step 2 : Determine how to approach the problem
We know that:
Closing Balance = P (1 + i · n)
Step 3 : Determine the closing balance at the end of the first year
Closing Balance = P (1 + i · n)
= R1 000(1 + 1 × 7%)
= R1 070
Step 4 : Determine the closing balance at the end of the second year
After the first year, we withdraw all the money and re-deposit it. The opening
balance for the second year is therefore R1 070, because this is the balance after the
first year.
Closing Balance = P (1 + i · n)
= R1 070(1 + 1 × 7%)
= R1 144,90
Step 5 : Determine the closing balance at the end of the third year
After the second year, we withdraw all the money and re-deposit it. The opening
balance for the third year is therefore R1 144,90, because this is the balance after
the first year.
Closing Balance = P (1 + i · n)
= R1 144,90(1 + 1 × 7%)
= R1 225,04
Step 6 : Write the final answer
The closing balance after withdrawing the all the money and re-depositing each year
for 3 years of saving R1 000 at an interest rate of 7% is R1 225,04.
In the two worked examples using simple interest, we have basically the same problem because
P =R1 000, i=7% and n=3 years for both problems. Except in the second situation, we end up
with R1 225,04 which is more than R1 210 from the first example. What has changed?
In the first example I earned R70 interest each year - the same in the first, second and third year.
But in the second situation, when I took the money out and then re-invested it, I was actually
earning interest in the second year on my interest (R70) from the first year. (And interest on
the interest on my interest in the third year!)
This more realistically reflects what happens in the real world, and is known as Compound
Interest. It is this concept which underlies just about everything we do - so we will look at more
closely next.
Definition: Compound Interest
Compound interest is the interest payable on the principal and its accumulated interest.
Compound interest is a double edged sword, though - great if you are earning interest on cash
you have invested, but crippling if you are stuck having to pay interest on money you have
borrowed!
In the same way that we developed a formula for Simple Interest, let us find one for Compound
Interest.
64
CHAPTER 8. FINANCE - GRADE 10
8.5
If our opening balance is P and we have an interest rate of i then, the closing balance at the
end of the first year is:
Closing Balance after 1 year = P (1 + i)
This is the same as Simple Interest because it only covers a single year. Then, if we take that
out and re-invest it for another year - just as you saw us doing in the worked example above then the balance after the second year will be:
Closing Balance after 2 years =
=
[P (1 + i)] × (1 + i)
P (1 + i)2
And if we take that money out, then invest it for another year, the balance becomes:
= [P (1 + i)2 ] × (1 + i)
= P (1 + i)3
Closing Balance after 3 years
We can see that the power of the term (1 + i) is the same as the number of years. Therefore,
Closing Balance after n years = P (1 + i)n
8.5.1
(8.3)
Fractions add up to the Whole
It is easy to show that this formula works even when n is a fraction of a year. For example, let
us invest the money for 1 month, then for 4 months, then for 7 months.
Closing Balance after 1 month =
Closing Balance after 5 months =
=
=
=
Closing Balance after 12 months =
1
P (1 + i) 12
Closing Balance after 1 month invested for 4 months more
4
1
[P (1 + i) 12 ] 12
1
4
P (1 + i) 12 + 12
5
P (1 + i) 12
Closing Balance after 5 month invested for 7 months more
5
7
=
[P (1 + i) 12 ] 12
=
P (1 + i) 12 + 12
=
=
P (1 + i) 12
P (1 + i)1
5
7
12
which is the same as investing the money for a year.
Look carefully at the long equation above. It is not as complicated as it looks! All we are doing
is taking the opening amount (P ), then adding interest for just 1 month. Then we are taking
that new balance and adding interest for a further 4 months, and then finally we are taking the
new balance after a total of 5 months, and adding interest for 7 more months. Take a look
again, and check how easy it really is.
Does the final formula look familiar? Correct - it is the same result as you would get for simply
investing P for one full year. This is exactly what we would expect, because:
1 month + 4 months + 7 months = 12 months,
which is a year. Can you see that? Do not move on until you have understood this point.
8.5.2
The Power of Compound Interest
To see how important this “interest on interest” is, we shall compare the difference in closing
balances for money earning simple interest and money earning compound interest. Consider an
amount of R10 000 that you have to invest for 10 years, and assume we can earn interest of 9%.
How much would that be worth after 10 years?
65
8.5
CHAPTER 8. FINANCE - GRADE 10
The closing balance for the money earning simple interest is:
Closing Balance =
=
=
P (1 + i · n)
R10 000(1 + 9% × 10)
R19 000
The closing balance for the money earning compound interest is:
Closing Balance =
=
=
P (1 + i)n )
R10 000(1 + 9%)10
R23 673,64
So next time someone talks about the “magic of compound interest”, not only will you know
what they mean - but you will be able to prove it mathematically yourself!
Again, keep in mind that this is good news and bad news. When you are earning interest on
money you have invested, compound interest helps that amount to increase exponentially. But
if you have borrowed money, the build up of the amount you owe will grow exponentially too.
Worked Example 14: Taking out a Loan
Question: Mr Lowe wants to take out a loan of R 350 000. He does not want
to pay back more than R625 000 altogether on the loan. If the interest rate he is
offered is 13%, over what period should he take the loan
Answer
Step 1 : Determine what has been provided and what is required
• opening balance, P = R350 000
• closing balance, A = R625 000
• interest rate, i = 13% peryear
We are required to find the time period(n).
Step 2 : Determine how to approach the problem
We know from (8.3) that:
Closing Balance,(A) = P (1 + i)n
We need to find n.
Therefore we covert the formula to:
A
= (1 + i)n
P
and then find n by trial and error.
Step 3 : Solve the problem
A
= (1 + i)n
P
625000
= (1 + 0,13)n
350000
1,785... = (1,13)n
Try n = 3 :
(1,13)3 = 1,44...
Try n = 4 :
Try n = 5 :
(1,13)4 = 1,63...
(1,13)5 = 1,84...
Step 4 : Write the final answer
Mr Lowe should take the loan over four years
66
CHAPTER 8. FINANCE - GRADE 10
8.5.3
8.5
Other Applications of Compound Growth
Worked Example 15: Population Growth
Question: South Africa’s population is increasing by 2,5% per year. If the current
population is 43 million, how many more people will there be in South Africa in two
year’s time ?
Answer
Step 1 : Determine what has been provided and what is required
• opening balance, P = 43 000 000
• period of time, n = 2 year
• interest rate, i = 2,5% peryear
We are required to find the closing balance(A).
Step 2 : Determine how to approach the problem
We know from (8.3) that:
Closing Balance,(A) = P (1 + i)n
Step 3 : Solve the problem
A =
=
=
P (1 + i)n
43 000 000(1 + 0,025)2
45 176 875
Step 4 : Write the final answer
There are 45 176 875 − 43 000 000 = 2 176 875 more people in 2 year’s time
Worked Example 16: Compound Decrease
Question: A swimming pool is being treated for a build-up of algae. Initially, 50m2
of the pool is covered by algae. With each day of treatment, the algae reduces by
5%. What area is covered by algae after 30 days of treatment ?
Answer
Step 1 : Determine what has been provided and what is required
• opening balance, P = 50m2
• period of time, n = 30 days
• interest rate, i = 5% perday
We are required to find the closing balance(A).
Step 2 : Determine how to approach the problem
We know from (8.3) that:
Closing Balance,(A) = P (1 + i)n
67
8.6
CHAPTER 8. FINANCE - GRADE 10
But this is compound decrease so we can use the formula:
Closing Balance,(A) = P (1 − i)n
Step 3 : Solve the problem
A =
=
=
P (1 − i)n
50(1 − 0,05)30
10,73m2
Step 4 : Write the final answer
Therefore the area still covered with algae is 10,73m2
Exercise: Compound Interest
1. An amount of R3 500 is invested in a savings account which pays compound
interest at a rate of 7,5% per annum. Calculate the balance accumulated by
the end of 2 years.
2. If the average rate of inflation for the past few years was 7,3% and your water
and electricity account is R 1 425 on average, what would you expect to pay
in 6 years time ?
3. Shrek wants to invest some money at 11% per annum compound interest. How
much money (to the nearest rand) should he invest if he wants to reach a sum
of R 100 000 in five year’s time ?
8.6
Summary
As an easy reference, here are the key formulae that we derived and used during this chapter.
While memorising them is nice (there are not many), it is the application that is useful. Financial
experts are not paid a salary in order to recite formulae, they are paid a salary to use the right
methods to solve financial problems.
8.6.1
P
i
n
Definitions
Principal (the amount of money at the starting point of the calculation)
interest rate, normally the effective rate per annum
period for which the investment is made
8.6.2
Equations

Closing Balance - simple interest 
Solve for i
= P (1 + i · n)

Solve for n
68
CHAPTER 8. FINANCE - GRADE 10
8.7

Closing Balance - compound interest 
Solve for i
= P (1 + i)n

Solve for n
Important: Always keep the interest and the time period in the same units of time (e.g.
both in years, or both in months etc.).
8.7
End of Chapter Exercises
1. You are going on holiday to Europe. Your hotel will cost e200 per night. How much will
you need in Rands to cover your hotel bill, if the exchange rate is e1 = R9,20.
2. Calculate how much you will earn if you invested R500 for 1 year at the following interest
rates:
(a) 6,85% simple interest
(b) 4,00% compound interest
3. Bianca has R1 450 to invest for 3 years. Bank A offers a savings account which pays
simple interest at a rate of 11% per annum, whereas Bank B offers a savings account
paying compound interest at a rate of 10,5% per annum. Which account would leave
Bianca with the highest accumulated balance at the end of the 3 year period?
4. How much simple interest is payable on a loan of R2 000 for a year, if the interest rate is
10%?
5. How much compound interest is payable on a loan of R2 000 for a year, if the interest rate
is 10%?
6. Discuss:
(a) Which type of interest would you like to use if you are the borrower?
(b) Which type of interest would you like to use if you were the banker?
7. Calculate the compound interest for the following problems.
(a) A R2 000 loan for 2 years at 5%.
(b) A R1 500 investment for 3 years at 6%.
(c) An R800 loan for l year at 16%.
8. If the exchange rate 100 Yen = R 6,2287 and 1 AUD = R 5,1094 , determine the exchange
rate between the Australian Dollar and the Japanese Yen.
9. Bonnie bought a stove for R 3 750. After 3 years she paid for it and the R 956,25 interest
that was charged for hire-purchase. Determine the simple rate of interest that was charged.
69
8.7
CHAPTER 8. FINANCE - GRADE 10
70
Appendix A
GNU Free Documentation License
Version 1.2, November 2002
c 2000,2001,2002 Free Software Foundation, Inc.
Copyright 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
Everyone is permitted to copy and distribute verbatim copies of this license document, but
changing it is not allowed.
PREAMBLE
The purpose of this License is to make a manual, textbook, or other functional and useful document “free” in the sense of freedom: to assure everyone the effective freedom to copy and
redistribute it, with or without modifying it, either commercially or non-commercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while
not being considered responsible for modifications made by others.
This License is a kind of “copyleft”, which means that derivative works of the document must
themselves be free in the same sense. It complements the GNU General Public License, which
is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software, because free
software needs free documentation: a free program should come with manuals providing the
same freedoms that the software does. But this License is not limited to software manuals; it
can be used for any textual work, regardless of subject matter or whether it is published as a
printed book. We recommend this License principally for works whose purpose is instruction or
reference.
APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium, that contains a notice placed
by the copyright holder saying it can be distributed under the terms of this License. Such a
notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under
the conditions stated herein. The “Document”, below, refers to any such manual or work. Any
member of the public is a licensee, and is addressed as “you”. You accept the license if you
copy, modify or distribute the work in a way requiring permission under copyright law.
A “Modified Version” of the Document means any work containing the Document or a portion
of it, either copied verbatim, or with modifications and/or translated into another language.
A “Secondary Section” is a named appendix or a front-matter section of the Document that deals
exclusively with the relationship of the publishers or authors of the Document to the Document’s
overall subject (or to related matters) and contains nothing that could fall directly within that
overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary
Section may not explain any mathematics.) The relationship could be a matter of historical
connection with the subject or with related matters, or of legal, commercial, philosophical,
ethical or political position regarding them.
619
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being
those of Invariant Sections, in the notice that says that the Document is released under this
License. If a section does not fit the above definition of Secondary then it is not allowed to be
designated as Invariant. The Document may contain zero Invariant Sections. If the Document
does not identify any Invariant Sections then there are none.
The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or
Back-Cover Texts, in the notice that says that the Document is released under this License. A
Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.
A “Transparent” copy of the Document means a machine-readable copy, represented in a format
whose specification is available to the general public, that is suitable for revising the document
straightforwardly with generic text editors or (for images composed of pixels) generic paint
programs or (for drawings) some widely available drawing editor, and that is suitable for input
to text formatters or for automatic translation to a variety of formats suitable for input to text
formatters. A copy made in an otherwise Transparent file format whose markup, or absence of
markup, has been arranged to thwart or discourage subsequent modification by readers is not
Transparent. An image format is not Transparent if used for any substantial amount of text. A
copy that is not “Transparent” is called “Opaque”.
Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo
input format, LATEX input format, SGML or XML using a publicly available DTD and standardconforming simple HTML, PostScript or PDF designed for human modification. Examples of
transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary
formats that can be read and edited only by proprietary word processors, SGML or XML for
which the DTD and/or processing tools are not generally available, and the machine-generated
HTML, PostScript or PDF produced by some word processors for output purposes only.
The “Title Page” means, for a printed book, the title page itself, plus such following pages as
are needed to hold, legibly, the material this License requires to appear in the title page. For
works in formats which do not have any title page as such, “Title Page” means the text near the
most prominent appearance of the work’s title, preceding the beginning of the body of the text.
A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely
XYZ or contains XYZ in parentheses following text that translates XYZ in another language.
(Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”,
“Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when
you modify the Document means that it remains a section “Entitled XYZ” according to this
definition.
The Document may include Warranty Disclaimers next to the notice which states that this
License applies to the Document. These Warranty Disclaimers are considered to be included by
reference in this License, but only as regards disclaiming warranties: any other implication that
these Warranty Disclaimers may have is void and has no effect on the meaning of this License.
VERBATIM COPYING
You may copy and distribute the Document in any medium, either commercially or non-commercially,
provided that this License, the copyright notices, and the license notice saying this License applies
to the Document are reproduced in all copies, and that you add no other conditions whatsoever
to those of this License. You may not use technical measures to obstruct or control the reading
or further copying of the copies you make or distribute. However, you may accept compensation
in exchange for copies. If you distribute a large enough number of copies you must also follow
the conditions in section A.
You may also lend copies, under the same conditions stated above, and you may publicly display
copies.
COPYING IN QUANTITY
If you publish printed copies (or copies in media that commonly have printed covers) of the
Document, numbering more than 100, and the Document’s license notice requires Cover Texts,
620
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: FrontCover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also
clearly and legibly identify you as the publisher of these copies. The front cover must present the
full title with all words of the title equally prominent and visible. You may add other material on
the covers in addition. Copying with changes limited to the covers, as long as they preserve the
title of the Document and satisfy these conditions, can be treated as verbatim copying in other
respects.
If the required texts for either cover are too voluminous to fit legibly, you should put the first
ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent
pages.
If you publish or distribute Opaque copies of the Document numbering more than 100, you must
either include a machine-readable Transparent copy along with each Opaque copy, or state in or
with each Opaque copy a computer-network location from which the general network-using public
has access to download using public-standard network protocols a complete Transparent copy of
the Document, free of added material. If you use the latter option, you must take reasonably
prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this
Transparent copy will remain thus accessible at the stated location until at least one year after
the last time you distribute an Opaque copy (directly or through your agents or retailers) of that
edition to the public.
It is requested, but not required, that you contact the authors of the Document well before
redistributing any large number of copies, to give them a chance to provide you with an updated
version of the Document.
MODIFICATIONS
You may copy and distribute a Modified Version of the Document under the conditions of
sections A and A above, provided that you release the Modified Version under precisely this
License, with the Modified Version filling the role of the Document, thus licensing distribution
and modification of the Modified Version to whoever possesses a copy of it. In addition, you
must do these things in the Modified Version:
1. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document,
and from those of previous versions (which should, if there were any, be listed in the History
section of the Document). You may use the same title as a previous version if the original
publisher of that version gives permission.
2. List on the Title Page, as authors, one or more persons or entities responsible for authorship
of the modifications in the Modified Version, together with at least five of the principal
authors of the Document (all of its principal authors, if it has fewer than five), unless they
release you from this requirement.
3. State on the Title page the name of the publisher of the Modified Version, as the publisher.
4. Preserve all the copyright notices of the Document.
5. Add an appropriate copyright notice for your modifications adjacent to the other copyright
notices.
6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the
Addendum below.
7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts
given in the Document’s license notice.
8. Include an unaltered copy of this License.
9. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating
at least the title, year, new authors, and publisher of the Modified Version as given on the
Title Page. If there is no section Entitled “History” in the Document, create one stating
the title, year, authors, and publisher of the Document as given on its Title Page, then
add an item describing the Modified Version as stated in the previous sentence.
621
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
10. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document
for previous versions it was based on. These may be placed in the “History” section. You
may omit a network location for a work that was published at least four years before the
Document itself, or if the original publisher of the version it refers to gives permission.
11. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the
section, and preserve in the section all the substance and tone of each of the contributor
acknowledgements and/or dedications given therein.
12. Preserve all the Invariant Sections of the Document, unaltered in their text and in their
titles. Section numbers or the equivalent are not considered part of the section titles.
13. Delete any section Entitled “Endorsements”. Such a section may not be included in the
Modified Version.
14. Do not re-title any existing section to be Entitled “Endorsements” or to conflict in title
with any Invariant Section.
15. Preserve any Warranty Disclaimers.
If the Modified Version includes new front-matter sections or appendices that qualify as Secondary
Sections and contain no material copied from the Document, you may at your option designate
some or all of these sections as invariant. To do this, add their titles to the list of Invariant
Sections in the Modified Version’s license notice. These titles must be distinct from any other
section titles.
You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements
of your Modified Version by various parties–for example, statements of peer review or that the
text has been approved by an organisation as the authoritative definition of a standard.
You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25
words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only
one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through
arrangements made by) any one entity. If the Document already includes a cover text for the
same cover, previously added by you or by arrangement made by the same entity you are acting
on behalf of, you may not add another; but you may replace the old one, on explicit permission
from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License give permission to use
their names for publicity for or to assert or imply endorsement of any Modified Version.
COMBINING DOCUMENTS
You may combine the Document with other documents released under this License, under the
terms defined in section A above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them
all as Invariant Sections of your combined work in its license notice, and that you preserve all
their Warranty Disclaimers.
The combined work need only contain one copy of this License, and multiple identical Invariant
Sections may be replaced with a single copy. If there are multiple Invariant Sections with the
same name but different contents, make the title of each such section unique by adding at the
end of it, in parentheses, the name of the original author or publisher of that section if known,
or else a unique number. Make the same adjustment to the section titles in the list of Invariant
Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled “History” in the various original
documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled
“Endorsements”.
622
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under
this License, and replace the individual copies of this License in the various documents with a
single copy that is included in the collection, provided that you follow the rules of this License
for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under
this License, provided you insert a copy of this License into the extracted document, and follow
this License in all other respects regarding verbatim copying of that document.
AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate and independent documents
or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the
copyright resulting from the compilation is not used to limit the legal rights of the compilation’s
users beyond what the individual works permit. When the Document is included an aggregate,
this License does not apply to the other works in the aggregate which are not themselves derivative
works of the Document.
If the Cover Text requirement of section A is applicable to these copies of the Document, then if
the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be
placed on covers that bracket the Document within the aggregate, or the electronic equivalent
of covers if the Document is in electronic form. Otherwise they must appear on printed covers
that bracket the whole aggregate.
TRANSLATION
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section A. Replacing Invariant Sections with translations requires
special permission from their copyright holders, but you may include translations of some or
all Invariant Sections in addition to the original versions of these Invariant Sections. You may
include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and
the original versions of those notices and disclaimers. In case of a disagreement between the
translation and the original version of this License or a notice or disclaimer, the original version
will prevail.
If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the
requirement (section A) to Preserve its Title (section A) will typically require changing the actual
title.
TERMINATION
You may not copy, modify, sub-license, or distribute the Document except as expressly provided
for under this License. Any other attempt to copy, modify, sub-license or distribute the Document
is void, and will automatically terminate your rights under this License. However, parties who
have received copies, or rights, from you under this License will not have their licenses terminated
so long as such parties remain in full compliance.
FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation
License from time to time. Such new versions will be similar in spirit to the present version, but
may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
623
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
Each version of the License is given a distinguishing version number. If the Document specifies
that a particular numbered version of this License “or any later version” applies to it, you have the
option of following the terms and conditions either of that specified version or of any later version
that has been published (not as a draft) by the Free Software Foundation. If the Document does
not specify a version number of this License, you may choose any version ever published (not as
a draft) by the Free Software Foundation.
ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of the License in the document
and put the following copyright and license notices just after the title page:
c YEAR YOUR NAME. Permission is granted to copy, distribute and/or
Copyright modify this document under the terms of the GNU Free Documentation License,
Version 1.2 or any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the
license is included in the section entitled “GNU Free Documentation License”.
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.”
line with this:
with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST,
and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other combination of the three,
merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these
examples in parallel under your choice of free software license, such as the GNU General Public
License, to permit their use in free software.
624
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement