Mathscape 9 Extension: An Essential Guide to Working Mathematically

First published 2004 by
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Copyright © Clive Meyers, Graham Barnsley, Lloyd Dawe, Lindsay Grimison 2004
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National Library of Australia
cataloguing in publication data
Meyers, Clive.
Mathscape 9 extension : working mathematically.
For secondary school students.
ISBN 0 7329 8085 2.
1. Mathematics – Textbooks. I. Grimison, Lindsay. II.
Barnsley, Graham J. III. Dawe, Lloyd. IV. Title.
510
Publisher: Ben Dawe
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Contents
Green indicates material is exclusively Stage 4. All other material is Stage 5.1/5.2/5.3.
Preface vi
How to use this book vii
Chapter 1 Rational numbers 1
1.1 Significant figures 2
1.2 The calculator 4
1.3 Estimation 8
Try this: Fermi problem 10
1.4 Recurring decimals 10
1.5 Rates 13
Try this: Desert walk 15
1.6 Solving problems with rates 15
Try this: Passing trains 19
Focus on working mathematically:
A number pattern from Galileo 1615 20
Language link with Macquarie 22
Chapter review 23
Chapter 2 Algebra 25
2.1 Describing simple patterns 26
Try this: Flags 31
2.2 Substitution 32
2.3 Adding and subtracting algebraic
expressions 33
2.4 Multiplying and dividing algebraic
expressions 36
Try this: Overhanging the overhang 38
2.5 The order of operations 38
2.6 The distributive law 40
2.7 The highest common factor 42
2.8 Adding and subtracting algebraic
fractions 44
2.9 Multiplying and dividing algebraic
fractions 47
2.10 Generalised arithmetic 49
Try this: Railway tickets 53
2.11 Properties of numbers 54
2.12 Generalising solutions to problems
using patterns 56
2.13 Binomial products 60
2.14 Perfect squares 63
Try this: Proof 66
2.15 Difference of two squares 67
2.16 Miscellaneous expansions 69
Focus of working mathematically:
A number pattern from Blaise Pascal
1654 71
Chapter review 74
Chapter 3 Consumer
arithmetic 78
3.1 Salaries and wages 79
3.2 Other methods of payment 83
3.3 Overtime and other payments 87
3.4 Wage deductions 90
3.5 Taxation 93
3.6 Budgeting 98
Try this: Telephone charges 101
3.7 Best buys 102
3.8 Discounts 104
Try this: Progressive discounting 107
3.9 Profit and loss 108
Sydney market prices in 1831 111
Chapter review 114
Chapter 4 Equations, inequations
and formulae 117
4.1 One- and two-step equations 118
4.2 Equations with pronumerals on both
sides 121
4.3 Equations with grouping symbols 123
4.4 Equations with one fraction 124
4.5 Equations with more than one fraction126
4.6 Inequations 129
4.7 Solving worded problems 134
Try this: A prince and a king 137
4.8 Evaluating the subject of a formula 138
4.9 Equations arising from substitution 141
Try this: Floodlighting by formula 143
4.10 Changing the subject of a formula 144
Splitting the atom 149
Chapter review 152
iii

M a t h s c a p e 9 E x t e n s i o n
Chapter 5 Measurement 155
5.1 Length, mass, capacity and time 156
5.2 Accuracy and precision 162
5.3 Pythagoras’ theorem 165
Try this: Pythagorean proof
by Perigal 170
5.4 Perimeter 170
5.5 Circumference 175
Try this: Command module 180
5.6 Converting units of area 181
5.7 Calculating area 183
Try this: The area of a circle 191
5.8 Area of a circle 192
5.9 Composite areas 195
Try this: Area 200
5.10 Problems involving area 200
The solar system 203
Chapter review 206
Chapter 6 Data representation
and analysis 211
6.1 Graphs 212
6.2 Organising data 219
6.3 Analysing data 225
6.4 Problems involving the mean 233
Try this: The English language 236
6.5 Cumulative frequency 236
6.6 Grouped data 242
World health 248
Chapter review 252
Chapter 7 Probability 256
7.1 Probability and its language 257
7.2 Experimental probability 260
Try this: Two-up 266
7.3 Computer simulations 266
Try this: The game of craps 271
7.4 Theoretical probability 272
Try this: Winning chances 275
A party game 276
Chapter review 279
Chapter 8 Surds 282
8.1 Rational and irrational numbers 283
8.2 Simplifying surds 288
Try this: Greater number 291
8.3 Addition and subtraction of surds 291
8.4 Multiplication and division of surds 294
Try this: Imaginary numbers 297
8.5 Binomial products with surds 298
8.6 Rationalising the denominator 301
Try this: Exact values 304
Fibonacci numbers and the
golden mean 305
Chapter review 309
Chapter 9 Indices 311
9.1 Index notation 312
9.2 Simplifying numerical expressions
using the index laws 313
9.3 The index laws 315
9.4 Miscellaneous questions on the
index laws 320
9.5 The zero index 322
Try this: Smallest to largest 323
9.6 The negative index 323
9.7 Products and quotients with negative
indices 326
Try this: Digit patterns 328
9.8 The fraction index 329
9.9 Scientific notation 333
9.10 Scientific notation on the calculator 335
Mathematics is at the heart of science 338
Chapter review 340
Chapter 10 Geometry 343
10.1 Angles 344
10.2 Parallel lines 350
10.3 Triangles 356
Try this: The badge of the
Pythagoreans 363
10.4 Angle sum of a quadrilateral 363
10.5 Special quadrilaterals 367
Try this: Five shapes 374
10.6 Polygons 374
iv

C o n t e n t s
Try this: How many diagonals in
a polygon? 379
Try this: An investigation of
triangles 380
10.7 Tests for congruent triangles 381
10.8 Congruent proofs 387
Try this: Triangle angles 392
10.9 Deductive reasoning and congruent
triangles 392
Does a triangle have a centre? 397
Chapter review 402
Chapter 11 The linear
function 408
11.1 The number plane 409
11.2 Graphing straight lines (1) 412
Try this: Size 8 417
11.3 Graphing straight lines (2) 417
11.4 Gradient of a line 422
Try this: Hanging around 427
11.5 The linear equation y = mx + b 427
Try this: Latitude and temperature 433
Television advertising 433
Chapter review 436
Chapter 12 Trigonometry 440
12.1 Side ratios in right-angled triangles 441
12.2 The trigonometric ratios 444
Try this: Height to base ratio 448
12.3 Trigonometric ratios using a
calculator 448
12.4 Finding the length of a side 451
12.5 Problems involving finding sides 456
Try this: Make a hypsometer 460
12.6 Finding the size of an angle 461
12.7 Problems involving finding angles 464
12.8 Angles of elevation and depression 467
Try this: Pilot instructions 470
12.9 Bearings 471
Try this: The sine rule 478
Finding your latitude from the Sun 479
Chapter review 484
Chapter 13 Simultaneous
equations 488
13.1 Equations with two unknowns 489
13.2 The graphical method 492
13.3 The substitution method 496
Try this: Find the values 498
13.4 The elimination method 499
Try this: A Pythagorean problem 502
13.5 Solving problems using simultaneous
equations 502
Exploring for water, oil and gas—
the density of air-filled porous rock 506
Chapter review 509
Chapter 14 Co-ordinate
geometry 511
14.1 The distance between two points 512
14.2 The midpoint of an interval 516
14.3 The gradient formula 520
Try this: A line with no integer
co-ordinates 525
14.4 General form of the equation of a line 525
14.5 The equation of a line given the
gradient and a point 530
14.6 The equation of a line given
two points 533
Try this: Car hire 536
14.7 Parallel lines 536
Try this: Temperature rising 540
14.8 Perpendicular lines 540
14.9 Regions in the number plane 544
14.10 Co-ordinate geometry problems 549
Finding the gradient of a ski run 554
Chapter review 559
Answers 563
v

Preface
Mathscape 9 Extension is a comprehensive teaching and learning resource that has been written to address the
new Stage 5.1/5.2/5.3 Mathematics syllabus in NSW. Our aim was to write a book that would allow more able
students to grow in confidence, to improve their understanding of Mathematics and to develop a genuine
appreciation of its inherent beauty. Teachers who wish to inspire their students will find this an exciting, yet
very practical resource. The text encourages a deeper exploration of mathematical ideas through substantial,
well-graded exercises that consolidate students’ knowledge, understanding and skills. It also provides
opportunities for students to explore the history of Mathematics and to address many practical applications in
contexts that are both familiar and relevant.
From a teaching perspective, we sought to produce a book that would adhere as strictly as possible to both the
content and spirit of the new syllabus. Together with Mathscape 10 Extension, this book allows teachers to
confidently teach the Stage 5.1/5.2/5.3 courses knowing that they are covering all of the mandatory outcomes.
Content from Stage 4 has been included in each chapter, where appropriate. This will allow teachers to
diagnose significant misconceptions and identify any content gaps. For those students who have achieved the
relevant Stage 4 outcomes, this material could be used as a review to introduce the Stage 5.1/5.2/5.3 topics, or
to revise important concepts when they occur. However, for those students who have not achieved these
outcomes by the start of Year 9, this material will be new work. All content is clearly listed as either Stage 4
or Stage 5.1/5.2/5.3 in the contents section at the front of the book. A detailed syllabus correlation grid has
been provided for teachers on the Mathscape 9/9 Extension School CD-ROM.
Mathscape 9 Extension has embedded cross-curriculum content, which will support students in achieving the
broad learning outcomes defined by the Board of Studies. The content also addresses the important key
competencies of the Curriculum Framework, which requires students to collect, analyse and organise
information; to communicate mathematical ideas; to plan and organise activities; to work with others in
groups; to use mathematical ideas and techniques; to solve problems; and to use technology.
A feature of each chapter which teachers will find both challenging and interesting for their students is the
‘Focus on working mathematically’ section. Although the processes of working mathematically are embedded
throughout the book, these activities are specifically designed to provoke curiosity and deepen mathematical
insight. Most begin with a motivating real-life context, such as television advertising, or the gradient of a ski
run, but on occasion they begin with a purely mathematical question. (These activities can also be used for
assessment purposes.)
In our view, there are many legitimate, time-proven ways to teach Mathematics successfully. However, if
students are to develop a deep appreciation of the subject, they will need more than traditional methods. We
believe that all students should be given the opportunity to appreciate Mathematics as an essential and relevant
part of life. They need to be given the opportunity to begin a Mathematical exploration from a real-life context
that is meaningful to them. To show interest and enjoyment in enquiry and the pursuit of mathematical
knowledge, students need activities where they can work with others and listen to their arguments, as well as
work individually. To demonstrate confidence in applying their mathematical knowledge and skills to the
solution of everyday problems, they will need experience of this in the classroom. If they are to learn to
persevere with difficult and challenging problems, they will need to experience these sorts of problems as well.
Finally, to recognise that mathematics has been developed in many cultures in response to human needs,
students will need experiences of what other cultures have achieved mathematically.
We have tried to address these values and attitudes in this series of books. Our best wishes to all teachers and
students who are part of this great endeavour.
Clive Meyers
Lloyd Dawe
Graham Barnsley
Lindsay Grimison
vi

How to use this book
Mathscape 9 Extension is a practical resource that can be used by teachers to supplement their teaching
program. The exercises in this book and the companion text (Mathscape 10 Extension) provide a complete and
thorough coverage of all content and skills in the Stage 5.1/5.2/5.3 course. The great number and variety of
questions allow for the effective teaching of more able students. Each chapter contains:
• a set of chapter outcomes directed to the student
• all relevant theory and explanations, with important definitions and formulae boxed and coloured
• step-by-step instructions for standard questions
• a large number of fully worked examples preceding each exercise
• extensive, thorough and well-graded exercises that cover each concept in detail
• chapter-related, problem-solving activities called ‘Try this’ integrated throughout
• a language skills section linked to the Macquarie Learners Dictionary
• novel learning activities focusing on the process of working mathematically
• a thorough chapter review.
Explanations and examples
The content and skills required to complete each exercise have been introduced in a manner and at a level that
is appropriate to the students in this course. Important definitions and formulae have been boxed and coloured
for easy reference. For those techniques that require a number of steps, the steps have been listed in point form,
boxed and coloured. Each exercise is preceded by several fully worked examples. This should enable the
average student to independently complete the majority of relevant exercises if necessary.
The exercises
The exercises have been carefully graded into three distinct sections:
• Introduction. The questions in this section are designed to introduce students to the most basic concepts
and skills associated with the outcome(s) being covered in the exercise. Students need to have mastered
these ideas before attempting the questions in the next section.
• Consolidation. This is a major part of the exercise. It allows students to consolidate their understanding of
the basic ideas and apply them in a variety of situations. Students may need to use content learned or skills
acquired in previous exercises or topics to answer some of these questions. The average student should be
able to complete most of the questions in this section, although the last few questions may be a little more
difficult.
• Further applications. Some questions presented in this section will be accessible to the average student;
however, the majority of questions are difficult. They might require a reverse procedure, the use of algebra,
more sophisticated techniques, a proof, or simply time-consuming research. The questions can be
open-ended, requiring an answer with a justification. They may also involve extension or off-syllabus
material. In some questions, alternative techniques and methods of solution other than the standard
method(s) may be introduced, which may confuse some students.
Teachers need to be selective in the questions they choose for their students. Some students may not need to
complete all of the questions in the Introduction or Consolidations sections of each exercise, while only the
most able students should usually be expected to attempt the questions in the Further applications section.
Those questions not completed in class might be set as homework at the teacher’s discretion. It is not intended
that any student would attempt to answer every possible question in each exercise.
Focus on working mathematically
The Working Mathematically strand of the syllabus requires a deeper understanding of Mathematics than do
the other strands. As such, it will be the most challenging strand for students to engage with and for teachers
to assess. The Working Mathematically outcomes listed in the syllabus have been carefully integrated into
each chapter of the book; however, we also decided to include learning activities in each chapter that will
vii

enable teachers to focus sharply on the processes of working mathematically. Each activity begins with a real-
life context and the Mathematics emerges naturally. Teachers are advised to work through them before using
them in class. Answers have not been provided, but notes for teachers have been included on the Mathscape
9/9 Extension School CD-ROM, with suggested weblinks. Teachers may wish to select and use the Learning
activities in ‘Focus on working mathematically’ for purposes of assessment. This too is encouraged. The
Extension activities will test the brightest students. Suggestions are also provided to assess the outcomes
regarding Communication and Reflection.
Problem solving
Each chapter contains a number of small, chapter-related, problem-solving activities called ‘Try this’. They
may be of some historical significance, or require an area outside the classroom, or require students to conduct
research, or involve the use of algebra, while others relate the chapter content to real-life context. Teachers are
advised to work through these exercises before using them in class.
Technology
The use of technology is a clear emphasis in the new syllabus. Innovative technology for supporting the growth
of understanding of mathematical ideas is provided on the Mathscape 9/9 Extension School CD-ROM, which
is fully networkable and comes free-of-charge to schools adopting Mathscape 9 Extension for student use.
Key features of the CD-ROM include:
• spreadsheet activities
• dynamic geometry
• animations
• executables
• student worksheets
• weblinks for ‘Focus on working mathematically’.
Language
The consistent use of correct mathematical terms, symbols and conventions is emphasised strongly in this
book, while being mindful of the students’ average reading age. Students will only learn to use and spell
correct mathematical terms if they are required to use them frequently in appropriate contexts. A language
section has also been included at the end of each chapter titled ‘Language link with Macquarie’, where students
can demonstrate their understanding of important mathematical terms. This might, for example, include
explaining the difference between the mathematical meaning and the everyday meaning of a word. Most
chapters include a large number of worded problems. Students are challenged to read and interpret the
problem, translate it into mathematical language and symbols, solve the problem, then give the answer in an
appropriate context.
Clive Meyers
Lloyd Dawe
Graham Barnsley
Lindsay Grimison
viii

1
Rational
numbers
This chapter at a glance
Stage 5.1/5.2/5.3
After completing this chapter, you should be able to:
evaluate numerical expressions using a calculator
estimate the result of a calculation
state the number of significant figures in a number
round off a number correct to a given number of significant figures
determine the effect of rounding during calculations on the accuracy
of the results
convert fractions to recurring decimals
convert recurring decimals to fractions
express a rate in its simplest form
convert rates from one set of units to another
solve problems involving rates. Rational
numbers
1

2
No quantity, such as length, mass or time, can be measured exactly. For a measurement to be
of use, we need to know how accurate it is. That is, we must be confident that each digit in the
measurement is significant.
When rounding off correct to a specified number of significant figures, choose the number that
is closest in value to the given number and which also contains the required number of
significant figures.
Example 1
State the number of significant figures in each number.
a 4.009 b 137.20 c 0.001 64 d 5000
Solutions
a In 4.009, the two non-zero digits (i.e. 4 and 9) are significant and the two zeros between
these digits are significant. ∴ The number has 4 significant figures.
b In 137.20, the four non-zero digits (i.e. 1, 3, 7 and 2) are significant and the zero at the end
of the decimal is significant. ∴ The number has 5 significant figures.
c In 0.001 64, the three non-zero digits (i.e. 1, 6 and 4) are significant; however, the zeros
at the beginning of the decimal are not significant. ∴ The number has 3 significant figures.
d In 5000, the non-zero digit (i.e. 5) is significant. Either some, all or none of the final zeros
could possibly be significant. This would need to be determined from the context in which
the number occurs. If we knew that the number had been rounded off correct to:
i 1 significant figure, then only the 5 would be significant
ii 2 significant figures, then only the 5 and the first zero would be significant
iii 3 significant figures, then only the 5 and the first two zeros would be significant
iv 4 significant figures, then all of the digits would be significant.
1.1 Significant figures
A significant figure is a number that is correct within
some stated degree of accuracy.
The rules for significant figures are:
All non-zero digits are significant.
Zeros between non-zero digits are significant.
Zeros at the end of a decimal are significant.
Zeros before the first non-zero digit in a decimal are not significant.
Zeros after the last non-zero digit in a whole number may or may not be
significant.
EG
+S

C h a p t e r 1 : Rational numbers 3
Example 2
Round off 47.503 correct to:
a 4 significant figures b 3 significant figures
c 2 significant figures d 1 significant figure
Solutions
a 47.503 = 47.50 (4 significant figures) b 47.503 = 47.5 (3 significant figures)
c 47.503 = 48 (2 significant figures) d 47.503 = 50 (1 significant figure)
Example 3
Round off 39.99 correct to:
a 3 significant figures b 2 significant figures c 1 significant figure
Solutions
a 39.99 = 40.0 (3 significant figures)
b 39.99 = 40 (2 significant figures; both the 4 and the 0 are significant figures)
c 39.99 = 40 (1 significant figure; only the 4 is significant)
1 State the number of significant figures in each of the following.
a 45 b 7281 c 859 d 132 494
e 607 f 3012 g 4001 h 20 809
2 State the number of significant figures in each decimal.
a 5.28 b 7.152 c 38.5 d 254.883
e 0.4 f 0.005 g 0.0371 h 0.003 469
i 5.062 j 13.007 k 58.0208 l 0.001 09
m 9.30 n 0.10 o 1.4700 p 0.004 080
q 3.030 r 32.0040 s 409.010 00 t 0.010 203 00
■ Consolidation
3 Round off each of the following correct to 1 significant figure.
a 83 b 27 c 65 d 94
e 136 f 415 g 250 h 3810
i 9450 j 26 449 k 539 499 l 850 000
4 Round off each of these numbers correct to 2 significant figures.
a 128 b 171 c 234 d 675
e 1459 f 4026 g 8350 h 12 042
i 45 718 j 76 285 k 285 195 l 644 003
EG
+S
EG
+S
Exercise 1.1

4
5 Round off each of the following decimals correct to the number of significant figures
indicated in the brackets.
a 3.67 [1] b 0.484 [1] c 0.0731 [2] d 6.2085 [4]
e 11.784 [2] f 0.3 [2] g 25.156 [3] h 49.066 28 [5]
i 91.045 [3] j 144.387 [2] k 7.3855 [4] l 10.9367 [2]
m 2018.68 [3] n 3693.21 [2] o 4002.142 [5] p 9187.549 [6]
6 Round off the following correct to:
i 1 significant figure ii 2 significant figures iii 3 significant figures
a 99.35 b 194.97 c 998.763 d 499.861
■ Further applications
7 Write down a possible number that is approximately equal to:
a 130, correct to 2 significant figures b 2.47, correct to 3 significant figures
As a wide variety of calculators is available, there are differences in the way they operate.
The examples here have been worked with the use of a direct logic calculator. That is, the
calculations are performed in the logical order in which they appear. For example, to evaluate
on a direct logic calculator, we press the square root key followed by the 9, then
press . For models that do not use direct logic, we enter the 9, then press the square root key.
You will need to familiarise yourself with how your calculator works.
Example 1
Evaluate each of the following.
a b −78 − 96 c 15.982 d
e f 3.524 g h
Solutions
Calculator steps: Answer:
a 6 7 5 2 3
b 78 96 –174
c 15.98 255.3604
d 69.4 8.330 666 24
e 41 3.448 217 24
f 3.52 4 153.522 012 2
g 5 117.3 2.593 340 858
h 0.274 3.649 635 036
1.2 The calculator
9
=
EG
+S
6
7
--
- 5
2
3
--
-
+ 69.4
41
3 117.3
5 1
0.274
------------
-
ab
c
--
- + ab
c
--
- ab
c
--
- = 611
21
-----
-
+/− − =
x2
=
=
3
=
xy
=
x =
x−1
=

Example 2
Evaluate each of these, correct to 2 decimal places, using the grouping symbols keys
and .
a b
Solutions
Calculator steps: Answer
a 86.9 213.7 5.6 8.3 6.47
b 342.5 114.8 15.09
Example 3
Use the memory function on the calculator to evaluate , correct to 3 decimal
places.
Solution
i Evaluate the denominator first and store the answer in the memory.
12.5 0.98
ii Evaluate the numerator, then divide the answer by the number stored in the memory.
72.6 153.9
Answer: 1.479 (3 decimal places)
1 a Evaluate × 12.43 correct to 3 decimal places, without rounding off during the
calculation.
b Round off to the nearest integer, then multiply by 12.43.
c Round off to 1, 2 and 3 decimal places then multiply by 12.43. What do you
notice?
d What effect does rounding off too early have on the accuracy of an answer?
2 Evaluate each of these using the fraction key , then give the answers as decimals,
correct to 2 decimal places.
a b c
3 Evaluate each of the following using the sign change key .
a −98 − 156 b −49 + 32 − 77 c −156 ÷ −12
4 Evaluate each of these correct to 4 significant figures using the square key .
a 7.82 b (–12.7)2 c
EG
+S ( )
86.9 213.7
+
5.6 8.3
×
-----------------------------
- 342.5 114.8
–
+ = ÷ ( × ) =
( − ) =
EG
+S
72.6 153.9
+
12.5
2
0.98
×
-----------------------------
-
x2 × = Min
+ = ÷ MR =
Exercise 1.2
72
72
72
ab
c
--
-
3
8
--
-
1
11
-----
-
+ 87
9
--
- 34
5
--
-
– 25
6
--
- 42
7
--
-
×
+/−
x2
47
8
--
-
( )
2

6
5 Evaluate each of these correct to the nearest tenth using the square root key and cube
root key .
a b c
d e f
6 Evaluate each of the following correct to 1 decimal place using the power key .
a 6.53 b 3.724 c 4.085
d e 1.857 × 4.3 f 8.94 − 3.15
7 Evaluate each of these correct to the nearest hundredth using the root key .
a b c
d e f
8 Evaluate each of the following correct to 3 significant figures using the reciprocal key
or .
a b c
d e f
9 Evaluate each of these correct to 2 decimal places using the pi key .
a π + 16.82 b 7π c
d π2 e f
10 Evaluate each of the following correct to the nearest tenth using the grouping symbols keys
and where necessary.
a b c
d e f
■ Consolidation
11 Find the value of each expression, correct to 2 decimal places.
a 10.652 × 8.3 b c
3
23 85 72.6
+ 90 16.45
×
70
3 110.4
3 2.96
÷ 36.7
3 152.6
+
xy
2 3
11
-----
-
( )
6
x
11
4 68.2
5 212.9
7
96
5 12.5
× 3 2.4
6
– 71
5
--
-
4
x−1
1
x
--
-
1
7
--
-
1
0.245
------------
-
1
0.065
2
---------------
1
3
------
-
1
51.4
3
--------------
-
1
1.98
4
------------
π
9π
2
-----
-
1
π
--
- π
5
( )
73 115
+
14
--------------------
-
172
8.5 3.1
×
--------------------
-
19.3 54.7
×
6.4 9.8
+
--------------------------
-
12 11 10
×
×
7 8 9
×
×
-----------------------------
-
9.4
3
5.1 7.25
×
-----------------------
-
135 18.7
+
11 π
–
-----------------------------
-
83
2.6
4
---------
101
7
--------
-

d e f 34 − 4.13
g h i
j k l
m n o
p q r
s (1.7 +1.16)6 t u
v w x
12 Evaluate, correct to the nearest tenth.
a b c
d e f
g h 4.6(19.83 − 7.12)3 i
j k l
13 Use the memory function on the calculator to evaluate each of these, correct to 1 decimal
place.
a b c
d e f
42 7.5
×
28
5
------
-
74.9 87.2
+
3 7.9
3
5 16.8
6
13.9
4
------------
10 20
+
15
-------------------------- 25 50.3 19.6
–
+
3 1
0.06 7
×
-----------------------
-
30
2 3
+
-------------------
-
250
–
5
2
8
2
–
---------------
- 82.6 16.1
×
4
24
2
23
2
+
16
2
15
2
–
----------------------
-
1
2
5
2
5
–
------------------
116.7 99.8
+
3
2.1
2
----------------------------------
18
2
7
3
+ 5
4
81
8
--
-
3
+
10 3
+
10 3
–
------------------
-
1
0.1
3 0.2
2
+
---------------------------
-
8.4
–
( )
3
6.3
– 11.4
+
----------------------------
π
3
14
×
1
3 5
–
---------------
- 13.6
( )
7
0.92
2.3
---------
-
18.9
5.14
---------
-
+
1
2
------
-
1
3
------
-
1
5
------
-
+ +
40.6
– 15.35
+
6.2 7.7
×
----------------------------------
17 18
+
17
3 18
3
+
--------------------------
-
1
0.86 0.29
2
–
--------------------------------
-
9
2
3
--
-
⎝ ⎠
⎛ ⎞
3
1
1
2
--
-
⎝ ⎠
⎛ ⎞
7
÷
100
10 10
3 10
4
+ +
-------------------------------------------
124.37 19.66
–
9.7 11.75
+
-----------------------------------
-
7.6
2
39
+
1.4
3
0.995
×
----------------------------
-
3
11.6
2.3
---------
-
⎝ ⎠
⎛ ⎞
5
9.47
1.02
---------
-
⎝ ⎠
⎛ ⎞
3
÷
8.1
1.9 2.64
+
-----------------------
-
13.4
7 0.16
2
×
---------------------
-
+
3.9 15.6
×
10.58 1.33
3
–
-------------------------------
-
21.4
6.09
---------
-
+
57.5
3
13.6
4
–
15 98.2
×
------------------------------
-
1
12.4
2
------------
×
17.5 5.3
×
6.7
-----------------------
-
4 1
0.075
2
---------------
3
–

8
Calculators do not make arithmetic errors. But sometimes we get incorrect answers when we
use a calculator. This is because we may have:
• left out a decimal point
• pressed the wrong key by mistake
• not pressed the equals key at the right time
• not understood the question
• set the calculator in the wrong mode
• not pressed the second function key.
By estimating the answer before using a calculator, we can work out whether the calculator
answer is reasonable. An estimate is more than a guess. It is an approximate answer that is
worked out logically. It does not have to be very close to the correct answer but it should be of
the same order of magnitude. That is, if the estimate is in the tens, the correct answer should
not be in the hundreds or the thousands.
For example, before evaluating 19.855 × 4.84 with a calculator, we could estimate that the
answer would be close to 20 × 5, that is, 100. If the calculator gives the answer as 9609.82, we
might have made an error when entering the data. In fact, a decimal point was omitted, since
the correct answer is 96.0982. It is also possible, of course, that our estimate is incorrect.
NOTE: Many different estimates can be given to calculations depending on the way that each
individual number is rounded off.
Example
Estimate the answer to each of these calculations.
a 386 × 19 b 154.5 ÷ 11.2 c 17.74 × 0.493 d
Solutions
1 Round off each number correct to 1 significant figure and hence estimate the value of:
a 48 × 33 b 385 × 11 c 69 × 114 d 19 952 × 9
e 223 ÷ 52 f 642 ÷ 22 g 38 840 ÷ 375 h 8445 ÷ 23
i 54 × 186 j 2751 ÷ 63 k 297 × 42 l 96 959 ÷ 4367
a 386 × 19
⯐ 400 × 20
= 8000
b 154.5 ÷ 11.2
⯐ 150 ÷ 10
= 15
c 17.74 × 0.493
⯐ 18 ×
= 9
d
⯐
= 6 × 20
= 120
1.3 Estimation
EG
+S 41.68 21.19
×
6.904
--------------------------------
-
1
2
--
-
41.68 21.19
×
6.904
--------------------------------
-
42 20
×
7
-----------------
-
Exercise 1.3

2 Estimate the answer, as an integer, to each of the following calculations.
a 8.7 + 19.4 + 12.1 b 96.5 − 27.3 + 15.046 c 24.2 × 3.75 × 5.3
d 24.8 × 3.88 e 32.42 ÷ 7.93 f 126.7 ÷ 9.82
g 5.34 × 11.92 × 8.15 h 53.5 ÷ 6.12 × 8.046 i 189.4 − 47.75 − 283.19
■ Consolidation
3 Estimate the answer to each of these.
a (14.797 + 32.88) ÷ 8.1 b (348.5 − 102.7) × 4.193 c 495.13 ÷ (9.96 × 10.02)
4 Find the approximate value of:
a 18.8 + 6.84 × 3.125 b 183.4 − 31.2 ÷ 5.17
c 20.4 ÷ 3.95 + 19.87 × 5.02 d 2117 − 12.13 × 8.4 × 4.96
5 Estimate:
a 16.45 × 0.482 b 43.65 × 0.252 c 13.82 × 1.55 d 8.094 × 1.26
6 Estimate the answer for each of these, giving the answer as an integer.
a b c d
7 Estimate the value of each calculation.
a b c d
8 The crowds at each day of a test cricket match played at the SCG between Australia and
England were as follows:
• Day 1—34 356 • Day 2—29 875 • Day 3—26 234
• Day 4—18 558 • Day 5—9063
Round off each day’s crowd to the nearest 5000 spectators and hence estimate the total
match attendance.
9 A group of 4 people having dinner in a restaurant ordered the following meals from the
menu:
• Tamara: spaghetti bolognaise $18.75
• Luke: steak Diane $21.75
• Amanda: fettuccine boscaiola $19.20
• Barry: veal parmigiana $20.60
They also ordered 2 bottles of wine at $11.45 each and 4 coffees at $3.25 each.
a Estimate the total cost of the meal, allowing for a small tip.
b Approximately how much would each person expect to pay if they shared the bill
equally?
10 Therese decided to re-carpet her lounge room using carpet squares of side length 50 cm.
The lounge room is rectangular in shape and measures 5.2 m by 6.8 m.
a Estimate the area of the room in square metres.
b How many carpet squares are needed to cover an area of 1 m2?
23.67 84.77 29.1
3 119.8
3
4.76 9.27
×
2.89
--------------------------
-
73.4 15.2
×
4.57
--------------------------
-
50.6 73.1
+
15.8 4.593
–
-----------------------------
-
106.2
7.046
2
----------------
-

10
c Estimate the number of carpet squares that are needed to cover the entire lounge room
floor.
d If the carpet squares are sold in packs of 40 at $385 per pack, estimate the total cost of
the re-carpeting.
e Should re-carpeting decisions be based on estimates or accurate measurements?
Explain.
11 a Evaluate and . Hence, find estimates for and , correct to 1 decimal place.
b Evaluate and . Hence, find estimates for , and , correct
to 1 decimal place.
12 Consider the statement 2n = 12.
a Show by substitution that:
i 3 ⬍ n ⬍ 4 ii 3.5 ⬍ n ⬍ 3.6
iii 3.58 ⬍ n ⬍ 3.59 iv 3.584 ⬍ n ⬍ 3.585
b Hence, estimate the value of n, correct to 3 decimal places.
13 By substituting and then refining estimates, find the approximate value of n in each of the
following, correct to 3 decimal places.
a 2n = 20 b 3n = 36 c 5n = 100
A recurring decimal has an infinite number of decimal places, with one or more of the digits
repeating themselves indefinitely. Recurring decimals are written with a dot above the first and
last digits in the repeating sequence.
For example: 0.444 444 … = 0.616 161 … =
0.329 329 … = 1.288 888 … =
A rational number is a number that can be written in the form where a and b are integers
(whole numbers) and b ≠ 0. Every recurring decimal can be expressed as a fraction, so recurring
decimals are rational numbers.
4 9 5 7
100 121 110 105 115
Fermi problem
A Fermi problem is a problem solved by making a good estimation.
Try these problems:
1 How many telephone calls are made in one day in Australia?
2 What would be the total value of all the books in every library in Australia?
1.4 Recurring decimals
0.4̇ 0.6̇1̇
0.3̇29̇ 1.28̇
a
b
--
-,
TRY THIS

Example 1
Convert each of these fractions to a recurring decimal.
a b c
a 0.5 5 5… b 0.6 3 6 3… c 0.08 3 3…
9 5.050505 11 7.04070407 12 1.0040404
∴ = 0. ∴ = 0. ∴ = 0.08
Example 2
Convert each recurring decimal to a fraction in simplest form.
a 0. b 0. c 0.2
Solutions
This exercise should be completed without the use of a calculator, unless otherwise indicated.
1 Write each of these as a recurring decimal.
a 0.222 … b 0.777 … c 0.6444 … d 0.3555 …
a Let x = 0. … Œ
∴ 10x = 8. …
Subtract Œ from
∴ 9x = 8
∴ x =
b Let x = 0. … Œ
∴ 100x = 15. …
Subtract Œ from
∴ 99x = 15
∴ x =
=
c Let x = 0.2 … Œ
∴ 10x = 2. …
∴ 100x = 24. …Ž
Subtract from Ž
∴ 90x = 22
∴ x =
=
To convert a fraction to a recurring decimal
divide the numerator by the denominator.
To convert a recurring decimal to a fraction:
let the decimal be x
multiply both sides by the smallest power of 10 so that the recurring part of the
decimal becomes a whole number
subtract the first equation from the second
solve the resulting equation.
EG
+S 5
9
--
-
7
11
-----
-
1
12
-----
-
Solutions
5
9
--
- 5̇
7
11
-----
- 6̇3̇
1
12
-----
- 3̇
EG
+S
8̇ 1̇5̇ 4̇
8̇
8̇
8
9
--
-
1̇5̇
1̇5̇
15
99
-----
-
5
33
-----
-
4̇
4̇
4̇
22
90
-----
-
11
45
-----
-
Exercise 1.4

12
e 0.272 727 … f 0.919 191 … g 0.484 848 … h 0.030 303 …
i 0.146 146 … j 0.029 029 … k 0.152 152 … l 0.698 698 …
m 1.666 … n 3.818 181 … o 8.274 274 … p 13.955 555 …
■ Consolidation
2 Use short division to convert each of these fractions to a recurring decimal.
a b c d
e f g h
i j k l
3 a Convert 1 to a decimal using a calculator.
b Does the calculator round off the answer at the last digit?
4 Express each of the following as a recurring decimal.
a b c d
5 a Write down the recurring decimal for .
b Hence, write down recurring decimals for , , and .
c What meaning should be given to ? Why?
6 Convert each of these recurring decimals to a fraction or mixed numeral, in simplest form.
a 0. b 0. c 0. d 0.
e 0. f 0. g 0. h 0.
i 0.1 j 0.4 k 0.7 l 0.9
m 2. n 1. o 7.8 p 3.41
7 a Write down the recurring decimal for .
b Hence, express and as recurring decimals.
8 a Express and as recurring decimals.
b Show that = by adding fractions.
c Show that = by adding decimals.
1
3
--
-
1
9
--
-
2
3
--
-
4
9
--
-
1
11
-----
-
3
11
-----
-
1
6
--
-
2
15
-----
-
5
12
-----
-
7
22
-----
-
5
6
--
-
11
12
-----
-
2
3
--
-
1
7
--
-
5
7
--
-
1
13
-----
-
4
13
-----
-
1
9
--
-
2
9
--
-
5
9
--
-
7
9
--
-
8
9
--
-
0.9̇
2̇ 7̇ 3̇ 6̇
1̇9̇ 3̇5̇ 2̇7̇ 7̇5̇
5̇ 8̇ 3̇ 4̇
1̇ 6̇0̇ 3̇ 6̇
1
3
--
-
1
30
-----
-
1
300
--------
-
11
30
-----
-
1
6
--
-
1
6
--
-
1
5
--
-
+
11
30
-----
-
1
6
--
-
1
5
--
-
+
11
30
-----
-

9 a Express as a recurring decimal.
b Use the fact that = = × to express as a recurring decimal.
A rate is a comparison of two unlike quantities. This is different from a ratio, in that a ratio
is a comparison of two or more like quantities. In particular, a rate is a measure of how one
quantity is changing with respect to another. In a ratio, units are not written, whereas in a rate,
the units must be written if the rate is to have any meaning.
Equivalent rates can be formed by changing the units in either or both quantities. For example,
a rate of 5 cm/s is equivalent to 50 mm/s since, in both cases, the object moves the same
distance (5 cm or 50 mm) in equal amounts of time (1 s).
To be in simplest form, a rate must be expressed as a quantity per 1 unit of another quantity.
For example, a rate of 60 km/h is in simplest form because it represents a change in distance of
60 km for every 1 hour of time.
Example 1
Express each of the following statements as a rate in simplest form.
a $150 in 3 hours b 48 L in 12 min
Solutions
a $150 in 3 hours b 48 L in 12 min
÷ 3 ÷ 3 ÷ 12 ÷ 12
= $50 in 1 hour = 4 in 1 min
= $50/h = 4 L/min
Example 2
Convert:
a 2.4 kg/day to g/day b 3.5 cm3/s to cm3/min c 18 m/s to km/h
Solutions
a 2.4 kg in 1 day b 3.5 cm3 in 1 s c 18 m in 1 s
= 2400 g in 1 day × 60 × 60 × 60 × 60
= 2400 g/day = 210 cm3 in 1 min = 1080 m in 1 min
= 210 cm3/min × 60 × 60
= 64 800 m in 1 h
= 64.8 km/h
2
3
--
-
1
15
-----
-
2
30
-----
-
2
3
--
-
1
10
-----
-
1
15
-----
-
1.5 Rates
A rate is a comparison of two unlike quantities.
EG
+S
EG
+S

14
1 Express each statement as a rate in simplest form.
a 30 m in 3 s b 80 km in 2 h c 45 L in 5 min
d 42 kg over 7 m2 e 32 g in 4 s f 200 trees in 8 h
g 108 km on 9 L h $180 in 4 h i 90c for 5 min
j $12 for 8 kg k 119 runs in 34 overs l 150 crates in 4 days
m 240 beats in 2 min n 72 kL in 1.5 h o 13 km on 1.25 L
2 Complete these equivalent rates.
a 3 cm/s = _____ cm/min b 5 g/min = _____ g/h c $2.30/kg = $_____ /t
d 7.5 L/h = _____ L/day e 0.9 km/min = _____ km/h f 0.4 kg/m2 = _____ kg/ha
a 2 L/min = _____ mL/min b 9 m/s = _____ cm/s
c 3.8 cm/s = _____ mm/s d $1.15/g = _____ c/g
e 14.6 t/day = _____ kg/day f 2.35 ha/week = _____ m2/week
a 70 mm/s = _____ cm/s b 850 cm/min = _____ m/min
c 4900 mL/day = _____ L/day d 24c/min = $ _____ /min
e 25 g/m3 = _____ kg/m3 f 59 600 L/year = _____ kL/year
■ Consolidation
5 Complete the following equivalent rates.
a 75 cm/s = _____ m/min b 8c/g = $ _____ /kg
c 9 m/mL = _____ km/L d 150 kg/h = _____ t/day
e 81.25 mL/h = _____ L/day f 142 m/min = _____ km/h
6 Complete the following equivalent rates.
a 25 m/s = _____ km/h b 40 mL/s = _____ L/h
c 27.5 g/s = _____ kg/h d 5 mm/min = _____ m/day
e 0.8 m/min = _____ km/day f 2.4c/mm = $ _____/m
g 72 km/h = _____ m/s h 12.24 t/day = _____ kg/min
7 Convert these annual interest rates to monthly rates.
a 12% p.a. b 6% p.a. c 18% p.a. d 4.2% p.a.
8 Convert these monthly interest rates to annual rates.
a 0.75% per month b 0.9% per month c 1.25% per month
9 Calculate the daily interest rate on a credit card if the annual rate is 15.33% p.a.
10 Convert:
a $240/week to an equivalent monthly rate
b $1352/month to an equivalent fortnightly rate
Exercise 1.5
1
2
--
-

c $2.80/week to an equivalent quarterly rate
d $44.20/quarter to an equivalent fortnightly rate.
a 5c/cm2 = $_____/m2 b 60 mL/m2 = _____ L/km2 c 1.2 g/cm3 = _____ t/m3
12 Complete this equivalent rate: $25/L = _____ c/cm3.
We use many different types of rates every day, often without realising it. For example:
• driving speed • bank interest rates • currency exchange rates
• petrol consumption rates • sporting strike rates • rates of pay
• electricity rates • pollution rates • medical recovery rates
As most adults drive a car, the concept of speed plays a very important role in our daily lives.
We need to know how fast to drive in order to reach a particular destination on time. It is also
important to know at what speed we can safely drive under various conditions, such as on
narrow roads, in wet weather, near pedestrian crossings and so on.
Informally, we think of speed as a measure of how fast an object is travelling. Formally,
however, speed is defined as the rate of change of distance with respect to time. If we know the
distance that an object has travelled from one point to another and the amount of time that it
took to get there, then we can calculate how fast it was travelling. You should already be
familiar with the following formulae relating speed, distance and time.
Desert walk
James is a cross-country walker. He comes
to a 60 km stretch of desert where there is
neither water nor food. He can walk 20 km
per day and he can carry enough food and
water for 2 days. How many days will it take
him to cross the desert, and how many
kilometres will he travel if he has to build up
depots of food and water?
Difficult part
If he was considering a 100 km trip across
the desert, how many days’ supply of food
would be necessary?
TRY THIS
1.6 Solving problems with rates

16
There is an important distinction that needs to be made between average speed and
instantaneous speed. The formulae above are usually associated with average speed, since the
speed of the object may vary at different times throughout its journey. It may start moving
slowly, speed up at times and slow down or even stop at other times. If, however, a speed
camera had been used to measure the speed of the object at a single moment in time, then it
would have measured the instantaneous speed of the object. The instantaneous speed at a split
second may therefore differ from the average speed over the entire journey.
The degrees and minutes key on the calculator can be used to simplify the working in some
questions, particularly when the time is given in hours and minutes or minutes and seconds.
Example 1
a The entry price to an amusement park is $7.50 per child. Find the total entry cost for a
group of 90 children.
b A farmer used 145 kg of super phosphate to cover an area of 5 ha. How many kilograms
were used per hectare?
Solutions
a The entry cost for 1 child = $7.50 b 145 kg covers an area of 5 ha
∴ cost for 90 children = 90 × $7.50 ÷ 5 ÷ 5
= $675 ∴ 29 kg covers an area of 1 ha
b 145 kg covers an area of 5 ha
÷ 5 ÷ 5
∴ 29 kg covers an area of 1 ha
Speed =
S =
Distance = Speed × Time
D = S × T
Time =
T =
Example 2
A car can travel 138 km on
15 L of petrol. How far can it
travel on a full tank of 35 L?
Solution
Using the unitary method,
138 km on 15 L
÷ 15 ÷ 15
= 9.2 km on 1 L
× 35 × 35
= 322 km on 35 L
∴ The car can travel 322 km on a full tank of 35 L of
petrol.
Distance
Time
--------------------
-
D
T
---
-
Distance
Speed
--------------------
-
D
S
---
-
EG
+S
EG
+S

Example 3
a Jenny ran 600 metres in 80 seconds. What was her running speed?
b A man drove at an average speed of 60 km/h for 7 hours. How far did he drive?
c Shona’s average walking speed is 5 km/h. How long would it take her to walk 20 km?
Solutions
1 a An author writes at a rate of 3 pages per hour. How many pages would she write in
6 hours?
b A shearer was able to shear 18 sheep per hour. How many sheep could he shear in
2 hours?
c If petrol costs 97.4 cents/L, find how much it would cost to fill the tank in a car if the
tank holds 42 L.
d A tap is dripping at the rate of 3 mL per minute. How many litres of water will be lost
in 2 days?
e The crew on a fishing boat put out the nets every 2 hours and catch an average of 240 kg
of fish. How many tonnes would the crew expect to catch if they fish for 10 hours?
2 a Trevor earns $15.20 per hour as a sales assistant. How many hours would he need to
work in order to earn $562.40?
b Janine has a typing speed of 54 words per minute. How long would it take her to type
a 1350 word article?
c A cricket side scored 243 runs in 50 overs during a limited overs cricket match.
Calculate the average scoring rate in runs per over.
d A plumber charged $200 for 2 hours labour to repair a broken pipe. Find the
plumber’s hourly rate.
e A machine prints 150 newspapers per minute. How long would it take to print 18 000
newspapers?
■ Consolidation
3 a Georgina drove 12 km in 10 minutes. At the same speed, how far would she drive in
30 minutes?
b Gino’s pulse rate was 100 beats per minute. How many times would his heart beat in
15 seconds?
a S =
=
= 7.5 m/s
b D = S × T
= 60 × 7
= 420 km
c T =
=
= 4 h
EG
+S
D
T
---
-
600
80
--------
-
D
S
---
-
20
5
-----
-
Exercise 1.6
1
2
--
-
1
2
--
-

18
c A fruit picker claimed that he could pick 1200 apples per hour. How many apples could
he pick in 20 minutes?
d A bank teller can serve 20 customers per hour. How many customers can she serve in
45 minutes?
e A tap drips 12 times in 20 seconds. How many times would it drip in 30 seconds?
4 Use the unitary method to answer the following questions.
a Dianne paid $3.75 for 3 kg of oranges. How much would she have paid for 7 kg?
b In a walking race, Paul took 40 minutes to walk 8 km. How long would it take him to
walk 13 km?
c Susan’s car uses petrol at the rate of 10.6 L/100 km. How much petrol would she use
on a journey of 250 km?
d If it takes 1 hours to remove 36 t of sugar from a silo, how long it would take to
remove 30 t?
e George delivered 400 pamphlets in 50 minutes. How many pamphlets would he deliver
in 2 hours?
f If sausages are being sold for $2.80 per kilogram, find the cost of purchasing 350 grams
of sausages.
5 The following currency conversions show the value of 1 Australian dollar (A$1) in US$,
euro and NZ$.
A$1 = US$0.6075 A$1 = 0.5636 euro A$1 = NZ$1.0887
Use these currency conversions to convert:
a A$20 into US$ b A$50 into euro c A$175 into NZ$
d A$250 into euro e A$600 into NZ$ f A$4500 into US$
6 Use the currency conversions in Q5 to convert the following amounts into Australian
dollars. Give your answers correct to the nearest cent.
a NZ$30 b US$95 c 110 euro d NZ$200
e US$565 f 782 euro g NZ$1400 h US$2378
7 a Dave drove 350 km in 5 hours. What was his average speed?
b A plane travelled 1960 km in 7 hours. What was the speed of the plane?
c Jennifer ran at a speed of 8 km/h for 1 hours. How far did she run?
d A ship sailed at 42 km/h for 25 hours. What distance did it sail?
e Morgan rode his motor bike a distance of 340 km at a speed of 85 km/h. How long was
the trip?
f A satellite orbits the Earth at a speed of 22 500 km/h. How long will it take for the
satellite to travel a distance of 78 750 km?
1
2
--
-
1
2
--
-
1
2
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8 Use the degrees and minutes key on your calculator to answer the following questions.
a How far will a bus travel in 4 h 25 min at an average speed of 90 km/h?
b Calculate the average speed of a battleship which sails 600 km in 11 h 45 minutes.
Answer correct to the nearest km/h.
c How long will it take for a plane to fly 615 km at a speed of 180 km/h? Answer correct
to the nearest minute.
9 The speed of ships and sometimes of aircraft is usually measured in knots. A knot is a speed
of 1 nautical mile per hour, where 1 nautical mile is equivalent to 1852 metres.
a Express 1 knot in km/h.
b If an aircraft is travelling at 120 knots, how long would it take to travel 5000 km?
c If another aircraft is travelling at 760 knots, how many kilometres will it travel in
6 hours?
10 The petrol consumption (C) of a car is measured in litres of petrol (L) used per 100 km (K)
travelled.
a Write down a formula connecting C, L and K.
b Calculate the petrol consumption of a car that travels 1038 km in a month and uses 95 L
of petrol.
c Meera is planning a tour of the Australian outback and expects to travel 10 000 km.
Her vehicle’s petrol consumption is expected to average 12 L/100 km. If the average
price of petrol in the outback is $1.12 per litre, calculate the expected cost of petrol for
this trip.
Passing trains
A slow train leaves Canberra at 9:17 am and arrives
at Goulburn at 12:02 pm. On the same day, the
express leaves Canberra at 9:56 am and arrives in
Goulburn at 11:36 am.
At what times does the express pass the slow train
if each is travelling at a constant speed?
HINT: A travel graph would give an approximate
time.
TRY THIS

20
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A NUMBER PATTERN FROM GALILEO 1615
Galileo looking through a telescope in his observatory
Introduction
Galileo Galilei (1564–1642), the famous Italian mathematician, is better known for his
scientific achievements than his mathematical ones. For example, in 1610 he made a series of
telescopes that enabled him to discover four of the moons of Jupiter, to see mountains on the
Moon, and to prove that the Milky Way was made up of stars. The four moons of Jupiter he
discovered centuries ago are today called the Galilean satellites in his honour. Their names are
Io, Europa, Ganymede and Callisto. We now know, thanks to space probes, that Jupiter has, in
fact, 16 moons, 13 of which have been discovered from Earth.
In this activity, however, you will investigate a number pattern for the fraction . In 1615,
Galileo wrote one of the earliest manuscripts describing this pattern, so we can see how
interested he was in pure mathematics. First, we search for a pattern among specific cases using
inductive reasoning, and then we use algebra to generalise the pattern using deductive reasoning.
FO C U S O N WO R K I N G MA T H E M A T I C A L L Y
0 F O C U S O N W 0 R K I N G M A T H E M A T I C A L L Y
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L E A R N I N G A C T I V I T I E S
1 Check that the following statement is true: =
2 Notice that the numbers in the numerator and denominator form the pattern of odd numbers
1, 3, 5 and 7.
3 Continue the pattern to obtain . Does it still equal ?
4 Write down the next term of the sequence and continue, checking that in each case the
fraction is equivalent to .
5 Why is this true? Don’t try a formal proof, but see if you can draw a diagram to show that
it must be. Use dots to represent the odd numbers and choose some specific cases. Ask for
help as needed.
C H A L L E N G E
This is suggested as a group activity for extension stage 5 classes as an exercise in collaborative
learning.
1 Investigate the pattern of odd numbers 1 + 3 + 5 + 7 + 9 + 11 + …
2 Notice that the partial sums 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, … are perfect squares.
3 See if you can find the pattern for the sum of 2 terms, 3 terms, 4 terms, …
4 Make a hypothesis about the sum of n terms.
5 Make a hypothesis about the sum of 2n terms.
6 If there were n terms in the numerator, how many would there be in the denominator? How
many altogether?
7 Look carefully at the following patterns:
1 + 3 + 5 = 32
and 7 + 9 + 11 = (1 + 3 + 5 + 7 + 9 + 11) − (1 + 3 + 5) = 62 − 32
So = = =
8 See if you can show that the next term is also using this same pattern:
= …
9 From the pattern of your results, see if you can write down an expression for the fraction
you would get if there were n terms on top. Ask your teacher for help if you need it, and
discuss the possibilities between yourselves. Check that the expression reduces to .
2
1
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1 3
+
5 7
+
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1 3 5
+ +
7 9 11
+ +
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1
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1 3 5
+ +
7 9 11
+ +
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32
62 32
–
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9
27
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1 3 5 7
+ + +
9 11 13 15
+ + +
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L E T ’ S C O M M U N I C A T E
Discuss what you have learned from this activity with a classmate or, perhaps, if you have
worked in a group for this activity, with the group members. Can you see the value of inductive
thinking in mathematics, that is, finding a pattern to suggest a general rule?
If you worked in a group, write a short account of whether you enjoyed collaborating with
others. Is it a good way to learn?
R E F L E C T I N G
Mathematical thinking can be inductive, searching for a pattern to suggest a general rule, or
deductive, reasoning in a chain of argument that leads to a mathematical proof. Both are very
important in learning mathematics and are often used together.
Think over how much of your learning in Year 9 is inductive and how much deductive. Discuss
with your teacher as to how the two go together in mathematics lessons.
E
%
1 What is a small word for magnitude?
2 Explain the difference between a guess
and an estimate.
3 What is a rational number?
4 When is a digit in a number significant?
5 Read the Macquarie Learners Dictionary
entry for rate:
rate noun 1. speed: to work at a steady rate | The car
was travelling at a rate of 100 kilometres an hour.
2. a charge or payment: The interest rate on the loan
is 10 per cent per year. 3. rates, the tax paid to the
local council by people who own land
–verb 4. to set a value on, or consider as: The council
rated the land at $20 000. | I rate him a very good
friend.
–phrase 5. at any rate, in any case: We enjoyed
ourselves at any rate.
6. at this rate, if things go on like this: At this rate
we will soon run out of money.
How is the word ‘rate’ used in this chapter?

C H A P T E R RE V I E W
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1 State the number of significant figures in:
a 406 b 7.2009
c 0.0031 d 12.0560
2 Round off each number correct to
1 significant figure.
a 76 b 150
c 4278 d 894 000
3 Round off each number correct to
2 significant figures.
a 341 b 725
c 15 049 d 369 412
4 Round off each number correct to the
number of significant figures shown in
the brackets.
a 198 [1] b 4316 [1]
c 18 209 [1] d 572 [2]
e 2154 [2] f 36 587 [2]
5 Round off each decimal correct to the
number of significant figures shown in
the brackets.
a 4.83 [1] b 0.0723 [2]
c 3.4661 [3] d 22.018 [3]
e 106.84 [2] f 8994.7 [1]
6 Evaluate each of these correct to
2 decimal places, using a calculator.
a 5 − 1 b −6.3 − 1.29
c 5.842 d
e f 2.715
g h
i
7 Evaluate each of the following, correct to
2 decimal places, using a calculator.
a b
c d 3.45 − (2 )4
e f
8 Estimate the value of each calculation.
a 9.84 × 15.2 + 18.77
b
c
9 Write each of these as a recurring
decimal.
a 0.333 333 … b 0.252 525 …
c 0.346 346 … d 5.918 181 …
10 Convert these fractions to recurring
decimals.
a b c 1
11 Convert these recurring decimals to
fractions.
a b c
12 Given that = , express each of the
following fractions as a recurring
decimal.
a b
13 Express each statement as a rate in
simplest form.
a 80 m in 10 s
b $45 for 9 min
c 72 L in 3 h
d 215 runs for 5 wickets
14 a A car uses 18 L of petrol to travel
150 km. How much petrol would be
needed to travel 350 km?
b A farmer spreads 25 kg of fertiliser
over an area of 4000 m2. How much
fertiliser would be needed to cover an
area of 1.5 ha?
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×
12.31 5.6
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×
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7.97 47.3 15.49
÷
+
194.7 259.2
×
53.6
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0.2̇ 0.7̇2̇ 0.13̇
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C H A P T E R RE V I E W
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15 Convert:
a 7 mm/min to mm/h
b 75 km/h to km/day
c 1.35 L/m2 to mL/m2
d 8.2 m/s to cm/s
16 Convert:
a 40 m/min to km/h
b 250 mL/h to L/day
c 13.5 g/m2 to kg/ha
d 5 m/s to km/h
17 a A plane flew 6000 km in 7 hours. At
what speed was the plane travelling?
b Karen walked 24 km at 5 km/h, for
how long did she walk?
c Jude drove at 80 km/h for 4 h 15 min.
What distance did he drive?
18 Daryl drove 527 km in 6 h 23 min. Find
his speed, correct to 1 decimal place.
1
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Algebra
This chapter at a glance
Stage 5.1/5.2/5.3
After completing this chapter, you should be able to:
use algebra to find rules for simple number patterns
use the method of finite differences to find rules for simple number patterns
evaluate algebraic expressions by substituting numbers for pronumerals
add and subtract algebraic expressions
multiply and divide algebraic expressions
simplify algebraic expressions using the order of operations
expand algebraic expressions that contain grouping symbols using the
distributive law
factorise algebraic expressions by removing the highest common factor
add and subtract algebraic fractions
multiply and divide algebraic fractions
link algebra with generalised arithmetic
use algebra to prove general properties of numbers
use algebra to generalise solutions to problems
expand binomial products
expand perfect squares using the special identities
determine whether a given expression is a perfect square
complete a perfect square
expand expressions using the difference of two squares identity
expand expressions that involve a combination of algebraic techniques.
Algebra
2

26
Many complex problems can often be solved more easily by using algebra. Algebra lets us
replace complex statements with short, simple expressions. Algebra also lets us generalise
results that are always true, or are true under certain conditions, so that we do not have to keep
solving the same types of problems over and over again.
■ Finite differences
It is not always easy to find the algebraic rule that describes the relationship between variables.
The method of finite differences is a simple technique that can be used to help us find this
relationship. Finite differences are the differences between the numbers in the bottom row of a
table of values.
For example, the numbers in the bottom row
of this table are increasing by 3. Therefore,
the finite differences in the table are all 3s.
NOTE: This method can only be used for linear relationships when the x-values are consecutive
integers (e.g. x = 1, 2, 3…).
Example
Find the rule that describes the relationship between x and y in this table of values.
Solution
Let the rule be in the form y = ∆x + ,
where ∆ is the difference between each
pair of consecutive y-values.
Now, the y-values are increasing by 5, ∴ ∆ = 5.
If y = 5x + and x = 0 when y = 7,
7 = (5 × 0) +
7 = 0 +
∴ = 7
∴ The rule is y = 5x + 7.
x 0 1 2 3
y 7 12 17 22
2.1 Describing simple patterns
x 1 2 3 4
y 13 16 19 22
+3 +3 +3
To find the rule that links the variables x and y in a linear relationship:
write the standard rule in the form y = ∆x +
find ∆, the finite differences between the bottom numbers in the table
find by substituting into the rule a pair of values from the table.
EG
+S
x 0 1 2 3
y 7 12 17 22
+5 +5 +5

C h a p t e r 2 : Algebra 27
1 Complete each table of values using the given rules.
2 For each table of values in Q1, compare the differences between the y-values and the
co-efficient of x in the rule. What do you notice?
3 Use the method of finite differences to find a rule for each table of values.
y = x + 3 y = 2x + 5
a x 1 2 3 4 b x 0 1 2 3
y y
y = 3x − 4 y = 5x − 7
c x 5 6 7 8 d x 2 3 4 5
y y
a x 1 2 3 4 b x 0 1 2 3
y 4 8 12 16 y 6 7 8 9
c x 4 5 6 7 d p 2 3 4 5
y 11 13 15 17 q 5 8 11 14
e p 1 2 3 4 f p 7 8 9 10
q 9 14 19 24 q 47 54 61 68
g a 4 5 6 7 h a 0 1 2 3
b 17 19 21 23 b 3 7 11 15
i a 3 4 5 6 j s 5 6 7 8
b 18 24 30 36 t 17 22 27 32
k s 1 2 3 4 l s 2 3 4 5
t 13 20 27 34 t 19 31 43 55
Exercise 2.1

28
■ Consolidation
4
a Copy and complete this table of values.
b Write down an algebraic rule that links the number of triangles (t) to the number of
pentagons (p).
c How many triangles would there be in a figure with 9 pentagons?
5
b Write down an algebraic rule that links the number of crosses (c) to the number of
squares (s).
c How many crosses would there be in a figure with 20 squares?
6
b Write down an algebraic rule that links the number of dots (d) to the number of
circles (c).
c How many dots would there be in a figure with 15 circles?
Number of pentagons (p) 1 2 3
Number of triangles (t)
Number of squares (s) 1 2 3
Number of crosses (c)
Number of circles (c) 1 2 3
Number of dots (d)

7
b Write down an algebraic rule that links the number of dots (d) to the number of large
rhombuses (r).
c How many dots would there be in a figure with 40 large rhombuses?
8
b Complete this rule that relates the number of dots to the number of squares: d = ∆s + .
9
b Complete this rule that relates the number of dots to the number of rectangles:
d = ∆r + .
Number of large rhombuses (r) 1 2 3
Number of dots (d)
Number of dots (d)
Number of rectangles (r) 3 4 5
Number of dots (d)

30
10
b Complete this rule that relates the number of dots to the number of circles: d = ∆c + .
11
b Complete this rule that relates the number of dots to the number of crosses: d = ∆c + .
12 Use the method of finite differences to find a rule linking the x- and y-values in each table.
13
Number of circles (c) 3 4 5
Number of dots (d)
Number of crosses (c) 2 3 4
Number of dots (d)
a x 1 2 3 4 b x 0 1 2 3
y −7 −14 −21 −28 y 5 4 3 2
c x 1 2 3 4 d x 3 4 5 6
y 7 5 3 1 y 11 8 5 2
e x −4 −3 −2 −1 f x −2 −1 0 1
y 10 9 8 7 y 13 10 7 4

b Write down an algebraic rule that links the number of dots (d) to the number of
squares (s).
c How many dots would there be in a figure with 64 squares?
14
b Write down an algebraic rule that links the total number of cans (c) to the number of
cans in the base (b).
c How many cans would there be in a pile with 10 cans in the base?
Number of dots (d)
Number of cans in base (b) 1 2 3
Total number of cans (c)
Flags
Consider the following diagrams, then complete the table.
1 2 3
Find a rule relating the number of squares in the flag to the pole length.
HINT: The rule is not linear.
Pole length 1 2 3 4 5 …n
Number of squares 3
TRY THIS

32
When we substitute for a pronumeral, we give the pronumeral the value of a number. An
algebraic expression can have a number of values, depending on the value(s) that are substituted
for each pronumeral.
Example 1
Evaluate each of the following when x = 3 and y = 7.
a 8x − 2y b 2x2 c d 6(x + y)
Example 2
Evaluate each of these when m = 2 and n = 5.
a m − n + 9 b 3m − 4n c mn(m − n)
Solutions
a m − n + 9 b 3m − 4n c mn(m − n)
= 2 − 5 + 9 = (3 × 2) − (4 × 5) = 2 × 5 × (2 − 5)
= −3 + 9 = 6 − 20 = 10 × (−3)
= 6 = −14 = −30
Example 3
Evaluate each of the following given that p = 4, q = −3 and r = −6.
a p + q − r b pqr c q(p − r)
Solutions
a p + q − r b pqr c q(p − r)
= 4 + (–3) − (–6) = 4 × (−3) × (−6) = −3(4 − −6)
= 4 − 3 + 6 = −12 × −6 = −3 × 10
= 1 + 6 = 72 = −30
= 7
a 8x − 2y
= (8 × 3) − (2 × 7)
= 24 − 14
= 10
b 2x2
= 2 × 32
= 2 × 9
= 18
c
=
=
= 5
d 6(x + y)
= 6(3 + 7)
= 6 × 10
= 60
2.2 Substitution
EG
+S x y
+
2
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Solutions
x y
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3 7
+
2
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10
2
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EG
+S
EG
+S

1 Evaluate each of the following when k = 5.
a k + 7 b k − 2 c 13 − k d 3k
e 7k + 8 f 12k − 23 g 30 − 4k h k2
i k3 j 3k2 k k2 + 3k l 2k2 − 9k
m n o p
■ Consolidation
2 Evaluate each of these when m = 7 and n = 3.
a 16 − m + n b mn − 8 c 6m − n d 2m + 5n
e 13n − 4m f 50 − 2mn g 3m + 6n − 11 h 100 − 5m − 3n
i n2 + 10 j 50 − m2 k m2 − n2 l 4n2
m 2m2 + 13 n n3 − 8m o mn2 p m2n − mn3
q 5(m + n) r 12(m − n + 6) s n(8m − 20) t 2n(5m + mn)
u v w x
3 Find the value of each expression using the substitutions r = 6, s = 2 and t = 11.
a s − r b r − t c −s + t d −t − r
e r − s − t f s − t + r g −r + s + t h −t + s − r
i 3s − t j −5t + 4r k −8r + st l 5s − rt
m 100 − rst n rs − st o r2 − 3rt p s2 − r2 + t2
q t − 5s2 r r(s − t) s 5(2t − 4r − 9s) t 3s(r2 − t2)
4 Evaluate each of the following given that a = −3, b = 8 and c = −6.
a a + b b b − c c c + a d a − b
e a − c + b f c + b + a g b − a − c h −b + c + a
i 4a − 2b − c j b + 5a + 2c k 3b − 5a + 10c l −4c + 3b − 7a
m b(a + c) n c(b − a) o 2a(c + b) p ac(b − 10)
q (b − a)(b + c) r a2b s ab − c3 t
u v w x
Algebraic terms with identical pronumerals are called like terms. Only like terms can be added
or subtracted.
Exercise 2.2
40
k
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4
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5k 11
+
2k 1
–
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m n
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4m 4n
+
5
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3m 2n
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n
2
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5n
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abc
b
2
c
2
+
ab
c
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–
a 1
–
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2 a
2
c
2
+
( )
ac
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Adding and subtracting
algebraic expressions
2.3

34
Some examples of:
• like terms are 3m and 5m, 7q and −2q, xy and yx, 4t2 and 9t2
• unlike terms are 4a and 4b, ef and fg, 6u2 and 11u.
Example 1
Simplify each of these.
a 7s + 3s b 12w − 4w c 6y − y
d 5r2 + 2r2 e 14gh − 9gh f 7pq + 6qp
a 7s + 3s = 10s b 12w − 4w = 8w c 6y − y = 5y
d 5r2 + 2r2 = 7r2 e 14gh − 9gh = 5gh f 7pq + 6qp = 13pq
Example 2
Simplify these expressions by collecting the like terms.
a 6e + 13 + 4e + 8 b 9v2 + 7v + v2 − 3v c 8x + 7y − 5x − 12y
Solutions
a 6e + 13 + 4e + 8 b 9v2 + 7v + v2 − 3v c 8x + 7y − 5x − 12y
= 6e + 4e + 13 + 8 = 9v2 + v2 + 7v − 3v = 8x − 5x + 7y − 12y
= 10e + 21 = 10v2 + 4v = 3x − 5y
1 a Simplify 7x + 3x.
b Verify your answer by substituting several values for x.
2 a Simplify 5n + 2n and 2n + 5n.
b Does 5n + 2n = 2n + 5n?
c Does it matter in which order algebraic expressions are added?
3 a Simplify 5s − 3s and 3s − 5s.
b Does 5s − 3s = 3s − 5s?
c Does it matter in which order algebraic expressions are subtracted?
4 Simplify each of the following.
a 4y + 5y b 12n − 8n c 2c + c d 7k − k
e 11z − 11z f 10b − 9b g 3a2 + 4a2 h 13g2 − 5g2
i 6pq + 5pq j 15xy − 8yx k 2abc + 6abc l 14m2n + 5m2n
To collect the like terms in an algebraic expression:
add or subtract the co-efficients
keep the same pronumeral(s).
EG
+S
Solutions
EG
+S
Exercise 2.3

m 3t − 7t n −2u + 12u o −13p + 4p p −8j − 7j
q 5pq − 11pq r −10yz + 9zy s e2 − 11e2 t −9rs2 + 7rs2
■ Consolidation
5 Simplify:
a 3a + 4a + 2a b 10b − 3b − b c 9k − 6k + 7k d 5m − 8m − 4m
e 3p − 10p + 15p f −6r + 4r + 9r g −x − 7x − 5x h −3c + 2c − 11c
i 4e2 − 7e2 − 10e2 j 8a2 − 12a2 + 4a2 k 5ab + ab − 9ab l −9pq + 6pq + 7pq
6 Collect the like terms in each expression.
a 4q + 3q + 2 b 5g + 8 + 9 c 15u − 7u − 3 d 13 + 6t − 5t
e 10c + 8c + d f 9j − 4k + 2j g 3a − 5a + 7 h 12 − 2n − 4n
i x2 + 4x + x j 8m + m2 − 10m k 3w2 + 2w2 + w l 4a2b + 6ab2 − 3ab2
7 Simplify these expressions by collecting the like terms.
a k + 2 + k + 3 b 7c + 4 + 5c + 1 c 8p + 3q + p + 7q
d 8m + 5n + m − 4n e 5t + 12 − 2t + 4 f 8u + 9v − 3u − v
g 10g + 4g − 3h + 6h h 11p + 2q − 6q − 4p i 3b − 5c + 2c − 8b
j 6s + 11 − 6s + 11 k 5y − 9 + 5y + 9 l 4m − 7n − 10m + 5n
m x + y − 4x − 7y n −6a + 2b + 5a + 10b o −5j − 12k + 15j − 4k
p x2 + 6x + 2x2 + 3x q 7a2 + a2 + a − 4a r 9u − 4u2 − u2 + 3u
s z2 − 2z + 5z2 − 3z t d2 + 7d + 5 − 4d u 4mn + 5m − 3mn − 9n
8 Find, in simplest form, an algebraic expression for the perimeter of each figure.
a b c
d e f
9 a Subtract 3x2 − 4x + 10 from 7x2 + 2x − 4.
b From 5a2 + 9, take a2 − 2a − 5.
c Find the difference between 5p + 3 and 2p2 + 6p + 3.
d By how much does 4k2 + 7k + 11 exceed k2 − 2k + 15?
e Take the sum of t2 − t + 4 and 2t2 + 17t + 9 from 4t2 + 9t + 20.
5k
8n
6n
m + 6
m
15 − x
x − 2
y − 5
y + 12
2c − 1
3c + 11
c + 4
c − 7

36
Any algebraic terms can be multiplied or divided. They do not have to be like terms.
Example 1
Simplify each of the following:
a b × 3 b 4r × 5s c × 24w
d 8a × 5a e 6xy × 7yz f −12u × (−5v)
Solutions
a b × 3 = 3b b 4r × 5s = 20rs c × 24w = 6w
d 8a × 5a = 40a2 e 6xy × 7yz = 42xy2z f −12u × (−5v) = 60uv
Example 2
Simplify each of the following:
a 15p ÷ 5p b 21ab ÷ 3a c 45t2 ÷ 9t d 64mn2 ÷ (−8mn)
Solutions
a 15p ÷ 5p b 21ab ÷ 3a c 45t2 ÷ 9t d 64mn2 ÷ (−8mn)
= = = =
= 3 = 7b = 5t = −8n
1 a Simplify 2a × 3b.
b Verify your answer by substituting several pairs of values for a and b.
2 a Does 5n × 4n equal 20n or 20n2?
b Substitute a value for n to verify your answer.
Multiplying and dividing
algebraic expressions
2.4
To multiply algebraic terms:
multiply the co-efficients
multiply the pronumerals.
To divide algebraic terms:
express the division as a fraction
divide the co-efficients
divide the pronumerals.
EG
+S 1
4
--
-
1
4
--
-
EG
+S
15p
5p
---------
21ab
3a
-----------
-
45t2
9t
---------
-
64mn2
8mn
–
---------------
-
Exercise 2.4

3 a Does 12y ÷ 2y equal 6 or 6y?
b Substitute a value for y to verify your answer.
4 a Simplify 5x × 3y and 3y × 5x.
b Does 5x × 3y = 3y × 5x?
c Does it matter in which order algebraic expressions are multiplied?
5 a Simplify 6p ÷ 12 and 12 ÷ 6p.
b Does 6p ÷ 12 = 12 ÷ 6p?
c Does it matter in which order algebraic expressions are divided?
6 Simplify these products.
a 5 × 3n b 6c × 4 c 9w × 7 d 11 × 8g
e u × 5v f 9m × n g 7a × 2b h 8x × 5y
i 4c × 9d j 10r × 7s k 5p × 12q l 9v × 9w
m a × a n 2e × e o 4k × 3k p 5h × 6h
q mn × mp r 6cd × 7c s 5fg × 4gh t 4vw × 8wx
u a × 14 v m × 12n w 24pq × r x 15c × cd
7 Simplify these quotients.
a 10b ÷ 2 b 21z ÷ 7 c 18k ÷ 3 d 40m ÷ 5
e 6w ÷ w f 32n ÷ 4n g ab ÷ b h pqr ÷ pr
i 50gh ÷ 5h j 42mn ÷ 6m k 30xy ÷ 3y l 54cde ÷ 9cd
m t2 ÷ t n 13v2 ÷ v o 6u2 ÷ 6u p 15a2 ÷ 5a
q 24m2 ÷ 3m r 72e2 ÷ 8e s 7a2b ÷ 7a t 60rs2 ÷ 12rs
■ Consolidation
8 Simplify:
a −3 × 7y b −8x × (−5) c 4g × (−12h) d −10b × (−c)
e −j × (−j) f −9v × 3v g −7ab × 5b h −8xy × (−12yz)
9 Simplify:
a b c d
e f g h
10 Simplify each of the following expressions.
a 3a × 2b × c b 4m × n × 7p c 5e ÷ 5 × 2f
d 4g × 3 ÷ 6g e 24k ÷ 3k ÷ 2 f 30ab ÷ 3a ÷ 2b
g 9pq ÷ 3p × 7q h 6m × 8n ÷ 12m i 10a2 × 4b ÷ 5ab
j 27y × 2yz ÷ 6y k 5c × 2d × 6cd l 72w2 ÷ 9w ÷ 4w
m −2x × (−3y) × 7 n 15p × (−3q) ÷ 9p o −50rs ÷ 5r × (−2s)
1
2
--
-
1
3
--
-
3
4
--
-
2
3
--
-
12c
–
3
-----------
-
49n
–
7
–
-----------
-
27k
9k
–
--------
-
36ef
–
4e
--------------
-
84mnp
–
12mp
–
-------------------
-
63k2
7k
–
----------
-
25t2
–
5t
–
------------
-
96u2
v
–
8uv
-----------------

38
11 Find the missing term in each of these.
a 3m × = 18m b × 4 = 28j c 12y ÷ = 3y
d ÷ 5t = 6 e × 6x = 24xy f 36pq ÷ = 12p
g 8e × = 40ef h ÷ 6k = 7m i 5a × = 15a2
j ÷ w = 5w k × 9h = 72h2 l 60c2 ÷ = 5c
m × −4p = −32pq n −25gh ÷ = 5g o ÷ 3x = −9x
12 Simplify, giving your answers in simplest fraction form.
a 5c ÷ 10 b 2 ÷ 2k c 9h ÷ 6 d 4ab ÷ 12a
e 12mn ÷ 20n f 14u ÷ 21uv g 25cd ÷ 35de h 42s2 ÷ 49s
i 18uv ÷ 27v2 j 35x2 ÷ 60xy k 36abc ÷ 44bcd l 72e2f ÷ 56ef 2
When simplifying expressions that contain several terms, follow the order of operations.
Overhanging the overhang
Place a ruler on the edge of a table. How
far will it overhang the edge of a table
before it topples?
Now move the ruler so that it overhangs
the table by 10 cm. Place another ruler on
top of this first ruler. How far can this
ruler overhang the first before it topples?
Now vary the bottom ruler each time.
Continue to see how far you can overhang
the top ruler.
Record your results.
Where should you place the two rulers so that you obtain the greatest possible
overhang?
Now try three rulers and repeat the procedure. If possible, try four rulers. What
conclusions can you make? Could you make a deduction if you had n rulers?
2.5 The order of operations
The order of operations is to:
simplify any expressions inside grouping symbols
simplify any multiplications and divisions, working from left to right
simplify any additions and subtractions, working from left to right.
TRY THIS

Example
Simplify:
a 42cd ÷ 7c × 5e b 40u − 9u × 3 + 5u c [25a − (3a + 12a)] ÷ 2a
Solutions
a 42cd ÷ 7c × 5e b 40u − (9u × 3) + 5u c [25a − (3a + 12a)] ÷ 2a
= × 5e = 40u − 27u + 5u = [25a − 15a] ÷ 2a
= 6d × 5e
= 13u + 5u =
= 30de
= 18u
= 5
1 Simplify:
a 3 × (4n + 2n) b (15q − 3q) ÷ 4 c (8j + 5j) × 2
d 12x − (5x + 3x) e 3t × (12t − 4t) f (s + 7s) × 4s
g 5p × (3q + 9q) h (11c − c) × 2d i 21b ÷ (5b + 2b)
j 36y2 ÷ (13y − 4y) k 63gh ÷ (3g × 3h) l 50cd ÷ (8d + 2d)
m 2 × (2f + 4f ) × 4 n 5 × (17t − 9t) ÷ 4t o (17a2 + 3a2) ÷ (9a − 4a)
■ Consolidation
2 Simplify these expressions by removing the innermost grouping symbols first.
a [11t + (3 × 4t)] + 2t b [17y − (27y ÷ 3)] − y c [40g − (7g × 5)] × 4
d 6c + [9c − (10c − 5c)] e 5 × [(15n + 6n) ÷ 7] f [8w + (4 × 10w)] ÷ 12w
g 32r − [12r + (45r ÷ 9)] h [(22f − 4f) ÷ 2] × 5f i −8k − [17k − (19k − 13k)]
3 Simplify each expression using the order of operations.
a 5k + 3k × 2 b 20z − 14z ÷ 2 c 4n × 2n + 7n2
d 25v2 − 6v × 4v e 22ab − 5a × 3b f 28pq ÷ 4p + 6q
g 18ef − 12ef ÷ 3 h 7y + 20xy ÷ 4x i 7 × 2s − 5s × 2
j 24a ÷ 8 + 4a × 2 k 8a × 4b + a2b ÷ a l 100x2 ÷ 2x − 8 × 5x
m 10g + 5g × 3 + 2g n 6x − 8 × 2x + x o 2k − 32k ÷ 4 − 3k
4 Express each of these in simplest form.
a b c d
5 Insert grouping symbols in each of these to make a true statement.
a 4 × 2s + 3s = 20s b 40pq ÷ 5p × 2q = 4 c 16a − 4a + 2a − 7a = 3a
d 24e2 − 6e2 ÷ 6e = 3e e 8 × 4n − 5n × 3 = 17n f 8w + 9w2 × 6 ÷ 3w = 26w
EG
+S
42cd
7c
-----------
-
10a
2a
--------
-
Exercise 2.5
10x 6
×
4 3x
×
-----------------
-
19u 9u
+
13u 6u
–
---------------------
-
8p 3q
×
12p 6p
–
---------------------
-
33rs 15sr
–
3r 2s
×
----------------------------

40
Algebraic expressions can be expanded by the use of the distributive law.
Example 1
Expand:
a 4(k + 5) b w(w − 1) c 6g(4g − 7h)
Solutions
a 4(k + 5) b w(w − 1) c 6g(4g − 7h)
= (4 × k) + (4 × 5) = (w × w) − (w × 1) = (6g × 4g) − (6g × 7h)
= 4k + 20 = w2 − w = 24g2 − 42gh
Example 2
Expand:
a −5(n + 2) b −7(e − 3) c −8z(3x − 4y)
Solutions
a −5(n + 2) b −7(e − 3) c −8z(3x − 4y)
= −5n − 10 = −7e + 21 = 24xz + 32yz
Example 3
Expand and simplify:
a 3(b + 2) + 10 b 12 + 4(a − 5) c 9(x + 5) − 4(x − 10)
Solutions
a 3(b + 2) + 10 b 12 + 4(a − 5) c 9(x + 5) − 4(x − 10)
= 3b + 6 + 10 = 12 + 4a − 20 = 9x + 45 − 4x + 40
= 3b + 16 = 4a − 8 = 5x + 85
×
1 Expand each of the following.
a 3(a + 4) b 5(p − 2) c 7(m + 1) d 8(5 − k)
e 4(5h + 7) f 6(2y − 3) g 5(3m + 7n) h 2(9y − 10z)
2.6 The distributive law
To expand an expression by using the distributive law:
multiply the term outside the grouping symbols by each term inside.
a(b + c) = ab + ac and a(b − c) = ab − ac
EG
+S
EG
+S
EG
+S
Exercise 2.6

i a(b + c) j p(q − r) k e(2f + g) l k(4m − 11n)
m 3t(u + v) n 6k(3m − 4) o 4f(5g − 7h) p 12r(3s + 5t)
q x(x + y) r b(1 − b) s 7n(2n − 7) t 9vw(3v − 8w)
2 Expand each of these.
a −2(n + 7) b −3(b − 6) c −9(k − 1) d −11(8 + u)
e −5(2j + 9) f −6(7 − 10y) g −x(y + z) h −t(3u − v)
i −c(5d + 2e) j −2n(p + q) k −9r(5s − 3) l −6h(4i − 11j)
m −s(s − t) n −j(1 + j) o −6y(5y − 12) p −4mn(2m + 5n)
3 Expand:
a (x + 5)6 b ( j − 2)7 c (k + 8)m d (2p + 3)4
e (c − d)d f (3a + 7b)5c g (5s − 2t)4s h (3m + 8n)2mn
■ Consolidation
4 Expand and simplify each of these expressions.
a 5(n + 7) + 6 b 4(c + 5) + 3c c 6(q + 4) − 11
d 12(3 + t) − 5t e −3(m + 2) + 10 f −7(2n − 3) − 5
g 10a + 4(6 − a) h 7 + 3(4x − 1) i 2q − 6(5 + 2q)
j 4m + 8(2m − 11) k 8 − (2x − 7) l 5c − 6(1 − 4c)
m 5(2m + 9) + m + 15 n 3k + 9 + 2(k − 4) o 12x + 17 − 2(3x − 5)
p 7(5t + 3) − 10t − 15 q 4y + 3(y + 7) + 8 r 5w − 4(w − 3) − 9
5 Expand each of these, then collect the like terms.
a 3(n + 4) + 5(n + 2) b 6(z + 5) + 4(z − 2) c 7(p − 2) + 8(p + 3)
d 5(w + 2) + 2(w − 5) e 4(x + 3) − 3(x − 5) f 3(n − 1) − 7(n − 2)
g 9(a + 6) − 7(3 − a) h −4(s − 5) − 6(s − 1) i 8(2b + 3) + 3(3b − 2)
j 6(3c − 4) − 5(4 − 3c) k −3(7y + 2) + 5(2y + 3) l −6(3k + 4) − 9(12 − 2k)
m x(x + 5) + 3(x + 9) n y(y − 2) + 6(y − 7) o 3a(a + 6) + 2a(a + 4)
p 4g(g + 3) − 6g(g − 2) q 8u(u − 2) − 5u(7 − u) r 10c(2d + e) + 5c(3d + 4e)
6 Are the following statements true (T) or false (F)? Explain.
a 6(2p + 5) = (2p + 5)6 b 7(3y + 2) = 21y + 2 c 5 + 4(x − 1) = 9(x − 1)
d −2(5v − 3) = −10v − 6 e ab(a + b) = a2b + ab2 f −(w − 2) = 2 − w
7 Find, in simplest form, an expression for the area of each figure.
a b c d
3a + 4
5 2mn
11m − 4n
6k
k + 8
7v
4w − 10

42
To factorise an expression means to write the expression as the product of its factors. This is
the same as reversing or undoing the expansion process.
Many expressions can be factorised in several different ways. For example, we can factorise
8n + 16 as 1(8n + 16) or 2(4n + 8) or 4(2n + 4) or 8(n + 2). However, by convention, we use
the highest common factor (HCF), that is, the greatest possible factor that is common to every
term in the expression, when factorising. In this example, the HCF of 8n and 16 is 8. Hence,
the correct factorisation of 8n + 16 is 8(n + 2).
NOTE:
• If the ﬁrst term of an expression is negative, then by convention, the HCF is also negative.
• Factorisations should be checked by expanding the answers.
Example 1
Factorise:
a 3x + 12 b 2r − 14 c 10p + 45
d a2 + 8a e 12t2 − 16tu f m2n + mn2 − mnp
Solutions
a 3x + 12 b 2r − 14 c 10p + 45
= 3 × x + 3 × 4 = 2 × r − 2 × 7 = 5 × 2p + 5 × 9
= 3(x + 4) = 2(r − 7) = 5(2p + 9)
d a2 + 8a e 12t2 − 16tu f m2n + mn2 − mnp
= a × a + a × 8 = 4t × 3t − 4t × 4u = mn × m + mn × n − mn × p
= a(a + 8) = 4t(3t − 4u) = mn(m + n − p)
2.7 The highest common factor
Expanding
a(b + c) = ab + ac
Factorising
To factorise an algebraic expression:
write the HCF of the terms outside the grouping symbols
divide each term in the expression by the HCF to find the terms inside the
grouping symbols.
ab + ac = a(b + c) and ab − ac = a(b − c)
EG
+S

Example 2
Factorise:
a −7g − 28 b −ab + bc
Solutions
a −7g − 28 b −ab + bc
= −7 × g − 7 × (+4) = −b × a − b × (−c)
= −7(g + 4) = −b(a − c)
1 Complete each of these factorisations.
a 2n + 6 = 2( ) b 3p − 15 = 3( ) c 7y + 7 = 7( )
d 4g + 10 = 2( ) e 12a − 9 = 3( ) f 15k − 25m = 5( )
g ax + ay = a( ) h pq − qr = q( ) i st − t = t( )
j m2 + 3m = m( ) k 4r − r2 = r( ) l ab + b2 = b( )
m 5d2 + 10d = 5d( ) n 12p2 − 21p = 3p( ) o 35yz + 28y2 = 7y( )
2 Factorise each of these expressions by taking out the highest common factor.
a 2c + 8 b 5y + 10 c 18 + 3q d 35 + 7p
e 2h − 14 f 6t − 30 g 33 − 3r h 48 − 4n
i 5c + 5d j 3x − 6y k 21g + 7h l 8m − 40n
m ab + ac n uv − uw o ef − fg p rs − r
q b2 + bc r k2 − 8k s 11n + n2 t a − a2
■ Consolidation
3 Factorise by removing the highest common factor.
a 6n + 9 b 10b + 25 c 10y + 12 d 12k − 8
e 21w − 35 f 18s − 21 g 16a + 24 h 18t − 30
i 30p + 27 j 14c + 49 k 30r − 80 l 22e − 99
m 35 − 55h n 90 + 63v o 39 + 26z p 24 − 60j
4 Factorise each expression completely.
a 3ab + 9bc b 2xy + 8xz c 4pq − 20qr d 7gh − 14hi
e 4uv + 6uw f 8ef + 20fg g 33rs − 77qr h 24mn − 20mp
i 7c2 + 21c j 24w2 − 6w k 10g2 − 22g l 15y + 40y2
m mnp + mnq n rst − rtu o a2b + ab2 p def − de2
q j2k − jk2m r 12tu + 15u2v s 4ab2 + 10a2bc t 49x2y2 − 42xyz
5 Factorise:
a 3a + 3b + 3c b pq + pr − ps c a2 − ab − ac
d 5r + 10s + 25 e 4x2 − 10x + 8xy f 6 + 24u − 18u2
g 42k2 − 14k + 21 h 3mn − m + mn2 i 2x2 + 2xy − 6x
j 30t − 15tu + 10t2 k 4cd + 28c2 − 20ce l 21f − 70fg − 56f 2
m a2b + ab2 + ab n 8pq − p2q + pq2 o u2vw − uv2w − uvw2
EG
+S
Exercise 2.7

44
6 Explain why each of these expressions has not been correctly or completely factorised.
a 8x + 12 = 2(4x + 6) b p2 + 7p = p(2 + 7)
c e2 + e = e(e + 0) d abc + abd = a(bc + bd)
e 7uv + 14u = 7u(v + 14u) f 3p + 3q + 15 = 3(p + q) + 15
7 Factorise by taking out the greatest negative common factor.
a −2p − 12 b −3x − 21 c −15g − 20 d −14u − 49
e −2t + 2 f −8w + 24 g −12k + 16 h −9r + 30
i −24 − 15m j −18 + 45q k −36 + 24y l −63 − 77c
m −ab + bc n −mn − km o −x2 − 2x p −4e + e2
q −9k2 + 12k r −20a − 28a2 s −25b + 55bc t −48x2y − 60y2
8 Factorise by taking out the binomial common factor.
a a(b + c) + 5(b + c) b m(x − y) + n(x − y) c p(p + 3) + 4(p + 3)
d x(a + 1) − 2(a + 1) e 3(m − 7) − n(m − 7) f a2(p + q) − 6(p + q)
g 5c(c + 4) + 2(c + 4) h 8(1 − k) − 3m(1 − k) i y(2s + 3) − z(2s + 3)
j 4g(3w − 5) + 9h(3w − 5) k x(x − 7) + (x − 7) l (7b + 2c) − 3d(2c + 7b)
Example 1
Simplify:
a b c d
Solutions
a b c d
= = = =
= = = =
=
Adding and subtracting
algebraic fractions
2.8
To add or subtract algebraic fractions:
express the fractions with a common denominator
add or subtract the numerators
simplify if possible.
EG
+S
11m
12
---------
-
5m
12
------
-
+
4
3c
-----
-
5
3c
-----
-
+
11k
10
--------
-
3k
5
-----
-
–
5h
6
-----
-
3h
4
-----
-
–
11m
12
---------
-
5m
12
------
-
+
4
3c
-----
-
5
3c
-----
-
+
11k
10
--------
-
3k
5
-----
-
–
5h
6
-----
-
3h
4
-----
-
–
16m
12
---------
-
9
3c
-----
-
11k
10
--------
-
6k
10
-----
-
–
10h
12
--------
-
9h
12
-----
-
–
4m
3
------
-
3
c
--
-
5k
10
-----
-
h
12
-----
-
k
2
--
-

Example 2
Simplify:
a b c
a b c
= = =
= = =
=
Example 3 Solution
Simplify:
=
=
=
1 Simplify:
a b c d
e f g h
i j k l
2 Simplify:
a b c d
EG
+S
1
a
--
-
5
2a
-----
-
+
13
20w
---------
-
2
5w
------
-
–
7x
12y
--------
-
5x
8y
-----
-
+
Solutions
1
a
--
-
5
2a
-----
-
+
13
20w
---------
-
2
5w
------
-
–
7x
12y
--------
-
5x
8y
-----
-
+
2
2a
-----
-
5
2a
-----
-
+
13
20w
---------
-
8
20w
---------
-
–
14x
24y
--------
-
15x
24y
--------
-
+
7
2a
-----
-
5
20w
---------
-
29x
24y
--------
-
1
4w
------
-
EG
+S
k 4
+
3
-----------
-
k 2
–
5
----------
-
+
k 4
+
3
-----------
-
k 2
–
5
----------
-
+
5 k 4
+
( )
15
-------------------
-
3 k 2
–
( )
15
-------------------
+
5k 20 3k 6
–
+ +
15
---------------------------------------
-
8k 14
+
15
-----------------
-
Exercise 2.8
3a
7
-----
-
2a
7
-----
-
+
5m
9
------
-
m
9
---
-
–
9h
13
-----
-
8h
13
-----
-
–
x
4
--
-
x
4
--
-
+
3n
8
-----
-
3n
8
-----
-
+
11k
12
--------
-
3k
12
-----
-
–
5c
3
-----
-
2c
3
-----
-
–
9d
10
-----
-
3d
10
-----
-
+
6b
7
-----
-
8b
7
-----
-
+
14w
15
---------
-
4w
15
------
-
–
19e
24
--------
-
9e
24
-----
-
–
13s
16
-------
-
9s
16
-----
-
+
5
x
--
-
2
x
--
-
+
8
p
--
-
7
p
--
-
–
10
3y
-----
-
4
3y
-----
-
+
12
7q
-----
-
3
7q
-----
-
–

46
e f g h
i j k l
■ Consolidation
3 Express these fractions with a common denominator, then simplify.
a b c d
e f g h
i j k l
m n o p
4 Explain why each of the following answers is not correct.
a b c
a b c d
e f g h
i j k l
6 Simplify:
a b c
d e f
g h i
1
2n
-----
-
1
2n
-----
-
+
3
4c
-----
-
5
4c
-----
-
+
11
5g
-----
-
9
5g
-----
-
+
13
12k
--------
-
4
12k
--------
-
–
17a
10r
--------
-
9a
10r
--------
–
4m
15b
--------
-
2m
15b
--------
-
+
7e
20v
--------
-
8e
20v
--------
-
+
19t
16z
-------
-
15t
16z
-------
-
–
n
2
--
-
n
4
--
-
+
a
3
--
-
a
9
--
-
–
k
3
--
-
k
12
-----
-
+
d
5
--
-
d
15
-----
-
–
y
5
--
-
y
2
--
-
+
t
3
--
-
t
4
--
-
–
b
4
--
-
b
7
--
-
–
h
12
-----
-
h
5
--
-
+
2c
5
-----
-
3c
10
-----
-
+
5m
12
------
-
m
3
---
-
–
3r
5
-----
4r
3
-----
+
3u
2
-----
-
6u
7
-----
-
–
w
4
---
-
5w
6
------
-
+
7x
6
-----
-
2x
9
-----
-
–
3 f
10
-----
-
5 f
8
-----
-
+
11s
12
-------
-
8s
9
----
-
–
5m
9
------
-
2m
9
------
-
+
7m
18
------
-
=
3w
5
------
-
2w
3
------
-
+
5w
8
------
-
=
4
5a
-----
-
3
5a
-----
-
+ 12
5
--
-a
=
1
x
--
-
3
2x
-----
-
+
2
3a
-----
-
1
6a
-----
-
+
17
20e
--------
-
2
5e
-----
-
–
13
12p
---------
2
3p
------
–
3
2u
-----
-
2
3u
-----
-
+
4
5 f
-----
-
3
4 f
-----
-
–
2
3t
----
-
4
7t
----
-
–
3
5h
-----
-
5
9h
-----
-
+
5c
4 j
-----
-
11c
6 j
--------
-
+
9m
8z
------
-
5m
6z
------
-
–
9a
10g
--------
-
3a
4g
-----
-
–
5k
12n
--------
-
7k
8n
-----
-
+
n 2
+
2
-----------
-
n 1
+
6
-----------
-
+
b 3
+
4
-----------
-
b 4
+
7
-----------
-
+
x 8
+
5
-----------
-
x 2
–
3
-----------
+
m 3
–
6
------------
-
m 6
+
7
------------
-
+
2w 5
–
12
---------------
-
w 1
–
4
------------
-
+
3s 2
+
9
--------------
-
2s 7
–
5
-------------
-
+
x 7
+
2
-----------
-
x 3
+
4
-----------
-
–
3c 10
+
5
-----------------
-
c 3
–
4
----------
-
–
7e 1
–
8
--------------
-
2e 5
–
3
--------------
-
–

NOTE: Any fractions can be multiplied or divided. They do not need to have a common
denominator.
Example 1
Simplify:
a b c
Solutions
a b c
= = =
Example 2
Simplify:
a b
Solutions
a b
= =
= =
Multiplying and dividing
algebraic fractions
2.9
To multiply algebraic fractions:
cancel any common factors between the numerators and the denominators
multiply the numerators
multiply the denominators.
To divide algebraic fractions:
change the division sign to a multiplication sign and take the reciprocal of the
second fraction
proceed as above for the multiplication of fractions.
EG
+S
m
3
---
-
n
4
--
-
×
15x
14y
--------
-
7
9x
-----
-
×
a2
bc2
-------
-
bc
a
-----
-
×
m
3
---
-
n
4
--
-
×
15x
14y
--------
-
7
9x
-----
-
×
5 1
2 3
a2
bc2
-------
-
bc
a
-----
-
×
mn
12
------
-
5
6y
-----
-
a
c
--
-
EG
+S
e
4
--
-
7
f
--
-
÷
9c
10d
--------
-
12c2
25de
-----------
-
÷
e
4
--
-
7
f
--
-
÷
9c
10d
--------
-
12c2
25de
-----------
-
÷
e
4
--
-
f
7
--
-
×
9c
10d
--------
-
25de
12c2
-----------
-
×
5
2
3
4
ef
28
-----
-
15e
8c
--------
-

48
1 Simplify:
a b c d
e f g h
2 Simplify:
a b c d
e f g h
■ Consolidation
3 Simplify each of the following by at first cancelling common factors.
a b c d
e f g h
i j k l
4 Express each of these as a multiplication, then simplify.
a b c d
e f g h
i j k l
5 Explain why each of the following solutions is incorrect.
a b
a b c d
e f g h
Exercise 2.9
a
3
--
-
b
2
--
-
×
u
3
--
-
u
4
--
-
×
a
b
--
-
c
d
--
-
×
1
p
--
-
1
q
--
-
×
1
x
--
-
1
4x
-----
-
×
4c
5
-----
-
d
3
--
-
×
9m
7
------
-
3n
4
-----
-
×
5
6x
-----
-
7
8x
-----
-
×
x
5
--
-
4
y
--
-
÷
v
2
--
-
6
v
--
-
÷
t
u
--
-
v
w
---
-
÷
1
g
--
-
1
h
--
-
÷
1
s
--
- 2s
÷
3e
7
-----
-
5 f
6
-----
-
÷
10a
11
--------
-
3b
4
-----
-
÷
4
5h
-----
-
3h
13
-----
-
÷
n
3
--
-
2
n
--
-
×
a
4
--
-
8
b
--
-
×
3
c
--
-
d
15
-----
-
×
3
x
--
-
2x
7
-----
-
×
ab
e
-----
-
cd
bc
-----
-
×
5a
3b
-----
-
b
10a
--------
-
×
8d
7c
-----
-
21
8e
-----
-
×
4e
10 f
--------
-
5 f
12e
--------
-
×
9t
14v
--------
-
7u
18tu
----------
-
×
15w
27y
---------
-
18x
25w
---------
-
×
11i
12h
--------
-
21h
22ij
---------
-
×
44r
35pq
------------
-
10p
99rs
----------
-
×
x
5
--
-
x
3
--
-
÷
m
2
---
-
n
6
--
-
÷
5
u
--
-
20
v
-----
-
÷
6
r
--
-
11s
3r
-------
-
÷
ef
g
----
-
eh
hi
-----
-
÷
10k
3
--------
-
5m
12
------
-
÷
4s
7t
----
-
16s
t
-------
-
÷
9w
28v
--------
-
27w
7v
---------
-
÷
4p
33q
--------
-
20pr
11r
-----------
-
÷
12c
45b
--------
-
16c
25a
--------
-
÷
12e
63d
--------
-
20ef
99d
-----------
÷
42xy
55x
-----------
-
49yz
60w
-----------
÷
5a
3b
-----
-
10
11b
--------
-
×
2a
33
-----
-
=
1 2
4c
7
-----
-
21
c
-----
-
÷
4
3
--
-
=
3
1
p2
q
----
-
q2
p
----
-
×
5m
4np
--------
-
2n
3m2
---------
-
×
2a2
3b2
-------
-
7b
5a
-----
-
×
ab2
p2
q
--------
-
pq
ab
------
×
r2
s
tu2
------
-
rs
tuv
-------
-
÷
8e
21f 2
-----------
-
24e2
35f
----------
-
÷
12x2
y
25ab
-------------
-
28xy2
15bc
-------------
-
÷
24tu2
33vw
-------------
36t 2
u
55wx
-------------
÷

7 Simplify:
a b c
d e f
8 Factorise each expression where possible, then simplify.
a b c
d e f
g h i
We use generalised arithmetic to form a general expression to describe any value in a situation.
For example, if Alicia is 10 years old, then:
• in 1 years time she will be (10 + 1) years old
• in 2 years time she will be (10 + 2) years old
• in k years time she will be (10 + k) years old.
Her exact age in any number of years time can be worked out simply by adding that number
to 10.
To form a general expression for a situation, choose a few numbers and look for a pattern in the
answers. For example, to find the number of centimetres in y m, consider:
1 m = (1 × 100) cm 2 m = (2 × 100) cm 3 m = (3 × 100) cm y m = (y × 100) cm
= 100 cm = 200 cm = 300 cm = 100y cm
Being able to form a general expression is an essential skill in mathematics.
Listed below are some common key words and phrases and their meaning.
• Addition—sum, increase, add, plus, total, more than
• Subtraction—difference, decrease, subtract, take away, reduce, less than
• Multiplication—product, times, multiply, double, multiple
• Division—quotient, divide, halve, share
NOTE: In additions and subtractions where the second term is a pronumeral, the words ‘sum’
and ‘difference’ are usually preferred to phrases such as ‘more than’ and ‘less than’.
Odd and even numbers both have the same general expression because both odd numbers and
even numbers increase by 2. So, if n is an odd number, then n + 2, n + 4, n + 6, … are all odd.
ab
bc
-----
-
cd
de
-----
-
ef
ag
-----
-
×
×
5m
7n
------
-
14p
15m
---------
-
9n
16q
--------
-
×
×
9r
20s
-------
-
15s
22u
--------
-
27r
11t
--------
÷
×
15w
7x
---------
-
40y
9x
--------
-
÷
16xy
45w
-----------
-
×
21a2
32bc
-----------
-
55e2
63ab
-----------
-
45e
24b2
c
-------------
-
÷
×
14pq2
9ru
--------------
-
49qr
18tu2
-------------
30stu
25r2
s
-------------
-
÷
÷
3x 12
+
12
-----------------
-
8
x 4
+
-----------
-
×
5m 30
+
3m 21
–
-------------------
-
9m 63
–
45
------------------
-
×
24m2
6k 42
+
-----------------
-
5k 35
+
18m
-----------------
-
×
12t 12
–
3u
-------------------
-
2u 8
+
8t 8
–
--------------
-
×
c2
c
+
3c 3
+
--------------
-
3c2
6c
+
6c2
-------------------
-
×
a2
2a
+
21x 21y
–
-----------------------
-
14x 14y
–
5a 10
+
-----------------------
-
×
25a2
b
18a 27b
–
-----------------------
-
35ab2
12a 18b
–
-----------------------
-
÷
15u 20v
+
24u 60v
–
-----------------------
-
30u 40v
+
16u 40v
–
-----------------------
-
÷
8bc 16c
–
6ab 30a
–
------------------------
-
4bc 8c
–
3ab 15a
+
------------------------
-
÷
2.10 Generalised arithmetic

50
However, if n is an even number, then n + 2, n + 4, n + 6, … are all even. Whether such
expressions are odd or even depends on whether n is odd or even.
Example 1
Write an algebraic expression for each of the following.
a five more than k b two less than y
c the sum of m and n d the difference between p and q
e the product of h and 3 f the quotient of d and e
g one-quarter of c h two-thirds of u
i the square of w j twice the cube of x
Solutions
a k + 5 b y − 2 c m + n d p − q e 3h
f g h i w2 j 2x3
Example 2
Write the meaning of each expression in words.
a 3m − 5 b c d 4(g + 2)
Solutions
a 5 less than the product of 3 and m
b 7 more than the quotient of x and y
c one-tenth of the difference between e and f
d 4 times the number which is 2 more than g
Example 3
Write down 3 consecutive numbers, the first of which is:
a n b n + 7 c n − 1
Solutions
a n, n + 1, n + 2 b n + 7, n + 8, n + 9 c n − 1, n, n + 1
Example 4
Write down 3 consecutive:
a even numbers, the first of which is t
b even numbers, the first of which is t + 5
c odd numbers, the first of which is 3t
d odd numbers, the first of which is t − 1
EG
+S
d
e
--
-
c
4
--
-
2u
3
-----
-
EG
+S x
y
-- 7
+
e f
–
10
-----------
-
EG
+S
EG
+S

Solutions
a t, t + 2, t + 4 b t + 5, t + 7, t + 9 c 3t, 3t + 2, 3t + 4 d t − 1, t + 1, t + 3
1 Write an algebraic expression for each of the following.
a 3 more than x b 5 less than t
c the sum of p and q d the difference between m and n
e the sum of x, y and 7 f the product of m and 4
g 9 times the number n h the product of a, 2 and b
i half of k j one-quarter of z
k two-thirds of w l the quotient of u and v
m the number of times that j divides into 4 n the square of k
o the cube of y p the square root of g
2 Write each expression in words.
a n + 4 b q − 6 c c + d d x − y
e 8u f 5ef g h
i j a2 k g3 l
■ Consolidation
3 Write an algebraic expression for each of these.
a 3 more than the product of 2 and x b 1 less than the product of y and 5
c the sum of 7 and the product of p and q d the difference between 4 and the square of u
e 6 more than half of c f 9 less than one-fifth of w
g 2 more than the quotient of e and f h 4 less than seven-tenths of r
i one-third of the sum of b and 1 j half the difference between g and h
k 3 times the number that is 12 more than a l 9 times the number that is 3 less than p
m 4 times the sum of c and d n 10 times the difference between r and s
o twice the square of y p 8 times the cube of x
q the quotient of 5 and the square of j r 1 more than half the cube of b
4 Write each of these algebraic expressions in words.
a 5x + 7 b 2n − 3 c gh + 4 d 9 − pq
e f g h
i 5(e + 2) j k 3r2 l 2s3 − 9
5 Write down an algebraic expression in simplest form for the number that is:
a 5 more than t + 2 b 4 less than p + 13
c 8 less than 3k − 2 d 6 more than 7y − 4
Exercise 2.10
h
3
--
-
3v
4
-----
-
m
n
---
- d
a 3
+
4
-----------
-
b
6
--
- 8
+
m n
–
7
------------
- u
v
w
---
-
–
2 c d
–
( )
3
-------------------
-

Mathscape 9 Extension: An Essential Guide to Working Mathematically

Mathscape 9 Extension: An Essential Guide to Working Mathematically

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