7th
EDITION
NEW SYLLABUS
MATHEMATICS
TEACHER’S RESOURCE BOOK
1
1
CONTENTS
1
CONTENTS
Syllabus Matching Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Scheme of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 1: Primes, Highest Common Factor and Lowest Common Multiple
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter 2: Integers, Rational Numbers and Real Numbers
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 3: Approximation and Estimation
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4: Basic Algebra and Algebraic Manipulation
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Revision Exercise A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Revision Exercise A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 5: Linear Equations and Simple Inequalities
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter 6: Functions and Linear Graphs
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter 7: Number Patterns
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Revision Exercise B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Revision Exercise B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter 8: Percentage
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Chapter 9: Ratio, Rate, Time and Speed
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2
1
Chapter 10: Basic Geometry
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Chapter 11: Triangles, Quadrilaterals and Polygons
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Chapter 12: Geometrical Constructions
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Revision Exercise C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Revision Exercise C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Chapter 13: Perimeter and Area of Plane Figures
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Chapter 14: Volume and Surface Area of Prisms and Cylinders
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Chapter 15: Statistical Data Handling
Teaching Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Revision Exercise D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Revision Exercise D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Problems in Real-World Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
1
3
Syllabus Matching Grid
Cambridge O Level Mathematics (Syllabus D) 4024/4029. Syllabus for examination in 2018, 2019 and 2020.
Theme or Topic
1.
Number
2.
Set language and notation
Subject Content
Identify and use:
• Natural numbers
• Integers (positive, negative and zero)
• Prime numbers
• Square numbers
• Cube numbers
• Common factors and common multiples
• Rational and irrational numbers (e.g. p, 2 )
• Real numbers
• Use set language, set notation and Venn diagrams to describe sets and
represent relationships between sets
• Definition of sets:
e.g. A = {x : x is a natural number},
B = {(x, y): y = mx + c},
C = {x : a < x < b},
D = {a, b, c, …}
2.
Squares, square roots, cubes and
cube roots
4.
Directed numbers
5.
Vulgar and decimal fractions
and percentages
6.
Ordering
• Order quantities by magnitude and demonstrate familiarity with the symbols
=, ≠, <, >, <, >.
7.
Standard form
• Use the standard form A × 10n, where n is a positive or negative integer, and
1 < A < 10.
8.
The four operations
9.
Estimation
10
Limits of accuracy
Calculate
• Squares
• Square roots
• Cubes and cube roots of numbers
• Use directed numbers in practical situations
• Use the language and notation of simple vulgar and decimal fractions and
percentages in appropriate contexts
• Recognise equivalence and convert between these forms
Use the four operations for calculations with:
• Whole numbers
• Decimals
• Vulgar (and mixed) fractions
including correct ordering of operations and use of brackets.
• Make estimates of numbers, quantities and lengths
• Give approximations to specified numbers of significant figures and decimal
places
• Round off answers to reasonable accuracy in the context of a given problem
• Give appropriate upper and lower bounds for data given to a specified
accuracy
• Obtain appropriate upper and lower bounds to solutions of simple problems
given to a specified accuracy
1
Reference
Book 1:
Chapter 1
Chapter 2
Book 2:
Chapter 14
Book 4:
Chapter 2
Book 1:
Chapter 1
Chapter 2
Book 1:
Chapter 2
Book 1:
Chapter 2
Book 1:
Chapter 2
Chapter 5
Book 3:
Chapter 4
Book 1:
Chapter 2
Book 1:
Chapter 3
Book 3:
Chapter 3
1
11. Ratio, proportion, rate
12. Percentages
13. Use of an electronic calculator
•
•
•
•
Demonstrate an understanding of ratio and proportion
Increase and decrease a quantity by a given ratio
Use common measures of rate
Solve problems involving average speed
•
•
•
•
Calculate a given percentage of a quantity
Express one quantity as a percentage of another
Calculate percentage increase or decrease
Carry out calculations involving reverse percentages
•
•
•
•
Use an electronic calculator efficiently
Apply appropriate checks of accuracy
Enter a range of measures including ‘time’
Interpret the calculator display appropriately
Book 1:
Chapter 9
Book 2:
Chapter 1
Book 1:
Chapter 8
Book 3:
Chapter 5
Book 1:
Chapter 2
Chapter 4
Book 2:
Chapter 11
Book 3:
Chapter 10
Book 4:
Chapter 4
14. Time
• Calculate times in terms of the 24-hour and 12-hour clock
• Read clocks, dials and timetables
15. Money
• Solve problems involving money and convert from one currency to another
Book 1:
Chapter 9
Book 3:
Chapter 5
16. Personal and small business
finance
• Use given data to solve problems on personal and small business finance
involving earnings, simple interest and compound interest
• Extract data from tables and charts
Book 3:
Chapter 5
17. Algebraic representation and
formulae
• Use letters to express generalised numbers and express arithmetic processes
algebraically
• Substitute numbers for words and letters in formulae
• Construct and transform formulae and equations
Book 1:
Chapter 4
Chapter 5
Book 2:
Chapter 2
Book 3:
Chapter 1
18. Algebraic manipulation
19. Indices
1
•
•
•
•
Manipulate directed numbers
Use brackets and extract common factors
Expand product of algebraic expressions
Factorise where possible expressions of the form:
ax + bx + kay + kby
a2x2 – b2y2
a2 + 2ab + b2
ax2 + bx + c
• Manipulate algebraic fractions
• Factorise and simplify rational expressions
• Understand and use the rules of indices
• Use and interpret positive, negative, fractional and zero indices
2
Book 1:
Chapter 4
Book 2:
Chapter 3
Chapter 4
Chapter 6
Book 3:
Chapter 4
20. Solutions of equations and
inequalities
•
•
•
•
Solve simple linear equations in one unknown
Solve fractional equations with numerical and linear algebraic denominators
Solve simultaneous linear equations in two unknowns
Solve quadratic equations by factorisation, completing the square or by use of
the formula
• Solve simple linear inequalities
Book 1:
Chapter 5
Book 2:
Chapter 2
Chapter 5
Book 3:
Chapter 1
Chapter 3
21. Graphical representation of
inequalities
• Represent linear inequalities graphically
22. Sequences
• Continue a given number sequence
• Recognise patterns in sequences and relationships between different sequences
• Generalise sequences as simple algebraic statements
Book 1:
Chapter 7
23. Variation
• Express direct and inverse variation in algebraic terms and use this form of
expression to find unknown quantities
Book 2:
Chapter 1
24. Graphs in practical situations
• Interpret and use graphs in practical situations including travel graphs and
conversion graphs
• Draw graphs from given data
• Apply the idea of rate of change to easy kinematics involving distance-time
and speed-time graphs, acceleration and deceleration
• Calculate distance travelled as area under a linear speed-time graph
Book 4:
Chapter 1
Book 1:
Chapter 6
Book 2:
Chapter 2
Book 3:
Chapter 7
25. Graphs in practical situations
• Construct tables of values and draw graphs for functions of the form axn
where a is a rational constant, n = –2, –1, 0, 1, 2, 3, and simple sums of not
more than three of these and for functions of the form kax where a is a positive
integer
• Interpret graphs of linear, quadratic, cubic, reciprocal and exponential
functions
• Solve associated equations approximately by graphical methods
• Estimate gradients of curve by drawing tangents
Book 1:
Chapter 6
Book 2:
Chapter 1
Chapter 2
Chapter 5
Book 3:
Chapter 1
Chapter 7
26. Function notation
• Use function notation, e.g. f(x) = 3x – 5; f : x 3x – 5, to describe simple
functions
• Find inverse functions f–1(x)
Book 2:
Chapter 7
Book 3:
Chapter 2
27. Coordinate geometry
• Demonstrate familiarity with Cartesian coordinates in two dimensions
• Find the gradient of a straight line
• Calculate the gradient of a straight line from the coordinates of two points on
it
• Calculate the length and the coordinates of the midpoint of a line segment
from the coordinates of its end points
• Interpret and obtain the equation of a straight line graph in the form y = mx + c
• Determine the equation of a straight line parallel to a given line
• Find the gradient of parallel and perpendicular lines
3
Book 1:
Chapter 6
Book 2:
Chapter 2
Book 3:
Chapter 6
1
28. Geometrical terms
• Use and interpret the geometrical terms: point; line; plane; parallel;
perpendicular; bearing; right angle, acute, obtuse and reflex angles; interior
and exterior angles; similarity and congruence
• Use and interpret vocabulary of triangles, special quadrilaterals, circles,
polygons and simple solid figures
• Understand and use the terms: centre, radius, chord, diameter, circumference,
tangent, arc, sector and segment
Book 1:
Chapter 10
Chapter 11
Book 2:
Chapter 8
Book 3:
Chapter 9 to
Chapter 13
29. Geometrical constructions
30. Similarity and congruence
31. Symmetry
32. Angles
33. Loci
34. Measures
1
• Measure lines and angles
• Construct a triangle, given the three sides, using a ruler and a pair of
compasses only
• Construct other simple geometrical figures from given data, using a ruler and
protractor as necessary
• Construct angle bisectors and perpendicular bisectors using a pair of
compasses as necessary
• Read and make scale drawings
• Use and interpret nets
• Solve problems and give simple explanations involving similarity and
congruence
• Calculate lengths of similar figures
• Use the relationships between areas of similar triangles, with corresponding
results for similar figures, and extension to volumes and surface areas of
similar solids
• Recognise rotational and line symmetry (including order of rotational
symmetry) in two dimensions
• Recognise symmetry properties of the prism (including cylinder) and the
pyramid (including cone)
• Use the following symmetry properties of circles:
(a) equal chords are equidistant from the centre
(b) the perpendicular bisector of a chord passes through the centre
(c) tangents from an external point are equal in length
• Calculate unknown angles and give simple explanations using the following
geometrical properties:
(a) angles at a point
(b) angles at a point on a straight line and intersecting straight lines
(c) angles formed within parallel lines
(d) angle properties of triangles and quadrilaterals
(e) angle properties of regular and irregular polygons
(f) angle in a semi-circle
(g) angle between tangent and radius of a circle
(h) angle at the centre of a circle is twice the angle at the circumference
(i) angles in the same segment are equal
(j) angles in opposite segments are supplementary
• Use the following loci and the method of intersecting loci for sets of points in
two dimensions which are:
(a) at a given distance from a given point
(b) at a given distance from a given straight line
(c) equidistant from two given points
(d) equidistant from two given intersecting straight line
• Use current units of mass, length, area, volume and capacity in practical
situations and express quantities in terms of larger or smaller units
4
Book 1:
Chapter 12
Chapter 14
Book 2:
Chapter 8
Book 4:
Chapter 8
Book 2:
Chapter 8
Book 3:
Chapter 11
Chapter 12
Book 2:
Chapter 13
Book 3:
Chapter 13
Book 1:
Chapter 10
Chapter 11
Book 3:
Chapter 13
Book 4:
Chapter 8
Book 1:
Chapter 13
Chapter 14
35. Mensuration
36. Trigonometry
37. Vectors in two dimensions
38. Matrices
39. Transformations
40. Probability
• Solve problems involving:
(a) the perimeter and area of a rectangle and triangle
(b) the perimeter and area of a parallelogram and a trapezium
(c) the circumference and area of a circle
(d) arc length and sector area as fractions of the circumference and area of a
circle
(e) the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid
and cone
(f) the areas and volumes of compound shapes
• Interpret and use three-figure bearings
• Apply Pythagoras’ theorem and the sine, cosine and tangent ratios for acute
angles to the calculation of a side or an angle of a right-angled triangles
• Solve trigonometrical problems in two dimensions involving angles of
elevation and depression
• Extend sine and cosine functions to angles between 90° and 180°
• Solve problems using the sine and cosine rules for any triangle and the
1
formula area of triangle =
ab sin C
2
• Solve simple trigonometrical problems in three dimensions
⎛ x⎞ →
• Describe a translation by using a vector represented by ⎜⎝ y ⎟⎠ , AB or a
• Add and subtract vectors
• Multiple a vector by a scalar
⎛ x⎞
• Calculate the magnitude of a vector ⎜⎝ y ⎟⎠ as x 2 + y 2
• Represent vectors by directed line segments
• Use the sum and difference of two vectors to express given vectors in terms of
two coplanar vectors
• Use position vectors
• Display information in the form of a matrix of any order
• Solve problems involving the calculation of the sum and product (where
appropriate) of two matrices, and interpret the results
• Calculate the product of a matrix and a scalar quantity
• Use the algebra of 2 × 2 matrices including the zero and identity 2 × 2
matrices
• Calculate the determinant |A| and inverse A–1 of a non-singular matrix A
• Use the following transformations of the plane: reflection (M), rotation (R),
translation (T), enlargement (E) and their combinations
• Identify and give precise descriptions of transformations connecting given
figures
• Describe transformations using coordinates and matrices
• Calculate the probability of a single event as either a fraction or a decimal
• Understand that the probability of an event occurring = 1 – the probability of
the event not occurring
• Understand relative frequency as an estimate of probability
• Calculate the probability of simple combined events using possibility
diagrams and tree diagrams where appropriate
5
Book 1:
Chapter 13
Chapter 14
Book 2:
Chapter 12
Book 3:
Chapter 10
Book 2:
Chapter 10
Chapter 11
Book 3:
Chapter 8
Chapter 9
Book 4:
Chapter 7
Book 4:
Chapter 5
Book 2:
Chapter 9
Book 4:
Chapter 6
Book 2:
Chapter 15
Book 4:
Chapter 3
1
41. Categorical, numerical and
grouped data
42. Statistical diagrams
1
• Collect, classify and tabulate statistical data
• Read, interpret and draw simple inferences from tables and statistical
diagrams
• Calculate the mean, median, mode and range for individual and discrete data
and distinguish between the purposes for which they are used
• Calculate an estimate of the mean for grouped and continuous data
• Identify the modal class from a grouped frequency distribution
• Construct and interpret bar charts, pie charts, pictograms, simple frequency
distributions, frequency polygons, histograms with equal and unequal
intervals and scatter diagrams
• Construct and use cumulative frequency diagrams
• Estimate and interpret the median, percentiles, quartiles and interquartile
range for cumulative frequency diagrams
• Calculate with frequency density
• Understand what is meant by positive, negative and zero correlation with
reference to a scatter diagram
• Draw a straight line of best fit by eye
6
Book 1:
Chapter 15
Book 2:
Chapter 17
Book 4:
Chapter 4
Book 1:
Chapter 15
Book 2:
Chapter 16
Book 4:
Chapter 4
Secondary 1 Mathematics Scheme of Work
Week
(5 classes
× 45 min)
1
Chapter
Section
1.1 Prime
1
Numbers
Primes, Highest
(pp. 3 – 9)
Common Factor
and Lowest
Common
Multiple
Specific Instructional
Objectives (SIOs)
• Explain what a prime
number is
• Determine whether
a whole number is
prime
• Express a composite
number as a product
of its prime factors
Syllabus Subject
Content
Identity and use prime
numbers
Activity
Investigation –
Classification of
Whole Numbers
(pp. 3 – 4)
Thinking Time
(p. 4)
Investigation –
Sieve of
Eratosthenes (p. 5)
ICT
Investigation –
Interesting Facts
about Prime
Numbers
(p. 8)
Additional
Resources
Reasoning,
Communication
and Connection
Thinking Time (p. 4)
Investigation –
Sieve of
Eratosthenes
(p. 5)
Journal Writing (p.
5)
Worked Example 2
(p. 7)
7
Journal Writing
(p. 5)
Practise Now 2
Q 1 – 2 (p. 7)
Investigation –
Interesting Facts
about Prime
Numbers (p. 7)
Thinking Time (p. 8)
Thinking Time
(p. 8)
1
1
Week
(5 classes
× 45 min)
1
Chapter
Section
1.2 Square Roots
and Cube
Roots
(pp. 9 – 14)
Specific Instructional
Objectives (SIOs)
• Find square roots
and cube roots using
prime factorisation,
mental estimation and
calculators
Syllabus Subject
Content
Identify and use square
numbers and cube
numbers
Activity
ICT
Additional
Resources
Thinking Time
(p. 12)
Reasoning,
Communication
and Connection
Attention (p. 10)
Attention (p. 11)
Thinking Time (p.
12)
Calculate squares,
square roots, cubes and
cube roots of numbers
8
Main Text –
‘Since √ 997 = 31.6
(to 1 d.p.), the largest
prime less than or
equal to √ 997 is 31.
To determine
whether 997 is a
prime, it is enough
to test whether 997
is divisible by 2, 3,
5, 7, ... or 31 (only
11 prime numbers
to test). We do not
have to test all the
167 prime numbers.
Why?’ (p. 13)
Ex 1A Q 11 – 12
(p. 14)
2
2
1.3 Highest
Common
Factor and
Lowest
Common
Multiple
(pp. 14 – 24)
Miscellaneous
• Find the highest
common factor
(HCF) and lowest
common multiple
(LCM) of two or
more numbers
• Solve problems
involving HCF and
LCM in real-world
contexts
Practise Now 11 Q 3
(p. 18)
Identify and use
common factors and
common multiples
Ex 1B Q6, 9(a) – (e),
10, 11(a) – (d), 13(i),
14(ii) (pp. 21 – 22)
Solutions for
Challenge
Yourself
Week
(5 classes
× 45 min)
2
Chapter
2
Integers,
Rational
Numbers and
Real Numbers
Section
2.1 Negative
Numbers
(pp. 27 – 30)
Specific Instructional
Objectives (SIOs)
• Use negative
numbers, rational
numbers and real
numbers in a realworld context
• Represent real
numbers on a number
line and order the
numbers
9
3
2.2 Addition and
Subtraction
involving
Negative
Numbers
(pp. 30 – 37)
• Perform operations
in real numbers,
including using the
calculator
Syllabus Subject
Content
Identify and use natural
numbers and integers
(positive, negative and
zero)
Activity
ICT
Thinking Time
(p. 28)
Use directed numbers in
Main Text
practical situations
(pp. 28 – 29)
Thinking Time
(p. 28)
Use the four operations
for calculations
with whole numbers
including correct
ordering of operations
and use of brackets.
Main Text
(pp. 31 – 32)
Class Discussion –
Addition involving
Negative Numbers
(p. 33)
Main Text
(p. 34)
Class Discussion –
Subtraction
involving
Negative Numbers
(p. 35)
Reasoning,
Communication
and Connection
Class Discussion –
Use of Negative
Numbers in the Real
World (p. 27)
Class Discussion –
Use of Negative
Numbers in the
Real World
(p. 27)
Order quantities
by magnitude and
demonstrate familiarity
with the symbols =, ≠,
<, >, ≤, ≥.
Additional
Resources
Main Text –
‘The number is
marked out on the
number line. Explain
how the point on
the number line is
obtained.’
(p. 29)
Main Text –
‘Alternatively,
you may visit
http://www.
shinglee.com.sg/
Student
Resources/
to access the
AlgeToolTM
software.’
(p. 30)
Class Discussion –
Addition involving
Negative Numbers
(p. 33)
Class Discussion –
Subtraction
involving Negative
Numbers
(p. 35)
1
1
Week
(5 classes
× 45 min)
3
Chapter
Section
Specific Instructional
Objectives (SIOs)
Syllabus Subject
Content
2.3 Multiplication
and Division
involving
Negative
Numbers
(pp. 37 – 43)
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Class Discussion –
Multiplication
involving Negative
Numbers (p. 39)
Main Text
(pp. 38 – 39)
Class Discussion –
Multiplication
involving
Negative Numbers
(p. 39)
Thinking Time
(p. 41)
Thinking Time
(p. 41)
4
10
2.4 Rational
Numbers and
Real Numbers
(pp. 44 – 54)
Use of an electronic
calculator efficiently
Main Text
(p. 42)
Apply appropriate
checks of accuracy
Practise Now 4b
Use the four operations
for calculations with
decimals, vulgar
(and mixed) fractions
including correct
ordering of operations
and use of brackets.
Identify and use rational Investigation –
Terminating,
and irrational numbers
Recurring and
(e.g. π, √2)
Non-Recurring
Decimals
(p. 50)
Thinking Time
(p. 49)
Investigation –
Some Interesting
Facts about the
Irrational Number
p
(p. 51)
Thinking Time
(p. 49)
Investigation –
Some Interesting
Facts about the
Irrational Number
p
(p. 51)
4
Miscellaneous
Solutions for
Challenge
Yourself
Week
(5 classes
× 45 min)
4
Chapter
Section
3
Approximation
and Estimation
3.1 Approximation
(pp. 59 – 62)
Specific Instructional
Objectives (SIOs)
Syllabus Subject
Content
Make estimates of
numbers, quantities and
lengths
Activity
Class Discussion –
Actual and
Approximated
Values
(p. 59)
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Class Discussion –
Actual and
Approximated
Values
(p. 59)
Practise Now 1 Q 2
(p. 60)
Practise Now 2 Q 2
(p. 61)
Ex 3A Q 5 – 7
(p. 62)
5
3.2 Significant
Figures
(pp. 63 – 67)
• Round off numbers
to a required number
of decimal places and
significant figures
Give approximations
to specified numbers of
significant figures and
decimal places
Investigation –
Rounding in Real
Life (p. 67)
Journal Writing
(p. 67)
Practise Now Q 2
(p. 64)
Practise Now 4 Q 2
(p. 66)
11
Investigation –
Rounding in Real
Life
(p. 67)
Journal Writing
(p. 67)
5
3.3 Rounding and
Truncation
Errors
(pp. 68 – 71)
• Explain the problem
of rounding and
truncation errors
Investigation –
The Missing 0.1%
Votes (p. 68)
Investigation –
The Missing 0.1%
Votes
(p. 68)
Thinking Time
(p. 70)
Ex 3B Q 3, 8 – 9,
10(iii) (pp. 70 – 71)
Thinking Time
(p. 69)
1
Investigation –
Rounding and
Truncation Errors
in Calculators
(p. 70)
1
Week
(5 classes
× 45 min)
Chapter
5
Section
3.4 Estimation
(pp. 71 – 77)
Specific Instructional
Objectives (SIOs)
• Estimate the results
of computations
• Apply estimation in
real-world contexts
Syllabus Subject
Content
Round off answers to
reasonable accuracy in
the context of a given
problem
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Worked Example 6
(p. 73)
Investigation –
Use of a Smaller
Quantity to
Estimate a Larger
Quantity
(p. 75)
Performance Task
(p. 76)
Miscellaneous
5
6
12
4
Basic Algebra
and Algebraic
Manipulation
4.1 Fundamental
Algebra
(pp. 81 – 91)
Solutions for
Challenge
Yourself
• Use letters to
represent numbers
• Express basic
arithmetical processes
algebraically
• Evaluate algebraic
expressions
• Add and subtract
linear expressions
Use letters to express
generalised numbers
and express arithmetic
processes algebraically
Substitute numbers for
words and letters in
formulae
Class Discussion
– Expressing
Mathematical
Relationships
using Algebra
(p. 83)
Investigation
– Comparison
between Pairs of
Expressions
(p. 84)
Construct and transform Main Text
formulae and equations (pp. 85 – 88)
Manipulate directed
numbers
Practise Now
(p. 89)
Investigation
–Comparison
between Pairs of
Expressions
(p. 84)
Journal Writing
(p. 85)
Practise Now
(p. 89)
Investigation –
Comparison between
Pairs of Expressions
(p. 84)
Journal Writing
(p. 85)
Week
(5 classes
× 45 min)
6
Chapter
Section
4.2 Expansion and
Simplification
of Linear
Expressions
(pp. 91 – 97)
Specific Instructional
Objectives (SIOs)
• Simplify linear
expressions
Syllabus Subject
Content
Expand product of
algebraic expressions
Activity
ICT
Main Text
(pp. 91 – 95)
Practise Now
(p. 92)
Practise Now
(p. 92)
Practise Now
(p. 92)
Practise Now
(p. 92)
Practise Now
(p. 95)
Additional
Resources
Reasoning,
Communication
and Connection
Practise Now
(p. 95)
Class Discussion
– The Distributive
Law
(p. 94)
Thinking Time
(p. 96)
Thinking Time
(p. 96)
13
7
7
7
4.3 Simplification
of Linear
Expressions
with Fractional
Coefficients
(pp. 98 – 99)
4.4 Factorisation
(pp. 100 – 102)
Miscellaneous
• Factorise algebraic
expressions by
extracting common
factors
Use brackets and extract Class Discussion
common factors
– Equivalent
Expressions
(p. 101)
Solutions for
Challenge
Yourself
1
1
Week
(5 classes
× 45 min)
8
Chapter
Section
5
Linear
Equations
and Simple
Inequalities
5.1 Linear
Equations
(pp. 109 – 119)
Specific Instructional
Objectives (SIOs)
Syllabus Subject
Content
• Explore the concepts Solve simple linear
equations in one
of equation and
unknown
inequality
• Solve linear equations
in one variable
• Solve fractional
equations that can
be reduced to linear
equations
Activity
ICT
Main Text
(pp. 110 – 113)
Practise Now
(p. 110)
Practise Now
(p. 110)
Practise Now
(p. 111)
Practise Now
(p. 111)
Practise Now
(p. 112)
Practise Now
(p. 112)
Practise Now
(p. 113)
Practise Now
(p. 113)
Additional
Resources
Reasoning,
Communication
and Connection
Main Text –
‘From Table 5.1,
discuss with your
classmate what a
linear equation is.’
(p. 109)
Journal Writing
(p. 113)
Thinking Time
(p. 115)
Journal Writing
(p. 113)
Thinking Time
(p. 115)
14
Solve fractional
equations with
numerical and linear
algebraic denominators
8
9
5.2 Formulae
(pp. 118 – 121)
5.3 Applications
of Linear
Equations in
Real-World
Contexts
(pp. 122 – 125)
Worked Example 2
(pp. 115 - 116)
Worked Example 3
(pp. 116 - 117)
• Evaluate an unknown
in a formula
• Formulate linear
equations to solve
word problems
Ex 5B Q 17(ii)
(p. 121)
Internet Resources
(p. 122)
Internet
Resources
(p. 122)
Week
(5 classes
× 45 min)
Chapter
9
Section
5.4 Simple
Inequalities
(pp. 125 – 129)
Specific Instructional
Objectives (SIOs)
• Solve simple linear
inequalities
Syllabus Subject
Content
Solve simple linear
inequalities
Activity
ICT
Additional
Resources
Journal Writing
(p. 128)
Reasoning,
Communication
and Connection
Investigation
– Properties of
Inequalities
(p. 126)
Ex 5D Q 7
(p. 129)
Investigation –
Properties of
Inequalities
(p. 126)
9
Miscellaneous
10
6
Functions and
Linear Graphs
6.1 Cartesian
Coordinates
(pp. 135 – 138)
Solutions for
Challenge
Yourself
• State the coordinates
of a point
• Plot a point in a
Cartesian plane
Demonstrate familiarity
with Cartesian
coordinates in two
dimensions
15
Class Discussion – Internet
Resources
Battleship Game
(p. 135)
(Two Players)
(p. 135)
Story Time
Internet Resources (p. 138)
(p. 135)
Class Discussion –
Ordered Pairs
(p. 136)
Class Discussion –
Ordered Pairs
(p. 136)
Journal Writing
(p. 137)
10
6.2 Functions
(pp. 139 – 145)
Investigation –
Function Machine
(pp. 139 – 142)
Internet
Resources
(p. 139)
Thinking Time
(p. 143)
Thinking Time
(p. 143)
10
6.3 Graphs
of Linear
Functions
(pp. 145 – 148)
• Draw the graph of a
linear function
Draw graphs from given Class Discussion –
data
Equation of a
Function (p. 147)
1
Thinking Time
(p. 147)
Class Discussion –
Equation of a
Function (p. 147)
Thinking Time
(p. 147)
1
Week
(5 classes
× 45 min)
Chapter
11
Section
6.4 Applications of
Linear Graphs
in Real-World
Contexts
(pp. 149 – 153)
Specific Instructional
Objectives (SIOs)
• Solve problems
involving linear
graphs in real-world
contexts
Syllabus Subject
Content
Interpret and use graphs
in practical situations
including travel graphs
and conversion graphs
Activity
ICT
Additional
Resources
Worked Example 2
(p. 149)
Thinking Time
(p. 151)
Ex 6C Q 3(ii)
(p. 153)
Thinking Time
(p. 151)
Interpret graphs of
linear functions
Miscellaneous
11
11
12
16
12
12
7
Number
Patterns
7.1 Number
Sequences
(pp. 159 – 161)
7.2 General Term
of a Number
Sequence
(pp. 161 – 165)
7.3 Number
Patterns
(pp. 165 – 167)
7.4 Number
Patterns in
Real-World
Contexts
(pp. 168 – 177)
Solutions for
Challenge
Yourself
Recognise simple
patterns from various
number sequences and
determine the next few
terms
Continue a given
number sequence
Class Discussion –
Number
Sequences (p. 159)
Determine the next few
terms and find a formula
for the general term of a
number sequence
Recognise patterns
in sequences and
relationships between
different sequences
Class Discussion –
Generalising
Simple Sequences
(p. 162)
Solve problems
involving number
squences and number
patterns
Generalise sequences
as simple algebraic
statements
Class Discussion –
The Triangular
Number Sequence
(p. 167)
Worked Example 5 Journal Writing
(p. 169)
(p. 170)
Investigation –
Fibonacci
Sequence
(pp. 168 – 169)
Journal Writing
(p. 169)
12
Miscellaneous
Reasoning,
Communication
and Connection
Solutions for
Challenge
Yourself
Week
(5 classes
× 45 min)
13
13
17
13
Chapter
Section
8
Percentage
8.1 Introduction to
Percentage
(pp. 185 – 192)
8.2 Percentage
Change and
Reverse
Percentage
(pp. 193 – 200)
Miscellaneous
Specific Instructional
Objectives (SIOs)
• Express a percentage
as a fraction and vice
versa
• Express a percentage
as a decimal and vice
versa
• Express one quantity
as a percentage of
another
• Compare two
quantities by
percentage
• Solve problems
involving percentage
change and reverse
percentage
Syllabus Subject
Content
Calculate a given
percentage of a quantity
Express one quantity as
a percentage of another
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Class Discussion –
Percentage in Real
Life (p. 185)
Class Discussion –
Percentage in Real
Life
(p. 185)
Class Discussion –
Expressing Two
Quantities in
Equivalent Forms
(p. 189)
Calculate percentage
increase or decrease
Thinking Time
(p. 198)
Carry out calculations
involving reverse
percentages
Internet Resources
(p. 198)
Just for Fun (p. 195)
Internet
Resources
(p. 198)
Attention (p. 195)
Thinking Time
(p. 198)
Solutions for
Challenge
Yourself
1
1
Week
(5 classes
× 45 min)
14
Chapter
Section
9.1 Ratio
9
(pp. 205 – 213)
Ratio, Rate,
Time and Speed
Specific Instructional
Objectives (SIOs)
• Find ratios involving
rational numbers
• Find ratios involving
three quantities
• Solve problems
involving ratio
Syllabus Subject
Content
Activity
Increase and decrease a Journal Writing
quantity by a given ratio (p. 208)
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Journal Writing
(p. 208)
Worked Example 5 Internet
Resources
(p. 210)
(p. 208)
Class Discussion –
Performance Task
Making Sense of
(p. 210)
the Relationship
between Ratios
and Fractions
(p. 207)
Worked Example 4
(p. 208)
Worked Example 6
(p. 211)
18
Internet Resources
(p. 208)
Investigation
– Golden Ratio
(p. 207)
Performance Task
(p. 210)
15
9.2 Rate
(pp. 214 – 217)
• Distinguish between
constant and average
rates
• Solve problems
involving rate
Use common measures
of rate
Investigation –
Average Pulse
Rate (p. 215)
Investigation –
Average Pulse Rate
(p. 215)
Thinking Time
(p. 216)
Thinking Time
(p. 216)
Week
(5 classes
× 45 min)
15
Chapter
Section
9.3 Time
(pp. 218 – 220)
Specific Instructional
Objectives (SIOs)
• Solve problems
involving time
Syllabus Subject
Content
Calculate times in terms
of the 24-hour and 12hour clock
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Main Text
(pp. 218 – 219)
Read clocks, dials and
timetables
16
9.4 Speed
(pp. 221 – 227)
• Discuss special types
of rates such as speed
and rate of rotation
• Solve problems
involving speed
Solve problems
Main Text
involving average speed (pp. 221 – 226)
Performance Task
(p. 225)
Just for Fun
(p. 221)
Internet
Resources
(p. 222)
Thinking Time
(p. 224)
Performance Task
(p. 225)
Thinking Time
(p. 224)
16
Miscellaneous
10
10.1 Points, Lines
Basic Geometry
and Planes
(pp. 233 – 234)
17
10.2 Angles
(pp. 235 – 266)
19
16
Solutions for
Challenge
Yourself
Use and interpret the
geometrical terms:
point; line; plane
• Identify various types
of angles
• Solve problems
involving angles on a
straight line, angles at
a point and vertically
opposite angles
Use and interpret the
geometrical terms:
parallel; perpendicular;
right angle, acute,
obtuse and reflex angles
Calculate unknown
angles and give simple
explanations using the
following geometrical
properties:
(a) angles at a point
(b) angles at a point on
a straight line and
intersecting straight
lines
Thinking Time
(p. 234)
Internet
Resources
(p. 234)
Thinking Time
(p. 234)
Just for Fun
(p. 235)
1
1
Week
(5 classes
× 45 min)
Chapter
Section
Specific Instructional
Objectives (SIOs)
10.3 Angles formed • Solve problems
involving angles
by Two Parallel
formed by two
Lines and a
parallel lines and
Transversal
a transversal, i.e.
(pp. 244 – 252)
corresponding angles,
alternate angles and
interior angles
17
Syllabus Subject
Content
Calculate unknown
angles and give simple
explanations using
angles formed within
parallel lines
Activity
Investigation –
Corresponding
Angles, Alternate
Angles and
Interior Angles
(pp. 245 – 246)
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Investigation –
Corresponding
Angles, Alternate
Angles and Interior
Angles
(pp. 245 – 246)
Investigation –
Corresponding
Angles, Alternate
Angles and
Interior Angles
(pp. 245 – 246)
Practise Now
(p. 246)
Ex 10B Q 1(b) – (c)
(p. 250)
Miscellaneous
17
18
11
Triangles,
Quadrilaterals
and Polygons
11.1 Triangles
(pp. 259 – 268)
Solutions for
Challenge
Yourself
20
• Identify different
types of triangles and
state their properties
• Solve problems
involving the
properties of triangles
Use and interpret the
geometrical terms:
interior and exterior
angles
Investigation –
Basic Properties of
a Triangle
(pp. 262 – 263)
Use and interpret
vocabulary of triangles
Calculate unknown
angles and give simple
explanations using
angle properties of
triangles
Thinking Time
(p. 260)
Investigation –
Basic Properties
of a Triangle
(pp. 262 – 263)
Thinking Time
(p. 260)
Investigation –
Basic Properties of a
Triangle
(pp. 262 – 263)
Week
(5 classes
× 45 min)
18
Chapter
Section
11.2 Quadrilaterals
(pp. 268 – 276)
Specific Instructional
Objectives (SIOs)
• Identify different
types of special
quadrilaterals and
state their properties
• Solve problems
involving the
properties of special
quadrilaterals
Syllabus Subject
Content
Calculate unknown
angles and give simple
explanations using
angle properties of
quadrilaterals
Activity
Investigation –
Properties
of Special
Quadrilaterals
(pp. 269)
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Investigation –
Properties
of Special
Quadrilaterals
(p. 269)
Thinking Time
(p. 271)
Class Discussion
– Naming of
Polygons (p. 277)
Main Text –
‘The shapes shown
in Fig. 11.13 are not
polygons. Why?’
(p. 276)
Just for Fun
(p. 271)
Use and interpret
vocabulary of
quadrilaterals
Thinking Time
(p. 271)
19
11.3 Polygons
(pp. 276 – 290)
21
• Identify different
types of polygons and
state their properties
• Solve problems
involving the
properties of
polygons
Use and interpret
vocabulary of polygons
Calculate unknown
angles and give simple
explanations using
angle properties of
regular and irregular
polygons
Investigation –
Sum of Interior
Angles of a
Polygon
(pp. 279 – 280)
Investigation –
Tessellation
(pp. 282 - 283)
Investigation –
Sum of Exterior
Angles of a
Pentagon
(pp. 284 – 285)
Main Text
(p. 285)
Internet
Resources
(p. 277)
Investigation –
Sum of Exterior
Angles of a
Pentagon
(pp. 284 – 285)
Thinking Time
(p. 277)
Journal Writing
(p. 278)
Investigation
– Sum of Exterior
Angles of a Pentagon
(p. 284 – 285)
1
1
Week
(5 classes
× 45 min)
Chapter
Section
Specific Instructional
Objectives (SIOs)
Syllabus Subject
Content
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Thinking Time
(p. 285)
Thinking Time
(p. 285)
Class Discussion –
Naming of
Polygons
(p. 277)
Ex 11C Q 19(ii) –
(iv)
(p. 290)
Thinking Time
(p. 277)
Investigation –
Properties of a
Regular Polygon
(p. 278)
Journal Writing
(p. 278)
Miscellaneous
19
22
20
20
20
12
Geometrical
Constructions
Solutions for
Challenge
Yourself
12.1 Introduction
to Geometrical
Constructions
(pp. 297 – 298)
12.2 Perpendicular • Construct
perpendicular
Bisectors and
bisectors and angle
Angle Bisectors
bisectors
(pp. 299 – 301)
• Apply properties
of perpendicular
bisectors and angle
bisectors
12.3 Construction
of Triangles
(pp. 301 – 306)
• Construct triangles
and solve related
problems
Measure lines and
angles
Construct angle
bisectors and
perpendicular bisectors
using a pair of
compasses as necessary
Construct a triangle,
given the three sides,
using a ruler and a pair
of compasses only
Investigation –
Property of a
Perpendicular
Bisector (p. 300)
Investigation –
Property of a
Perpendicular
Bisector (p. 300)
Investigation –
Property of an
Angle Bisector
(p. 301)
Internet
Resources
(p. 300)
Investigation –
Property of an
Angle Bisector
(p. 301)
Just for Fun (p. 303)
Ex 12A Q 15 – 16
(p. 306)
Week
(5 classes
× 45 min)
Chapter
Section
Specific Instructional
Objectives (SIOs)
12.4 Construction of • Construct
quadrilaterals
Quadrilaterals
and solve related
(pp. 306 – 311)
problems
20
Syllabus Subject
Content
Activity
ICT
Additional
Resources
Worked Example 6
(pp. 306 – 307)
Internet
Resources
(p. 309)
Construct other simple
geometrical figures
from given data, using
a ruler and protractor as
necessary
Reasoning,
Communication
and Connection
Practise Now 6 Q
1–2
(p. 307)
Practise Now 7 Q 2
(p. 308)
Ex 12B Q 1 – 3, 5,
8 – 9, 11(ii), 15
(pp. 310 – 311)
Miscellaneous
20
21
23
21
13
Perimeter and
Area of Plane
Figures
13.1 Conversion of
Units
(p. 317)
13.2 Perimeter and
Area of Basic
Plane Figures
(pp. 318 – 323)
Solutions for
Challenge
Yourself
• Convert between cm2
and m2
• Find the perimeter
and area of squares,
rectangles, triangles
and circles
• Solve problems
involving the
perimeter and area of
composite figures
Use current units of
mass, length and area
in practical situations
and express quantities
in terms of larger or
smaller units
Class Discussion –
International
System of Units
(p. 317)
Solve problems
involving the perimeter
and area of a rectangle
and triangle, and the
circumference and area
of a circle
Practise Now
(p. 318)
Class Discussion
–
International
System of Units
(p. 317)
1
1
Week
(5 classes
× 45 min)
22
Chapter
Section
13.3 Perimeter
and Area of
Parallelograms
(pp. 324 – 327)
Specific Instructional
Objectives (SIOs)
• Find the perimeter
and area of
parallelograms
• Solve problems
involving the
perimeter and area of
composite figures
Syllabus Subject
Content
Activity
Solve problems
involving the
perimeter and area of a
parallelogram
Investigation –
Formula for Area
of a Parallelogram
(p. 325)
ICT
Additional
Resources
Thinking Time
(p. 325)
Reasoning,
Communication
and Connection
Investigation –
Formula for Area of
a Parallelogram
(p. 325)
Thinking Time
(p. 325)
Practise Now
(p. 324)
Thinking Time
(p. 325)
22
13.4 Perimeter
and Area of
Trapeziums
(pp. 328 – 333)
• Find the perimeter
and area of
trapeziums
• Solve problems
involving the
perimeter and area of
composite figures
Solve problems
involving the perimeter
and area of a trapezium
Investigation –
Formula for Area of
a Trapezium
(pp. 328 – 329)
Investigation
– Formula
for Area of a
Trapezium
(pp. 328 – 329)
Thinking Time
(p. 329)
Practise Now
(p. 328)
24
Thinking Time
(p. 329)
22
Miscellaneous
Solutions for
Challenge
Yourself
Week
(5 classes
× 45 min)
23
23
23
25
24
Chapter
Section
14
Volume and
Surface Area
of Prisms and
Cylinders
14.1 Conversion of
Units
(pp. 339 – 340)
Specific Instructional
Objectives (SIOs)
• Convert between cm3
and m3
14.2 Nets
(pp. 341 – 342)
14.3 Volume and
Surface Area
of Cubes and
Cuboids
(pp. 343 – 347)
• Find the volume and
surface area of cubes
and cuboids
• Find the volume and
14.4 Volume and
surface area of prisms
Surface Area of
Prisms
(pp. 348 – 353)
Syllabus Subject
Content
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Class Discussion
–
Measurements in
Daily Lives
(p. 339)
Use current units of
volume and capacity
in practical situations
and express quantities
in terms of larger or
smaller units
Class Discussion –
Measurements in
Daily Lives
(p. 329)
Use and interpret nets
Investigation –
Cubes, Cuboids,
Prisms and
Cylinders
(pp. 341 – 342)
Solve problems
involving the surface
area and volume of a
cuboid
Class Discussion –
Surface Area
of Cubes and
Cuboids
(p. 345)
Class Discussion –
Surface Area of
Cubes and Cuboids
(p. 345)
Thinking Time
(p. 349)
Thinking Time
(p. 349)
Solve problems
involving the surface
area and volume of a
prism
Ex 14A Q 18(ii)
(p. 347)
Main Text –
‘Can you find a
relationship between
the volume of a
prism and the area of
its cross section?’
(p. 349)
1
1
Week
(5 classes
× 45 min)
24
Chapter
Section
14.5 Volume and
Surface Area of
Cylinders
(pp. 354 – 360)
Specific Instructional
Objectives (SIOs)
• Find the volume
and surface area of
cylinders
Syllabus Subject
Content
Solve problems
involving the surface
area and volume of a
cylinder
Activity
ICT
Additional
Resources
Thinking Time
(p. 354)
Thinking Time
(p. 354)
Investigation –
Comparison
between a
Cylinder and a
Prism
(p. 355)
Just for Fun
(p. 356)
Thinking Time
(p. 358)
Class Discussion –
Total Surface Area
of Other Types of
Cylinders
(p. 358)
Thinking Time
(p. 358)
Class Discussion –
Total Surface Area
of Other Types of
Cylinders
(p. 358)
24
26
24
14.6 Volume and
Surface Area
of Composite
Solids
(pp. 361 – 363)
Miscellaneous
• Solve problems
involving the volume
and surface area of
composite solids
Reasoning,
Communication
and Connection
Solve problems
involving the areas and
volumes of compound
shapes
Solutions for
Challenge
Yourself
Week
(5 classes
× 45 min)
25
25
Chapter
Section
15
Statistical Data
Handling
15.1 Introduction to
Statistics
(p. 369)
Specific Instructional
Objectives (SIOs)
15.2 Pictograms and • Collect, classify and
tabulate data
Bar Graphs
(pp. 369 – 374) • Construct and
interpret data from
pictograms and bar
graphs
Syllabus Subject
Content
Activity
ICT
Additional
Resources
Reasoning,
Communication
and Connection
Story Time
(p. 369)
Collect, classify and
tabulate statistical data
Main Text
(p. 370)
Read, interpret and
draw simple inferences
from tables and
statistical diagrams
Construct and interpret
pictograms and bar
charts
Main Text –
‘Two levels in the
school are selected
as the sample group
for the survey
conducted by the
school canteen
vendor. Are they
representative of
the entire school?
Explain your
answer.’
(p. 369)
27
Main Text –
‘If the canteen
vendor decides to
sell three types of
fruits to the students,
which three should
he choose? Explain
your answer.’
(p. 370)
Thinking Time
(p. 371)
Practise Now Q 2(d)
(ii), (e)
(p. 372)
Ex 15A Q 4(e), 5(iv),
6(iii)
(p. 374)
1
Thinking Time
(p. 371)
1
Week
(5 classes
× 45 min)
Chapter
Section
25
15.3 Pie Charts
(pp. 375 – 377)
25
15.4 Line Graphs
(pp. 377 – 379)
25
28
26
Specific Instructional
Objectives (SIOs)
• Construct and
interpret data from
pie charts
• Construct and
interpret data from
line graphs
• Evaluate the purposes
and appropriateness
of the use of different
statistical diagrams
Activity
Miscellaneous
Reasoning,
Communication
and Connection
Practise Now 1 Q
2(iii)
(p. 377)
Read, interpret and
draw simple inferences
from tables and
statistical diagrams
Worked Example
2 Q (ii)
(p. 378)
Worked Example 2
Q (iv)
(p. 378)
Class Discussion –
Comparison of
Various Statistical
Diagrams
(p. 379)
Practise Now 2 Q
(iv)
(p. 379)
Class Discussion –
Evaluation of
Statistics
(pp. 382 – 383)
Ex 15B Q 10 – 13
(pp. 385 – 386)
26
Additional
Resources
Main Text
(pp. 375 – 376)
Performance Task
(p. 381)
Explain why some
statistical information or
diagrams can lead to a
misinterpretation of data
ICT
Construct and interpret
pie charts
Main Text
(pp. 380 – 381)
15.5 Statistics in
Real-World
Contexts
(pp. 380 – 381)
15.6 Evaluation of
Statistics
(pp. 382 – 386)
Syllabus Subject
Content
Class Discussion –
Comparison of
Various Statistical
Diagrams
(p. 379)
Performance Task
(p. 381)
Internet
Resources
(p. 381)
Performance Task
(p. 381)
Class Discussion –
Evaluation of
Statistics
(pp. 382 – 383)
Class Discussion
–
Evaluation of
Statistics
(pp. 382 – 383)
Ex 15B Q 8(iv),
10 – 11, 12(iii), 13
(pp. 385 – 386)
Solutions for
Challenge
Yourself
Chapter 1 Primes, Highest Common Factor and Lowest Common Multiple
TEACHING NOTES
Suggested Approach
Students have learnt only whole numbers in primary school (they will only learn negative numbers and integers in Chapter 2).
They have also learnt how to classify whole numbers into two groups, i.e. odd and even numbers. Teachers can introduce prime
numbers as another way in which whole numbers can be classified (see Section 1.1). Traditionally, prime numbers apply to
positive integers only, but the syllabus specifies whole numbers, which is not an issue since 0 is not a prime number. Teachers
can also arouse students’ interest in this topic by bringing in real-life applications (see chapter opener on page 2 of the textbook).
Section 1.1:
Prime Numbers
Teachers can build upon prerequisites, namely, factors, to introduce prime numbers by classifying whole numbers
according to the number of factors they have (see Investigation: Classification of Whole Numbers). Since the
concept of 0 may not be easily understood, it is dealt with separately in the last question of the investigation.
Regardless of whether 0 is classified in the same group as 1 or in a new fourth group, 0 and 1 are neither prime
nor composite. Teachers are to take note that 1 is not a prime number ‘by choice’, or else the uniqueness of prime
factorisation will fail (see Information on page 8 of the textbook). Also, 0 is not a composite number because it
cannot be expressed as a product of prime factors unlike e.g. 40 = 23 × 5.
To make practice more interesting, a game is designed in Question 2 of Practise Now 1. Teachers can also tell
students about the largest known prime number (there is no largest prime number since there are infinitely many
primes) and an important real-life application of prime numbers in the encryption of computer data (see chapter
opener and Investigation: Interesting Facts about Prime Numbers) in order to arouse their interest in this topic.
Section 1.2:
Square Roots and Cube Roots
Teachers can build upon what students have learnt about squares, square roots, cubes and cube roots in primary
school. Perfect squares are also called square numbers and perfect cubes are also called cube numbers. Perfect
numbers are not the same as perfect squares or perfect cubes. Perfect numbers are numbers which are equal
to the sum of its proper factors, where proper factors are factors that are less than the number itself,
e.g. 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are the only two perfect numbers less than 100 (perfect numbers
are not in the syllabus). After students have learnt negative numbers in Chapter 2, there is a need to revisit square
roots and cube roots to discuss negative square roots and negative cube roots (see page 40 of the textbook).
Teachers can impress upon students that the square root symbol √
refers to the positive square root only.
A common debate among some teachers is whether 0 is a perfect square. There is an argument that 0 is not a
perfect square because 0 can multiply by any number (not necessarily itself) to give 0. However, this is not the
definition of a perfect square. Since 0 is equal to 0 multiplied by itself, then 0 (the first 0, not the second 0, in this
sentence) is a perfect square. Compare this with why 4 is a perfect square (4 is equal to the integer 2 multiplied
by itself). Similarly, 0 is a perfect cube.
Section 1.3:
Highest Common Factor and Lowest Common Multiple
Teachers can build upon prerequisites, namely, common factors and common multiples, to develop the concepts
of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) respectively (HCF and LCM are no
longer in the primary school syllabus although some primary school teachers teach their students HCF and LCM).
Since the listing method (see pages 15 and 18 of the textbook) is not an efficient method to find the HCF and the
LCM of two or more numbers, there is a need to learn the prime factorisation method and the ladder method (see
Methods 1 and 2 in Worked Example 9 and in Worked Example 11). However, when using the ladder method to
find the LCM of two or three numbers (see Worked Examples 11 and 12), we stop dividing when there are no
common prime factors between any two numbers. The GCE O-level examinations emphasise on the use of the
prime factorisation method.
29
1
Challenge Yourself
Some of the questions (e.g. Questions 4 and 5) are not easy for average students while others (e.g. Question 2)
should be manageable if teachers guide them as follows:
Question 2: The figure consists of 3 identical squares but students are to divide it into 4 identical parts. Teachers
can guide students by asking them to find the LCM of 3 and 4, which is 12. Thus students have to divide the
figure into 12 equal parts before trying to regroup 3 equal parts to form each of the 4 identical parts.
Questions 4 and 5: Teachers can get students to try different numerical examples before looking for a pattern
in order to generalise. In both questions, it is important that students know whether m and n are co-primes,
m
i.e. HCF(m, n) = 1. If m and n are not co-primes, they can be built from the ‘basic block’ of HCF( m , n ) and
n
HCF( m , n ) , which are co-primes.
1
30
WORKED SOLUTIONS
Investigation (Sieve of Eratosthenes)
Investigation (Classification of Whole Numbers)
1.
1.
Number
Working
Factors
1
1 is divisible by 1 only .
1
2
2=1×2
1, 2
3
3=1×3
1, 3
4
4=1×4=2×2
1, 2, 4
5
5=1×5
1, 5
6
6=1×6=2×3
1, 2, 3, 6
7
7=1×7
1, 7
8
8=1×8=2×4
1, 2, 4, 8
9
9=1×9=3×3
1, 3, 9
10
10 = 1 × 10 = 2 × 5
1, 2, 5, 10
11
11 = 1 × 11
1, 11
12
12 = 1 × 12 = 2 × 6 = 3 × 4
1, 2, 3, 4, 6, 12
13
13 = 1 × 13
1, 13
14
14 = 1 × 14 = 2 × 7
1, 2, 7, 14
15
15 = 1 × 15 = 3 × 5
1, 3, 5, 15
16
16 = 1 × 16 = 2 × 8 = 4 × 4
1, 2, 4, 8, 16
17
17 = 1 × 17
1, 17
18
18 = 1 × 18 = 2 × 9 = 3 × 6
1, 2, 3, 6, 9, 18
19
19 = 1 × 19
1, 19
20
20 = 1 × 20 = 2 × 10 = 4 × 5
1, 2, 4, 5, 10, 20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
2. (a)
(b)
(c)
(d)
The smallest prime number is 2.
The largest prime number less than or equal to 100 is 97.
There are 25 prime numbers which are less than or equal to 100.
No, not every odd number is a prime number, e.g. the number
9 is an odd number but it is a composite number.
(e) No, not every even number is a composite number, e.g. the
number 0 is an even number but it is neither a prime nor a
composite number.
(f) For a number greater than 5, if its last digit is 0, 2, 4, 6 or 8, then
the number is a multiple of 2, thus it is a composite number; if
its last digit is 0 or 5, then the number is a multiple of 5, thus it
is a composite number. Hence, for a prime number greater than
5, its last digit can only be 1, 3, 7 or 9.
Journal Writing (Page 5)
Table 1.1
2. Group A: 1
Group B: 2, 3, 5, 7, 11, 13, 17, 19
Group C: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20
3. 0 is divisible by 1, 2, 3, 4, …
0 has an infinite number of factors.
1. Yes, the product of two prime numbers can be an odd number, e.g.
the product of the two prime numbers 3 and 5 is the odd number 15.
2. Yes, the product of two prime numbers can be an even number, e.g.
the product of the two prime numbers 2 and 3 is the even number 6.
3. No, the product of two prime numbers P1 and P2 cannot be a prime
number since P1P2 has at least 3 distinct factors, i.e. 1, P1 and P1 P2 .
Thinking Time (Page 4)
Investigation (Interesting Facts about Prime Numbers)
1. A prime number is a whole number that has exactly 2 different
factors, 1 and itself.
A composite number is a whole number that has more than 2 different
factors. A composite number has a finite number of factors.
Since 0 has an infinite number of factors, it is neither a prime nor a
composite number.
Since 1 has exactly 1 factor, it is also neither a prime nor a composite
number.
2. No, I do not agree with Michael. Consider the numbers 0 and 1.
They are neither prime numbers nor composite numbers.
The 1 000 000th prime number is 15 485 863.
The last digit of the largest known prime number is 1.
Thinking Time (Page 8)
The index notation is useful for writing complicated and repetitive
expressions in a more compact form, e.g. we can write
3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 as 38 .
31
1
Thinking Time (Page 12)
Practise Now 3
If no brackets are used in pressing the sequence of calculator keys in
Worked Example 7, the value obtained would be 60.0416 (to 4 d.p.).
The mathematical statement that would have been evaluated is
1. 126 = 2 × 32 × 7
2. 539 = 72 × 11
82 +
50
–
73
3
Practise Now 4
63 .
1. 784 = 2 × 2 × 2 × 2 × 7 × 7
=
(2 × 2 × 7) × (2 × 2 × 7)
=
(2 × 2 × 7)2
Practise Now 1
1. 537 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 537 is 5 + 3 + 7 = 15 which is divisible
by 3, therefore 537 is divisible by 3 (divisibility test for 3).
\ 537 is a composite number.
784 = 2 × 2 × 7
= 28
Alternatively,
784 = 2 × 2 × 2 × 2 × 7 × 7
= 24 × 72
\
59 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 59 is 5 + 9 = 14 which is not divisible
by 3, then 59 is not divisible by 3.
The last digit of 59 is neither 0 nor 5, so 59 is not divisible by 5.
A calculator may be used to test whether 59 is divisible by prime
numbers more than 5.
Since 59 is not divisible by any prime numbers less than 59, then
59 is a prime number.
49
183
147
93
121
236
201
261
150
11
131
5
89
291
117
153
End
57
0
61
192
231
27
1
111
100
149
17
103
43
7
127
51
53
83
33
32
105
29
71
37
7056 = 2 4 × 32 × 7 2
= 22 × 3 × 7
= 84
2.
Practise Now 5
2.
135
784 = 2 4 × 7 2
= 22 × 7
= 28
\
1. 2744 = 2 × 2 × 2 × 7 × 7 × 7
=
(2 × 7) × (2 × 7) × (2 × 7)
=
(2 × 7)3
\
3
\
3
2744 = 2 × 7
= 14
Alternatively,
2744 = 2 × 2 × 2 × 7 × 7 × 7
= 23 × 73
2.
Start
Practise Now 2
1. Since 31 is a prime number, then 1 and 31 are its only two factors.
It does not matter whether p or q is 1 or 31 as we only want to find
the value of p + q .
\ p + q = 1 + 31 = 32
2. Since n × (n + 28) is a prime number, then n and n + 28 are its
only two factors.
Since 1 has to be one of its two factors, then n = 1 .
\ n × (n + 28) = 1 × (1 + 28)
= 1 × 29
= 29
3
3
9261 = 33 × 7 3
=3×7
= 21
3
Practise Now 6
123 ≈ 121
= 11
3
123 ≈ 3 125
=5
Practise Now 7
1. (a) 232 +
(b)
1
2744 = 2 3 × 7 3
=2×7
= 14
32
2025 – 73 = 231
32 ×
53 –
3
20
2013
= 0.3582 (to 4 d.p.)
Practise Now 11
2. Length of each side of poster = 987
= 31.42 cm (to 2 d.p.)
Perimeter of poster = 4 × 31 .42
= 125.7 cm (to 1 d.p.)
1. Method 1:
24 = 23 × 3
90 = 2 × 32 × 5
LCM of 24 and 90 = 23 × 32 × 5
= 360
Method 2:
Practise Now 8
2013 = 44.9 (to 1 d.p.), so the largest prime number less than or equal
to 2013 is 43 .
2013 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 2013 is 2 + 1 + 3 = 6 which is divisible by
3, therefore 2013 is divisible by 3 (divisibility test for 3).
\ 2013 is a composite number.
2 24, 90
3 12, 45
4, 15
HCF of 24 and 90 = 2 × 3 × 4 × 15
= 360
2. Smallest whole number that is divisible by both 120 and 126
=
LCM of 120 and 126
= 23 × 32 × 5 × 7
= 2520
3. 6 = 2 × 3
24 = 23 × 3
Smallest value of n = 23
=8
2017 = 44.9 (to 1 d.p.), so the largest prime number less than or equal
to 2017 is 43 .
Since 2017 is not divisible by any of the prime numbers 2, 3, 5, 7, …,
43, then 2017 is a prime number.
Practise Now 9
1. Method 1:
56 = 23 × 7
84 = 22 × 3 × 7
HCF of 56 and 84 = 22 × 7
= 28
Method 2:
Practise Now 12
9 = 32
30 = 2 × 3 × 5
108 = 22 × 33
LCM of 9, 30 and 108 = 22 × 33 × 5
= 540
2 56, 84
2 28, 42
Practise Now 13
7 14, 21
2, 3
1. 15 = 3 × 5
16 = 24
36 = 22 × 32
LCM of 15, 16 and 36 = 24 × 32 × 5
= 720
720 minutes = 12 hours
\ The three bells will next toll together at 2.00 a.m.
2. (i) 140 = 22 × 5 × 7
168 = 23 × 3 × 7
210 = 2 × 3 × 5 × 7
HCF of 140, 168 and 210 = 2 × 7
= 14
Greatest possible length of each of the smaller pieces of rope
=
14 cm
(ii) Number of smallest pieces of rope he can get altogether
HCF of 56 and 84 = 2 × 2 × 7
= 28
2. 28 = 22 × 7
70 = 2 × 5 × 7
Largest whole number which is a factor of both 28 and 70
=
HCF of 28 and 70
=2×7
= 14
3. Greatest whole number that will divide both 504 and 588 exactly
=
HCF of 504 and 588
= 22 × 3 × 7
= 84
Practise Now 10
90 = 2 × 32 × 5
135 = 33 × 5
270 = 2 × 33 × 5
HCF of 90, 135 and 270 = 32 × 5
= 45
140
168
210
+
+
14
14
14
= 10 + 12 + 15
= 37
=
33
1
Exercise 1A
Alternatively,
576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
= 26 × 32
1. (a) 87 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 87 is 8 + 7 = 15 which is divisible
by 3, therefore 87 is divisible by 3 (divisibility test for 3).
\ 87 is a composite number.
(b) 67 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 67 is 6 + 7 = 13 which is not
divisible by 3, then 67 is not divisible by 3.
The last digit of 67 is neither 0 nor 5, so 67 is not divisible
by 5 .
A calculator may be used to test whether 67 is divisible by prime
numbers more than 5.
Since 67 is not divisible by any prime numbers less than 67,
then 67 is a prime number.
(c) 73 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 73 is 7 + 3 = 10 which is not
divisible by 3, then 73 is not divisible by 3.
The last digit of 73 is neither 0 nor 5, so 73 is not divisible
by 5 .
A calculator may be used to test whether 73 is divisible by prime
numbers more than 5.
Since 73 is not divisible by any prime numbers less than 73,
then 73 is a prime number.
(d) 91 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 91 is 9 + 1 = 10 which is not
divisible by 3, then 91 is not divisible by 3.
The last digit of 91 is neither 0 nor 5, so 91 is not divisible
by 5 .
A calculator may be used to test whether 91 is divisible by
prime numbers more than 5.
Since 91 is divisible by 7, therefore 91 is a composite number.
2. (a) 72 = 23 × 32
(b) 187 = 11 × 17
4
(c) 336 = 2 × 3 × 7
(d) 630 = 2 × 32 × 5 × 7
3. (a) 1764 = 2 × 2 × 3 × 3 × 7 × 7
=
(2 × 3 × 7) × (2 × 3 × 7)
=
(2 × 3 × 7)2
576 = 2 6 × 32
= 23 × 3
= 24
(c) 3375 = 3 × 3 × 3 × 5 × 5 × 5
=
(3 × 5) × (3 × 5) × (3 × 5)
=
(3 × 5)3
\
\
3375 = 3 × 5
= 15
Alternatively,
3375 = 3 × 3 × 3 × 5 × 5 × 5
= 33 × 53
3375 = 3 33 × 5 3
=3×5
= 15
(d) 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
=
(2 × 2 × 3) × (2 × 2 × 3) × (2 × 2 × 3)
=
(2 × 2 × 3)3
4.
5.
1764 = 2 × 3 × 7
= 42
Alternatively,
1764 = 2 × 2 × 3 × 3 × 7 × 7
= 22 × 32 × 72
\
1764 = 2 2 × 32 × 7 2
=2×3×7
= 42
(b) 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
=
(2 × 2 × 2 × 3) × (2 × 2 × 2 × 3)
=
(2 × 2 × 2 × 3)2
3
\
3
\
3
1728 = 3 2 6 × 33
= 22 × 3
= 12
9801 = 34 × 112
= 32 × 11
= 99
3
3 6
3
21 952 = 2 × 7
2
=2 ×7
= 28
6. (a)
66 ≈ 64
=8
(b)
80 ≈ 81
=9
(c)
3
218 ≈ 3 216
=6
(d)
3
730 ≈ 3 729
=9
7. (a) 72 –
(b)
576 = 2 × 2 × 2 × 3
= 24
1
\
1728 = 2 × 2 × 3
= 12
Alternatively,
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 26 × 33
\
\
3
(c)
34
361 + 213 = 9291
555 + 5 2
2 ×
3
3
43 +
222
3
= 1.0024 (to 4 d.p.)
4913 = 9
Exercise 1B
8. Length of each side of photo frame = 250
= 15.81 cm (to 2 d.p.)
Perimeter of photo frame = 4 × 15 .81
= 63.2 cm (to 1 d.p.)
1. (a) 12 = 22 × 3
30 = 2 × 3 × 5
HCF of 12 and 30 = 2 × 3
=6
(b) 84 = 22 × 3 × 7
156 = 22 × 3 × 13
HCF of 84 and 156 = 22 × 3
= 12
(c) 15 = 3 × 5
60 = 22 × 3 × 5
75 = 3 × 52
HCF of 15, 60 and 75 = 3 × 5
= 15
(d) 77 = 7 × 11
91 = 7 × 13
143 = 11 × 13
HCF of 77, 91 and 143 = 1
2. (a) 24 = 23 × 3
30 = 2 × 3 × 5
LCM of 24 and 30 = 23 × 3 × 5
= 120
(b) 42 = 2 × 3 × 7
462 = 2 × 3 × 7 × 11
LCM of 42 and 462 = 2 × 3 × 7 × 11
= 462
(c) 12 = 22 × 3
18 = 2 × 32
81 = 34
LCM of 12, 18 and 81 = 22 × 34
= 324
(d) 63 = 32 × 7
80 = 24 × 5
102 = 2 × 3 × 17
LCM of 63, 80 and 102 = 24 × 32 × 5 × 7 × 17
= 85 680
3. 42 = 2 × 3 × 7
98 = 2 × 72
Largest whole number which is a factor of both 42 and 98
=
HCF of 42 and 98
=2×7
= 14
4. Greatest whole number that will divide both 792 and 990 exactly
=
HCF of 792 and 990
= 2 × 32 × 11
= 198
5. Smallest whole number that is divisble by both 176 and 342
=
LCM of 176 and 342
= 24 × 32 × 11 × 19
= 30 096
9. Length of each side of box = 3 2197
= 13 cm
Area of one side of box = 132
= 169 cm2
10. (a)
667 = 25.8 (to 1 d.p.), so the largest prime number less than or
equal to 667 is 23 .
.
667 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 667 is 6 + 6 + 7 = 19 which is not
divisible by 3, then 667 is not divisible by 3.
The last digit of 667 is neither 0 nor 5, so 667 is not divisible
by 5 .
A calculator may be used to test whether 667 is divisible by
prime numbers more than 5.
Since 667 is divisible by 23, therefore 667 is a composite
number.
(b)
677 = 26.0 (to 1 d.p.), so the largest prime number less
than or equal to 677 is 23 .
Since 677 is not divisible by any of the prime numbers
2, 3, 5, 7, …, 23, then 677 is a prime number.
(c)
2021 = 45.0 (to 1 d.p.), so the largest prime number less than
or equal to 2021 is 43 .
2021 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 2021 is 2 + 0 + 2 + 1 = 5 which
is not divisible by 3, then 2021 is not divisible by 3.
The last digit of 2021 is neither 0 nor 5, so 2021 is not divisible
by 5 .
A calculator may be used to test whether 2021 is divisible by
prime numbers more than 5.
Since 2021 is divisible by 43, therefore 2021 is a composite
number.
(d)
2027 = 45.0 (to 1 d.p.), so the largest prime number less
than or equal to 2027 is 43 .
Since 2027 is not divisible by any of the prime numbers
2, 3, 5, 7, …, 43, then 2027 is a prime number.
11. Since 37 is a prime number, then 1 and 37 are its only two factors.
It does not matter whether p or q is 1 or 37 as we only want to find
the value of p + q .
\ p + q = 1 + 37 = 38
12. Since n × (n + 42) is a prime number, then n and n + 42 are its only
two factors.
Since 1 has to be one of its two factors, then n = 1 .
\ n × (n + 42) = 1 × (1 + 42)
= 1 × 43
= 43
35
1
(c) True. If 18 is a multiple of a whole number n, then 18 = nk for
18
some whole number k. Thus
= k is a whole number, i.e. 18
n
is divisible by n .
(d) True. Since m is a multiple of p, by the same reasoning as in
(c), then m is divisible by p. Similarly, m is divisible by q .
12. (i) 64 = 26
48 = 24 × 3
HCF of 64 and 48 = 24
= 16
Length of each square = 16 cm
64
48
(ii) Number of squares that can be cut altogether =
×
16
16
=4×3
= 12
13. (i) Let the number of boys in the class be n .
Then 15 × n = 3 × 5 × n is divisible by 21 = 3 × 7 .
Thus the possible values of n are multiples of 7.
Hence, n = 14 since 14 + 20 = 34 students is the only possibility
where the number of students in the class is between 30 and 40.
\ Number of students in the class = 34
(ii) Number of chocolate bars their form teacher receive
6. 15 = 3 × 5
45 = 32 × 5
Smallest value of n = 32
=9
7. (i) 171 = 32 × 19
63 = 32 × 7
27 = 33
HCF of 171, 63 and 27 = 32
=9
Largest number of gift bags that can be packed = 9
(ii) Number of pens in a gift bag = 171 ÷ 9
= 19
Number of pencils in a gift bag = 63 ÷ 9
=7
Number of erasers in a gift bag = 27 ÷ 9
=3
8. (i) 60 = 22 × 3 × 5
80 = 24 × 5
LCM of 60 and 80 = 24 × 3 × 5
= 240
It will take 240 s for both cars to be back at the starting point at
the same time.
(ii) 5 × 240 s = 1200 s
=
20 minutes
It will take 20 minutes for the faster car to be 5 laps ahead of
the slower car.
9. (a) True.
If 6 is a factor of a whole number n, then n = 6k for some whole
number k .
We have n = 6k = 2(3k). Since 3k is a whole number, then 2 is
a factor of n .
We also have n = 6k = 3(2k). Since 2k is a whole number, then
3 is a factor of n .
(b) True. Since 2 and 3 are distinct prime factors of the whole
number, then the prime factorisation of the whole number will
contain both of these prime factors.
(c) False, e.g. 2 and 4 are factors of 4, but 8 is not a factor of 4.
(d) True. If f is a factor of n, then n = fk for some whole number
n
k. Thus
= k is a whole number. Since n can be written as
f
n
n
a product of the whole numbers
and f, then
is a factor
f
f
of n .
(e) True. Since h is a factor of both p and q, then both p and q are
divisible by h .
10. 9 = 32
12 = 22 × 3
252 = 22 × 32 × 7
Possible values of n = 7, 3 × 7 or 32 × 7
= 7, 21 or 63
11. (a) True. If 6 is a multiple of a whole number n, then 6 = nk for
some whole number k . We have 12 = 2nk = n(2k). Since 2k is
a whole number, then 12 is a multiple of n .
(b) False, e.g. 12 is a multiple of 4, but 6 is not a multiple of 4.
1
15 × 14
21
= 10
14. (i) 126 = 2 × 32 × 7
108 = 22 × 33
HCF of 126 and 108 = 2 × 33
= 18
Length of each square = 18 cm
Least number of square patterns that could be formed on the
sheet of paper
=
126
108
×
18
18
=7×6
= 42
(ii) To fit the sheet of paper perfectly, the patterns can be rectangular,
triangular or trapeziums with two right angles, etc.
15. (i) 45 = 32 × 5
42 = 2 × 3 × 7
LCM of 45 and 42 = 2 × 32 × 5 × 7
= 630
Number of patterns needed to form the smallest square
=
630
630
×
45
42
= 14 × 15
= 210
(ii) 630 mm = 0.63 m
Area of smallest square that can be formed = 0.632
= 0.3969 m2
By trial and error,
Area of largest square that can be formed
= 0 .3969 × 22
=
1.5876 m2 < 1.6 m2
=
\ Length of largest square that can be formed = 1.5876
= 1.26 m
36
5. 6 = 2 × 3
Review Exercise 1
12 = 22 × 3
660 = 22 × 3 × 5 × 11
Possible values of n = 5 × 11, 2 × 5 × 11, 3 × 5 × 11,
22 × 5 × 11, 2 × 3 × 5 × 11
or 22 × 3 × 5 × 11
= 55, 110, 165, 220, 330 or 660
6. (i) 108 = 22 × 33
81 = 34
54 = 2 × 33
HCF of 108, 81 and 54 = 33
= 27
Largest number of baskets that can be packed = 27
(ii) Number of stalks of roses in a basket = 108 ÷ 27
=4
Number of stalks of lilies in a basket = 81 ÷ 27
=3
Number of stalks of orchids in a basket = 54 ÷ 27
=2
7. Time taken for Khairul to run 1 round = 360 s
= 6 minutes
Time taken for Devi to cycle 1 round = 4 ÷ 2
= 2 minutes
18 = 2 × 32
6=2×3
2=2
LCM of 18, 6 and 2 = 2 × 32
= 18
All three of them will next meet at 6.03 p.m.
8. (i) By counting, they will next have the same day off on
7 May.
(ii) 4 = 22
6=2×3
LCM of 4 and 6 = 22 × 3
= 12
Subsequently, they will have the same day off every 12 days.
1. (a) 1225 = 5 × 5 × 7 × 7
=
(5 × 7) × (5 × 7)
=
(5 × 7)2
1225 = 5 × 7
= 35
Alternatively,
1225 = 5 × 5 × 7 × 7
= 52 × 72
\
\
1225 = 5 2 × 7 2
=5×7
= 35
(b) 13 824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
=
(2 × 2 × 2 × 3) × (2 × 2 × 2 × 3) × (2 × 2 × 2 × 3)
=
(2 × 2 × 2 × 3)3
\
3
\
3
13 824 = 2 × 2 × 2 × 3
= 24
Alternatively,
13 824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 29 × 33
2. (a)
(b)
3. (a)
13 824 = 3 2 9 × 33
= 23 × 3
= 24
63 ≈ 64
=8
3
345 ≈ 3 343
=7
753 = 27.4 (to 1 d.p.), so the largest prime number
less than or equal to 753 is 23 .
753 is an odd number, so it is not divisible by 2.
Since the sum of the digits of 753 is 7 + 5 + 3 = 15 which is
divisible by 3, therefore 753 is divisible by 3 (divisibility test
for 3).
\ 753 is a composite number.
(b)
757 = 27.5 (to 1 d.p.), so the largest prime number less
Challenge Yourself
than or equal to 757 is 23 .
Since 757 is not divisible by any of the prime numbers
2, 3, 5, 7, …, 23, then 757 is a prime number.
4. (i) Greatest whole number that will divide both 840 and 8316
exactly
=
HCF of 840 and 8316
= 22 × 3 × 7
= 84
(ii) Smallest whole number that is divisible by both 840 and 8316
=
LCM of 840 and 8316
= 23 × 33 × 5 × 7 × 11
= 83 160
1. (i) The six adjacent numbers are 11, 12, 1, 2, 3, 4.
(ii) The other six numbers are 5, 6, 7, 8, 9, 10.
Make a list where the sum of each of the pairs of numbers is a
prime number:
• 4 + 7 = 11; 4 + 9 = 13
• 5 + 6 = 11; 5 + 8 = 13
• 6 + 5 = 11; 6 + 7 = 13; 6 + 11 = 17
• 7 + 4 = 11; 7 + 6 = 13; 7 + 10 = 17
• 8 + 5 = 13; 8 + 9 = 17; 8 + 11 = 19
• 9 + 4 = 13; 9 + 8 = 17; 9 + 10 = 19
• 10 + 7 = 17; 10 + 9 = 19
• 11 + 6 = 17; 11 + 8 = 19
37
1
(iii) Yes, the result in (ii) can be generalised for any two numbers.
Proof:
Consider two numbers x and y .
Then x = HCF × p,
— (1)
y = HCF × q,
— (2)
where the HCF of p and q is 1 .
(1) × q: x × q = HCF × p × q — (3)
(2) × p: y × p = HCF × p × q — (4)
(3) × (4): x × y × p × q = HCF × p × q × HCF × p × q
x × y = HCF × HCF × p × q
Since the HCF of p and q is 1, we cannot take out a factor greater
than 1 in the product p × q, thus HCF × p × q = LCM.
\ x × y = HCF × LCM
(iv) No, the result in (ii) cannot be generalised for any three numbers.
For example, consider the numbers 10, 20 and 25.
10 = 2 × 5
20 = 22 × 5
25 = 52
HCF of 10, 20 and 25 = 5
LCM of 10, 20 and 25 = 22 × 52
= 100
HCF × LCM = 5 × 100
= 500
≠ 10 × 20 × 25
4. Number of squares passed through by a diagonal of a m-by-n
rectangle
= m + n – HCF(m, n)
Notice that 5 and 10 are the only numbers that can be adjacent
to two numbers only:
• 5 can be adjacent to 6 and 8 only, i.e. 6 – 5 – 8 or 8 – 5 – 6;
• 10 can be adjacent to 7 and 9 only, i.e. 7 – 10 – 9 or
9 – 10 – 7 .
Since 4 and 11 are adjacent to another number on one side, then
• the only two possibilities for the other side of 4 are 7 and
9;
• the only two possibilities for the other side of 11 are 6 and
8 .
With the above information, we can narrow down the possible
arrangements to only two ways:
11 12
11 12
1
2
8
3
5
6
4
7
10
9
1
2
6
3
5
8
4
9
10
7
2. LCM of 3 and 4 = 12
\ We divide the 3 identical squares into 12 equal parts.
Thus we have:
5. (i) Fraction of a sausage each person gets =
12
18
2
=
3
\ Least number of cuts required = 12
(ii) Least number of cuts required = n – HCF(m, n)
3. (i) 120 = 23 × 3 × 5
126 = 2 × 32 × 7
HCF of 120 and 126 = 2 × 3
=6
LCM of 120 and 126 = 23 × 32 × 5 × 7
= 2520
(ii) HCF × LCM = 6 × 2520
= 15 120
= 120 × 126 (Shown)
120 = 23 × 3 × 5
126 = 2 × 32 × 7
To obtain the HCF of 120 and 126, we choose the power of
each of the common prime factors with the smaller index,
i .e . 2 and 3 .
On the other hand, to obtain the LCM of 120 and 126, we choose
the power of each of the common prime factors with the higher
index, i.e. 23 and 32, and the remaining factors, i.e. 5 and 7 .
Since each term in the prime factorisation of 120 and 126 is
used to find either their HCF or their LCM, the product of the
HCF and LCM of 120 and 126 is equal to the product of 120
and 126 .
1
38
Chapter 2 Integers, Rational Numbers and Real Numbers
TEACHING NOTES
Suggested Approach
Although the concept of negative numbers is new to most students as they have not learnt this in primary school, they do
encounter negative numbers in their daily lives, e.g. in weather forecasts. Therefore, teachers can get students to discuss
examples of the use of negative numbers in the real world to bring across the idea of negative numbers (see Class Discussion:
Uses of Negative Numbers in the Real World). The learning experiences in the new syllabus specify the use of algebra discs.
In this chapter, only number discs (or counters) showing the numbers 1 and –1 are needed. Since many Secondary 1 students
are still in the concrete operational stage (according to Piaget), the use of algebra discs can help them to learn the concepts
more easily. However, there is still a need to guide students to move from the ‘concrete’ to the ‘abstract’, partly because they
cannot use algebra discs in examinations, and partly because they cannot use algebra discs to add or subtract large negative
integers, and decimals (see Section 2.2).
Section 2.1:
Negative Numbers
Teachers should teach students to read the negative number –2 as negative 2, not minus 2 (‘negative’ is a state
while ‘minus’ is an operation). For example, if you have $5 and you owe your friend $2, how much do you
have left? Since nothing is mentioned about you returning money to your friend, you have $5 left. Thus $2 is
a state of owing money. However, if you return $2 to your friend, you have $5 + (–$2) = $5 – $2 = $3 left,
i.e. 5 minus 2 is an operation of returning money.
Students should also learn about the absolute value of a negative number (see page 29 of the textbook) because
they will need it in Section 2.2.
In primary school, students have only learnt the terms ‘less than’ and ‘more than’, so there is a need to teach
them how to use the symbols ‘<’ and ‘>’ when comparing numbers. It is not necessary to teach them about ‘less
than or equal to’ and ‘more than or equal to’ now.
Section 2.2:
Addition and Subtraction involving Negative Numbers
Algebra discs cannot be used to add or subtract large negative integers, and decimals, so there is a need to help
students consolidate what they have learnt in the class discussions on pages 33 and 35 of the textbook by moving
away from the ‘concrete’ to the following two key ‘abstract’ concepts:
Key Concept 1: Adding a negative number is the same as subtracting the absolute value of the number,
e.g. 5 + (−2) = 5 − 2.
Key Concept 2: Subtracting a negative number is the same as adding the absolute value of the number,
e.g. 5 − (−2) = 5 + 2.
To make the key concepts less abstract, numerical examples are used. Do not use algebra now because students are
still unfamiliar with algebra even though they have learnt some basic algebra in primary school. Avoid teaching
students ‘− × − = +’ now because the idea behind 5 − (−2) is subtraction, not multiplication. To make practice
more interesting, a puzzle is designed on page 36 of the textbook.
Section 2.3:
Multiplication and Division involving Negative Numbers
The idea of flipping over a disc to obtain the negative of a number, e.g. –(−3) = 3, is important in teaching
multiplication involving negative numbers. Since algebra discs cannot be used to teach division involving negative
numbers, another method is adopted (see page 40 of the textbook).
There is a need to revisit square roots and cube roots in this section to discuss negative square roots and negative
cube roots (see page 40 of the textbook). Teachers can impress upon students that the square root symbol √
refers to the positive square root only.
39
1
Section 2.4:
1
Rational Numbers and Real Numbers
Traditionally, real numbers are classified as either rational or irrational numbers. Another way to classify real
numbers is according to whether their decimal forms are terminating, recurring, or non-recurring (see page 50 of
the textbook). If teachers show students the first million digits of p (see page 51 of the textbook), many students
may be surprised that p has so many digits! This suggests that students do not know that p has an infinite number
of decimal places. Teachers may wish to celebrate Pi Day with students on March 14 by talking about p or singing
the Pi song.
40
WORKED SOLUTIONS
Thinking Time (Page 28)
Class Discussion (Uses of Negative Numbers in the Real
World)
•
•
•
•
•
–5
One of the most common uses of negative numbers is in the
measurement of temperature, where negative numbers are used
to show temperatures below the freezing point of water, i.e. 0 °C.
Absolute zero, defined as 0 Kelvin, is the theoretical lowest possible
temperature. 0 Kelvin is equivalent to a temperature of –273.15 °C,
therefore the theoretical lowest possible temperature is 273.15 °C
below 0 °C.
The elevation of a location commonly refers to its height with
reference to Earth’s sea level and can be represented by a positive
or a negative number. Given a point with an elevation of –200 m,
we can deduce that the point is 200 m below sea level. The lowest
elevation on Earth that is on dry land is the Dead Sea shore in
Asia with an elevation of – 423 m, i.e. the shore of the Dead Sea is
423 m below sea level.
Negative numbers are also used to tell time zones, which are based
on Greenwich Mean Time (GMT). A country which is in the time
zone of GMT –2 means that the time in that country is 2 hours behind
the GMT. For example, Honolulu, Hawaii is in the time zone of
GMT –10, while Liverpool, United Kingdom is in the time zone of
GMT 0, therefore when it is 10 a.m. in Liverpool, it is 12 midnight
in Honolulu.
Latitude and longitude are a set of coordinates that allow for the
specification of a geographical location on the Earth’s surface and
can be represented by positive and/or negative numbers. The latitude
of a point is determined with reference to the equatorial plane; the
North Pole has a latitude of +90°, which means that it is 90° north of
the equator while the South Pole has a latitude of –90°, which means
that it is 90° south of the equator. The longitude of a point gives
its east-west position relative to the Prime Meridian (longitude 0°);
a location with a longitude of +50° means that it is 50° east of the
Prime Meridian while a location with a longitude of –50° means that
it is 50° west of the Prime Meridian. The latitude and longitude of
Rio Grande, Mexico are approximately –32° and –52° respectively,
which means that it is 32° south of the equator and 52° west of the
Prime Meridian.
The use of negative numbers can also be seen in scoring systems, such
as in golf. Each hole has a par score, which indicates the number of
strokes required and a golfer’s score for that hole is calculated based
on the number of strokes played. A score of +3 on a hole shows that
the golfer played three strokes above par, while a score of –3 on a
hole shows that the golfer played three strokes under par.
–4
–3
–2
–1
0
1
2
3
4
5
(a) Since –3 is on the left of 2, we say ‘–3 is less than 2’ and we write
‘–3 < 2’.
(b) Since –3 is on the right of –5, we say ‘–3 is more than –5’ and we
write ‘–3 > –5’.
Class Discussion (Addition involving Negative Numbers)
Part I
1. (a)
(b)
2. (a)
(b)
3. (a)
(b)
7 + (–3) = 4
6 + (– 4) = 2
(–7) + 3 = – 4
(–6) + 4 = –2
(–7) + (–3) = –10
(– 6) + (– 4) = –10
Note:
• If we add a positive number and a negative number,
(i) we take the difference between the absolute values of the two
numbers, and
(ii) the sign of the answer follows the sign of the number with the
greater absolute value,
e.g. 5 + (–2) = 3 and (–5) + 2 = –3.
• If we add two negative numbers,
(i) we take the sum of the absolute values of the two numbers, and
(ii) the answer is negative,
e.g. (–5) + (–2) = –7.
Class Discussion (Subtraction involving Negative Numbers)
Part I
1. (a) 7 – (–3) = 7 + 3
=
10
(b) 6 – (– 4) = 6 + 4
=
10
2. (a) (–7) – 3 = (–7) + (–3)
=
–10
(b) (– 6) – 4 = (– 6) + (– 4)
=
–10
3. (a) (–7) – (–3) = (–7) + 3
=
–4
(b) (– 4) – (– 6) = (– 4) + 6
=
2
4. (a) 3 – 7 = 3 + (–7)
=
–4
(b) 4 – 6 = 4 + (– 6)
=
–2
Teachers may wish to note that the list is not exhaustive.
41
1
Note:
• If we take the difference of a positive number and a negative number,
(i) we add the absolute values of the two numbers, and
(ii) the sign of the answer follows the sign of the number with the
greater absolute value,
e.g. 5 – (–2) = 7 and (–5) – 2 = –7.
• If we take the difference of two negative numbers or two positive
numbers,
(i) we take the difference between the absolute values of the two
numbers, and
(ii) the sign of the answer depends on whether the first number is
greater than or smaller than the second number,
e.g. (–5) – (–2) = –3 but (–2) – (–5) = 3;
2 – 5 = –3 but 5 – 2 = 3.
Investigation (Terminating, Recurring and Non-Recurring
Decimals)
Group 1
1
= –3.125
8
63
= 0.984 375
64
–
123
= –1.242 424 242 – 3 5 = –1.709 975 947
99
22
= 3.142 857 143
7
8
–√ 5
–3
–2
–3
–4
3
–1
–
Note: In general,
p = 3.141 592 654
123
99
0
64
1
4
2
1
√2
p
3
4
22
7
Fig. 2.7
positive number × negative number = negative number,
negative number × positive number = negative number,
negative number × negative number = positive number.
5.
9 63 1 1
123
22
1
, ,–
, p, ,
,
, – 3 5 , –3
4 64
99
7
8
2 3
Investigation (Some Interesting Facts about the Irrational
Number p)
1. The 1 000 000th digit of p is 1.
2. The 5 000 000 000 000th digit of p is 2.
3. Lu Chao, a graduate student from China, took 24 hours and 4 minutes
to recite p to 67 890 decimal places in 2005.
Thinking Time (Page 41)
It is not possible to obtain the square roots of a negative number,
e.g. ± –16 , because the square of any number is more than or equal to 0.
Teachers may wish to take this opportunity to highlight to
higher-ability students that even though ± –16 is not defined in the
set of real numbers, it is defined in the set of complex numbers.
Practise Now (Page 27)
1. (i) 2013, 6
(ii) –5, –17
Thinking Time (Page 49)
(iii) 2013, 1.666,
2
m
(a) Any integer m can be expressed in the form
, e.g. 2 =
and
1
1
–3
–3 =
.
1
3
,6
4
1
2
(iv) –5, – , –3.8, –17, –
2
3
2. (a) – 43.6 °C
(b) – 423 m
(c) –1
(d) –$10 000
0
In particular, the integer 0 can be expressed in the form , where
n
n is any integer except 0.
1
= 0.707 106 781 2
22
.
7
2. For each of the numbers in Group 2, some digits after the decimal
point repeat themselves indefinitely. The numbers in Group 2 are
rational numbers.
3. For each of the numbers in Group 1, the digits after the decimal
point terminate. The numbers in Group 1 are rational numbers.
For each of the numbers in Group 3, the digits after the decimal
point do not repeat but they continue indefinitely. The numbers in
Group 3 are irrational numbers.
9
1 63
4.
1
3
1 × (– 4) = – 4
2 × (– 4) = –8
3 × (– 4) = –12
(–1) × 4 = – 4
(–2) × 4 = –8
(–3) × 4 = –12
(–1) × (– 4) = 4
(–2) × (– 4) = 8
(–3) × (– 4) = 12
e.g. 0.5 =
2
1. Based on the calculator values, p is not equal to
Part I
(b) There is more than one way to express a decimal in the form
1
Table 2.1
Class Discussion (Multiplication involving Negative
Numbers)
1. (a)
(b)
(c)
2. (a)
(b)
(c)
3. (a)
(b)
(c)
Group 3
1
= 0.333 333 333 3
3
9
= 2.25
4
–3
Group 2
a
,
b
2
3
1
2
3
1
=
= … and 0.333… =
=
= …
6
9
2
4
6
3
42
Practise Now (Page 39)
Practise Now (Page 29)
1. (b) <
2.
–5
–5
(c) >
–1
–3.8
–4
–3
\ –5, –3.8, –1
1
2
–2
(d) >
0
–1
0
3
4 1.666
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
4
2
3
4
5
1
3
, 0, , 1.666, 4
2
4
Practise Now (Page 33)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
9 + (–2) = 7
–7 + 4 = –3
3 + (–5) = –2
– 6 + (–8) = –14
27 + (–13) = 14
–25 + 11 = –14
14 + (–16) = –2
–12 + (–15) = –27
2 × (–6) = –12
–5 × 4 = –20
–1 × (–8) = 8
–3 × (–7) = 21
–(–10) = 10
–9(–2) = 18
15 × (–2) = –30
–3 × 12 = –36
– 4 × (–10) = 40
–2(–100) = 200
Practise Now (Page 40)
(a) –8 ÷ 2 = – 4
(b) 15 ÷ (–3) = –5
(c) –8 ÷ (– 4) = 2
–6
= –2
3
20
(e)
= –4
–5
–12
(f)
=4
–3
(d)
Practise Now 1
Temperature in the morning = –8 °C + 2 °C
= – 6 °C
Practise Now 3a
Practise Now (Page 35)
(a) Square roots of 64 = ± 64
= ±8
(a) 9 – (–2) = 9 + 2
=
11
(b) –7 – 4 = –11
(c) –3 – (–5) = –3 + 5
=
2
(d) –8 – (–6) = –8 + 6
=
–2
(e) 4 – 8 = – 4
(f) 27 – (–13) = 27 + 13
=
40
(g) –25 – 11 = –36
(h) –14 – (–16) = –14 + 16
=
2
(i) –15 – (–12) = –15 + 12
=
–3
(j) 10 – 28 = –18
(b) Negative square root of 9 = – 9
= –3
36 = 6
(c)
Practise Now 3b
(a) (–3)3 = –27
(c)
3
216 = 6
(b) (– 4)3 = – 64
(d)
3
–8 = –2
Practise Now 4a
(a) –3 × (15 – 7 + 2) = –3 × (8 + 2)
= –3 × 10
= –30
(b) 43 – 7 × [16 – ( 3 64 – 5)] = 64 – 7 × [16 – (4 – 5)]
= 64 – 7 × [16 – (–1)]
= 64 – 7 × (16 + 1)
= 64 – 7 × 17
= 64 – 119
= –55
Practise Now 2
1. Point A shows –5 °C.
Point B shows 23 °C.
Difference in temperature = 23 °C – (–5 °C)
= 23 °C + 5 °C
= 28 °C
2. Altitude at D = –165 m
Difference in altitude = 314 m – (–165 m)
= 314 m + 165 m
= 479 m
Practise Now 4b
(a) –3 × (15 – 7 + 2) = –30
(b) 43 – 7 × [16 – ( 3 64 – 5)] = –55
43
1
Practise Now 5
(b) 1
(a) 7
3
1
1
3
+ –3 = 7 – 3
5
2
2
5
5
6
=7
–3
10
10
5
6
10
=
6 + 10 + 10 – 3 10
9
=3
10
(b) –2
5
2
3
3
5
2
+ – – – = –2 –
+
6
3
4
4
6
3
11
5
2
=–
–
+
4
6
3
33 10
8
=–
–
+
12
12
12
–33 – 10 + 8
=
12
– 43 + 8
=
12
Practise Now 7b
Practise Now 5
(a) 7
–35
12
11
= –2
12
(a) 2
2
93
8
2
9
×
=
×
41
3
4
1 3
=
2×3
=
6
1
5
25
5
÷
=
÷
6
2
6
2
(a) 5
4
1
7
÷ –2 = –1
5
4
8
(b) 1
6
1
3
9
× + – = 1
5
2
4
40
Practise Now 8
1
25
2
=
×
51
3 6
(a)
13 . 5 6
2.4
5 424
+ 27 12
3 2. 5 4 4
×
5
=
3
2
=
1
3
\ 13.56 × 2.4 = 32.544
Practise Now 7a
(b)
14
4
1
21
÷ –2 =
÷ –
5
4
4
5
3
5
21
= × –
4
14
2
13 7 . 8
× 0 . 35
6 890
+ 41 34
4 8. 2 3 0
\ 137.8 × 0.35 = 48.23
15
=–
8
7
= –1
8
1
2
9
×
=6
3
4
Practise Now 7a
5
(a) 5
5
3
11
+ – – – 2 = –2
4
12
6
3
Practise Now 6
Practise Now 6
(b) 4
9
3
1
+ –3 = 3
10
2
5
(b) –2
=
(a) 2
6
1 7 6 1
3
× + – =
×
–
5
2 4 5 2
4
7 12 5
= ×
–
4 10 10
7
7
= ×
4 10
49
=
40
9
=1
40
44
(b) 4
1
5
2
÷
=1
6
2
3
Practise Now 9
Exercise 2A
0.92
(a) 0.92 ÷ 0.4 =
0.4
1. (i) 10 001, 4
(ii) –12, –2017
1
5
1
(iii) , 4.33, 10 001, 4
(iv) – 0.3, – , –12, –1 , –2017
5
7
2
2. (a) 30 m above sea level
(b) –35
(c) An anticlockwise rotation of 30°
(d) A speed of 45 km/h of a car travelling West
3. (a) <
(b) <
(c) <
(d) >
(e) <
(f) >
4. (a)
9.2
=
4
2.3
4 ) 9.2
–8
12
–12
0
\ 0.92 ÷ 0.4 = 2.3
1.845
(b) 1.845 ÷ 0.15 =
0.15
184.5
=
15
12 . 3
15 ) 1 8 4 . 5
–15
34
–30
45
–45
0
–4
– 2.8
–4
–3
(b)
2
0
–2
0
–1
–
–2
1
10
–1
2
1
2
5
6
3
2
5
4
4
0
1
2
3
4
–3
–2
–1
0
1
2
3
–3
–2
–1
0
1
2
3
–2
6
– 0.55
(c)
\ 1.845 ÷ 0.15 = 12.3
–4
Practise Now 10
(d) 1
(a) 32 – (–1.6) = 32 + 1.6
=
33.6
(b) 1.3 + (–3.5) = –2.2
–0.23
0.12
1.2 –0.23
(c)
×
=
×
0.6
0.6
0.4
4
0
1
–13
–20
–0.23
= 1 3 ×
6 2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
10
0
20
40
80
100 120 140 160 180 200 220 240
\ –13, –3, 23, 30, 230
–
(b)
3
20
–10
– 0.65
60
30
5
45
– 27
230
23
–3
= –0.115
45
4.5
(d) – 0.32 ×
– 0.65
– 0.65 = – 0.32 ×
–27
–2.7
0.03
2
5. (a)
–0.23
= 0.3 ×
0.6
= – 0.09 ×
4
15
150
31
= 0.15 – 0.65
= – 0.5
–20
0
20
40
60
80
100 120 140 160
– 0.5
\ –10, – 0.5, –
Practise Now 11
π × 0.7 2
= 0.583 (to 3 d.p.)
3
3
2.4 + 1
10
6. (a)
(b)
7. (a)
(c)
45
–273.15 °C
–86 m
>
<
3
, 15, 150
20
(b) >
(d) >
1
8. (a)
–
1
3
0.11
1
1
3
8
2.5
4.
– 0.5
0
0.5
1
1.5
2.5
2
– 0.2
(b)
1
0
3
1 2 3 4
(c)
5
7
5
6 7
2
3
2
3
(d)
0
9
8
1
2
1
2
15
17
19
9 10 11 12 13 14 15 16 17 18 19 20
5
4
13
11
5
5.
7
6
7
8
9
10
4
3
4
5
Exercise 2B
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
3. (a)
(b)
(c)
(d)
(e)
(f)
(g)
6 + (–2) = 4
–5 + 8 = 3
4 + (–10) = – 6
–1 + (–7) = –8
9 + (–3) = 6
–11 + (–5) = –16
–10 + 2 = –8
1 + (–8) = –7
–(–7) = 7
5 – (–3) = 5 + 3
=
8
– 4 – 7 = –11
–8 – (–2) = –8 + 2
=
–6
–1 – (–10) = –1 + 10
=
9
6 – 9 = –3
–8 – 3 = –11
2 – (–7) = 2 + 7
=
9
4 + (–7) – (–3) = 4 + (–7) + 3
=
0
–3 – 5 + (–9) = –17
1 – 8 – (–8) = 1 – 8 + 8
=
1
–2 + (–1) – 6 = –9
8 – (–9) + 1 = 8 + 9 + 1
=
18
–5 + (–3) + (–2) = –10
6 + (–5) – (–8) = 6 + (–5) + 8
=
9
1
6.
7.
8.
9.
(h) 2 – (–7) – 8 = 2 + 7 – 8
=
1
(a) 23 + (–11) = 12
(b) –19 + 12 = –7
(c) 17 + (–29) = –12
(d) –21 + (–25) = – 46
(e) –13 + 18 = 5
(f) –24 + (–13) = –37
(g) 16 + (–27) = –11
(h) –26 + 14 = –12
(a) 22 – (–13) = 22 + 13
=
35
(b) –14 – 16 = –30
(c) –19 – (–11) = –19 + 11
=
–8
(d) –18 – (–22) = –18 + 22
=
4
(e) 17 – 23 = –6
(f) –20 – 15 = –35
(g) 12 – (–17) = 12 + 17
=
29
(h) –21 – 17 = –38
Temperature in the morning = –11 °C + 7 °C
= – 4 °C
Point A shows –7 °C.
Point B shows 16 °C.
Difference in temperature = 16 °C – (–7 °C)
= 16 °C + 7 °C
= 23 °C
Altitude of town = –51 m
Difference in altitude = 138 m – (–51 m)
= 138 m + 51 m
= 189 m
(i) Difference between –2 and 3 = 3 – (–2)
=3+2
=5
(ii) The timeline for BC and AD does not have a zero while the
number line has a zero.
(iii) There are 4 years between 2 BC and 3 AD.
Note: As there is no zero on the timeline, we cannot use
3 – (–2) to find the difference between 2 BC and 3 AD. In fact,
the calculation should be 3 – (–2) – 1, provided one year is in
BC and the other year is in AD. If both are in BC, or both are
in AD, the calculation is the same as that in (i).
(iv) A real-life example is the floors in a building, i.e. we can consider
B1 (Basement 1) as –1 but there is no floor with the number 0.
Exercise 2C
1. (a)
(b)
(c)
(d)
46
3 × (–9) = –27
–8 × 4 = –32
–7 × (–5) = 35
–1 × (–6) = 6
(e)
(f)
2. (a)
(b)
(c)
(g) 3 × (–3)2 – (7 – 2)2 = 3 × (–3)2 – 52
= 3 × 9 – 25
= 27 – 25
=2
(h) 5 × [3 × (–2) – 10] = 5 × (–6 – 10)
= 5 × (–16)
= –80
2
(i) –12 ÷ [2 – (–2)] = –12 ÷ [4 – (–2)]
= –12 ÷ (4 + 2)
= –12 ÷ 6
= –2
–2(–7) = 14
–6 × 0 = 0
–21 ÷ 7 = –3
16 ÷ (–2) = –8
–8 ÷ (–2) = 4
–14
= –7
2
15
(e)
= –3
–5
–18
(f)
=6
–3
(d)
3. (a) Square roots of 81 = ± 81
= ±9
(j)
=
(b) Square roots of 16 = ± 16
= ±4
(c) Square roots of 25 = ± 25
= ±5
8. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(d) Square roots of 100 = ± 100
= ±10
81 = 9
4. (a)
4 =2
(b)
(c) – 9 = –3
(d)
5. (a)
(b)
(c)
(d)
6. (a)
Not possible
(–2)3 = –8
(–5)3 = –125
(–10)3 = –1000
(– 6)3 = –216
3
27 = 3
(b) – 64 = – 4
3
10 – (–6)
10 + 6
= 16
=4
–55 + (–10) – 10 = –75
–12 – [(–8) – (–2)] + 3 = –3
–100 + (– 45) + (–5) + 20 = –130
–2 + 3 × 15 = 43
(–5 – 2) × (–3) = 21
–25 × (– 4) ÷ (–12 + 32) = 5
3 × (–3)2 – (7 – 2)2 = 2
5 × [3 × (–2) – 10] = –80
–12 ÷ [22 – (–2)] = –2
(j)
10 – 3 × (–2) = 4
9. (a) 24 × (–2) × 5 ÷ (– 6) = – 48 × 5 ÷ (–6)
= –240 ÷ (–6)
= 40
(b) 4 × 10 – 13 × (–5) = 40 – (–65)
= 40 + 65
= 105
(c) (16 – 24) – (57 – 77) ÷ (–2) = (–8) – (–20) ÷ (–2)
= (–8) – 10
= –18
(d) 160 ÷ (– 40) – 20 ÷ (–5) = – 4 – (– 4)
= –4 + 4
=0
(e) [(12 – 18) ÷ 3 – 5] × (– 4) = (–6 ÷ 3 – 5) × (– 4)
= (–2 – 5) × (– 4)
= (–7) × (– 4)
= 28
(f) {[(–15 + 5) × 2 + 8] – 32 ÷ 8} – (–7)
=
{[(–10) × 2 + 8] – 32 ÷ 8} – (–7)
=
[(–20 + 8) – 32 ÷ 8] – (–7)
=
[(–12) – 32 ÷ 8] – (–7)
=
[(–12) – 4] – (–7)
=
(–16) – (–7)
=
(–16) + 7
=
–9
3
(c)
10 – 3 × (–2) =
8 =2
(d) –216 = –6
7. (a) –55 + (–10) – 10 = –65 – 10
= –75
(b) –12 – [(–8) – (–2)] + 3 = –12 – [(–8) + 2] + 3
= –12 – (–6) + 3
= –12 + 6 + 3
= –6 + 3
= –3
(c) –100 + (– 45) + (–5) + 20 = –145 + (–5) + 20
= –150 + 20
= –130
(d) –2 + 3 × 15 = –2 + 45
=
43
(e) (–5 – 2) × (–3) = (–7) × (–3)
= 21
(f) –25 × (– 4) ÷ (–12 + 32) = –25 × (– 4) ÷ 20
= 100 ÷ 20
=5
3
47
1
(g) (5 – 2)3 × 2 + [– 4 + (–7)] ÷ (–2 + 4)2 = 33 × 2 + (–11) ÷ 22
= 27 × 2 + (–11) ÷ 4
3
= 54 + –2
4
= 51
(b) 3
1
2
8
=
2 + 8 + 8 – 8
7
=
2
8
1
1
2
5
(c) 5 – 4 = 5
–4
5
2
10
10
2
5
10
+
=
–4
4+
10
10 10
7
=
10
2
1
1
2
(d) –3 + – 4 = –3 – 4
3
6
6
3
1
4
= –3 – 4
6
6
5
= –7
6
3
1
1
2. (a) – + – = –1
2
4
4
1
4
(h) {–10 – [12 + (–3)2] + 33} ÷ (–3)
=
[–10 – (12 + 9) + 33] ÷ (–3)
=
(–10 – 21 + 33) ÷ (–3)
=
(–10 – 21 + 27) ÷ (–3)
=
(–31 + 27) ÷ (–3)
=
(– 4) ÷ (–3)
10. (a)
(b)
(c)
(d)
(e)
(f)
1
=
1
3
24 × (–2) × 5 ÷ (–6) = 40
4 × 10 – 13 × (–5) = 105
(16 – 24) – (57 – 77) ÷ (–2) = –18
160 ÷ (– 40) – 20 ÷ (–5) = 0
[(12 – 18) ÷ 3 – 5] × (– 4) = 28
{[(–15 + 5) × 2 + 8] – 32 ÷ 8} – (–7) = –9
(g) (5 – 2)3 × 2 + [– 4 + (–7)] ÷ (–2 + 4)2 = 51
(h) {–10 – [12 + (–3)2] + 33} ÷ (–3) = 1
11.
3
1 1
1
1
+ –
= 3 –
8 4
8
4
1
2
=
3 –
8
8
1
4
1
3
–2 × (–6.5) – [–2 × (–3) + 8 × (–2) – 8 × 2] + 5 2
3
=
–2 × (–6.5) – [6 + (–16) – 16] + 5 2
(b) 3
1
1
7
+ – =2
4
8
8
(c) 5
1
1
7
–4 =
5
2
10
2
1
5
+ – 4 = –7
3
6
6
5
15
41
5
3. (a)
×
=
31
2
2 8
(d) –3
3
–2 × (–6.5) – (–10 – 16) + 5 2
=
3
–2 × (–6.5) – (–26) + 5 2
=
1
=
2
2
3 –2 × (–6.5) – (–26) + 25
=
3
=
13 – (–26) + 25
(b) 2
3
=
13 + 26 + 25
3
15
×
=
5
26
3
=
39 + 25
3
=
64
=
4
(c)
15
5
÷
4
2
Exercise 2D
1. (a) –
1 3
1
3
= – –
+ –
2 4
2
4
2
3
=
– –
4
4
–2 – 3
=
4
–5
=
4
1
= –1
4
1
(d) 1
7
4
÷
9
3
15
13
×
5
26 2
1
3
=
2
1
=
1
2
3
15
=
2 4
3
=
2
1
=
1
2
16
=
÷
9
4
16
=
3 9
4
=
3
1
=
1
3
48
3
1
×
21
51
4
3
×
31
41
8
5. (a)
5
7. (a)
4
1
=2
3
2
15
1
×
=1
26
2
5
1
÷
=1
2
2
4
1
÷
=1
3
3
×
\ 14.72 × 1.2 = 17.664
(b)
3
64
8
× – =–
5
81
15
3
5
\ 130.4 × 0.15 = 19.56
31
4
× –
10 5
5 15
2
=
–
25
7
3
49
1
3
(c) – 6 ×
=–
×
8
14
8
14 2
21
=
–
16
5
=
–1
16
4
÷ – 10 =
15
3
2
(c)
×
(d)
11
(d) –2
1
22
5
1
2
×4 =–
×
51
2
5
1 2
(e) –1
1
3
5
3
÷
=– ÷
4
8
4
8
5
82
=
–
×
3
1 4
10
=
–
3
1
=
–3
3
2
8
8
5
(f) – ÷ –1 = – ÷ –
3
3
9
9
2.7
3 ) 8.1
–6
21
–2 1
0
(e)
(f)
\ 0.81 ÷ 0.3 = 2.7
1.32
(b) 1.32 ÷ 0.12 =
0.12
(d)
\ 0.27 × 0.08 = 0.0216
0 . 25
× 1 . 96
150
225
+ 25
0 . 4900
8.1
=
3
31
8
=
–
× –
9
5
3
8
=
15
64
3
3
6. (a)
× – = –1
8
15
5
(c)
0 . 27
0 . 08
0. 0 2 1 6
\ 0.25 × 1.96 = 0.49
0.81
8. (a) 0.81 ÷ 0.3 =
0.3
=
–11
(b)
13 0 . 4
0 . 15
6 520
+13 0 4
1 9. 5 6 0
×
1
= –1
(b)
14 . 72
1.2
2 944
+ 14 72
17.664
×
132
=
12
=
11
4
2
÷ – 10 = –
15
25
3
1
3
5
–6 ×
= –1
8
14
16
1
2
–2 × 4 = –11
2
5
1
3
1
–1 ÷
= –3
4
8
3
8
8
2
– ÷ –1 =
3
9
15
(c) 3.426 ÷ 0.06 =
3.426
0.06
15
8
3
(b) 2
5
15
(c)
4
7
(d) 1
9
4. (a)
342.6
=
6
6
5 7.1
) 3 4 2.6
–30
42
–42
6
– 6
0
49
\ 3.426 ÷ 0.06 = 57.1
1
4.35
1.5
4
1
4
1
(d) – 4 + –3 + – = – 4 – 3 –
3
8
3
8
1
1
= –4 – 3 – 1
8
3
3
8
= –4 – 3
–1
24
24
11
= –8
24
7
1
1
1
1
7
(e) – + 2 + – = – + 2 –
5
4
5
4
2
2
1
9
7
=– +
–
5
4
2
4
45
70
=–
+
–
20
20
20
– 4 + 45 – 70
=
20
–29
=
20
9
= –1
20
(d) 4.35 ÷ 1.5 =
43.5
=
15
2.9
15 ) 4 3.5
–30
135
–135
0
\ 4.35 ÷ 1.5 = 2.9
9. (a) 4.3 – (–3.9) = 4.3 + 3.9
=
8.2
(b) 2.8 + (–1.5) = 1.3
(c) –5.9 + 2.7 = –3.2
(d) – 6.7 – 5.4 = –12.1
10. (a) –
(b) 6
(c) 4
1
8
1
8
1
1
– –2 –
= – +2 –
5
2
5
4
2
4
8
9
1
=– +
–
5
4
2
32
45
10
=–
+
–
20
20
20
–32 + 45 – 10
=
20
3
=
20
11. (a) –
1
3
1
1
3
1
– – + – 4 = 6 +
–4
4
5
5
4
10
10
31
3
41
=
+
–
5
4
10
124
15
82
=
+
–
20
20
20
124 + 15 – 82
=
20
57
=
20
17
=2
20
1
3
1
17
– – + –4 = 2
10
4
5
20
(c) 4
1
4
2
6
+ –6 3 – – = –1
21
7
7
(e) –
12. (a) –
7
1
1
9
+ 2 + – = –1
2
5
4
20
28 5
2
5
5
28
× –
+
+1 = – × –
15 3
3
7
7
15
=–
28 25
5
+
× –
15 15
7
=–
5
3
× –
7
15
=–
1
=
1
5
× –
7
51
1
7
1 1
(b) – 1 – – 1 ÷ 1 – 1 = – + ÷
4 3
4
3
4
3
3
4
= – +
12 12
90 – 133 + 4
21
=
–39
21
–13
=
7
6
= –1
7
1
(b) 6
1
11
4
(d) – 4 + – 3 + – = –8
24
8
3
4
1
2
2
1
4
+ –6 – – = 4 – 6 +
21
3
7
7
3
21
30
19
4
=
–
+
7
3
21
90
133
4
=
–
+
21
21
21
=
1
8
1
3
– –2 –
=
4
5
2
20
=
=
1 1
4 – 3
÷ 3 – 4
12 12
1
1
÷ –
12
12
1
1
× (– 12 )
1 12
= –1
50
(c) 10 –
3
1
15
1
× ÷ 4 + –
2
4
8
2
=
10 –
3 9
15
1
× ÷ + –
2 2
4
8
=
10 –
13
21
15
1
×
×
+ –
9 3
4
8
12
(b) – 1 – – 1 ÷ 1 – 1 = –1
4
3
4 3
5
2
(e)
2
× –0.16
= 0.3
1.2
–0.16
= 13 ×
12 4
= – 0.04
0.027
1.4
1.4 2.7
=
(b)
×
×
0.03
–0.18
–0.18
3
× 1.4
= 0.9
–0.18
–1.3
(c) – 0.42 × 0.8
1
4
1
1
1
4
1
+
× – =
+
×
9
3
9
3
41
2
1
1
=
+
3
9
3
1
=
+
9
9
4
=
9
14
= 19 ×
– 18 2
= –7
–13
– 0.62 = – 0.42 × 8 – 0.62
0.02
–13
= – 0.16 ×
– 0.62
81
= 0.26 – 0.62
= – 0.36
27
270
(d) (– 0.2)3 ×
+ 0.105 = (– 0.2)3 ×
+ 0.105
1.6
16
2
1
1
4
4
+
× – =
3
9
9
2
3
1
1
1
(f) ×
– 2 = –5
3
10
2
15
–0.16 1.5
–0.16
=
14. (a) 0.15 ×
×
1.2
1.2
5
0.5
2
(e)
2
3
3
3
1
3
(d) – + – = –1
16
4
4
2
15
1
1
+ –
×
8
4
31
1
5
+ –
=
10 –
4
8
5
1
–
=
10 –
8
4
5
2
=
10 –
–
8
8
8
5
2
=
9 + 8 – 8 – 8
1
=
9
8
3
2
1
3
3
3
1
9
(d) – + – =
–
+ –
2
4
4
4
8
16
1
9
3
=
–
–
8
16
4
2
9
12
=
–
–
16
16
16
2 – 9 – 12
=
16
–19
=
16
3
= –1
16
=
10 –
1
1
3
15
1
× ÷ 4 + – =9
2
4
2
8
8
(c) 10 –
0.001
= – 0.008 ×
2
= – 0.135 + 0.105
= – 0.03
1 3
7
1
1
3
–2 = ×
–
(f) ×
3 2
15
15 3
2
35
1
3
–
= ×
15 15
2
2
1
π + 52
15. (a)
= 16.934 (to 3 d.p.)
–2.1
2
3
34
= × –
2
15
17
9 34
×–
15 5
2 4
51
=–
10
1
= –5
10
5
1
13. (a) – × – 28 + 1 2 =
7
7
3
15
=
3
135
270
+ 0.105
16 8 1
2
(b) –
π2 +
7–
3
(c)
(d)
51
2
3
4
= –2.085 (to 3 d.p.)
14 2 + 19 2
= –5.842 (to 3 d.p.)
π – 4.55
1
4.6 2 + 8.32 – 6
2
2 × 4.6 – 8.3
2
= 7.288 (to 3 d.p.)
1
(b) 2.36 – 10.58 = –8.22
–11.97 – (–2.69) = –11.97 + 2.69
= –9.28
\ 2.36 – 10.58 > –11.97 – (–2.69)
(c) –5 × 1.5 = –7.5
50 ÷ (–8) = – 6.25
\ –5 × 1.5 < 50 ÷ (–8)
3
1
1
3
(d) 7 – –3 = 7 + 3
10
5
5
10
2
3
=
7
+3
10
10
5
= 10
10
1
= 10
2
1
2
2
1
19 + –8 = 19 – 8
10
5
5
10
4
1
= 19
–8
10
10
3
= 11
10
1
3
1
2
\ 7 – –3 < 19 + –8
10
10
5
5
29
2. (a)
3
16. Amount of time Nora spent on visiting old folks’ homes
4
1
=× 8
7
16
1
4
= × 129
7
16 4
129
=
28
17
=
4
hours
28
7
23
3
5
3
5
23
7
17. 5 – 2 + – – – 4 = 5 – 2 –
+4
10
15
4
6
4
6
15
10
23 17
23
47
=
–
–
+
4
6
15
10
345
170
92
282
=
–
–
+
60
60
60
60
345 – 170 – 92 + 282
=
60
365
=
60
73
=
12
1
=6
12
18. Fraction of sum of money left after Farhan has taken his share
=
1–
4
=
5
Fraction of sum of money left after Khairul has taken his share
1
4
=
1 – 3 × 5
2
4
=×
3
5
8
=
15
Fraction of sum of money left after Huixian has taken her share
–3
–2
\ 5.5, 4,
–1
33
4
0
–8
–2p
–8
–6
3. (a)
3
82
= ×
4
15 5
1
(b)
2
=
5
(c)
1
2
Fraction of sum of money taken by Jun Wei = 1 – ×
7
5
6
2
=
×
7
5
12
=
35
(d)
4. (a)
(b)
Review Exercise 2
(c)
1. (a) –7 – 38 = – 45
8 + (–55) = – 47
\ –7 – 38 > 8 + (–55)
52
2
5
8
–4
3
4
5
–2
0
10
5.855
2
1
5
, 5.855, , –2p, –8
2
8
13 – (–54) = 13 + 54
=
67
(–74) – (– 46) = –74 + 46
= –28
11 + (–33) – (–7) = –22 – (–7)
= –22 + 7
= –15
–13 + (–15) + (–8) = –28 + (–8)
= –36
–12 × 7 = –84
4 × (–5) × (– 6) = –20 × (– 6)
= 120
– 600 ÷ 15 = – 40
\ 10
1
1
5.5
4
6
29
3
, – , –2.365
33
4
(b)
1
8
=
1 – 4 × 15
1
–
–2.365
1
5
4
6
8
10
1
2
12
50
÷ (–5)
8
25
=–
÷ (–5)
4
(d) 50 ÷ (–8) ÷ (–5) = –
6.
–18 – 3 –(33 × 5 3 ) – (– 6)2
–18 – 3 –3375 – (– 6)2
=
4+9
4+9
–18 – [–(3 × 5) – 36]
2+9
–18 – (–15 – 36)
=
11
–18 – (–51)
=
11
–18 + 51
=
11
33
=
11
=3
=
25 1
=–
×–
4
51
5
=
4
1
=1
4
(–3 – 5) × (–3 – 4) = (–8) × (–7)
= 56
4 × (–5) ÷ (–2) = –20 ÷ (–2)
= 10
–5 × 6 – 18 ÷ (–3) = –30 – (– 6)
= –30 + 6
= –24
2 × (–3)2 – 3 × 4 = 2 × 9 – 3 × 4
= 18 – 12
=6
–3 × (–2) × (2 – 5)2 = –3 × (–2) × (–3)2
= –3 × (–2) × 9
=6×9
= 54
(–2)2 – (–2) × 3 + 2 × 32 = 4 – (–2) × 3 + 2 × 9
= 4 – (– 6) + 18
= 4 + 6 + 18
= 10 + 18
= 28
(– 4)2 ÷ (–8) + 3 × (–2)3 = 16 ÷ (–8) + 3 × (–8)
= (–2) + (–24)
= –26
4 × 32 ÷ (–6) – (–1)3 × (–3)2 = 4 × 9 ÷ (–6) – (–1) × 9
= 36 ÷ (–6) – (–1) × 9
= – 6 – (–9)
= –6 + 9
=3
–2 × (–2)3 × (–2) × 3 + (–2) × 3 × (–1)2
=
–2 × (–8) × (–2) × 3 + (–2) × 3 × 1
=
16 × (–2) × 3 + (– 6)
=
–32 × 3 + (– 6)
=
–96 + (– 6)
=
–102
5 – {12 × [(–5)2 – 7] ÷ 3} = 5 – [12 × (25 – 7) ÷ 3]
= 5 – (12 × 18 ÷ 3)
= 5 – (216 ÷ 3)
= 5 –72
= – 67
5
5. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
4
2
4
2
3
3
+1 – – =3 +1 +
7
7
5
7
5
7
25
7
3
=
+
+
7
5
7
125
49
15
=
+
+
35
35
35
125 + 49 + 15
=
35
189
=
35
27
=
5
2
=5
5
2
3
2
2
3
2
(b)
– –3
+3
–
+ – =
20
5
3
3
20
5
2
63
2
=
+
–
3
20
5
40
189
24
=
+
–
60
60
60
40 + 189 – 24
=
60
205
=
60
41
=
12
5
=3
12
4
3
5
16
27
20
(c) – 6 – 3 – 3 = – 6
–3
–3
9
4
9
36
36
36
63
= –12
36
7
= –12
4
3
= –13
4
7. (a) 3
53
1
=
–12
14
1
5
5
+
÷ (– 4) –
× –
2
3
7
5
=
–12
1
5
1
+
× – –
2
3
4
1
5
+ – – (–2)
2
12
1
5
=
–12 –
+2
2
12
6
5
=
–12
–
+2
12
12
11
=
–10
12
2
3
1
2
13
8 2 15
1
3
× 1 × –1 = –
×
×–
13
4
5
51 13 1
1 4
=6
1
1
1
1
3
× – – ÷ –2 + 1
4 6
4
5
3
8.
4
2
– 7 – – 5
–
64
÷
625
3
–
3
8
125
=
598
1225
9. (a) –12.8 – 88.2 = –101
(b) 500.3 – (–200.2) – 210.1 = 500.3 + 200.2 – 210.1
= 700.5 – 210.1
= 490.4
1.44
(c) 1.44 ÷ 1.2 × (– 0.4) =
× (– 0.4)
1.2
2
4
3
3
3
–
÷ –2 + 1
=× –
12
12 12
12
5
1
3
5
= × – ÷ –1
12
12
5
=
14.4
× (– 0.4)
12
= 1.2 × (– 0.4)
= – 0.48
(d) (– 0.3)2 ÷ (– 0.2) + (–2.56) = 0.09 ÷ (– 0.2) + (–2.56)
0.09
=
+ (–2.56)
–0.2
3
5
13
= × – ÷ –
12
5
12
1
1
5 12
3
= ×–
×–
13
12 1
1 5
3
=
13
3
7
9
3
1
57
19
1
(g) –3
÷1
–
× –1 = –
÷
–
× –
4
4
16
16
3
16
16
3
3
=–
=
1
0.9
+ (–2.56)
–2
= – 0.45 + (–2.56)
= – 0.45 – 2.56
= –3.01
57
16
1 7
×
– ×–
19 1 3 4
1 16
7
= –3 – –
12
7
= –3 +
12
5
= –2
12
Challenge Yourself
1. Since
x – 3 and (y + 2)2 cannot be negative,
x – 3 = 0 and
x – 3 = 0 and
\ x = 3 and
2. (a)
324
×
57
2268
+1620
18468
(y + 2)2 = 0
y+2 =0
y = –2
(b)
3. (b) (3 + 3) ÷ 3 + 3 – 3 = 2
(d) (3 + 3 + 3 + 3) ÷ 3 = 4
(f) 3 + 3 + (3 – 3) × 3 = 6
1
14 2
5
× –
51
1 7
1
=
–12
(e) –3
(f)
4
1
2
5
+ 1 ÷ (– 4) –
× –2
5
2
3
7
(h) –12
1 1
1 1
1
(d) – + + + – + –
4
3
2 3
20
1
1
1
1
1
=
– +
+
–
–
2
3
4
3
20
1
1
1
1
1
=
– +
–
+
–
2
3
3
4
20
1
1
1
=
– +
–
2
4
20
10
5
1
=
–
+
–
20
20
20
–10 + 5 – 1
=
20
–6
=
20
3
=
–
10
54
48
28 ) 1 3 4 4
–112
224
–224
0
(c) 3 + 3 – 3 – 3 + 3 = 3
(e) 3 + 3 ÷ 3 + 3 ÷ 3 = 5
Chapter 3 Approximation and Estimation
TEACHING NOTES
Suggested Approach
Teachers can give students a real-life example when an approximated or estimated value is used before getting them to discuss
occasions when they use approximation and estimation in their daily lives. In this chapter, they will first learn the five rules
to identify the digits which are significant in a number before learning how to round off numbers to a specified number of
significant figures. Students will also learn how to carry out estimation through worked examples that involve situations in
real-world contexts.
Section 3.1:
Approximation
To make learning of mathematics relevant, students should know some reasons why they need to use approximations
in their daily lives (see Class Discussion: Actual and Approximated Values).
Teachers should do a recap with students on what they have learnt in primary school, i.e. how to round off
numbers to the nearest tenth, whole number and 10 etc.
Section 3.2:
Significant Figures
Through the example on measuring cylinders on page 63 of the textbook, students will learn that a number is
more accurate when it is given to a greater number of significant figures.
After learning how to round off numbers to a specified number of significant figures, teachers can arouse students’
interest in this topic by bringing in real-life situations where they cannot just round off a number using the rules
they have learnt (see Investigation: Rounding in Real Life). The journal writing on page 67 of the textbook
requires students to cite examples of such situations.
Section 3.3:
Rounding and Truncation Errors
Teachers should tell students that the general instructions for O-level Mathematics examinations state, ‘If the
degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three
significant figures. Give answers in degrees to one decimal place.’ The investigation on page 68 of the textbook
highlights the importance of giving intermediate values correct to four significant figures if we want the final
answer to be accurate to three significant figures. Otherwise, a rounding error may occur.
Students should also learn that there is a difference between ‘approximately 2.5 million’ and ‘equal to 2.5 million
(to 2 s.f.)’ (see the thinking time on page 69 of the textbook).
Teachers should tell students the difference between rounding off a number to, say, 3 significant figures and
truncating the same number to 3 significant figures. The investigation on page 70 of the textbook enables students
to find out more about rounding and truncation errors in calculators.
Section 3.4:
Estimation
Teachers can impress upon students that there are differences between approximation and estimation. Since
students need to be aware when an answer is obviously wrong, estimation allows them to check the reasonableness
of an answer obtained from a calculator (see Worked Example 6).
Students will also learn an important estimation strategy: use a smaller quantity to estimate a larger quantity (see
Investigation: Use of a Smaller Quantity to Estimate a Larger Quantity).
Teachers should get students to work in groups to estimate quantities in a variety of contexts, compare their
estimates and share their estimation strategies with one another. (see the performance task on page 76 of the
textbook).
55
1
WORKED SOLUTIONS
Journal Writing (Page 67)
Class Discussion (Actual and Approximated Values)
•
1. The actual values indicated in the article include ‘42 038 777
passengers’, ‘13 .0%’, ‘24 awards’ and ‘four terminals’ while
approximated values include ‘over 360 awards’ and ‘73 million
passengers’ . Actual values are exact numbers while approximated
values are values which are usually rounded off .
2. (a) It is not necessary to specify the actual number of awards won,
as an approximation is sufficient to show that Changi Airport
has won many awards .
(b) A headline serves as a brief summary of the article to draw
readers’ attentions, thus it is more appropriate to use an
approximated value instead of the actual value .
•
Investigation (Rounding in Real Life)
Teachers may wish to note that the list is not exhaustive.
Scenario 1
Total number of passengers = 215 + 5
= 220
Investigation (The Missing 0.1% Votes)
1. The percentage of votes for each candidate given is correct to
3 significant figures. Due to rounding errors in the intermediate
steps, there is a follow-through error, resulting in the missing 0 .1%
of the votes. If the final answer is correct to 2 significant figures, we
will obtain 100%. Hence, the final answer can only be accurate to
2 significant figures.
Number of buses required = 220 ÷ 30
=7
1
3
1
is 7 . However, 7 buses are not enough
3
to carry 220 passengers, thus we round up to find the number of buses
required to carry all the passengers .
\ The number of buses required is 8 .
The nearest whole number to 7
188
× 100%
301
= 62 .5% (to 3 s .f .)
2. Percentage of votes for Vishal =
52
× 100%
301
= 17 .3% (to 3 s .f .)
Scenario 2
Maximum mass of lift = 897 kg
= 900 kg (to the nearest 100 kg)
Percentage of votes for Rui Feng =
61
× 100%
301
= 20 .3% (to 3 s .f .)
Total percentage of votes = 62 .5% + 17 .3% + 20 .3%
= 100 .1%
Percentage of votes for Huixian =
If the maximum mass of the lift is given as 900 kg, it means that the
lift is able to carry a mass of 900 kg . However, the maximum mass
allowed is only 897 kg .
\ The maximum mass of the lift should be given as 800 kg .
The percentage of votes for each candidate given is correct to
3 significant figures. Due to rounding errors in the intermediate
steps, which results in a follow through error, the total percentage of
votes is 100.1%. If the final answer is correct to 2 significant figures,
we will obtain 100%. Hence, the final answer can only be accurate
to 2 significant figures.
Scenario 3
In Singapore, the issue of 1-cent coins has ceased since 2002; while
the coins are legal tender and are still in circulation, most shops have
stopped accepting 1-cent coins . As such, when people wish to pay for
their purchases in cash, the prices of their purchases have to be rounded
off to the nearest 5 cents which is now considered to be the smallest
denomination of currency in Singapore .
Thinking Time (Page 69)
Teachers may wish to ask students to explain why when other methods
of payment are used, it is not necessary to round off the prices of their
purchases to the nearest 5 cents.
1
A developer wants to build a house on a plot of land that has a height
restriction of 10 m. The height from the floor to the ceiling of each
level is about 2 .6 m .
Number of levels the developer can build = 10 m ÷ 2 .6 m
= 3 .85 (to 3 s .f .)
The nearest whole number to 3 .85 is 4 . However, a house with
4 levels will be taller than 10 m, and thus will go against the
height restrictions . Hence, the maximum number of levels that the
developer can build is 3 .
The boiling point of oxygen, i.e. the temperature at which liquid
oxygen boils to form gaseous oxygen, is –183 °C. The maximum
temperature, correct to the nearest 10 °C, at which liquid oxygen
can be stored is –190 °C as oxygen will be in its gaseous state at a
temperature of –180 °C.
1. (i) When the population of City A is approximately 2 .5 million, it
is possible for the exact population size to be 2 .47 million .
(ii) When the population of City A is approximately 2 .5 million, it
is possible for the exact population size to be 2 .6 million .
56
2. (i) When the population of City B is equal to 2 .5 million (to 2 s .f .),
it is possible for the exact population size to be 2 .47 million as it
is equal to 2.5 million when rounded off to 2 significant figures.
(ii) When the population of City B is equal to 2 .5 million (to 2 s .f .),
it is not possible for the exact population size to be 2 .6 million
as it is still equal to 2.6 million when rounded off to 2 significant
figures.
Performance Task (Estimation in Our Daily Lives)
1. Use surveys, questionnaires or verbal questioning to find out the
number of hours spent surfing the Internet by each student in the
class on a weekday and on a Saturday or Sunday . Ensure that students
have a common understanding of the phrase ‘surfing the Internet’.
Calculate the total number of hours spent surfing the Internet by all
the students in the class on a weekday and on a Saturday or Sunday .
Note: There is a difference between ‘approximately 2 .5 million’ and
‘equal to 2 .5 million (to 2 s .f .)’ .
Total amount of time spent surfing the Internet by all the students
in the class on a weekday = x hours
Investigation (Rounding and Truncation Errors in
Calculators)
Total amount of time spent surfing the Internet by all the students
in the class on a Saturday or Sunday = y hours
For this activity, the calculator model used is SHARP EL-509VM.
(a) 1. 0 .727 922 061
2. 7 .27 922 061 3
3. 2 .7 922 061 3
The calculator stores 12 digits .
Estimate the total number of hours spent surfing the Internet by all
the students in the class in a month . Assume that the average number
of weekdays and the average number of Saturdays and Sundays in
a month are 22 and 8 respectively .
Total amount of time spent surfing the Internet by all the students
in the class in a month ≈ (22x + 8y) hours
The calculator truncates the value of 162 at the 12th digit to give
12 .727 922 061 3, instead of rounding 162 to 12 .727 922 061 4 .
(b) 5. 6 .999 999 999
2. Assume that there are 8 slices in a large pizza . Use verbal questioning
to find out the number of slices needed to feed one class (e .g . about
40 students) in the school when they go for an excursion .
Investigation (Use of a Smaller Quantity to Estimate a
Larger Quantity)
Number of slices needed to feed one class in the school = x
For this investigation, the smaller box used is of length 9.2 cm, width
5.6 cm and height 2.7 cm.
x
8
Find out the number of classes in the school . Ensure that there
is approximately the same number of students in each class,
e .g . 40 students .
Number of pizzas needed to feed one class in the school =
Three trials are carried out to find the average number of 10¢ coins that
can fill the box. The result of each trial is shown in the table.
Trial
Number of 10¢ coins
1
294
2
280
3
284
Number of classes in the school = y
Estimate the amount of pizza needed to feed all the students in the
school during an excursion .
Total number of pizzas needed to feed all the students in the school
xy
≈
8
Average number of 10¢ coins that can fill the smaller box
294 + 280 + 284
3
858
=
3
= 286
=
3. Find out the opening hours of the drinks stall on a weekday and
determine the durations of the peak (e .g . recess and lunchtime) and
non-peak periods respectively .
Duration of peak period = x hours
Volume of smaller box = 9 .2 × 5 .6 × 2 .7
= 139 .104 cm3
Duration of non-peak period = y hours
Volume of tank = 50 × 23 × 13
= 14 950 cm3
Find out the amount of money collected by the drinks stall in half an
hour during the peak period and half an hour during the non-peak
period .
Number of 10¢ coins that can fill the tank =
286
× 14 950
139.104
= 30 737 (to the nearest
whole number)
\ Amount of money in the tank = 30 737 × 10¢
= $3073 .70
Amount of money collected by drinks stall in half an hour during
peak period = $p
Amount of money collected by drinks stall in half an hour during
non-peak period = $q
57
1
Estimate the total amount of money collected for both the peak and
non-peak periods .
(c) The number 3.0021 has 5 significant figures.
(d) The number 70.8001 has 6 significant figures.
Total amount of money collected by drinks stall during peak period
≈ $2px
Practise Now (Page 64)
1. (a) The number 0.10 has 2 significant figures.
(b) The number 0.500 has 3 significant figures.
(c) The number 41.0320 has 6 significant figures.
(d) The number 6.090 has 4 significant figures.
2. 4.10 cm is more accurate because 4.10 cm is measured to 3 significant
figures, while 4.1 cm is measured to 2 significant figures.
Total amount of money collected by drinks stall during non-peak
period ≈ $2qy
Hence,
Total amount of money collected by drinks stall on a weekday
≈ $(2px + 2qy)
Practise Now (Page 65)
(a) The number 0.021 has 2 significant figures.
(b) The number 0.603 has 3 significant figures.
(c) The number 0.001 73 has 3 significant figures.
(d) The number 0.1090 has 4 significant figures.
Practise Now 1
1. (a) 3 409 725 = 3 409 730 (to the nearest 10)
(b) 3 409 725 = 3 409 700 (to the nearest 100)
(c) 3 409 725 = 3 410 000 (to the nearest 1000)
(d) 3 409 725 = 3 410 000 (to the nearest 10 000)
2. Largest possible number of overseas visitors = 11 649 999
Smallest possible number of overseas visitors = 11 550 000
Practise Now (Page 65)
(a) 3800 m, which is corrected to the nearest 10 m, has 3 significant
figures.
(b) 25 000 km, which is corrected to the nearest km, has 5 significant
figures.
(c) 100 000 g, which is corrected to the nearest 10 000 g, has 2 significant
figures.
Practise Now 2
1. (a) 78 .4695 = 78 .5 (to 1 d .p .)
(b) 78 .4695 = 78 (to the nearest whole number)
(c) 78 .4695 = 78 .47 (to the nearest hundredth)
(d) 78 .4695 = 78 .470 (to the nearest 0 .001)
2. No, I do not agree with Jun Wei . 8 .40 is rounded off to 2 decimal
places which is more accurate than 8 .4 which is rounded off to
1 decimal place .
Practise Now 4
3748 = 3750 (to 3 s .f .)
0 .004 709 89 = 0 .004 710 (to 4 s .f .)
4971 = 5000 (to 2 s .f .)
0 .099 99 = 0 .10 (to 2 s .f .)
0 .099 99 = 0 .100 (to 3 s .f .)
2. Since 67 0X1 (to 3 s .f .), then the possible values of X are 5, 6, 7, 8
or 9 .
If 67 0X1 is a perfect square, then by trial and error, X = 8 .
1. (a)
(b)
(c)
(d)
Practise Now 3
Cost of 450 kWh of electricity = 450 × $0 .29
= $130 .50
Cost of 38 m3 of water = 38 × $1 .17
= $44 .46
Practise Now 5
Total amount of money the household has to pay = $130 .50 + $44 .46
= $174 .96
= $175 (to the nearest
dollar)
(i) Length of square = 105
= 10 .2 m (to 3 s .f .)
(ii) Perimeter of square = 10 .25 × 4
= 41 .0 m (to 3 s .f .)
Practise Now (Page 64)
(a)
(b)
(c)
(d)
The number 192 has 3 significant figures.
The number 83.76 has 4 significant figures.
The number 3 has 1 significant figure.
The number 4.5 has 2 significant figures.
Practise Now 6
1. 798 × 195 ≈ 800 × 200
= 160 000
\ Nora’s answer is not reasonable .
2. (a) 5712 ÷ 297 ≈ 5700 ÷ 300
= 19
Using a calculator, 5712 ÷ 297 = 19 .2 (to 3 s .f .) .
\ The estimated value is close to the actual value .
Practise Now (Page 64)
(a) The number 506 has 3 significant figures.
(b) The number 1.099 has 4 significant figures.
1
58
(b)
63 ×
3
129 ≈ 64 ×
=8×5
= 40
3
5. No, I do not agree with Kate . She needs to put a ‘0’ in the ones place
as a place holder after dropping the digit ‘2’, i .e . 5192 .3 = 5190 (to
the nearest 10) .
6. Largest possible value of Singapore’s population = 5 077 499
Smallest possible value of Singapore’s population = 5 076 500
7. No, I do not agree with Farhan . 27 .0 is rounded off to 1 decimal
place which is more accurate than 27 which is rounded off to the
nearest whole number .
125
Using a calculator, 63 × 3 129 = 40 .1 (to 3 s .f .) .
\ The estimated value is close to the actual value .
250
80
240
≈
hours
80
3. Time taken to drive from Singapore to Malacca =
Exercise 3B
1. (a) The number 39 018 has 5 significant figures.
(b) The number 0.028 030 has 5 significant figures.
(c) 2900, which is corrected to the nearest 10, has 3 significant
figures.
2. (a) 728 = 730 (to 2 s .f .)
(b) 503 .88 = 503 .9 (to 4 s .f .)
(c) 0 .003 018 5 = 0 .003 019 (to 4 s .f .)
(d) 6396 = 6400 (to 2 s .f .)
6396 = 6400 (to 3 s .f .)
(e) 9 .9999 = 10 .0 (to 3 s .f .)
(f) 8 .076 = 8 .08 (to 3 s .f .)
3. Possible values of x = 4, 5 or 6
Practise Now 7
Rp 10 000 ≈ S$1 .50, so Rp 20 000 ≈ S$3, Rp 5000 ≈ S$0 .75
\ The price of the pair of earrings is Rp 25 000 ≈ S$3 .75 .
Practise Now 8
For option A, 300 ml costs about $9 .
Thus 100 ml will cost about $3, and 50 ml will cost about $1 .50 .
\ For option A, 350 ml will cost about $9 + $1 .50 = $10 .50 .
For option B, 350 ml costs $10 .40 which is $0 .10 cheaper than option A .
However, for option A, 300 ml actually costs $8 .80 which is less than $9 .
Thus for option A, 350 ml will cost at least $0 .20 less than the estimated
$10 .50 .
\ Option A is better value for money .
1
= 0 .010 10 (to 4 s .f .)
99
(b) 871 × 234 = 203 814
= 200 000 (to 2 s .f .)
4. (a)
Practise Now 9
2
× 100%
3
2
= 66 %
3
Percentage of shaded region =
(c)
212
= 2013 .698 63
0.219
= 2013 .7 (to 5 s .f .)
3.913 – 2.1
= 9 .0 (to 2 s .f .)
6.41
5. Greatest number of sweets that can be bought
$2
=
$0.30
= 6 (to the nearest whole number)
(d)
Exercise 3A
1. (a)
(b)
(c)
2. (a)
(b)
(c)
3. (i)
(ii)
4. (a)
(b)
(c)
698 352 = 698 400 (to the nearest 100)
698 352 = 698 000 (to the nearest 1000)
698 352 = 700 000 (to the nearest 10 000)
45 .7395 = 45 .7 (to 1 d .p .)
45 .7395 = 46 (to the nearest whole number)
45 .7395 = 45 .740 (to 3 d .p .)
Perimeter of land = 2(28 .3 + 53 .7)
= 2(82)
= 160 m (to the nearest 10 m)
Area of grass needed to fill up the entire plot of land
= 28 .3 × 53 .7
= 1519 .71 m2
= 1500 m2 (to the nearest 100 m2)
4 .918 m = 4 .9 m (to the nearest 0 .1 m)
9 .71 cm = 10 cm (to the nearest cm)
$10 .982 = $11 .00 (to the nearest ten cents)
(d) 6 .489 kg = 6 .49 kg (to the nearest
6. (i) Length of square = 264
= 16 .2 cm (to 3 s .f .)
(ii) Perimeter of square = 16 .25 × 4
= 65 .0 cm (to 3 s .f .)
136
2p
= 21 .6 m (to 3 s .f .)
(ii) Area of circle = p(21 .65)2
= 1470 m2 (to 3 s .f .)
8. Since 21 X09 = 22 000 (to 2 s .f .), then the possible values of X are
5, 6, 7, 8 or 9 .
If 21 X09 is a perfect square, then by trial and error, X = 6 .
9. Largest possible number of people at the concert = 21 249
Smallest possible number of people at the concert = 21 150
10. (i) 987 654 321 + 0 .000 007 – 987 654 321 = 0 .000 007
(ii) 987 654 321 + 0 .000 007 – 987 654 321 = 0
7. (i) Radius of circle =
1
kg)
100
59
1
10. KRW 900 ≈ S$1
\ Price of handbag = KRW 26 700
≈ KRW 27 000
= 30 × KRW 900
≈ 30 × S$1
= S$30
(iii) No, the answers for (i) and (ii) are different . This is because the
calculator truncates the value of 987 654 321 + 0 .000 007 to
give 987 654 321 . Hence, the answer for (ii) is 0 .
Exercise 3C
1. 218 ÷ 31 ≈ 210 ÷ 30
=7
\ Priya’s answer is not reasonable .
Using a calculator, 218 ÷ 31 = 7 .03 (to 3 s .f .) .
\ The estimated value is close to the actual value .
2. (a) 2013 × 39 ≈ 2000 × 40
= 80 000
Using a calculator, 2013 × 39 = 78 507 .
\ The estimated value is close to the actual value .
(b)
145.6 ÷
3
65.4 ≈ 144 ÷
= 12 ÷ 4
=3
3
Review Exercise 3
6479 .952 = 6500 (to the nearest 100)
6479 .952 = 6000 (to the nearest 1000)
6479 .952 = 6480 .0 (to the nearest tenth)
4 .793 = 4 .8 (to 2 s .f .)
39 .51 = 40 (to 2 s .f .)
(ii) 4 .793 ÷ 39 .51 ≈ 4 .8 ÷ 40
= 0 .12 (to 2 s .f .)
3. Smallest possible mass of chocolate truffle = 0.0245 kg
4. Rp 10 000 ≈ S$1 .50, so Rp 30 000 ≈ S$4 .50, Rp 5000 ≈ S$0 .75 .
\ The price of the toy is Rp 35 000 ≈ S$5 .25 .
5. Total mass = 3 × 109 + 2 × 148 + 5 × 84
≈ (3 × 110 + 2 × 150 + 5 × 80) g
1. (a)
(b)
(c)
2. (i)
64
Using a calculator, 145.6 ÷ 3 65.4 = 2 .99 (to 3 s .f .) .
\ The estimated value is close to the actual value .
3. (i) 3 .612 = 3 .6 (to 2 s .f .)
29 .87 = 30 (to 2 s .f .)
(ii) 3 .612 ÷ 29 .87 ≈ 3 .6 ÷ 30
= 0 .12 (to 2 s .f .)
28.2
4.03
28
≈
4
6. Number of batteries required =
274
9.1
270
≈
l
9
Ratio of area of shaded region to that of unshaded region = 1 : 2
Total amount of money that the shopkeeper has to pay
= 32 × $18 + 18 × $8 + 47 × $26 + 63 × $23 + 52 × $9
≈ 30 × $20 + 20 × $10 + 50 × $30 + 60 × $20 + 50 × $10
= $600 + $200 + $1500 + $1200 + $500
= $4000 (to the nearest hundred dollars)
RM10 ≈ S$4, so RM20 ≈ S$8, RM5 ≈ S$2 .
\ The price of the bag is RM25 ≈ S$10 .
For option A, 300 g costs about $6 .
Thus 100 g will cost about $2 .
\ For option A, 500 g will cost about 5 × $2 = $10 .
For option B, 500 g costs $9 .90 which is $0 .10 cheaper than
option A .
However, for option A, 300 g actually costs $5 .80 which is $0 .20
less than $6 .
Thus for option A, 500 g will cost at least $0 .20 less than the
estimated $10 .
\ Option A is better value for money .
80
Price of dress in Shop A after a 20% discount =
× $79 .50
100
80
≈
× $80
100
80
× $85 .05
100
80
≈
× $85
100
4. Amount of petrol used =
5.
6.
7.
8.
9.
7. Price of hard disk in Store A after a 20% discount =
90
× $76 .05
100
90
≈
× $76
100
Price of hard disk in Store B after a 10% discount =
8. For option A, 250 ml costs about $15 .
Thus 50 ml will cost about $3, and 100 ml will cost about $6 .
\ For option A, 300 ml will cost about 3 × $6 = $18 .
Furthermore, for option A, 250 ml actually costs $15 .20 which is
$0 .20 more than $15 .
Thus for option A, 300 ml will cost at least $0 .20 more than the
estimated $18 .
\ Option B is better value for money .
Challenge Yourself
1. 987 × 123 is more than 988 × 122 because 987 × 123
= 987 × (122 + 1), i .e . there is an additional 987 × 1; but
988 × 122 = (987 + 1) × 122, i .e . there is only an additional
1 × 122 . In fact, 987 × 123 – 988 × 122 = 987 – 122 = 865 .
2. This question tests students’ sense of mass . The mass of an ordinary
car is likely to be 2000 kg .
Teachers may wish to get students to give examples of objects with
masses of 20 kg, 200 kg and 20 000 kg, e.g. 2 10-kg bags of rice
have a total mass of 20 kg, 5 Secondary 1 students have a total mass
of about 200 kg and a rocket has a mass of about 20 000 kg.
90
× $69 .50
100
90
≈
× $70
100
Price of dress in Shop B after a 10% discount =
1
60
Chapter 4 Basic Algebra and Algebraic Manipulation
TEACHING NOTES
Suggested Approach
Some students are still unfamiliar with algebra even though they have learnt some basic algebra in primary school. Thus for the
lower ability students, teachers should teach this chapter as though they do not know algebra at all. The learning experiences in
the new syllabus specify the use of algebra discs. In addition to the algebra discs showing the numbers 1 and –1 which students
have encountered in Chapter 2, algebra discs showing x, –x, y and –y are needed. Since many Secondary 1 students are still in
the concrete operational stage (according to Piaget), the use of algebra discs can help them to learn the concepts more easily.
However, there is still a need to guide students to move from the ‘concrete’ to the ‘abstract’, partly because they cannot use
algebra discs in examinations, and partly because they cannot use algebra discs to manipulate algebraic expressions which
consist of algebraic terms that have large or fractional coefficients (see Section 4.1, 4.2 and 4.3).
Section 4.1:
Fundamental Algebra
Teachers should teach students how to use letters to represent numbers and interpret basic algebraic notations
such as ab = a × b. Teachers should illustrate the definitions of mathematical terms such as ‘algebraic term’,
‘coefficient’, ‘algebraic expression’ and ‘linear expression’ using appropriate examples.
In the class discussion on page 83 of the textbook, students are required to use algebraic expressions to express
mathematical relationships.
To make learning more interactive, students are given the opportunity to use a spreadsheet to explore the concept
of variables (see Investigation: Comparison between Pairs of Expressions). Through this investigation, students
should be able to observe that evaluating an algebraic expression means finding the value of the expression
when the variables take on certain values. This investigation also provides students with an intuitive sense of the
difference between pairs of expressions such as 2n and 2 + n, n2 and 2n, and 2n2 and (2n)2. Students are expected
to give a more rigorous mathematical explanation for the difference between such a pair of expressions in the
journal writing on page 85 of the textbook.
Algebra discs cannot be used to add or subtract algebraic terms with large coefficients, so there is a need to help
students consolidate what they have learnt in Worked Example 2. For the lower ability students, before going
through Worked Example 2(d) and (e), teachers should revisit the procedure for simplifying ordinary numerical
fractions, e.g. 1 + 1 .
2
Section 4.2:
3
Expansion and Simplification of Linear Expressions
The idea of flipping over a disc to obtain the negative of a number or variable, e.g. –(−x) = x, is needed to teach
students how to obtain the negative of a linear expression. Algebra discs cannot be used to manipulate algebraic
expressions which consist of algebraic terms that have large coefficients, so there is a need to help students
consolidate what they have learnt in the class discussion on page 94 of the textbook by moving away from the
‘concrete’ to the following ‘abstract’ concept:
Distributive Law: a(b + c) = ab + ac
Teachers should emphasise the importance of the rules by which operations are performed when an algebraic
expression involves brackets by using the thinking time on page 96 of the textbook.
Section 4.3:
Simplification of Linear Expressions with Fractional Coefficients
After going through Worked Example 5 and 6, students should observe that the procedure for simplifying linear
expressions with fractional coefficients is similar to that of simplifying ordinary numerical fractions.
61
1
Section 4.4:
Factorisation
Students should learn how to appreciate the factorisation process, i.e. it is the reverse of expansion. Teachers
should tell students the difference between ‘complete’ and ‘incomplete’ factorisation. In Secondary 1, students
only need to know how to factorise algebraic expressions by extracting the common factors.
The class discussion on page 101 of the textbook requires students to work in pairs to select and justify pairs of
equivalent expressions. Teachers should make use ofthis opportunity to highlight some common errors made by
students when manipulating algebraic expressions.
1
62
(iii) When n = 8,
2n = 2 × 8
= 16
When n = 9,
2n = 2 × 9
= 18
When n = 10,
2n = 2 × 10
= 20
WORKED SOLUTIONS
Class Discussion (Expressing Mathematical Relationships
using Algebra)
1.
In words
Algebraic expression
(a) Sum of 2x and 3z
2x + 3z
(b) Product of x and 7y
7xy
3ab
2c
(c) Divide 3ab by 2c
3.
(d) Subtract 6q from 10z
10z – 6q
1
(e) Subtract the product of x and y
from the sum of p and q
(p + q) – xy
2
(f)
Divide the sum of 3 and y by 5
(g) Subtract the product of 2 and c
from the positive square root of b
(h) There are three times as many
girls as boys in a school . Find an
expression, in terms of x, for the
total number of students in the
school, where x represents the
number of boys in the school .
3+ y
5
b – 2c
It is given that x represents the
number of boys .
\ 3x represents the number of girls .
Total number of students = x + 3x
= 4x
(i)
The age of Nora’s father is thrice
hers . The age of Nora’s brother
is 5 years more than hers . Find
an expression, in terms of y, for
the sum of their ages, where y
represents Nora’s age .
It is given that y represents Nora’s
age .
\ Nora’s father is 3y years old .
Nora’s brother is (y + 5) years old .
Sum of their ages = y + 3y + y + 5
= (5y + 5) years
(j)
The length is three times as long
as the breadth of the rectangle .
Find an expression, in terms of b,
for the perimeter and the area of
the rectangle, where b represents
the breadth of the rectangle .
It is given that b represents the
breadth of the rectangle in m .
3b represents the length of the
rectangle in m .
Perimeter of rectangle = 2(3b + b)
= 2(4b)
= 8b m
Area of rectangle = 3b × b
= 3b2 m2
A
B
C
D
E
F
n
2n
2+n
n2
2n2
(2n)2
3
1
2
3
1
2
4
4
2
4
4
4
8
16
5
3
6
5
9
18
36
6
4
8
6
16
32
64
7
5
10
7
25
50
100
4. •
2n and 2 + n
Referring to columns B and C on the spreadsheet, the
expressions 2n and 2 + n are equal only when n = 2 . When n < 2,
2n < 2 + n . When n > 2, 2n > 2 + n .
n2 and 2n
Referring to columns B and D on the spreadsheet, the expressions
n2 and 2n are equal when n = 2 . By observation, they are also
equal when n = 0 . When n < 0 or n > 2, n2 > 2n . When 0 < x < 2,
n2 < 2n .
2n2 and (2n)2
By observation, the expressions 2n2 and (2n)2 are equal when
n = 0 . For any value of n ≠ 0, (2n)2 > 2n2 .
•
•
Journal Writing (Page 85)
By observation, the expressions 5 + n and 5n are equal only when
Table 4.3
n=1
1
1
1
. When n < 1 , 5n < 5 + n . When n > 1 , 5n > 5 + n .
4
4
4
Investigation (Comparison between Pairs of Expressions)
A
2.
B
C
D
E
Class Discussion (The Distributive Law)
F
1. (a)
(b)
(c)
(d)
1
2
n
2n
3
1
2
4
2
4
5
3
6
6
4
8
7
5
10
2+n
n
2
2n
2
2
(2n)
2(–x – 4) = –2x – 8
–2(–x – 4) = 2x + 8
3(y – 2x) = 3y – 6x
–3(y – 2x) = –3y + 6x
2. a(b + c) = ab + ac
(i) The value of 2n changes as n changes .
(ii) We multiply the given value of n by 2 to obtain the
corresponding value of 2n .
63
1
3. G and N
Thinking Time (Page 96)
3( x + 3)
4(2 x + 3) 9( x + 3) – 16(2 x + 3)
–
=
3
4
12
9 x + 27 – 32 x – 48
=
12
9 x – 32 x + 27 – 48
=
12
–23 x – 21
=
12
Students may mistakenly match B and G due to an error in their
working as shown:
–(x – 5) + 6x – (7x – 2) + 12 = –x + 5 + 6x – 7x + 2 + 12
= –x + 6x – 7x + 5 + 2 + 12
= –2x + 19
Possible ways:
• –(x – 5) + 6x – 7x – (2 + 12) = –(x – 5) + 6x – 7x – 14
= –x + 5 + 6x – 7x – 14
= –x + 6x – 7x + 5 – 14
= –2x– 9
• –x – (5 + 6x) – (7x – 2) + 12 = –x – 5 – 6x – 7x + 2 + 12
= –x – 6x – 7x – 5 + 2 + 12
= –14x + 9
• –x – (5 + 6x) – 7x – (2 + 12) = –x – (5 + 6x) – 7x – 14
= –x – 5 – 6x – 7x – 14
= –x – 6x – 7x – 5 – 14
= –14x – 19
3( x + 3)
4(2 x + 3) 9( x + 3) – 16(2 x + 3)
–
=
3
4
12
9 x + 27 – 32 x + 48
=
12
9 x – 32 x + 27 + 48
=
12
–23 x + 75
=
12
4. I and M
2x – 3[5x – y – 2(7x – y)] = 2x – 3(5x – y – 14x + 2y)
= 2x – 3(5x – 14x – y + 2y)
= 2x – 3(–9x + y)
= 2x + 27x – 3y
= 29x – 3y
Students may mistakenly match L and M due to errors in their
working as shown:
2x – 3[5x – y – 2(7x – y)] = 2x – 3(5x – y – 14x – 2y)
= 2x – 3(5x – 14x – y – 2y)
= 2x – 3(–9x – 3y)
= 2x – 27x – 9y
= –25x – 9y
5. C and J, C and K or J and K
7ay – 49y = 7(ay – 7y) = 7y(a – 7)
Class Discussion (Equivalent Expressions)
The five pairs of equivalent expressions are as follows:
1. D and F
3(x – 2y) – 2(3x – y) = 3x – 6y – 6x + 2y
= 3x – 6x – 6y + 2y
= –3x – 4y
Students may mistakenly match D and O due to an error in their
working as shown:
3(x – 2y) – 2(3x – y) = 3x – 6y – 6x – 2y
= 3x – 6x – 6y – 2y
= –3x – 8y
2. A and E
x–3
2x – 5
3( x – 3) – 2(2 x – 5)
–
=
6
2
3
3 x – 9 – 4 x + 10
=
6
3 x – 4 x – 9 + 10
=
6
–x + 1
=
6
1– x
=
6
Students may mistakenly match E and H due to an error in their
working as shown:
Teachers may wish to get students to indicate the expression which
is obtained when the expression 7ay – 49y is factorised completely.
Practise Now 1
1. (a) 5y – 4x = 5(4) – 4(–2)
= 20 + 8
= 28
x–3
2x – 5
3( x – 3) – 2(2 x – 5)
–
=
6
2
3
3 x – 9 – 4 x – 10
=
6
(b)
3 x – 4 x – 9 – 10
6
– x – 19
=
6
=
1
64
1
1
–y+3=
–4+3
x
−2
1
=– –4+3
2
1
= –4 + 3
2
1
= –1
2
2. (i) 2p – 5q + 7r – 4p + 2q – 3r = 2p – 4p – 5q + 2q + 7r – 3r
= –2p – 3q + 4r
2
1
2. p2 + 3q2 = – + 3(–2)2
2
1
1
, q = – , r = 4,
2
3
1
1
–2p – 3q + 4r = –2 – 3 – + 4(4)
2
3
= –1 + 1 + 16
= 0 + 16
= 16
1
+ 3(4)
4
1
=
+ 12
4
1
= 12
4
(ii) When p =
=
Practise Now (Page 87)
(a)
(b)
(c)
(d)
Practise Now (Page 92)
3x + 4x = 7x
3x + (– 4x) = –x
–3x + 4x = x
–3x + (– 4x) = –7x
(a)
(b)
(c)
(d)
–(3x + 2) = –3x – 2
–(3x – 2) = –3x + 2
–(–3x – 2) = 3x + 2
–(2x + y – 4) = –2x – y + 4
Practise Now (Page 88)
Practise Now (Page 92)
(a) 4x – 3x = x
(b) 4x – (–3x) = 4x + 3x
= 7x
(c) – 4x – 3x = –7x
(d) – 4x – (–3x) = – 4x + 3x
= –x
(a) x + 1 + [–(3x – 1)] = x + 1 – 3x + 1
= x – 3x + 1 + 1
= –2x + 2
(b) 5x – 3 + [–(4x + 1)] = 5x – 3 – 4x – 1
= 5x – 4x – 3 – 1
=x–4
(c) 3x + 2y + [–(–y + 2x)] = 3x + 2y + y – 2x
= 3x – 2x + 2y + y
= x + 3y
(d) – 4x + 2y + [–(–x – 5y)] = – 4x + 2y + x + 5y
= – 4x + x + 2y + 5y
= –3x + 7y
Practise Now (Page 89)
(a) x + 2 + 5x – 4 = x + 5x + 2 – 4
= 6x – 2
(b) 2x + (–3) – 3x + 5 = 2x – 3x + (–3) + 5
= –x + 2
(c) –x – y – (–2x) + 4y = –x – y + 2x + 4y
= –x + 2x – y + 4y
= x + 3y
(d) –3x – 7y + (–2y) – (– 4x) = –3x – 7y + (–2y) + 4x
= –3x + 4x – 7y + (–2y)
= x – 9y
Practise Now (Page 93)
(a)
(b)
(c)
(d)
3(5x) = 15x
3(–5x) = –15x
–3(5x) = –15x
–3(–5x) = 15x
Practise Now 2
Practise Now 3
1. (a) 2x – 5y + 4y + 8x = 2x + 8x – 5y + 4y
= 10x – y
(b) 11x – (–5y) – 14x – 2y = 11x + 5y – 14x – 2y
= 11x – 14x + 5y – 2y
= –3x + 3y
(c) –9x – (–y) + (–3x) – 7y = –9x + y – 3x – 7y
= –9x – 3x + y – 7y
= –12x – 6y
(a) 3(x + 2) = 3x + 6
(b) –5(x – 4y) = –5x + 20y
(c) –a(x + 2y) = –ax – a(2y)
= –ax – 2ay
Practise Now (Page 95)
(a) x + 7 + 3(x – 2) = x + 7 + 3x – 6
= x + 3x + 7 – 6
= 4x + 1
(b) 3(x + 2) + 2(–2x + 1) = 3x + 6 – 4x + 2
= 3x – 4x + 6 + 2
= –x + 8
(c) 2(–x – y) – (2x – y) = –2x – 2y – 2x + y
= –2x – 2x – 2y + y
= – 4x – y
1
1
3
2
(d)
x– x= x– x
2
3
6
6
1
= x
6
14
7
5
5
(e)
y– y=
y– y
4
8
8
8
9
= y
8
65
1
(d) –(x + 4y) – 2(3x – y) = –x – 4y – 6x + 2y
= –x – 6x – 4y + 2y
= –7x – 2y
Practise Now 6
1. (a)
Practise Now 4
1. (a) 6(4x + y) + 2(x – y) = 24x + 6y + 2x – 2y
= 24x + 2x + 6y – 2y
= 26x + 4y
(b) x – [y –3(2x – y)] = x – (y – 6x + 3y)
= x – (–6x + y + 3y)
= x – (–6x + 4y)
= x + 6x – 4y
= 7x – 4y
(c) 7x – 2[3(x – 2) – 2(x – 5)] = 7x – 2(3x – 6 – 2x + 10)
= 7x – 2(3x – 2x – 6 + 10)
= 7x – 2(x + 4)
= 7x – 2x – 8
= 5x – 8
2. (i) Michael’s present age = (p + 5) years
(ii) Vishal’s present age = 3(p + 5)
= (3p + 15) years
(iii) Sum of their ages in 6 years’ time
= p + p + 5 + 3p + 15 + 3 × 6
= p + p + 5 + 3p + 15 + 18
= p + p + 3p + 5 + 15 + 18
= (5p + 38) years
(iv) Sum of their ages 3 years ago = p + p + 5 + 3p + 15 – 3 × 3
= p + p + 5 + 3p + 15 – 9
= p + p + 3p + 5 + 15 – 9
= (5p + 11) years
Alternatively,
Sum of their ages 3 years ago = 5p + 38 – 3 × 9
= 5p + 38 – 27
= (5p + 11) years
(b)
1
1
2
1
1
x+ y– y– x = x–
2
4
5
3
2
3
= x–
6
1
= x–
6
1
x+
3
2
x+
6
3
y
20
x–2
2x – 7
3( x – 2) 4(2 x – 7)
–
=
–
12
4
3
12
3( x – 2) – 4(2 x – 7)
=
12
3 x – 6 – 8 x + 28
=
12
3 x – 8 x – 6 + 28
=
12
–5 x + 22
=
12
6
x –1
2 x – 3 4( x – 1)
3(2 x – 3)
1
+
–
=
+
–
12
12
2
3
4
12
4( x – 1) + 6 – 3(2 x – 3)
=
12
4x – 4 + 6 – 6x + 9
=
12
4x – 6x – 4 + 6 + 9
=
12
–2 x + 11
=
12
x–4
2 x – 5 9(2 x )
x–4
3(2 x – 5)
(b) 2x +
–
=
+
–
9
3
9
9
9
9(2 x ) + x – 4 – 3(2 x – 5)
=
9
18 x + x – 4 – 6 x + 15
=
9
18 x + x – 6 x – 4 + 15
=
9
13 x + 11
=
9
2. (a)
Practise Now 5
(a)
x–3
2 x – 5 3( x – 3) 2(2 x – 5)
+
=
+
6
2
3
6
3( x – 3) + 2(2 x – 5)
=
6
3 x – 9 + 4 x – 10
=
6
3 x + 4 x – 9 – 10
=
6
7 x – 19
=
6
1
2
y– y
4
5
5
8
y–
y
20
20
Practise Now 7
(a) –10x + 25 = –5(2x – 5)
(b) 18a – 54ay + 36az = 9a(2 – 6y + 4z)
1
1
(b)
[–y – 3(16x – 3y)] = (–y – 48x + 9y)
8
8
1
= (–y + 9y – 48x)
8
1
= (8y – 48x)
8
= y – 6x
Exercise 4A
1. (a) ab + 5y
(c) 6kq
(e) 3x – 4 z
1
66
(b) f 3 – 3
2w
(d)
3 xy
2p
(f)
5q
(d) 6x – 20y + 7z – 8x + 25y – 11z
= 6x – 8x – 20y + 25y + 7z – 11z
= –2x + 5y – 4z
5. (a) Required answer = 2x + 4y + (–5y)
= 2x + 4y – 5y
= 2x – y
(b) Required answer = –b – 4a + 7b – 6a
= – 4a – 6a – b + 7b
= –10a + 6b
(c) Required answer = 6d – 4c + (–7c + 6d)
= 6d – 4c – 7c + 6d
= – 4c – 7c + 6d + 6d
= –11c + 12d
(d) Required answer = 3pq – 6hk + (–3qp + 14kh)
= 3pq – 6hk – 3qp + 14kh
= 3pq – 3qp – 6hk + 14kh
= 8hk
2. (a) 4x – 7y = 4(6) – 7(–4)
= 24 + 28
= 52
(b)
5x
5(6)
+x=
+6
3y
3(– 4)
30
+6
–12
1
= –2 + 6
2
1
=3
2
=
(c) 2x2 – y3 = 2(6)2 – (– 4)3
= 72 – (–64)
= 72 + 64
= 136
x
6
(d) 3x +
– y2 = 3(6) +
– (– 4)2
y
–4
= 18 – 1
= 16
1
– 16
2
6. (a) (a + b)2 – 3 3 xy
(b) Total value = (20x + 500y) cents
3a – b
3(3) – (–2)
3a – c 3(3) – (– 4)
7. (a)
+
=
+
2(–2)
c–b
2c
–2 – (– 4)
9+4
9+2
=
+
–4
–2 + 4
13
11
+
=
–4
2
1
1
= –3 + 5
4
2
1
=2
4
5(3) + 4(–2)
2(–2) – 3
5a + 4c
2c – a
(b)
–
=
–
–2 – 3
3(–2) + (– 4)
c–a
3c + b
–4 – 3
15 – 8
=
–
–6 – 4
–5
−7
7
=
–
–10
–5
7
2
=
+1
10
5
1
=2
10
1
– 16
2
1
2
3. (a) a(3c – b) = 3[3(6) – (–5)]
= 3(18 + 5)
= 3(23)
= 69
(b) ab2 – ac = 3(–5)2 – 3(6)
= 3(25) – 18
= 75 – 18
= 57
=
(c)
(d)
4. (a)
(b)
(c)
b
c –5
6
– =
–
a
b
3
–5
2
1
= –1 + 1
3
5
7
=–
15
b+c
a+c
–5 + 6
3+ 6
+
=
+
a
b
3
−5
9
1
=
+
3
–5
1
4
=
–1
3
5
7
= –1
15
5x + 22 – 6x – 23 = 5x – 6x + 22 – 23
= –x – 1
x + 3y + 6x + 4y = x + 6x + 3y + 4y
= 7x + 7y
6xy + 13x – 2yx – 5x = 6xy – 2yx + 13x – 5x
= 4xy + 8x
(c)
67
3 + (– 4) + 2(–2)
5(–2)
a + b + 2c
5c
–
=
–
4(– 4)
3(–2) – 3 – (– 4)
3c – a – b
4b
–10
3– 4 – 4
=
–
–16
–6 – 3 + 4
–5
5
=
–
8
–5
5
=1–
8
3
=
8
1
(d)
11. (i) Raj’s age 5 years later = (12m + 5) years
(ii) Present age of Raj’s son = 12m – 9m
= 3m years
Age of Raj’s son 5 years later = (3m + 5) years
Sum of their ages in 5 years’ time = 12m + 5 + 3m + 5
= 12m + 3m + 5 + 5
= (15m + 10) years
12. Amount of money Huixian had at first
= 8 × $w + 7 × $m + $(3w + 5m)
= $8w + $7m + $(3w + 5m)
= $(8w + 3w + 7m + 5m)
= $(11w + 12m)
5
13. (a) Number of people who order plain prata = a
2
b–c
bc ac
÷
+
b
a
3c + 4 b
=
– 4 – (–2)
(– 4)(–2) 3(–2)
+
÷
3
– 4
3(–2) + 4(– 4)
=
–4 + 2
8 –6
÷ 3 + –4
–6 – 16
2
1
–2
÷ 2 + 1
2
–22
3
1
1
=
÷4
6
11
6
=
275
15x + (–7y) + (–18x) + 4y = 15x – 7y – 18x + 4y
= 15x – 18x – 7y + 4y
= –3x – 3y
–3x + (–5y) – (–10y) – 7x = –3x – 5y + 10y – 7x
= –3x – 7x – 5y + 10y
= –10x + 5y
9x – (–2y) – 8x – (–12y) = 9x + 2y – 8x + 12y
= 9x – 8x + 2y + 12y
= x + 14y
–7x – (–15y) – (–2x) + (–6y) = –7x + 15y + 2x – 6y
= –7x + 2x + 15y – 6y
= –5x + 9y
=
8. (a)
(b)
(c)
(d)
9. (a)
(b)
(c)
(d)
10. (i)
2
b
5
2
(c) Number of people who order egg prata = c
7
(b) Number of people who order egg prata =
Exercise 4B
3
4
1
1
x+ x =
x+
x
4
3
12
12
7
=
x
12
6
5
2
1
y– y =
y–
y
5
3
15
15
1
=
y
15
15
21
3
3
– a+ a =–
a+
a
7
5
35
35
6
=
a
35
27
16
9
4
b– b=
b–
b
4
3
12
12
11
=
b
12
3p + (–q) – 7r – (–8p) – q + 2r = 3p – q – 7r + 8p – q + 2r
= 3p + 8p – q – q – 7r + 2r
= 11p – 2q – 5r
3.
4.
1
, r = –5,
2
1
11p – 2q – 5r = 11(2) – 2 –1 – 5(–5)
2
= 22 + 3 + 25
= 25 + 25
= 50
5.
(ii) When p = 2, q = –1
6.
1
–(x + 5) = –x – 5
–(4 – x) = – 4 + x
2(3y + 7) = 6y + 14
8(2y – 5) = 16y – 40
8(3a – 4b) = 24a – 32b
–3(c + 6) = –3c – 18
– 4(d – 6) = – 4d + 24
2a(x – y) = 2ax – 2ay
5(a + 2b) – 3b = 5a + 10b – 3b
= 5a + 7b
(b) 7(p + 10q) + 2(6p + 7q) = 7p + 70q + 12p + 14q
= 7p + 12p + 70q + 14q
= 19p + 84q
(c) a + 3b – (5a – 4b) = a + 3b – 5a + 4b
= a – 5a + 3b + 4b
= – 4a + 7b
(d) x + 3(2x – 3y + z) + 7z = x + 6x – 9y + 3z + 7z
= 7x – 9y + 10z
Present age of Khairul’s uncle = 4(x + 5)
= (4x + 20) years
Total cost = 4x + 6(x – y)
= 4x + 6x – 6y
= (10x – 6y) cents
Total cost of skirts Devi bought
= 7 × $x + n × $12 + (2n + 1) × $15 + 4 × $3x
= $7x + $12n + $15(2n + 1) + $12x
= $7x + $12n + $(30n + 15) + $12x
= $(7x + 12n + 30n + 15 + 12x)
= $(7x + 12x + 12n + 30n + 15)
= $(19x + 42n + 15)
(a) 4u – 3(2u – 5v) = 4u – 6u + 15v
= –2u + 15v
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2. (a)
68
9. Average monthly salary of the female employees
(b) –2a – 3(a – b) = –2a – 3a + 3b
= –5a + 3b
(c) 7m – 2n – 2(3n – 2m) = 7m – 2n – 6n + 4m
= 7m + 4m – 2n – 6n
= 11m – 8n
(d) 5(2x + 4) – 3(– 6 – x) = 10x + 20 + 18 + 3x
= 10x + 3x + 20 + 18
= 13x + 38
(e) – 4(a – 3b) – 5(a – 3b) = – 4a + 12b – 5a + 15b
= – 4a – 5a + 12b + 15b
= –9a + 27b
(f) 5(3p – 2q) – 2(3p + 2q) = 15p – 10q – 6p – 4q
= 15p – 6p – 10q – 4q
= 9p – 14q
(g) x + y – 2(3x – 4y + 3) = x + y – 6x + 8y – 6
= x – 6x + y + 8y – 6
= –5x + 9y – 6
(h) 3(p – 2q) – 4(2p – 3q – 5) = 3p – 6q – 8p + 12q + 20
= 3p – 8p – 6q + 12q + 20
= –5p + 6q + 20
(i) 9(2a + 4b – 7c) – 4(b – c) – 7(–c – 4b)
= 18a + 36b – 63c – 4b + 4c + 7c + 28b
= 18a + 36b – 4b + 28b – 63c + 4c + 7c
= 18a + 60b – 52c
(j) – 4[5(2x + 3y) – 4(x + 2y)] = – 4(10x + 15y – 4x – 8y)
= – 4(10x – 4x + 15y – 8y)
= – 4(6x + 7y)
= –24x – 28y
7. (a) Required answer = 2x – 5 – (–6x – 3)
= 2x – 5 + 6x + 3
= 2x + 6x – 5 + 3
= 8x – 2
(b) Required answer = 10x – 2y + z – (6x – y + 5z)
= 10x – 2y + z – 6x + y – 5z
= 10x – 6x – 2y + y + z – 5z
= 4x – y – 4z
(c) Required answer = – 4p – 4q + 15sr – (8p + 9q – 5rs)
= – 4p – 4q + 15sr – 8p – 9q + 5rs
= – 4p – 8p – 4q – 9q + 15sr + 5rs
= –12p – 13q + 20rs
(d) Required answer = 10a – b – 4c – 8d – (8a – 3b + 5c – 4d)
= 10a – b – 4c – 8d – 8a + 3b – 5c + 4d
= 10a – 8a – b + 3b – 4c – 5c – 8d + 4d
= 2a + 2b – 9c – 4d
8. (a) –2{3a – 4[a – (2 + a)]} = –2[3a – 4(a – 2 – a)]
= –2[3a – 4(–2)]
= –2(3a + 8)
= –6a – 16
(b) 5{3c – [d – 2(c + d)]} = 5[3c – (d – 2c – 2d)]
= 5[3c – (–2c – d)]
= 5(3c + 2c + d)
= 5(5c + d)
= 25c + 5d
2000( m + f ) – m (b + 200)
=$
f
2000 m + 2000 f – mb – 200 m
= $
f
2000 m – 200 m – mb + 2000 f
= $
f
1800 m – mb + 2000 f
= $
f
Exercise 4C
1. (a)
(b)
(c)
(d)
2. (a)
(b)
69
1
1
1
1
1
1
1
1
x+ y– x–
y = x– x+ y–
y
4
5
6
10
4
6
5
10
3
2
2
1
=
x–
x+
y–
y
10
10
12
12
1
1
=
x+
y
12
10
2
1
3
2
1
3
a – b + 2a – b = a + 2a – b – b
3
7
5
3
7
5
6
5
21
2
= a+ a–
b–
b
3
3
35
35
8
26
= a–
b
3
35
5
3
7
4
5
7
3
4
c+ d– c– d = c– c+ d– d
9
4
8
3
9
8
4
3
40
63
9
16
=
c–
c+
d–
d
72
72
12
12
23
7
=–
c–
d
72
12
28
5
9
1
5
2f – h + k – f –
k+ h
3
4
2
4
5
28
1
5
5
9
= 2f – f – h + h + k –
k
2
3
4
4
5
4
20
15
45
112
1
= f– f–
h+
h+
k–
k
2
2
12
12
20
20
67
3
5
= f–
h–
k
2
12
20
3
3
5a + 4b – 3c – 2 a – b + c
2
2
3
3
= 5a + 4b – 3c – 2a + b – c
2
2
3
3
= 5a – 2a + 4b + b – 3c – c
2
2
8
6
3
3
= 3a + b + b – c – c
2
2
2
2
11
9
= 3a +
b– c
2
2
1
1
[2x + 2(x – 3)] = (2x + 2x – 6)
2
2
1
= (4x – 6)
2
= 2x – 3
1
(c)
(d)
3. (a)
(b)
(c)
(d)
(e)
2
2
[12p – (5 + 2p)] = (12p – 5 – 2p)
5
5
2
= (12p – 2p – 5)
5
2
= (10p – 5)
5
= 4p – 2
(f)
1
1
[8x + 10 – 6(1 – 4x)] = (8x + 10 – 6 + 24x)
2
2
1
= (8x + 24x + 10 – 6)
2
1
= (32x + 4)
2
= 16x + 2
(g)
x
2x
5x
4x
+
=
+
2
5
10
10
9
=
x
10
a
a
4a
3a
–
=
–
3
4
12
12
1
=
a
12
2h
h +1
10 h
7( h + 1)
+
=
+
7
5
35
35
10 h + 7( h + 1)
=
35
10 h + 7 h + 7
=
35
17 h + 7
=
35
3x
x+2
3x
2( x + 2)
–
=
–
8
4
8
8
3 x – 2( x + 2)
=
8
3x – 2 x – 4
=
8
x–4
=
8
4x + 1
2(4 x + 1)
5(3 x – 1)
3x – 1
+
=
+
2
5
10
10
2(4 x + 1) + 5(3 x – 1)
=
10
8 x + 2 + 15 x – 5
=
10
8 x + 15 x + 2 – 5
=
10
23 x – 3
=
10
1
(h)
4. (a)
(b)
(c)
(d)
(e)
3y – 1 2 y – 3
3(3 y – 1) 2(2 y – 3)
–
=
–
12
4
6
12
3(3 y – 1) – 2(2 y – 3)
=
12
9y – 3 – 4y + 6
=
12
9y – 4y – 3 + 6
=
12
5y + 3
=
12
a–2
a+7
2( a – 2)
a+7
–
=
–
4
8
8
8
2( a – 2) – ( a + 7)
=
8
2a – 4 – a – 7
=
8
2a – a – 4 – 7
=
8
a – 11
=
8
3 p – 2q 4 p – 5q
4(3 p – 2 q ) 3(4 p – 5 q )
–
=
–
3
4
12
12
4(3 p – 2 q ) – 3(4 p – 5 q )
=
12
12 p – 8 q – 12 p + 15 q
=
12
12 p – 12 p – 8 q + 15 q
=
12
7
=
q
12
12x – 9 = 3(4x – 3)
–25y – 35 = –5(5y + 7)
27b – 36by = 9b(3 – 4y)
8ax + 12a – 4az = 4a(2x + 3 – z)
4m – 6my – 18mz = 2m(2 – 3y – 9z)
5. (a) y –
2
(9x – 3y) = y – 2(3x – y)
3
= y – 6x + 2y
= – 6x + y + 2y
= – 6x + 3y
1
{6(p + q) – 3[p – 2(p – 3q)]}
3
1
= – [6(p + q) – 3(p – 2p + 6q)]
3
1
= – [6(p + q) – 3(–p + 6q)]
3
1
= – (6p + 6q + 3p – 18q)
3
1
= – (6p + 3p + 6q – 18q)
3
1
= – (9p – 12q)
3
= –3p + 4q
(b) –
70
6. (a)
(b)
(c)
(d)
(e)
7( x + 3)
5(2 x – 5)
21( x + 3)
10(2 x – 5)
+
=
+
2
3
6
6
21( x + 3) + 10(2 x – 5)
=
6
21x + 63 + 20 x – 50
=
6
21x + 20 x + 63 – 50
=
6
41x + 13
=
6
3x – 4
3( x – 1)
2(3 x – 4)
15( x – 1)
–
=
–
5
2
10
10
2(3 x – 4) – 15( x – 1)
=
10
6 x – 8 – 15 x + 15
=
10
6 x – 15 x – 8 + 15
=
10
–9 x + 7
=
10
3( z – 2)
4(2 z – 3)
15( z – 2) 16(2 z – 3)
–
=
–
4
5
20
20
15( z – 2) – 16(2 z – 3)
=
20
15 z – 30 – 32 z + 48
=
20
15 z – 32 z – 30 + 48
=
20
–17 z + 18
=
20
2( p – 4 q )
3(2 p + q )
4( p – 4 q )
9(2 p + q )
–
=
–
3
2
6
6
4( p – 4 q ) – 9(2 p + q )
=
6
4 p – 16 q – 18 p – 9 q
=
6
4 p – 18 p – 16 q – 9 q
=
6
–14 p – 25 q
=
6
2b
3( a – 2 b )
10 b
9( a – 2 b )
–
–
=–
–
3
5
15
15
–10 b – 9( a – 2 b )
=
15
–10 b – 9 a + 18 b
=
15
–9 a – 10 b + 18 b
=
15
–9 a + 8 b
=
15
(f)
(g)
(h)
(i)
(j)
71
2( x + 3)
3x – 4
8( x + 3) 10
5(3 x – 4)
1
–
+
=
–
+
2
5
4
20
20
20
8( x + 3) – 10 + 5(3 x – 4)
=
20
8 x + 24 – 10 + 15 x – 20
=
20
8 x + 15 x + 24 – 10 – 20
=
20
23 x – 6
=
20
a +1
a+3
5a – 2
–
–
2
3
4
6( a + 1)
4( a + 3)
3(5 a – 2)
=
–
–
12
12
12
6( a + 1) – 4( a + 3) – 3(5 a – 2)
=
12
6 a + 6 – 4 a – 12 – 15 a + 6
=
12
6 a – 4 a – 15 a + 6 – 12 + 6
=
12
13
=–
a
12
x +1
x+3
5x – 1
3( x + 1)
2( x + 3)
5x – 1
+
–
=
+
–
2
3
6
6
6
6
3( x + 1) + 2( x + 3) – (5 x – 1)
=
6
3x + 3 + 2 x + 6 – 5 x + 1
=
6
3x + 2 x – 5 x + 3 + 6 + 1
=
6
10
=
6
5
=
3
2
=1
3
2( a – b )
2 a + 3b
a+b
–
+
7
14
2
4( a – b )
2 a + 3b
7( a + b )
=
–
+
14
14
14
4( a – b ) – (2 a + 3b ) + 7( a + b )
=
14
4 a – 4 b – 2 a – 3b + 7 a + 7 b
=
14
4 a – 2 a + 7 a – 4 b – 3b + 7 b
=
14
9
=
a
14
x+3
5(3 x + 4)
2( x + 3)
5(3 x + 4)
6
+
+1 =
+
+
3
6
6
6
6
2( x + 3) + 5(3 x + 4) + 6
=
6
2 x + 6 + 15 x + 20 + 6
=
6
2 x + 15 x + 6 + 20 + 6
=
6
17 x + 32
=
6
1
7. (a) –39b2 – 13ab = –13b(3b + a)
(b) 5x + 10x(b + c) = 5x[1 + 2(b + c)]
= 5x(1 + 2b + 2c)
(c) 3xy – 6x(y – z) = 3x[y – 2(y – z)]
= 3x(y – 2y + 2z)
= 3x(–y + 2z)
(d) 2x(7 + y) – 14x(y + 2) = 2x[7 + y – 7(y + 2)]
= 2x(7 + y – 7y – 14)
= 2x(y – 7y + 7 – 14)
= 2x(–6y – 7)
(e) –3a (2 + b) + 18a(b – 1) = 3a[–(2 + b) + 6(b – 1)]
= 3a(–2 – b + 6b – 6)
= 3a(–b + 6b – 2 – 6)
= 3a(5b – 8)
(f) – 4y(x – 2) – 12y(3 – x) = 4y[–(x – 2) – 3(3 – x)]
= 4y(–x + 2 – 9 + 3x)
= 4y(–x + 3x + 2 – 9)
= 4y(2x – 7)
=
=
=
=
=
Review Exercise 4
1. (a) 4a + 5b = 4(–2) + 5(7)
= –8 + 35
= 27
(b) 2a2 = 2(–2)2
=8
(c) (2a)2 = [2(–2)]2
= (– 4)2
= 16
(d) a(b – a) = (–2)[7 – (–2)]
= (–2)(7 + 2)
= (–2)(9)
= –18
(e) b – a2 = 7 – (–2)2
=7–4
=3
(f) (b – a)2 = [7 – (–2)]2
= (7 + 2)2
= 92
= 81
3(3) – 5(– 4)2 – 2(3)(– 4)(2)
3 x – 5 y 2 – 2 xyz
2.
=
2
(– 4)2
3
x y
–
–
2
–4
y
z
9 – 80 + 48
=
3 16
– –
4
2
5( p – q )
2q – p
2( p + q )
–
–
2
14
7
35( p – q )
2q – p
4( p + q )
=
–
–
14
14
14
35( p – q ) – (2 q – p ) – 4( p + q )
=
14
35 p – 35 q – 2 q + p – 4 p – 4 q
=
14
35 p + p – 4 p – 35 q – 2 q – 4 q
=
14
32
p
–
41
q
=
14
3( a – 3b ) 4( a + 2 b )
2a + b
–
(b) –
–
2
5
3
8. (a)
2a + b
3( a – 3b )
4( a + 2 b )
–
+
3
2
5
10(2 a + b )
45( a – 3b )
24( a + 2 b )
=–
–
+
30
30
30
–10(2 a + b ) – 45( a – 3b ) + 24( a + 2 b )
=
30
–20 a – 10 b – 45 a + 135 b + 24 a + 48 b
=
30
–20 a – 45 a + 24 a – 10 b + 135 b + 48 b
=
30
– 41a + 173b
=
30
3( f – h )
7( h + k )
5( k – f )
(c)
–
+
4
6
2
9( f – h ) 14( h + k )
30( k – f )
=
–
+
12
12
12
9( f – h ) – 14( h + k ) + 30( k – f )
=
12
9 f – 9 h − 14 h – 14 k + 30 k – 30 f
=
12
9 f – 30 f – 9 h – 14 h – 14 k + 30 k
=
12
–21 f – 23h + 16 k
=
12
=–
1
x–y
3( y + 4 z )
5( x + 3z )
–
+
3
4
8
96
8( x – y ) 18( y + z )
15( x + 3z )
–
–
+
24
24
24
24
96 – 8( x – y ) – 18( y + z ) + 15( x + 3z )
24
96 – 8 x + 8 y – 18 y – 18 z + 15 x + 45 z
24
96 – 8 x + 15 x + 8 y – 18 y – 18 z + 45 z
24
96 + 7 x – 10 y + 27 z
24
(d) 4 –
=
–23
3 32
– –
4
4
–23
35
–
4
22
=2
35
3. (a) 3ab – 5xy + 4ab + 2yx = 3ab + 4ab – 5xy + 2yx
= 7ab – 3xy
(b) 4(3p – 5q) + 6(2q – 5p) = 12p – 20q + 12q – 30p
= 12p – 30p – 20q + 12q
= –18p – 8q
=
72
(c) 2a + 3[a – (b – a)] + 7(2b – a) = 2a + 3(a – b + a) + 7(2b – a)
= 2a + 3(a + a – b) + 7(2b – a)
= 2a + 3(2a – b) + 7(2b – a)
= 2a + 6a – 3b + 14b – 7a
= 2a + 6a – 7a – 3b + 14b
= a + 11b
(d) –2[3x – (4x – 5y) – 2(3x – 4y)] = –2(3x – 4x + 5y – 6x + 8y)
= –2(3x – 4x – 6x + 5y + 8y)
= –2(–7x + 13y)
= 14x – 26y
(e) 4{h – 3[ f – 6( f – h)]} = 4[h – 3( f – 6f + 6h)]
= 4[h – 3(–5f + 6h)]
= 4(h + 15f – 18h)
= 4(15f + h – 18h)
= 4(15f – 17h)
= 60f – 68h
(f) 5(x + 5y) – {2x – [3x – 3(x – 2y) + y]}
= 5(x + 5y) – [2x – (3x – 3x + 6y + y)]
= 5(x + 5y) – (2x – 7y)
= 5x + 25y – 2x + 7y
= 5x – 2x + 25y + 7y
= 3x + 32y
(d)
(e)
2x
5–x
8x
3(5 – x )
+
=
+
3
4
12
12
8 x + 3(5 – x )
=
12
8 x + 15 – 3 x
=
12
8 x – 3 x + 15
=
12
5 x + 15
=
12
x–y
3x – 2 y
3( x – y )
2(3 x – 2 y )
(b)
–
=
–
8
12
24
24
3( x – y ) – 2(3 x – 2 y )
=
24
3x – 3y – 6 x + 4 y
=
24
3x – 6 x – 3y + 4 y
=
24
−3 x + y
=
24
4(2 a – b )
2(3a + b )
20(2 a – b )
6(3a + b )
(c)
–
=
–
3
5
15
15
20(2 a – b ) − 6(3a + b )
=
15
40 a – 20 b – 18 a – 6 b
=
15
40 a – 18 a – 20 b – 6 b
=
15
22 a – 26 b
=
15
4. (a)
(f)
5. (a)
(b)
6. (a)
(b)
h+ f
f +k
4h – k
–
+
3
2
5
10( h + f ) 15( f + k )
6(4 h – k )
=
–
+
30
30
30
10( h + f ) – 15( f + k ) + 6(4 h – k )
=
30
10 h + 10 f – 15 f – 15 k + 24 h – 6 k
=
30
10 f – 15 f + 10 h + 24 h – 15 k – 6 k
=
30
–5 f + 34 h – 21k
=
30
4 p – 3q
q – 4p
3q –
–
5
6
90 q
6(4 p – 3q ) 5( q – 4 p )
=
–
–
30
30
30
90 q – 6(4 p – 3q ) – 5( q – 4 p )
=
30
90 q – 24 p + 18 q – 5 q + 20 p
=
30
–24 p + 20 p + 90 q + 18 q – 5 q
=
30
– 4 p + 103q
=
30
4( x – 5)
5( x – y ) 7 x – y
–
+
6
21
7
4( x – 5)
5( x – y )
7x – y
=
–
–
7
6
21
24( x – 5)
35( x – y )
2(7 x – y )
=
–
–
42
42
42
24( x – 5) – 35( x – y ) – 2(7 x – y )
=
42
24 x – 120 – 35 x + 35 y – 14 x + 2 y
=
42
24 x – 35 x – 14 x + 35 y + 2 y – 120
=
42
–25 x + 37 y – 120
=
42
21pq + 14q – 28qr = 7q(3p + 2 – 4r)
4x – 8(y – 2z) = 4[x – 2(y – 2z)]
= 4(x – 2y + 4z)
Total value of 5-cent coins = 5x cents
Total value of 10-cent coins = (3x × 10) cents
= 30x cents
3
x
7
3
5 x + 7 x × 10 cents
30
5 x + 7 x cents
30
35
7 x + 7 x cents
65
x cents
7
(c) Number of 10-cent coins =
Total value of coins =
=
=
=
73
1
x
3 × 60
x
=
km
180
xy
Distance Farhan can cycle in y minutes =
km
180
8. (a) Required difference = 3y × 60 – 25y
= 180y – 25y
= 155y seconds
(b) Required sum = 50(3z – 2) × 60 + 4(z + 1) × 3600
= 3000(3z – 2) + 14 400(z + 1)
= 9000z – 6000 + 14 400z + 14 400
= 9000z + 14 400z – 6000 + 14 400
= (23 400z + 8400) seconds
9. (i) Total amount Shirley earned = $[(25 – 5) × x + 5 × 1 .5x]
= $(20 × x + 5 × 1 .5x)
= $(20x + 7 .5x)
= $27 .5x
Total amount Kate earned = $[(18 – 4) × y + 4 × 1 .5y]
= $(14 × y + 4 × 1 .5y)
= $(14y + 6y)
= $20y
Total amount they earned = $(27 .5x + 20y)
(ii) Amount Kate was paid per hour = $5 .50 + $0 .50
= $6
Total amount they earned = $[27 .5(5 .5) + 20(6)]
= $(151 .25 + 120)
= $271 .25
10. (i) Total score obtained by Michael in the first two papers
= p – 3q + 13 + 3p + 5q – 4
= p + 3p – 3q + 5q + 13 – 4
= (4p + 2q + 9) marks
(ii) Score obtained by Michael in the third paper
= 10p + 5q – (4p + 2q + 9)
= 10p + 5q – 4p – 2q – 9
= 10p – 4p + 5q – 2q – 9
= (6p + 3q – 9) marks
(iii) 6p + 3q – 9 = 3(2p + q – 3)
3. Let the two numbers be xy and xz , where y + z = 10 .
xy = 10x + y
xz = 10x + z
\ xy × xz = (10x + y)(10x + z)
= 10x(10x + z) + y(10x + z)
= 100x2 + 10xz + 10xy + yz
= 100x2 + 10x(y + z) + yz
= 100x2 + 10x(10) + yz
= 100x2 + 100x + yz
= 100x(x + 1) + yz
7. Distance Farhan can cycle in 1 minute =
Challenge Yourself
1. Let the number of heads up in the pile of 5 be x .
Then the number of tails up in the pile of 5 is 5 – x,
the number of heads up in the pile of 7 is 5 – x .
After the teacher flips over all the coins in the pile of 5,
the number of heads up in that pile is 5 – x .
Hence, both piles now have the same number of heads up . (shown)
2. The only possible set of values is {x = 2, y = 3, z = 6} .
Proofs
1
1
1
+
+ < 1 .
z
y
x
1
1
1
If x 3, then y, z > 3 and
+
+
< 1 .
z
y
x
If x = 2 and y 4, then z 5 and
1
74
628
6.8 × 60
630
≈
m/s
7 × 60
a 2 bd
12 (2)(–3)
7. (a)
=
3ac – d
3(1)(0) – (–3)
–6
=
0+3
–6
=
3
= –2
Revision Exercise A1
6. Average speed =
1. (a) 42 = 2 × 3 × 7
66 = 2 × 3 × 11
78 = 2 × 3 × 13
HCF of 42, 66 and 78 = 2 × 3
=6
(b) 7 = 7
13 = 13
14 = 2 × 7
LCM of 7, 13 and 14 = 2 × 7 × 13
= 182
2. (i) Greatest whole number which is a factor of both 405 and 1960
= HCF of 405 and 1960
=5
(ii) Smallest whole number that is divisible by both 405 and 1960
= LCM of 405 and 1960
= 23 × 34 × 5 × 72
= 158 760
3. (i) 105 = 3 × 5 × 7
126 = 2 × 32 × 7
HCF of 105 and 126 = 3 × 7
= 21
Greatest number of students that the refreshment can cater to
= 21
(ii) Number of bags of crisps each student will receive = 105 ÷ 21
=5
(iii) Number of packets of fruit juice each student will receive
= 126 ÷ 21
=6
4. (i) Pairs of cards that have a sum of 4 = {–2, 6}, {–1, 5}, {1, 3}
(ii) Pairs of cards that have a product of 2 = {–2, –1}, {1, 2}
(iii) Groups of three cards that have a sum of 10
= {–1, 5, 6}, {1, 3, 6}, {1, 4, 5}, {2, 3, 5}
5. (a) 101 ×
(b)
(c)
3
bc + d 2
2(0) + (–3)2
=
a+b
1+ 2
0+9
=
3
9
=
3
=3
(c) a2 + b2 – c2 + d2 = 12 + 22 – 02 + (–3)2
=1+4–0+9
= 14
3
3
3
3
(d) –a – b + c – d = –13 – 23 + 03 – (–3)3
= –1 – 8 + 0 – (–27)
= –9 + 0 – (–27)
= –9 – (–27)
= –9 + 27
= 18
8. Cost of a pear = (a + b) cents
(b)
10 a 12( a + b )
Total cost = $
+
100
100
10 a + 12( a + b )
=$
100
10 a + 12 a + 12 b
= $
100
22 a + 12 b
= $
100
80.7 ≈ 100 × 81
= 100 × 9
= 900
2(11a + 6 b )
=$
100
26 × 502 ÷ 49 ≈ 3 27 × 500 ÷ 50
= 3 × 500 ÷ 50
= 1500 ÷ 50
= 30
65 ×
3
= $ 11a + 6 b
50
63 ÷ 17 ≈ 64 × 3 64 ÷ 16
= 8 × 4 ÷ 16
= 32 ÷ 16
=2
75
1
6. Number of buttons in Box B after Kate transfers 15 buttons from
Box A to Box B
= 35 + 15
= 50
Number of buttons in Box A after Kate transfers 15 buttons from
Box A to Box B
Revision Exercise A2
1. (a) 54 = 2 × 33
126 = 2 × 32 × 7
342 = 2 × 32 × 19
HCF of 54, 126 and 342 = 2 × 32
= 18
(b) 16 = 24
28 = 22 × 7
44 = 22 × 11
68 = 22 × 17
LCM of 16, 18, 44 and 68 = 24 × 7 × 11 × 17
= 20 944
10
2
2. (a) 9216 = 2 × 3
= 50 ÷
=
1
9
×2
10
10
≈ (4 × 3) m2
(ii) Cost of carpet ≈ 4 × 3 × $89 .75
≈ $(4 × 3 × 90)
8. Required answer = –8x + 9 + 15 – 4x – (–7x + 4 + 5x + 7)
= –8x + 9 + 15 – 4x – (–7x + 5x + 4 + 7)
= –8x + 9 + 15 – 4x – (–2x + 11)
= –8x + 9 + 15 – 4x + 2x – 11
= –8x – 4x + 2x + 9 + 15 – 11
= –10x + 13
3
1
4
2
2
1
4
– –3
+3
–
+ – =
3 20
20
5
5 3
61
2
4
=
+
–
3
5
20
40
183
48
=
+
–
60
60
60
40 + 183 – 48
=
60
175
=
60
35
=
12
11
=2
12
(b) (i) [– 4 .749 – 6 .558 × (–2 .094)3] ÷
3
–1.999
= – 44 .030 (to 3 d .p .)
2
5
8
1
(ii) – 3
– (–0.375)3 × [–p ÷ (–6 .5)]
× –
6
33
3
= 0 .313 (to 3 d .p .)
1
7
51
7. (i) Area of carpet = 4
3
3 6
8000 = 2 × 5
= 22 × 5
= 20
2
2
3. 1764 = 2 × 3 × 72
36 = 22 × 32
8820 = 22 × 32 × 5 × 72
\ Value of p = 22 × 32 × 5
= 180
4. (i) Temperature of town at 6 p .m . = – 6 °C + 8° C – 4 °C
= –2 °C
(ii) Overall increase = –2 °C – (– 6 °C)
= –2 °C + 6 °C
= 4 °C
5. (a)
50 ×
= 70
Initial number of buttons in Box A = 70 + 15
= 85
\ – 9216 = – 210 × 32
= –(25 × 3)
= –96
(b) 8000 = 26 × 53
\
10
5
7
76
Chapter 5 Linear Equations and Simple Inequalities
TEACHING NOTES
Suggested Approach
Since many Secondary 1 students are still in the concrete operational stage (according to Piaget), teaching students
how to solve linear equations in one variable with the use of algebra discs on a balance can help them to learn
the concepts more easily. However, there is still a need to guide students to move from the ‘concrete’ to the
‘abstract’, partly because they cannot use this approach in examinations, and partly because they cannot use this
approach to solve linear equations which consist of algebraic terms that have large or fractional coefficients (see
Section 5.1). After students learn how to solve linear equations, they will learn how to evaluate an unknown in a formula
and formulate linear equations to solve problems in real-world contexts. Since the concept of inequality is harder than that of
equation, students only learn how to solve inequalities towards the end of this chapter.
Section 5.1:
Linear Equations
Students have learnt how to complete mathematical sentences such as 7 +
= 13 in primary school. Teachers
can introduce equations by telling students that when we replace
with x, we have 7 + x = 13, which is an
equation. Teachers should illustrate the meaning of ‘solving an equation’ using appropriate examples. Students
should know the difference between linear expressions and linear equations.
Teachers can use the ‘Balance Method’ to show how to solve linear equations which do not involve any brackets
before illustrating how to solve those which involve brackets. As this approach cannot be used to solve linear
equations which consist of algebraic terms that have large or fractional coefficients, so there is a need to help
students consolidate what they have learnt in Worked Examples 1, 2 and 3. The thinking time on page 115
of the textbook reinforces students’ understanding of the concept of equation. For example, since x + 3 = 6,
2x + 3 = 9 and 10x – 4 = 5x + 11 are equivalent equations that can be obtained from x = 3, then the value of x in
each of the equations is 3.
Section 5.2:
Formulae
Teachers can use simple formulae such as A = lb, where A, l and b are the area, the length and the breadth of
the rectangle respectively, to let students understand that a formula makes use of variables to write instructions
for performing a calculation. Teachers may get students to provide examples of formulae which they have
encountered in mathematics and the sciences.
Section 5.3:
Applications of Linear Equations in Real-World Contexts
Teachers should illustrate how a word problem is solved using the model method before showing how the same
problem can be solved using the algebraic method. Students should observe how the algebraic method is linked
to the model method. Also, students should be aware why they need to learn the algebraic method. In this section,
students are given ample opportunities to formulate linear equations to solve problems in real-world contexts.
Section 5.4:
Simple Inequalities
In the investigation on page 126 of the textbook, students are required to work with numerical examples before
generalising the conclusions for some properties of inequalities. In Secondary 1, students only need to know how
to solve linear inequalities of the form ax b, ax b, ax < b and ax > b, where a and b are integers and a > 0.
Teachers should get students to formulate inequalities based on real-world contexts (see the journal writing on
page 128 of the textbook).
77
1
WORKED SOLUTIONS
Journal Writing (Page 128)
Journal Writing (Page 113)
•
1. To solve an equation in x means to find the value of x so that the
values on both sides of the equation are equal, i.e. x satisfies the
equation.
2. The operations should be applied to both sides of the equation such
that the equation is simplified to the form ax = b, where a and b are
b
constants. Thus x = .
a
•
Teachers may wish to point out common mistakes that students
may make in solving a linear equation in order to extract their
understanding of the process.
Thinking Time (Page 115)
Some equivalent equations that have the solution y = –1:
• y = –1
• y+1=0
• y – 1= –2
• 3y + 8 = 5
• 2y – 1 = –3
• 2y – 1 = –3
• 10y + 2 = 13y + 5
• 2(2y – 3) = 5(y – 1)
Teachers may wish to note that the list is not exhaustive.
Practise Now (Page 110)
x+3
x+3–3
\x
(b)
x–7
x–7+7
\x
(c)
x+3
x+3–3
\x
(d)
x–2
x–2+2
\x
(a)
Investigation (Properties of Inequalities)
1.
Cases
Working
Multiplication LHS = 10 × 5
by a positive
= 50
number on
RHS = 6 × 5
both sides of
= 30
the inequality
10 > 6
Division by
a positive
number on
both sides of
the inequality
10 > 6
LHS = 10 ÷ 5
=2
Inequality
50 > 30
Is the
inequality
Conclusion
sign
reversed?
No
If x > y and
c > 0, then
cx > cy.
2x – 5 = 5
2x – 5 + 5 = 5 + 5
2x = 10
\x =5
(b)
3x + 4 = 7
3x + 4 – 4 = 7 – 4
3x = 3
\x =1
(c)
–3x + 3 = 9
–3x + 3 – 3 = 9 – 3
–3x = 6
3x = –6
\ x = –2
(d)
–5x – 2 = 13
–5x – 2 + 2 = 13 + 2
–5x = 15
5x = –15
\ x = –3
(a)
2 > 1.2
No
RHS = 6 ÷ 5
= 1.2
If x > y and
c > 0, then
x y
> .
c c
2. Yes, the conclusions drawn from Table 5.3 apply to 10 6.
The following conclusions hold for x y:
If x y and c > 0, then cx cy and
The following conclusions hold for x < y:
•
If x < y and c > 0, then cx < cy and
•
If x y and c > 0, then cx cy and
x
y
.
c
c
x
y
< .
c
c
The following conclusions hold for x y:
1
=7
=7–3
=4
=6
=6+7
= 13
= –7
= –7 – 3
= –10
= –3
= –3 + 2
= –1
Practise Now (Page 111)
Table 5.3
•
A bowl of rice contains 5 g of protein. A teenager needs a minimum
of 49 g of protein each day. It is given that he only eats rice on a
particular day. The inequality which we need to set up to find the
least number of bowls of rice he needs to eat in order to meet his
minimum protein requirement that day is:
5x 49,
where x represents the number of bowls of rice he needs to eat that
day.
The flag-down fare of a taxi is $5. The taxi charges $0.30 for each
385 m it travels. A person has not more than $50 to spend on his taxi
ride. The inequality which we need to set up to find the maximum
distance that he can travel on the taxi is:
30x 4500,
where x is the number of blocks of 385 m.
x
y
.
c
c
78
(d)
Practise Now (Page 112)
3x + 4 = x – 10
3x – x + 4 = x – x – 10
2x + 4 = –10
2x + 4 – 4 = –10 – 4
2x = –14
\ x = –7
(b)
4x – 2 = x + 7
4x – x – 2 = x – x + 7
3x – 2 = 7
3x – 2 + 2 = 7 + 2
3x = 9
\x=3
(c)
3x – 2 = –x + 14
3x + x – 2 = –x + x + 14
4x – 2 = 14
4x – 2 + 2 = 14 + 2
4x = 16
\x =4
(d)
–2x – 5 = 5x – 12
–2x – 5x – 5 = 5x – 5x – 12
–7x – 5 = –12
–7x – 5 + 5 = –12 + 5
–7x = –7
7x = 7
x=1
(a)
–2(3x – 4)
–6x + 8
–6x – 8x + 8
–14x + 8
–14x + 8 – 8
–14x
14x
= 4(2x + 5)
= 8x + 20
= 8x – 8x + 20
= 20
= 20 – 8
= 12
= –12
\x =–
6
7
Practise Now 1
x+9=4
x+9–9=4–9
\ x = –5
(b)
3x – 2 = 4
3x – 2 + 2 = 4 + 2
3x = 6
1. (a)
3x
6
=
3
3
\x =2
(c)
7x + 2 = 2x – 13
7x – 2x + 2 = 2x – 2x – 13
5x + 2 = –13
5x + 2 – 2 = –13 – 2
5x = –15
5x
–15
=
5
5
\ x = –3
(d)
3(3y + 4) = 2(2y + 1)
9y + 12 = 4y + 2
9y – 4y + 12 = 4y – 4y + 2
5y + 12 = 2
5y + 12 – 12 = 2 – 12
5y = –10
Practise Now (Page 113)
2(x – 3) = –3x + 4
2x – 6 = –3x + 4
2x + 3x – 6 = –3x + 3x + 4
5x – 6 = 4
5x – 6 + 6 = 4 + 6
5x = 10
\x=2
(b)
2(x + 3) = 5x – 9
2x + 6 = 5x – 9
2x – 5x + 6 = 5x – 5x – 9
–3x + 6 = –9
–3x + 6 – 6 = –9 – 6
–3x = –15
3x = 15
\x=5
(c)
–2(x + 2) = 3x – 9
–2x – 4 = 3x – 9
–2x – 3x – 4 = 3x – 3x – 9
–5x – 4 = –9
–5x – 4 + 4 = –9 + 4
–5x = –5
5x = 5
\x=1
(a)
5y
–10
=
5
5
\ y = –2
(e) 2(y – 1) + 3(y – 1) = 4 – 2y
2y – 2 + 3y – 3 = 4 – 2y
2y + 3y – 2 – 3 = 4 – 2y
5y – 5 = 4 – 2y
5y + 2y – 5 = 4 – 2y + 2y
7y – 5 = 4
7y – 5 + 5 = 4 + 5
7y = 9
7y 9
=
7
7
2
\y=1
7
79
1
x + 0.7 = 2.7
x + 0.7 – 0.7 = 2.7 – 0.7
\x=2
(b)
2y – 1.3 = 2.8
2y – 1.3 + 1.3 = 2.8 + 1.3
2y = 4.1
2. (a)
Practise Now 3
(a)
2y
4.1
=
2
2
\ y = 2.05
Practise Now 2
(a)
=5–9
= –4
= 2 × (– 4)
= –8
5
1
1
y+2= y+3
7
2
4
5
1
1
1
1
y– y+2= y– y+3
7
2
2
2
4
3
1
y+2=3
4
14
3
1
y+2–2 =3 –2
4
14
3
1
y=1
4
14
14
3
14
1
×
×1
y=
4
3
14
3
5
\y=5
6
3z – 1
z–4
(c)
=
2
3
3z – 1
z–4
6×
=6×
2
3
3(3z – 1) = 2(z – 4)
9z – 3 = 2z – 8
9z – 2z – 3 = 2z – 2z – 8
7z – 3 = –8
7z – 3 + 3 = –8 + 3
7z = –5
7z
5
=–
7
7
5
\z =–
7
(b)
1
=4
= (2x – 3) × 4
= 4(2x – 3)
= 8x – 12
=8
= 8 + 12
= 20
20
8x
=
8
8
1
\x =2
2
y–3
3
=
(b)
y+4
2
y–3
3
2(y + 4) ×
= 2(y + 4) ×
y+4
2
2(y – 3) = 3(y + 4)
2y – 6 = 3y + 12
2y – 3y – 6 = 3y – 3y + 12
–y – 6 = 12
–y – 6 + 6 = 12 + 6
–y = 18
\ y = –18
x
+9 =5
2
x
+9–9
2
x
2
x
2×
2
\x
8
2x – 3
8
(2x – 3) ×
2x – 3
8
8
8x – 12
8x – 12 + 12
8x
Practise Now 4
F = ma
(a) When m = 1000, a = 0.05,
F = 1000(0.05)
= 50 N
Net force acting on body = 50 N
(b) When F = 100, a = 0.1,
100 = m(0.1)
100
0.1
= 1000 kg
Mass of body = 1000 kg
\m=
80
2. Let the number of marks Lixin obtains be x.
Then the number of marks Devi obtains is x + 15.
x + 15 = 2x
x – 2x = –15
–x = –15
\ x = 15
Lixin obtains 15 marks.
Practise Now 5
1.
x
2 x + y – 3z
=
2y
y + 3x
When x = 1, y = 4,
2 (1) + 4 – 3z
4 + 3 (1)
2 + 4 – 3z
4+3
6 – 3z
7
8(6 – 3z)
48 – 24z
–24z
–24z
1
2(4)
1
=
8
1
=
8
=7
=7
= 7 – 48
= – 41
=
Practise Now 8
Let the number be x.
1
7
=7
x+3
5
10
1
7
x =7–3
5
10
1
3
x =3
5
10
3
\x =5×3
10
1
= 16
2
1
The number is 16 .
2
− 41
–24
17
=1
24
\z =
v–u
a
2. t =
When t = 3, v = 2
1
1
,u=1 ,
2
3
1
1
–1
2
3
3=
a
1
1
3= 6
a
2
Practise Now 9
1. (a) 15x > 75
x>
1
6
1
\a=1 ÷3
6
7
=
18
3a = 1
75
15
\x>5
4
5
6
7
–6
–5
–4
–3
8
9
(b) 4x –16
–16
x
4
x –4
Practise Now 6
1
(i) A = pr2
2
(ii) When r = 5,
2. 6x > 7
7
6
1
x >1
6
\ The smallest integer value of x is 2.
x>
1
(3.142)(5)2
2
= 39.275 cm2
Area of semicircle = 39.275 cm2
A =
Practise Now 10
Practise Now 7
Let the number of buses that are needed to ferry 520 people be x.
Then 45x 520
1. Let the smaller number be x.
Then the larger number is 5x.
x + 5x = 24
6x = 24
520
45
5
x 11
9
\ The minimum number of buses that are needed to ferry 520 students
is 12.
x
24
6
=4
\ The two numbers are 4 and 20.
x=
81
1
Exercise 5A
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
2. (a)
(f)
x + 8 = 15
x + 8 – 8 = 15 – 8
\x =7
x + 9 = –5
x + 9 – 9 = –5 – 9
\ x = –14
x – 5 = 17
x – 5 + 5 = 17 + 5
\ x = 22
y – 7 = –3
y – 7 + 7 = –3 + 7
\y =4
y + 0.4 = 1.6
y + 0.4 – 0.4 = 1.6 – 0.4
\ y = 1.2
y – 2.4 = 3.6
y – 2.4 + 2.4 = 3.6 + 2.4
y =6
–2.7 + a = – 6.4
–2.7 + 2.7 + a = – 6.4 + 2.7
\ a = –3.7
4x = –28
7 y 21
=
7
7
\y =3
(g)
4y – 1.9 = 6.3
4y – 1.9 + 1.9 = 6.3 + 1.9
4y = 8.2
4y
8.2
=
4
4
\ y = 2.05
(h)
–3y – 7.8 = –9.6
–3y – 7.8 + 7.8 = –9.6 + 7.8
–3y = –1.8
3y = 1.8
3y
1.8
=
3
3
\ y = 0.6
3
1
=
4
2
3
3
1
3
7y – 2 + 2 =
+2
4
4
2
4
1
7y = 3
4
7y
1
=3 ÷7
7
4
13
\y =
28
1
1
(j)
1 – 2y =
2
4
1
1
1
1
1 – 1 – 2y =
–1
2
2
4
2
1
–2y = –1
4
1
2y = 1
4
2y
1
=1 ÷2
2
4
5
\y =
8
3. (a)
3x – 7 = 4 – 8x
3x + 8x – 7 = 4 – 8x + 8x
11x – 7 = 4
11x – 7 + 7 = 4 + 7
11x = 11
(i)
4x
–28
=
4
4
\ x = –7
(b) –24x = –144
24x = 144
24 x 144
=
24
24
\x =6
(c)
3x – 4 = 11
3x – 4 + 4 = 11 + 4
3x = 15
3x
15
=
3
3
\x =5
(d)
9x + 4 = 31
9x + 4 – 4 = 31 – 4
9x = 27
9x
27
=
9
9
\x =3
(e)
12 – 7x = 5
12 – 12 – 7x = 5 – 12
–7x = –7
7x = 7
7y – 2
11x
11
=
11
11
\x =1
(b)
4x – 10 = 5x + 7
4x – 5x – 10 = 5x – 5x + 7
–x – 10 = 7
–x – 10 + 10 = 7 + 10
–x = 17
7x
7
=
7
7
1
\x=1
3 – 7y = –18
3 – 3 – 7y = –18 – 3
–7y = –21
7y = 21
82
(c)
(d)
4. (a)
(b)
(c)
(d)
\ x = –10
2(2x – 2.2) = 4.6
4x – 4.4 = 4.6
4x – 4.4 + 4.4 = 4.6 + 4.4
4x = 9
4x
9
=
4
4
\ x = 2.25
(f)
4(3y + 4.1) = 7.6
12y + 16.4 = 7.6
12y + 16.4 – 16.4 = 7.6 – 16.4
12y = –8.8
\ x = –17
30 + 7y = –2y – 6
30 + 7y + 2y = –2y + 2y – 6
30 + 9y = – 6
30 – 30 + 9y = – 6 – 30
9y = –36
9y
–36
=
9
9
\ y = –4
2y – 7 = 7y – 27
2y – 7y – 7 = 7y – 7y – 27
–5y – 7 = –27
–5y – 7 + 7 = –27 + 7
–5y = –20
5y = 20
5y
20
=
5
5
\y =4
2(x + 3) = 8
2x + 6 = 8
2x + 6 – 6 = 8 – 6
2x = 2
2x
2
=
2
2
\x=1
5(x – 7) = –15
5x – 35 = –15
5x – 35 + 35 = –15 + 35
5x = 20
5x
20
=
5
5
\x =4
7(–2x + 4) = – 4x
–14x + 28 = – 4x
–14x + 4x + 28 = – 4x + 4x
–10x + 28 = 0
–10x + 28 – 28 = 0 – 28
–10x = –28
10x = 28
10 x
28
=
10
10
4
\x =2
5
3(2 – 0.4x) = 18
6 – 1.2x = 18
6 – 6 – 1.2x = 18 – 6
–1.2x = 12
1.2x = –12
1.2 x
–12
=
1.2
1.2
(e)
(g)
(h)
(i)
(j)
12 y
–8.8
=
12
12
11
\y =–
15
3(2y + 3) = 4y + 3
6y + 9 = 4y + 3
6y – 4y + 9 = 4y – 4y + 3
2y + 9 = 3
2y + 9 – 9 = 3 – 9
2y = –6
–6
2y
=
2
2
\ y = –3
3(y + 1) = 4y – 21
3y + 3 = 4y – 21
3y – 4y + 3 = 4y – 4y – 21
–y + 3 = –21
–y + 3 – 3 = –21 – 3
–y = –24
\ y = 24
3(y + 2) = 2(y + 4)
3y + 6 = 2y + 8
3y – 2y + 6 = 2y – 2y + 8
y+6=8
y+6–6=8–6
\y=2
5(5y – 6) = 4(y – 7)
25y – 30 = 4y – 28
25y – 4y – 30 = 4y – 4y – 28
21y – 30 = –28
21y – 30 + 30 = –28 + 30
21y = 2
21y
2
=
21
21
2
\y =
21
(k) 2(3b – 4) = 5(b + 6)
6b – 8 = 5b + 30
6b – 5b – 8 = 5b – 5b + 30
b – 8 = 30
b – 8 + 8 = 30 + 8
83
1
(l)
\ b = 38
3(2c + 5) = 4(c – 3)
6c + 15 = 4c – 12
6c – 4c + 15 = 4c – 4c – 12
2c + 15 = –12
2c + 15 – 15 = –12 – 15
2c = –27
(d)
y
–8+8
4
y
4
y
4×
4
\y
2c
–27
=
2
2
1
\ c = –13
2
(m)
9(2d + 7) = 11(d + 14)
18d + 63 = 11d + 154
18d – 11d + 63 = 11d – 11d + 154
7d + 63 = 154
7d + 63 – 63 = 154 – 63
7d = 91
(c)
6. (a)
1
x =7
3
1
3× x=3×7
3
\ x = 21
3
x = –6
4
4
3
4
× x=
× (– 6)
3
4
3
\ x = –8
(b)
1
x+3=4
3
1
x + 3 – 3= 4 – 3
3
1
x=1
3
1
3× x=3×1
3
\x=3
1
=6
=4×6
= 24
=2
=2–3
= –1
=1
=4×1
=4
2
(f)
15 – y = 11
5
2
15 – 15 – y = 11 – 15
5
2
– y = –4
5
2
y=4
5
5 2
5
x y=
×4
2 5
2
\ y = 10
7f
–13
=
7
7
6
\ f = –1
7
(b)
= –2 + 8
1
(e)
3– y
4
1
3–3– y
4
1
– y
4
1
y
4
1
4× y
4
\y
7d
91
=
7
7
\ d = 13
(n)
5(7f – 3) = 28( f – 1)
35f – 15 = 28f – 28
35f – 28f – 15 = 28f – 28f – 28
7f – 15 = –28
7f – 15 + 15 = –28 + 15
7f = –13
5. (a)
y
– 8 = –2
4
84
1
x
3
1
1
1
x + x = 12 – x + x
3
3
3
4
x = 12
3
3
4
3
× x=
× 12
4
3
4
\x =9
x = 12 –
3
x
5
3
1
x– x
2
5
1
x
10
1
10 ×
x
10
\x
1
1
x+
2
2
1
1
1
= x– x+
2
2
2
1
=
2
1
= 10 ×
2
=5
=
(c)
y
2
5y
6
y
2
y
+
3
5y
6
1
–
5
1
5
1
–
5
1
–
5
1
+
5
5y
6
6
5y
×
5
6
–
–3(2 – x) = 6x
– 6 + 3x = 6x
– 6 + 3x – 6x = 6x – 6x
– 6 –3x = 0
– 6 + 6 – 3x = 0 + 6
–3x = 6
3x = – 6
3x
–6
=
3
3
\ x = –2
(b)
5 – 3x = – 6(x + 2)
5 – 3x = – 6x – 12
5 – 3x + 6x = – 6x + 6x – 12
5 + 3x = –12
5 – 5 + 3x = –12 – 5
3x = –17
3x
–17
=
3
3
2
\ x = –5
3
(c)
–3(9y + 2) = 2(– 4y – 7)
–27y – 6 = –8y – 14
–27y + 8y – 6 = –8y + 8y – 14
–19y – 6 = –14
–19y – 6 + 6 = –14 + 6
–19y = –8
19y = 8
19 y
8
=
19
19
8
\y =
19
(d)
–3(4y – 5) = –7(–5 – 2y)
–12y + 15 = 35 + 14y
–12y – 14y + 15 = 35 + 14y – 14y
–26y + 15 = 35
–26y + 15 – 15 = 35 – 15
–26y = 20
26y = –20
8. (a)
y
3
y
y
=2–
+
3
3
=2–
=2
=2+
=2
1
5
1
5
6
1
×2
5
5
16
\y=2
25
2
3
5
(d)
y–
= 2y +
3
4
8
2
3
5
y – 2y –
= 2y – 2y +
3
4
8
4
3
5
– y–
=
3
4
8
4
3
3
5
3
– y–
+
=
+
3
4
4
8
4
4
3
– y =1
3
8
4
3
y = –1
3
8
3
4
3 –1 3
×
y=
×
8
4
3
4
1
\ y = –1
32
2
4
7. (a)
=
x
5
2
4
5x ×
= 5x ×
x
5
10 = 4x
4x = 10
=
4x
10
=
4
4
1
\x =2
2
12
2
(b)
=
y –1
3
12
2
3(y – 1) ×
= 3(y – 1) ×
y –1
3
36 = 2(y – 1)
36 = 2y – 2
2y – 2 = 36
2y – 2 + 2 = 36 + 2
2y = 38
2y
38
=
2
2
\ y = 19
26 y
–20
=
26
26
10
\y =–
13
(e) 3(5 – h) – 2(h – 2) = –1
15 – 3h – 2h + 4 = –1
15 + 4 – 3h – 2h = –1
19 – 5h = –1
19 – 19 – 5h = –1 – 19
–5h = –20
5h = 20
5h
20
=
5
5
\h =4
85
1
9. (a)
5x + 1
3
5x + 1
3×
3
5x + 1
5x + 1 – 1
5x
=7
(e)
=3×7
= 21
= 21 – 1
= 20
5x
20
=
5
5
\x =4
(b)
2x – 3
4
2x – 3
12 ×
4
3(2x – 3)
6x – 9
6x – 4x – 9
2x – 9
2x – 9 + 9
2x
x–3
3
x–3
= 12 ×
3
= 4(x – 3)
= 4x – 12
= 4x – 4x – 12
= –12
= –12 + 9
= –3
=
(f)
2x
–3
=
2
2
1
\ x = –1
2
3x – 1
x –1
(c)
=
5
3
3x – 1
x –1
15 ×
= 15 ×
5
3
3(3x – 1) = 5(x – 1)
9x – 3 = 5x – 5
9x – 5x – 3 = 5x – 5x – 5
4x – 3 = –5
4x – 3 + 3 = –5 + 3
4x = –2
4x
–2
=
4
4
1
\x =–
2
1
1
(d)
(5y + 4) = (2y – 1)
4
3
1
1
12 × (5y + 4) = 12 × (2y – 1)
4
3
3(5y + 4) = 4(2y – 1)
15y + 12 = 8y – 4
15y – 8y + 12 = 8y – 8y – 4
7y + 12 = – 4
7y + 12 – 12 = – 4 – 12
7y = –16
9 y 22
=
9
9
4
\y=2
9
2y + 3
y–5
=0
+
4
6
2y + 3
y–5 y–5
5–y
–
=0–
+
4
6
6
6
2y + 3
5–y
=–
4
6
2y + 3
y–5
=
4
6
2y + 3
y–5
12 ×
= 12 ×
4
6
3(2y + 3) = 2(5 – y)
6y + 9 = 10 – 2y
6y + 2y + 9 = 10 – 2y + 2y
8y + 9 = 10
8y + 9 – 9 = 10 – 9
8y = 1
8y
1
=
8
8
1
\y=
8
10. (a)
12
x+3
12
(x + 3) ×
x+3
12
12
2x + 6
2x + 6 – 6
2x
=2
= (x + 3) × 2
= 2(x + 3)
= 2x + 6
= 12
= 12 – 6
=6
6
2x
=
2
2
\x =3
7y
–16
=
7
7
2
\ y = –2
7
1
2y – 1
y+3
–
=0
5
7
2y – 1
y+3
y+3
y+3
–
+
=0+
5
7
7
7
2y – 1 y + 3
=
5
7
2y – 1
y+3
35 ×
= 35 ×
5
7
7(2y – 1) = 5(y + 3)
14y – 7 = 5y + 15
14y – 5y – 7 = 5y – 5y + 15
9y – 7 = 15
9y – 7 + 7 = 15 + 7
9y = 22
86
(b)
y+5
5
=
y–6
4
y+5
5
= 4(y – 6) ×
4(y – 6) ×
y–6
4
4(y + 5) = 5(y – 6)
4y + 20 = 5y – 30
4y – 5y + 20 = 5y – 5y – 30
–y + 20 = –30
–y + 20 – 20 = –30 – 20
–y = –50
\ y = 50
2y + 1
4
(f)
=
3y – 5
7
2y + 1
4
= 7(3y – 5) ×
7(3y – 5) ×
3y – 5
7
7(2y + 1) = 4(3y – 5)
14y + 7 = 12y – 20
14y – 12y + 7 = 12y – 12y – 20
2y + 7 = –20
2y + 7 – 7 = –20 – 7
2y = –27
11
=4
2x – 1
11
(2x – 1) ×
= (2x – 1) × 4
2x – 1
11 = 4(2x – 1)
11 = 8x – 4
8x – 4 = 11
8x – 4 + 4 = 11 + 4
8x = 15
(e)
8 x 15
=
8
8
7
\x=1
8
32
1
(c)
–3 =
2x – 5
4
32
1
–3+3 =
+3
2x – 5
4
32
13
=
2x – 5
4
32
13
4(2x – 5) ×
= 4(2x – 5) ×
2x – 5
4
128 = 13(2x – 5)
128 = 26x – 65
26x – 65 = 128
26x – 65 + 65 = 128 + 65
26x = 193
2y
–27
=
2
2
1
\ y = –13
2
2
3
(g)
=
y–2
y+6
2
3
(y – 2)(y + 6) ×
= (y – 2)(y + 6) ×
y–2
y+6
26 x
193
=
26
26
11
\x =7
26
1
1
(d)
=
–1
2
x+2
1
1
–1 =
x+2
2
1
1
+1
–1+1 =
x+2
2
1
3
=
x+2
2
1
3
2(x + 2) ×
= 2(x + 2) ×
x+2
2
2 = 3(x + 2)
2 = 3x + 6
3x + 6 = 2
3x + 6 – 6 = 2 – 6
3x = – 4
2(y + 6) = 3(y – 2)
2y + 12 = 3y – 6
2y – 3y + 12 = 3y – 3y – 6
–y + 12 = – 6
–y + 12 – 12 = – 6 – 12
–y = –18
\ y = 18
2
3
(h)
=
7y – 3
9y – 5
2
3
(7y – 3)(9y – 5) ×
= (7y – 3)(9y – 5) ×
7y – 3
9y – 5
2(9y – 5)
18y – 10
18y – 21y – 10
–3y – 10
–3y – 10 + 10
–3y
3y
–4
3x
=
3
3
1
\ x = –1
3
= 3(7y – 3)
= 21y – 9
= 21y – 21y – 9
= –9
= –9 + 10
=1
=1
3y
–1
=
3
3
1
\y =–
3
87
1
11. (a)
5x + 4
3
3(10 x ) – (5 x + 4)
3
30 x – 5 x – 4
3
25 x – 4
3
25 x – 4
3×
3
25x – 4
25x – 4 + 4
25x
10x –
(d)
=7
=7
=7
=7
=3×7
= 21
= 21 + 4
= 25
25 x 25
=
25
25
\x =1
(b)
4x
x −1
–
3
2
2(4 x ) – 3( x – 1)
6
8 x – 3x + 3
6
5x + 3
6
5x + 3
12 ×
6
2(5x + 3)
10x + 6
10x + 6 – 6
10x
1
4
=1
5
4
5
=
4
5
=
4
=
= 12 ×
13 y
1
=
13
13
1
\y=
13
6( y – 2)
2( y – 7)
(e)
– 12 =
7
3
6( y – 2) – 84
2( y – 7)
=
7
3
6 y – 12 – 84
2( y – 7)
=
7
3
6 y – 96 2( y – 7)
=
7
3
6 y – 96
2( y – 7)
21 ×
= 21 ×
7
3
3(6y – 96) = 14(y – 7)
18y – 288 = 14y – 98
18y – 14y – 288 = 14y – 14y – 98
4y – 288 = –98
4y – 288 + 288 = –98 + 288
4y = 190
5
4
= 15
= 15
= 15 – 6
=9
10 x
9
=
10
10
9
\x=
10
x –1
x+3
(c)
–
= –1
3
4
4( x – 1) – 3( x + 3)
= –1
12
4 x – 4 – 3x – 9
= –1
12
4 x – 3x – 4 – 9
= –1
12
x – 13
= –1
12
x – 13
12 ×
= 12 × (–1)
12
x – 13 = –12
x – 13 + 13 = –12 + 13
\ x=1
1
y+5
3( y – 1)
=
3
4
3 – ( y + 5)
3( y – 1)
=
3
4
3– y–5
3( y – 1)
=
3
4
–y + 3 – 5
3( y – 1)
=
3
4
–y – 2
3( y – 1)
=
3
4
–y – 2
3( y – 1)
12 ×
= 12 ×
3
4
4(–y – 2) = 9(y – 1)
– 4y – 8 = 9y – 9
– 4y – 9y – 8 = 9y – 9y – 9
–13y – 8 = –9
–13y – 8 + 8 = –9 + 8
–13y = –1
13y = 1
1–
4 y 190
=
4
4
1
\ y = 47
2
88
(f)
7 – 2y
2
1
– (2 – y) = 1
2
5
4
5(7 – 2 y ) – 4(2 – y ) 5
=
10
4
5
35 – 10 y – 8 + 4 y
=
10
4
5
–10 y + 4 y + 35 – 8
=
10
4
5
–6 y + 27
=
10
4
5
–6 y + 27
20 ×
= 20 ×
10
4
2(–6y + 27) = 25
–12y + 54 = 25
–12y + 54 – 54 = 25 – 54
–12y = –29
12y = 29
14.
\
\x=
= 4(7x – 4y) ×
3
4
= 3(7x – 4y)
= 21x – 12y
= 21x – 21x – 12y
= –12y
= –12y + 20y
= 8y
= –8y
x
8
=–
y
9
3
x + 26
5
When x = 12,
1. y =
3
(12) + 26
5
1
= 33
5
y 2 – xz
2. a =
5
When x = 2, y = –1, z = –3,
y=
1 19
5
+
3 20
6
19
5
+
60
6
=1
3
4
Exercise 5B
19
12. When x =
,
20
19
3
LHS = 2 –
20
4
9
3
=1
–
10
4
3
= 1
20
=
=
9x
–8 y
=
9
9
8y
x =–
9
1
1 8y
×x =
× –
y
y 9
12 y
29
=
12
12
5
\y=2
12
RHS =
3x – 5 y
7x – 4y
3x – 5 y
4(7x – 4y) ×
7x – 4y
4(3x – 5y)
12x – 20y
12x – 21x – 20y
–9x – 20y
–9x – 20y + 20y
–9x
9x
(–1)2 – 2(–3)
5
1+ 6
=
5
7
=
5
2
=1
5
3. S = 4pr2
1
(i) When r = 10 ,
2
2
22 1
S = 4 7 10 2
a=
3
= LHS
20
19
is the solution of the equation
20
5
3
1
= x+ .
4
3
6
13.
4x + y = 3x + 5y
4x – 3x + y = 3x – 3x + 5y
x + y = 5y
x + y – y = 5y – y
x = 4y
3
3
×x =
× 4y
16 y
16 y
3x
3
\
=
16 y
4
2x –
= 1386
89
1
(ii) When S = 616,
7. U = p(r + h)
22
616 = 4 r2
7
88 2
616 =
r
7
When U = 16
16
1
2
22
(r + 2)
7
88 2
r = 616
7
7
r2 =
× 616
88
2
r = 49
r+2
r+2
\ r = ± 49
\r
= ±7
= 7 or –7 (N.A. since r > 0)
1
, h = 2,
2
22
=
(r + 2)
7
1
= 16
2
7
1
=
× 16
2
22
1
=5
4
1
=5 –2
4
1
4
=3
1
bh
2
(i) When b = 20, h = 45,
4. A =
8. v2 = u2 + 2gs
When v = 25, u = 12, g = 10,
252 = 122 + 2(10)s
625 = 144 + 20s
144 + 20s = 625
20s = 625 – 144
20s = 481
1
(20)(45)
2
= 450 cm2
Area of triangle = 450 cm2
(ii) When A = 30, b = 10,
A=
1
(10)h
2
30 = 5h
5h = 30
\h =6
Height of triangle = 6 cm
5. (a) P = xyz
(b) S = p2 + q3
30 =
481
20
1
= 24
20
a
2c
9.
–d=
b
b
When a = 3, b = 4, d = –5,
\s =
2c
3
– (–5) =
4
4
3
c
5 =
4
2
c
3
=5
2
4
m+n+ p+q
(c) A =
4
(d) T = 60a + b
p + 2q
3
When k = 7, q = 9,
6. k =
\c =2×5
p + 2(9)
3
p + 18
7=
3
3 × 7 = p + 18
21 = p + 18
p + 18 = 21
\ p = 21 – 18
=3
7=
= 11
10.
1
2
1
1
1
1
–
= +
a
b
c
d
1
1
1
When a = , b = , d = – ,
2
4
5
1
1
1
1
–
= +
1
1
–1
c
5
4
2
2–4 =
–2
–2 + 5
3
3c
\c
1
90
3
4
1
–5
c
1
=
–5
c
1
=
c
1
=
c
=1
1
=
3
11. N =
m
x+q
14.
When N = 1
4
, m = 9, x = 2,
5
9
4
1 =
2
+
q
5
9
9
=
2+q
5
1
1
=
2+q
5
2+q =5
\q =5–2
=3
d–e
a
12. c =
–
f –d
b
When a = 3, b = 4, c = – 6, d = –5 and e = 2,
6[ n (–2)2 – (–3)]
= 5n
–5
6(4 n + 3)
= 5n
–5
6(4n + 3) = –25n
24n + 18 = –25n
24n + 25n = –18
49n = –18
18
49
15. (i) Let the smallest odd number be n.
The next odd number will be n + 2.
The greatest odd number will be (n + 2) + 2 = n + 4.
\ S = n + (n + 2) + (n + 4)
=n+n+n+2+4
= 3n + 6
(ii) When the greatest odd number is –101,
n + 4 = –101
\ n = –101 – 4
= –105
\ S = 3(–105) + 6
= –309
\n =–
–5 – 2
3
–
f – (–5)
4
–7
3
–
–6 =
f +5
4
7
3
–6 =
+
f +5
4
–6 =
3
4
3
–6
4
27
–
4
–27( f + 5)
–27( f + 5)
–27f – 135
–27f
–27f
–6 –
m ( nx 2 – y )
= 5n
z
When m = 6, x = –2, y = –3, z = –5,
7
f +5
7
=
f +5
7
=
f +5
=4×7
= 28
= 28
= 28 + 135
= 163
=
16. (i) T = c × d + e ×
ef
100
–145 c
(ii) e =
4–c
When e = 150,
= cd +
163
–27
1
= –6
27
150(4 – c)
600 – 150c
–150c + 145c
–5c
b
c–b
When a = 3, c = 10,
13. a =
3(10 – b)
30 – 3b
–3b – b
– 4b
\b
– 600
–5
= 120
\c =
b
=
10 – b
=b
=b
= –30
= –30
–30
=
–4
=7
–145 c
4–c
= –145c
= –145c
= – 600
= –600
150 =
\f =
3
f
100
f +5
50
When d = 3,
d=
f +5
50
50 × 3 = f + 5
150 = f + 5
f + 5 = 150
\ f = 150 – 5
= 145
3=
1
2
\ T = 120(3) +
= 577.50
91
150(145)
100
1
3. Let Priya’s age be x years old.
Then Amirah’s age is (x + 4) years,
Shirley’s age is (x – 2) years.
x + (x + 4) + (x – 2) = 47
x + x + x + 4 – 2 = 47
3x + 2 = 47
3x = 47 – 2
3x = 45
45
x=
3
= 15
\ Priya is 15 years old, Amirah is 15 + 4 = 19 years old and
Shirley is 15 – 2 = 13 years old.
4. Let the greater number be x.
2
Then the smaller number is x.
3
17. y = (x – 32) ×
5
9
(i) When x = 134,
y = (134 – 32) ×
5
9
= 56.7 (to 3 s.f.)
Required temperature = 56.7 °C
(ii) When x = 0,
5
y = (0 – 32) ×
9
= –17.8 (to 3 s.f.)
Since 0 °F = –17.8 °C, it is less common for the temperature
to fall below 0 °F because 0 °F is much lower than 0 °C.
(iii) When y = –62.1,
– 62.1 = (x – 32) ×
(x – 32) ×
5
9
5
= – 62.1
9
x – 32 = –62.1 ×
x+
9
5
x – 32 = –111.78
\ x = –111.78 + 32
= –79.78
= –79.8 (to 3 s.f.)
\ The smaller number is
Required temperature = –79.8 °F
5. Let the number be x.
3x = x + 28
3x – x = 28
2x = 28
Exercise 5C
1. Let the mass of the empty lorry be x kg.
Then the mass of the bricks is 3x kg.
x + 3x = 11 600
4x = 11 600
2
(27) = 18.
3
28
2
= 14
The number is 14.
6. Let the number of people going on the holiday be x.
15x = 84 + 12x
15x – 12x = 84
3x = 84
\x =
11 600
4
= 2900
\ The mass of the bricks is 3(2900) = 8700 kg.
2. Let the smallest odd number be n.
The next odd number will be n + 2.
Then the next odd number will be (n + 2) + 2 = n + 4.
The greatest odd number will be (n + 4) + 2 = n + 6.
n + (n + 2) + (n + 4) + (n + 6) = 56
n + n + n + n + 2 + 4 + 6 = 56
4n + 12 = 56
4n = 56 – 12
4n = 44
44
n=
4
= 11
\ The greatest of the 4 numbers is 11 + 6 = 17.
x=
1
2
x = 45
3
5
x = 45
3
3
x=
× 45
5
= 27
84
3
= 28
There are 28 people going on the holiday.
7. Let the number of boys who play badminton be x.
Then the number of boys who play soccer is 3x.
3x – 12 = x + 12
3x – x = 12 + 12
2x = 24
\ x = 24
2
= 12
There are 12 boys who play badminton.
\x=
92
13. Let Kate’s average speed for the first part of her journey be x km/h.
Then her average speed for the second part of her journey is
(x – 15) km/h.
8. Let the number be x.
1
9
x + 49 = x
2
4
1
9
x – x = – 49
2
4
7
– x = – 49
4
4
\ x = – × (– 49)
7
= 28
The number is 28.
9. Let the number be x.
68 – 4x = 3(x + 4)
68 – 4x = 3x + 12
– 4x – 3x = 12 – 68
–7x = –56
–56
\x =
–7
=8
The number is 8.
10. Let the son’s age be x years.
Then the man’s age is 6x years.
6x + 20 = 2(x + 20)
6x + 20 = 2x + 40
6x – 2x = 40 – 20
4x = 20
20
x=
4
=5
\ The man was 6(5) – 5 = 25 years old when his son was born.
11. Let the cost of a mooncake with one egg yolk be $x.
Then the cost of a mooncake with two egg yolks is $(x + 2).
6(x + 2) + 5x = 130.8
6x + 12 + 5x = 130.8
6x + 5x = 130.8 – 12
11x = 118.8
Time taken for first part of journey =
350
hours.
x
470 – 350
Time taken for second part of journey =
x – 15
120
=
hours.
x – 15
350
120
=
x
x – 15
350(x – 15) = 120x
350x – 5250 = 120x
350x – 120x = 5250
230x = 5250
5250
x=
230
19
= 22
23
\ Kate’s average speed for the second part of her journey is
19
19
– 15 = 7
km/h.
23
23
14. Let the denominator of the fraction be x.
Then the numerator of the fraction is x – 5.
22
\ The fraction is
x – 5 +1
x +1
x–4
x +1
3(x – 4)
3x – 12
3x – 2x
x
x–5
.
x
2
3
2
=
3
= 2(x + 1)
= 2x + 2
= 2 + 12
= 14
=
\ The fraction is
14 – 5
9
=
.
14
14
15. Let the number in the tens place be x.
Then the number in the ones place is 2.5x.
\ The number is 10x + 2.5x = 12.5x.
\ The number obtained when the digits are reversed is
10(2.5x) + x = 25x + x = 26x.
26x – 12.5x = 27
13.5x = 27
118.8
11
= 10.8
\ The cost of a mooncake with two egg yolks is
$(10.8 + 2) = $12.80.
12. Let the number of 20-cent coins Jun Wei has be x.
Then the number of 10-cent coins he has is x + 12.
10(x + 12) + 20x = 540
10x + 120 + 20x = 540
10x + 20x = 540 – 120
30x = 420
x=
27
13.5
=2
The number is 12.5(2) = 25.
x=
420
30
= 14
\ Jun Wei has 14 + (14 + 12) = 40 coins.
x=
93
1
Exercise 5D
4.
1. (a) If x > y, then 5x > 5y.
(b) If x < y, then
x
y
<
.
20
20
(c) If x y, then 3x > 3y.
1
\ The smallest rational value of y is 1 .
7
5. 20x > 33
(d) If x y, then
x
y
.
10
10
(e) If 15 > 5 and 5 > x, then 15 > x.
(f) If x < 50 and 50 < y, then x < y.
2. (a)
x
x
62
4
2
4
3
14
15 15 1 16
2
–36
3
y < –12
5y > –24
–16
126
12
1
\ x > 10
2
(g) 2y < –5
7
–105
3
x < –35
\ The greatest odd integer value of x is –37.
2y > – 6
7. 5y < 20
and
17
19
18
–14
–15
–13
–12
–11
–6
–5 –4
4
–4
5
–3
–2
1. (a)
1
2
(h) 9y > –20
1
x
2
1
x =1
2
1
x =1
2
\x =2
(b) 2(x – 1) + 3(x + 1) = 4(x + 4)
2x – 2 + 3x + 3 = 4x + 16
5x + 1 = 4x + 16
5x – 4x = 16 – 1
\ x = 15
(c) 2y – [7 – (5y – 4)] = 6
2y – (7 – 5y + 4) = 6
2y – (11 – 5y) = 6
2y – 11 + 5y = 6
7y – 11 = 6
7y = 6 + 11
7y = 17
4
5
6
7
8
10
–10
1 11
2
12
13
–5
–4
–3
–2
1 –2
2
20
9
2
–3 2 –2
–1
0
\ y > –2
–2
9
9
3. Let the number of vans that are needed to ferry 80 people be x.
Then 12x 80
y >–
17
7
3
=2
7
\y =
80
12
2
x6
3
\ The minimum number of vans that are needed to ferry 80 people
is 7.
x
1
x–1 =
x–
5
2
y < –2
y–
Review Exercise 5
–1
x>
y<–
6
2
y –3
20
5
y<4
\ The possible values are –3, –2, –1, 0, 1, 2 and 3.
y>
28
x<
4
\x<7
(f) 12x > 126
6
y<
\y<
–24
5
4
\ y > –4
5
(e) 4x < 28
5
x<
1
\ x 15
2
(c) 3y < –36
(d)
33
20
13
x>1
20
\ The smallest value of x if x is a prime number is 2.
6. 3x < –105
x>
3x 18
18
3
\x6
(b) 4x 62
8 7y
7y 8
8
y
7
1
y 1
7
94
(d)
3
x – 5 = 0.5x
4
(i)
3
x – 0.5x = 5
4
1
x =5
4
\x =4×5
= 20
(e)
2y + 7
4
2y + 7
2y + 7
2y
2y
= 12
= 4 × 12
= 48
= 48 – 7
= 41
41
\y =
2
1
= 20
2
4y – 1
5
(f)
=
5y + 1
7
7(4y – 1) = 5(5y + 1)
28y – 7 = 25y + 5
28y – 25y = 5 + 7
3y = 12
12
\y =
3
=4
(g)
(h)
(j)
2c
c –1
c+3
–
=
9
6
12
2(2 c ) – 3( c – 1) c + 3
=
18
12
4 c – 3c + 3 c + 3
=
18
12
c+3
c+3
=
18
12
12(c + 3) = 18(c + 3)
2(c + 3) = 3(c + 3)
2c + 6 = 3c + 9
2c – 3c = 9 – 6
–c = 3
\ c = –3
2(3 – 4 d )
3( d + 7)
1
–
= 5d +
3
2
6
4(3 – 4 d ) – 9( d + 7) 6(5 d ) + 1
=
6
6
12 – 16 d – 9 d – 63 30 d + 1
=
6
6
–25 d – 51
30 d + 1
=
6
6
–25d – 51 = 30d + 1
–25d – 30d = 1 + 51
–55d = 52
\d = –
a +1
a –1
=4
+
4
3
3( a + 1) + 4( a – 1)
=4
12
3a + 3 + 4 a – 4
=4
12
7a – 1
=4
12
7a – 1 = 12 × 4
7a – 1 = 48
7a = 48 + 1
7a = 49
49
\a =
7
=7
52
55
2. (a) 18x < –25
–25
18
7
\ x < –1
18
(b) 10y –24
x<
24
10
2
\ y > –2
5
3. 3(x – 1) – 5(x – 4) = 8
3x – 3 – 5x + 20 = 8
–2x + 17 = 8
–2x = 8 – 17
–2x = –9
y>–
b–4
2b + 1
5b – 1
–
=
3
6
2
2(b – 4) – (2 b + 1)
5b – 1
=
6
2
2b – 8 – 2b – 1
5b – 1
=
6
2
–9
5b – 1
=
6
2
–3
5b – 1
=
2
2
–3 = 5b – 1
5b – 1 = –3
5b = –2
2
\b=–
5
–9
–2
1
=4
2
1
1
1
\x–5 =4 –5
2
2
2
= –1
4. 4x 11
x=
11
4
3
x2
4
\ The smallest integer value of x is 3.
x
95
1
5.
10. Let the smaller odd number be n.
Then the greater number is n + 2.
n + 2 + 5n = 92
6n + 2 = 92
6n = 92 – 2
6n = 90
90
n=
6
= 15
\ The two consecutive odd numbers are 15 and 17.
11. Let the mass of Object B be x kg.
Then the mass of Object A is (x + 5) kg,
the mass of Object C is 2(x + 5) kg.
(x + 5) + x + 2(x + 5) = 255
x + 5 + x + 2x + 10 = 255
4x + 15 = 255
4x = 255 – 15
4x = 240
240
x=
4
= 60
\ The mass of Object C is 2(60 + 5) = 130 kg.
12. Let Farhan’s present age be x years.
Then Farhan’s cousin’s present age is (38 – x) years.
x – 7 = 3(38 – x – 7)
x – 7 = 3(31 – x)
x – 7 = 93 – 3x
x + 3x = 93 + 7
4x = 100
100
\x =
4
= 25
Farhan is 25 years old now.
13. Let Raj’s present age be x years.
Then Nora’s present age is 2x years,
Ethan’s present age be is 2(2x) years.
2(2x) + 22 = 2(x + 22)
4x + 22 = 2x + 44
4x – 2x = 44 – 22
2x = 22
22
x=
2
= 11
\ Nora is 2(11) = 22 years old now.
14. Let the number of sweets that the man has to give to his son be x.
55 + x = 4(25 – x)
55 + x = 100 – 4x
x + 4x = 100 – 55
5x = 45
45
\x =
5
=9
The man has to give 9 sweets to his son.
3y < –24
–24
y<
3
y < –8
\ The greatest integer value of y is –9.
6. 5x < 125
125
5
x < 25
\ The greatest value of x if x is divisible by 12 is 24.
7. 5y 84
x<
y
84
5
4
5
\ The smallest value of y if y is a prime number is 17.
y 16
4 3
pr
3
(i) When r = 7,
4 22
V = (7)3
3 7
1
= 1437
3
1
(ii) When V = 113 ,
7
4 22 3
1
113 = r
3 7
7
1
88 3
r
113 =
7
21
88 3
1
r = 113
21
7
21
1
r3 =
× 113
88
7
r3 = 27
\ r = 3 27
=3
8. V =
3y – n
m
When y = 5, m = –3,
9. n – 2y =
n – 2(5) =
n – 10
–3(n – 10)
–3n + 30
–3n + n
–2n
3(5) – n
–3
15 – n
=
–3
= 15 – n
= 15 – n
= 15 – 30
= –15
–15
–2
1
=7
2
\n =
1
96
19. Let the number of tickets Jun Wei can buy be x.
Then 12.50x < 250
15. Let the original price of each apple be x cents.
24x = (24 + 6)(x – 5)
24x = 30(x – 5)
24x = 30x – 150
24x – 30x = –150
– 6x = –150
–150
\x =
–6
= 25
The original price of each apple is 25 cents.
16. Let the distance between Town A and Town B be x km.
45 minutes =
45
60
x
x
+
4
6
6x + 4x
24
10 x
24
5x
12
hour =
250
12.5
x < 20
\ The maximum number of tickets Jun Wei can buy with $250 is 20.
x<
20. Let the first integer be x.
Then the second integer will be (x + 1).
x + (x + 1) < 42
2x < 41
3
hour
4
41
2
x < 20.5
\ The largest possible integer x can be is 20.
20 + 1 = 21
212 = 441
\ The square of largest possible integer is 441.
x<
3
4
3
=
4
3
=
4
3
=
4
=
5x = 12 ×
5x = 9
x=
21. Let Nora’s age be x years.
Then Kate’s age is (x – 4) years.
x + (x – 4) < 45
2x < 45 + 4
3
4
49
2
x < 24.5
\ Maximum possible age of Nora is 24 years.
24 – 4 = 20
\ The maximum possible age of Kate is 20 years.
x<
9
5
=1
4
5
\ The main travels a total distance of 2 × 1
4
3
= 3 km.
5
5
17. Let the denominator of the fraction be x.
Then the numerator of the fraction is x – 2.
\ The fraction is
22. Let the number of ships needed to carry 400 passengers be .
60x > 400
x–2
.
x
400
60
2
x>6
3
\ The minimum number of ships needed to carry 400 passengers
is 7.
x>
x–2–3
x–3
x–5
x–3
4(x – 5)
4x – 20
4x – 3x
x
3
=
4
3
=
4
= 3(x – 3)
= 3x – 9
= –9 + 20
= 11
11 – 2
9
\ The fraction is
=
.
11
11
18. Let the number of sets of multimedia equipment that can be bought
with $35 000 be x.
Then 1900x 35 000
23. Let the number of pencils that can be bought with $27 be x.
2.50x < 27
27
2.5
4
x < 10
5
\ The maximum number of pencils that can be bought with $27 is
10.
x<
35 000
1900
8
x 18
19
\ The maximum number of sets of multimedia equipment that can
be bought with $35 000 is 18.
x
97
1
4. A × B = 8
B × C = 28
C × D = 63
B × D = 36
Challenge Yourself
1.
x +2 =0
x = –2
There is no solution since x cannot be a negative number.
2. Since (x + 2)2 and (y – 3)2 cannot be negative.
(x + 2)2 = 0
and (y – 3)2 = 0
x+2 =0
and
y–3=0
x = –2
and
y=3
\ x + y = –2 + 3
=1
3. A + B = 8 — (1)
B + C = 11 — (2)
B + D = 13 — (3)
C + D = 14 — (4)
(2) – (3): B + C – B – D = 11 – 13
C – D = –2 — (5)
(4) + (5): C + D + C – D = 14 + (–2)
2C = 12
12
\C = 2
(2) ÷ (3):
28
B×C
=
63
C×D
Since C cannot be equal to 0, then
B
4
= , — (5)
D
9
(4) × (5): B × D ×
B
4
= 36 ×
D
9
Since D cannot be equal to 0, then B2 = 16.
\ B = ± 16
= 4 or – 4 (N.A. since B > 0)
Substitute B = 4 into (1): A × 4 = 8
8
4
=2
Substitute B = 4 into (2): 4 × C = 28
\A=
28
4
=7
Substitute B = 4 into (4): 4 × D = 36
\C =
=6
Substitute C = 6 into (4): 6 + D = 14
\ D = 14 – 6
=8
Substitute C = 6 into (2): B + 6 = 11
\ B = 11 – 6
=5
Substitute B = 5 into (1): A + 5 = 8
\A =8–5
=3
1
— (1)
— (2)
— (3)
— (4)
36
4
=9
\D =
98
Chapter 6 Functions and Linear Graphs
TEACHING NOTES
Suggested Approach
Although the topic on functions and linear graphs is new to most students, they do encounter examples of their applications in
their daily lives, e.g. maps show the usage of Cartesian coordinates; escalators and moving walkways illustrate the concept of
steepness. Teachers can get students to discuss about in detail these real-life examples. When students are able to appreciate
their uses, teachers can proceed to introduce the concept of functions and linear graphs.
Section 6.1:
Cartesian Coordinates
Teachers can build upon prerequisites, namely number lines to introduce the horizontal axis (x-axis) and the
vertical axis (y-axis). Teachers can introduce this concept by playing a game (see Class Discussion: Battleship
Game (Two Players)) to arouse students’ interest.
Teachers should teach students not only on how to draw horizontal and vertical axes and plot the given
points, but also to determine the position of points. Teachers can impress upon students that the first number in
each ordered pair is with reference to the horizontal scale while the second number is with reference to the vertical
scale. As such, students need to take note that the point (3, 4) has a different position compared to the point (4, 3).
Section 6.2:
Functions
Teachers can use the Function Machine (see Investigation: Function Machine) to explore the concept of a function
with the students and show that when a function is applied to any input x, it will produce exactly one output y.
Once the students have understood the relationship between the input x and the output y, they are then able to
represent the function using an equation, a table and a graph.
Section 6.3:
Graphs of Linear Functions
Teachers should illustrate how a graph of a linear function is drawn on a sheet of graph paper. Teachers can
impress upon students that when they draw a graph, the graph has to follow the scale stated for both the x-axis
and y-axis and the graph is only drawn for the values of x stated in the range.
Section 6.4:
Applications of Linear Graphs in Real-World Contexts
Teachers can give examples of linear graphs used in many daily situations and explain what each of the graphs
is used for. Through Worked Example 2, students will learn how functions and linear graphs are applied in realworld contexts and solve similar problems
The thinking time on page 151 of the textbook requires students to think further and consider if a negative value
is possible or logical in the real world. Teachers should get the students to apply the answer of it to the other
problems in real-world contexts.
Challenge Yourself
To further guide pupils to better understand the concept, teachers may modify the question to giving the
x-coordinate of C.
99
1
WORKED SOLUTIONS
Class Discussion (Ordered Pairs)
Class Discussion (Battleship Game (Two Players))
1. A single number is not sufficient to describe the exact position of a
student in the classroom seating plan. For example, when the
number 1 is used to indicate the position of a student in the classroom,
the student could be either in row 1 or column 1. From Fig. 6.2, we
can see that there are 11 possible positions of the student.
Similarly, the location of a seat in a cinema cannot be represented
by a single number. An example of a seating plan of a cinema is as
shown:
The purpose of this Battleship Game is to introduce students to the use
of 2D Cartesian coordinates to specify points through an interesting
and engaging activity.
Teachers may wish to emphasise to students that they should call out
a location on the grid by calling the letter before calling the number,
e.g. D7 instead of 7D.
Teachers may wish to use the grids provided (similar to that in Fig. 6.1)
to conduct this activity.
A
B
1
C
2
D
3
E
4
F
5
G
6
H
7
I
8
J
1
9
B
C
D
E
F
G
H
I
3
4
5
6
7
8
9
10 11 12
From the seating plan shown, both the number and the letter are
required to represent the location of a seat in the cinema.
2. The order in which two numbers are written are important,
i.e. (5, 3) and (3, 5) do not indicate the same position.
10
A
2
J
Self
Journal Writing (Page 137)
1
1. Guiding Questions:
• How do you determine the locations of your house, a bus stop
and a shopping mall in your neighbourhood on the map?
• How can you obtain the approximate distances between your
house, a bus stop and a shopping mall in your neighbourhood?
2. Guiding Questions:
• What types of shops can normally be found on the ground floor
of a shopping mall?
• What is the size of each shop? How many spaces on the map
should each shop occupy?
• Are there any other considerations, e.g. walkways and
washrooms, when designing the map?
3. Guiding Questions:
• What types of horizontal and vertical scales are commonly used
for the seating plan of a cinema in Singapore?
• What are the different types of seats, e.g. wheelchair berths and
couple seats, which can be found in a cinema?
2
3
4
5
6
7
8
9
10
A
B
C
D
E
F
G
H
I
J
Opponent
1
100
1. y = x + 3
2. (a) Input x = 4 → Output y = 4 + 3 = 7
(b) Input x = –7 → Output y = –7 + 3 = – 4
3. (a) Input x = 9 – 3 = 6 → Output y = 9
(b) Input x = 0 – 3 = –3 → Output y = 0
4.
y
9.
Investigation (Function Machine)
1
0
–1
–1
x
1
2
3
–2
x
–7
–3
2
4
6
y
–4
0
5
7
9
–3
–4
–5
Table 6.1
–6
y
5.
–7
y = –2x – 1
y=x+3
Fig. 6.8
9
10. Every input x produces exactly one output y .
8
7
6
Thinking Time (Page 143)
5
1. y2 = x is not the equation of a function because
• there are two values of y for every positive value of x,
e.g. if the input x = 9, then the output y = ±3,
• there is no value for the output y if the input x is negative.
2. It is possible for a function to have two input values x with the same
output value y. Consider the equation of the function y = x2. If the
input x = –3 or 3, then the output y = 9 .
4
3
2
1
–7 – 6 –5 – 4 –3 –2 –1 0
–1
x
1
2
3
4
5
6
–2
Class Discussion (Equation of a Function)
–3
1. Since the point A lies on the graph of the function y = 2x,
its coordinates satisfy the equation of the function y = 2x .
Since the point B do not lie on the graph of the function y = 2x,
its coordinates do not satisfy the equation of the function y = 2x .
2. Examples of coordinates of points that satisfy the equation of the
function y = 2x include (2, 4), (3, 6), (4, 8), (0.5, 1) and (1.25, 2.5).
3. Amirah is correct to say that ‘the coordinates of every point on the
line satisfy the equation of the function y = 2x’. Since a graph is
a way to display a function, the coordinates of every point on the
graph satisfy the equation of the function.
–4
Fig. 6.6
The coordinates of every point on the straight line in Fig. 6.6 satisfy
the equation of the function y = x + 3 .
6. Every input x produces exactly one output y .
7. y = –2x – 1
8.
x
–1
–0 .5
0
2
3
y
1
0
–1
–5
–7
Table 6.2
Thinking Time (Page 147)
(i) The y-coordinate of each point that lies on the line y = 2x is twice
its x-coordinate, i.e. the coordinates are given by (x, 2x)
101
1
(ii)
(ii) When x = 35,
y = 100 – 35 × 3
= 100 – 105
= –5
This means that 35 days after Nora receives her monthly allowance,
she has –$5. Logically speaking, she should not have a negative
amount of money. However, in the real world, it is possible for her
to have –$5 as she may have borrowed $5 from her friends.
y
y=x+3
9
8
7
6
5
4
Practise Now (Page 138)
3
y
2
1
–7 – 6 –5 – 4 –3 –2 –1 0
–1
B(–2, 3)
x
1
2
3
4
5
6
3
A(2, 2)
2
–2
1
–3
–2 –1 0
–1
–4
C(–1, –2)
y
–2
x
1
2
3
D(3, –1)
Practise Now (Page 143)
1
0
–1
–1
1. (i) When x = 4,
y = 2(4) – 3
=8–3
=5
(ii) When y = –5,
–5 = 2x – 3
–5 + 3 = 2x
–2 = 2x
\ x = –1
2. (i) When x = 0,
x
1
2
3
–2
–3
–4
–5
–6
–7
y = –2x – 1
Yes, I agree with Lixin. As shown above, the graphs of the functions
y = x + 3 and y = –2x – 1 are straight lines. Hence, they are linear.
1
2
(0) –
3
5
2
=0–
5
2
=–
5
2
(ii) When y = – ,
3
2
1
2
–
=– x–
3
3
5
2
2
1
– +
=– x
3
5
3
4
1
–
=– x
15
3
4
\x =
5
y=–
Thinking Time (Page 151)
(i) When x = –2,
y = 100 – (–2) × 3
= 100 + 6
= 106
This means that 2 days before Nora receives her monthly allowance,
she has $106. However, in the real world, it is not possible for her to
have more money before she receives her monthly allowance than
when she receives her monthly allowance.
1
102
(ii) Amount of money the passenger has to pay if taxi travels 6 km
=
$3 + 6 × $0.50
=
$3 + $3
=
$6
(iii) Amount of money the passenger has to pay if taxi travels 10 km
=
$3 + 10 × $0.50
=
$3 + $5
=
$8
Practise Now 1
x
0
2
4
y = 2x + 1
1
5
9
1. (i)
y
y = 2x + 1
9
8
(b)
7
(ii)
6
5
x
3
6
10
y
4 .50
6
8
y
(c)
4
3
8
2
1
7
x
0
1
2
3
4
6
(ii) From the graph in (i),
when y = 6,
q = x = 2 .5
2.
5
x
–2
0
2
y = 3x
–6
0
6
4
3
x
–2
0
2
y = 2 – 2x
6
2
–2
2
y
1
y = 3x
y = 2 – 2x
6
0
5
x
1
2
3
4
5
6
7
8
9
10
4
3
Exercise 6A
2
1. A(– 4, –3), B(–2, 4), C(3, 4), D(4, 2), E(1, 1), F(3, –3)
y
2.
1
–2
–1
0
–1
x
1
2
A(2, 5)
5
–2
4
–3
F(–1, 2)
–4
–5
3
2
B(1, 2)
1
–6
–3 –2 –1 0
–1
C(–2, –1)
–2
Practise Now 2
x
1
2
3
E(3, –2)
4
5
6
D(6, –2)
3. (i) When x = 3,
y = 4(3) + 5
= 12 + 5
= 17
(a) (i) Amount of money the passenger has to pay if taxi travels 3 km
=
$3 + 3 × $0.50
=
$3 + $1.50
=
$4.50
103
1
(ii) When x = –2,
y = 4(–2) + 5
= –8 + 5
= –3
4. (i) When y = 34,
34 = 25 – 3x
3x = 25 – 34
3x = –9
\ x = –3
(ii) When y = –5,
–5 = 25 – 3x
3x = 25 + 5
3x = 30
\ x = 10
y
(c)
8
(0, 8)
7
6
5
(5, 4)
4
3
y
5. (a)
(– 6, 4)
(6, 4)
4
2
3
1
2
1
– 6 –5 – 4 –3 –2 –1 0
–1
1
2
3
4
5
x
0
(0, 0)
x
6
1
2
4
y
(d)
–3
(–1, 4)
–4
4
(6, – 4)
2
1
y
5
(0, 5)
x
1
–2
(–5, –2)
3
The figure shows a quadrilateral.
2
1
– 6 –5 – 4 –3 –2 –1 0
–1
(1, 0)
–5 – 4 –3 –2 –1 0
–1
4
(– 6, 0)
(0, 3)
3
The figure shows a rectangle.
(b)
5
The figure shows an isosceles triangle.
–2
(– 6, – 4)
3
(6, 0)
1
2
3
4
5
y
(e)
(–1, 3)
3
x
6
(5, 2)
2
–2
1
–3
–1 0
–1
–4
–5
(0, –5)
–2
The figure shows a rhombus.
x
1
2
3
4
5
(5, –2)
–3
(–1, –3)
The figure shows a trapezium.
1
104
(b) (i) When y = 1,
y
6.
8
C(1, 8)
2
1
x+
3
3
2
= x
3
2
= x
3
=1
1 =
1
3
2
3
\x
7
1–
6
5
4
3
(ii) When y = –
2
1
A(1, 0)
0
1
B(7, 0)
2
3
4
5
6
1
6
1
1
– –
6
3
1
–
2
–
x
7
Area of ABC =
1
×6×8
2
= 24 units2
\x
y
7.
(–3, 7)
1
,
6
2
1
= x+
3
3
2
= x
3
2
= x
3
3
=–
4
Exercise 6B
7
x
0
2
4
5
y = 2x + 8
8
12
16
4
y = 2x + 2
2
6
10
3
y = 2x – 3
–3
1
5
2
y = 2x – 6
–6
–2
2
1. (a)
6
(–2, 5)
(–1, 3)
1
–3 –2 –1 0
–1
(0, 1)
y
x
1
y = 2x + 8
2 3
(1, –1)
16
14
–2
–3
(2, –3)
12
–5
y = 2x + 2
10
–4
8
(3, –5)
y = 2x – 3
6
The points lie on a straight line.
8. (a) (i) When x = –3,
4
y = 2x – 6
2
2
1
(–3) +
3
3
1
= –2 +
3
2
= –1
3
1
(ii) When x = 1 ,
2
2 1
1
y = 1 +
3 2
3
1
=1+
3
1
=1
3
y=
0
–2
x
1
2
3
4
–4
–6
(b) They are parallel lines.
105
1
x
–4
0
4
y = 3x + 7
–5
7
y = 3x + 5
–7
5
y = 3x – 3
–15
y = 3x – 6
–18
2. (a)
x
–4
0
4
19
y = –2x + 5
13
5
–3
17
y = –2x + 3
11
3
–5
–3
9
y = –2x – 4
4
–4
–12
–6
6
y = –2x – 7
1
–7
–15
3. (a)
y
y
y = –2x + 5
y = 3x + 7
20
18
12
y = 3x + 5
16
y = –2x + 3
10
8
14
6
12
y = –2x – 4
y = 3x – 3
10
8
y = –2x – 7
4
2
y = 3x – 6
6
4
0
– 4 –3 –2 –1
–2
2
–4
0
– 4 –3 –2 –1
–2
–6
x
1
2
3
4
–8
–4
–10
–6
–12
–8
–14
–10
–16
–12
(b) They are parallel lines.
–14
–16
–18
(b) They are parallel lines.
1
14
106
x
1
2
3
4
4. (a)
x
–4
0
4
y = – 4x + 8
24
8
–8
y = – 4x + 2
18
2
–14
y = – 4x – 3
13
–3
–19
y = – 4x – 6
10
–6
–22
x
–3
0
3
y = 6 – 3x
15
6
–3
5. (i)
y
y = 6 – 3x
15
14
y
13
y = – 4x + 8
12
24
y = – 4x + 2
22
11
20
10
18
9
16
8
14
y = – 4x – 3
y = – 4x – 6
7
(ii)
12
6
10
5
8
4
6
3
4
2
1
2
0
– 4 –3 –2 –1
–2
(ii)
x
1
2
3
–3
4
–2
–1
0
–1
–4
–2
–6
–3
–8
x
1
2
3
(ii) From the graph in (i),
when y = 0,
a=x=2
From the graph in (i),
when x = –2,
b = y = 12
From the graph in (i),
when y = 1.5,
c = x = 1 .5
–10
–12
–14
–16
–18
–20
–22
(b) y = mx + c1, y = mx + c2, y = mx + c3, y = mx + c4,
where m, c1, c2, c3 and c4 are constants
107
1
x
–2
0
y = 2x + 4
0
4
8
y = 2 – 3x
8
2
–4
6.
2. (a) (i) Distance car can travel if it has 3 l of petrol = 27 km
(ii) Distance car can travel if it has 5.2 l of petrol = 47 km
(b) Amount of petrol required to travel 36 km = 4 l
\ Cost of petrol required to travel 36 km = 4 × $1.40
=
$5.60
2
y
y = 2 – 3x
y = 2x + 4
3. (i)
8
7
5
4
3
30
50
70
C
100
200
300
400
Review Exercise 6
2
y
1. (a)
1
–1
10
(ii) There is a fixed overhead of $50.
(iii) Amount of money Devi has to pay for 68 T-shirts = $390
(iv) Number of T-shirts Devi can order with $410 = 72
6
–2
N
(–2, 6)
x
0
–1
1
2
5
–2
4
–3
3
–4
(–2, 2)
Exercise 6C
x
3
6
10
y
105
90
70
2
3
5
4
(–2, 2)
3
(6, 2)
2
1
–2 –1 0
–1
–2
x
1
2
3
(2, –2)
The figure shows a square.
100
90
80
70
60
50
40
30
20
10
x
1
2
3
4
5
6
4
(2, 6)
6
110
1
x
1
y
(b)
120
0
(4, 2)
1
The figure shows a rectangle.
y
(c)
2
–2 –1 0
1. (a) (i) Amount of money left after 3 days = $120 – 3 × $5
=
$120 – $15
=
$105
(ii) Amount of money left after 6 days = $120 – 6 × $5
=
$120 – $30
=
$90
(iii) Amount of money left after 10 days = $120 – 10 × $5
=
$120 – $50
=
$70
(b)
(4, 6)
6
7
8
9
10
108
4
5
6
y
(c)
(iii) When x = –
(6, 8)
8
1
1
y = 4 – – 1
2
2
1
=–2–1
2
1
= –3
2
4. (i) When y = 150,
150 = 250 – 20x
150 – 250 = –20x
–100 = –20x
\x =5
(ii) When y = 450,
450 = 250 – 20x
450 – 250 = –20x
200 = –20x
\ x = –10
(iii) When y = –1150,
–1150 = 250 – 20x
–1150 – 250 = –20x
–1400 = –20x
\ x = 70
7
6
5
(–2, 4)
(8, 4)
4
3
2
1
–2 –1 0
–1
x
1
2
3
4
5
6
7
8
–2
–3
–4
(2, – 4)
The figure shows a trapezium.
y
(d)
7
(0, 7) (2, 7)
6
5
(2, 5)
4
1
,
2
x
–3
0
3
1
y=2 x+3
2
– 4 .5
3
10 .5
5. (i)
3
2
(– 4, 1)
y
1
– 4 –3 –2 –1 0
x
1
1
y=2 x+3
2
11
2
10
The figure shows a kite.
2. (a) A(–5, 0), B(– 4, 3), C(–3, 4), D(0, 5), E(3, 4), F(4, 3), G(4, –3),
H(3, – 4), I(–3, – 4), J(– 4, –3)
(b) (i) H
(ii) G
3. (i) When x = 12,
9
8
7
6
5
1
y = 4(12) – 1
2
1
= 48 – 1
2
1
= 46
2
1
(ii) When x = 2 ,
2
1
1
y = 4 2 – 1
2
2
1
= 10 – 1
2
1
=8
2
4
3
2
1
–3
–2
(ii)
–1
0
–1
x
1
2
3
–2
–3
–4
–5
109
1
(ii) From the graph in (i),
when x = –2,
a = y = –2
From the graph in (i),
when y = 3,
b=x=0
Challenge Yourself
y
6
B
4
2
A
C2
–6
–4
–2
0
C1
2
4
6
x
8
Teachers can guide students to first draw the line y = 1 on a piece of
graph paper.
The base of the triangle will be AC and its height will be the perpendicular
height from B to AC i.e 4 units. Hence, the base has to be 6 units. Counting
6 units to the right of A will give the point C1(7, 1), and counting 6 units
to the left of A will give the point C2(–5, 1).
1
110
Chapter 7 Number Patterns
TEACHING NOTES
Suggested Approach
Students have done word problems involving number sequences and patterns in primary school. These word problems required
the students to recognise simple patterns from various number sequences and determine either the next few terms or a specific
term. However, they were not taught to use algebra to solve problems involving number patterns. Teachers can arouse students’
interest in this topic by bringing in real-life applications (see chapter opener on page 157 and Investigation: Fibonacci Sequence).
Section 7.1:
Number Sequences
In primary school, students were only asked how to find the next few terms and a specific term of number sequences
but they have not been taught how to state the rule. Teachers can build upon this by getting students to work
in pairs to state the rules of number sequences and then write down the next few terms (see Class Discussion:
Number Sequences). Students should learn that they can add, subtract, multiply or divide or use a combination
of arithmetic operations to get the next term of a number sequence.
Section 7.2:
General Term of a Number Sequence
Teachers can build upon what students have learnt in Chapter 4 (Basic Algebra and Manipulation) and teach students how
to observe a number sequence and look for a pattern so that they can use algebra and find a formula for the general term,
Tn = nth term.
Teachers can get students to work in pairs to find a formula for the general term and hence find a specific term
for different number sequences (see Class Discussion: Generalising Simple Sequences). After the students have
learnt how to generalise simple sequences, they should know that the aim is not to simply solve the problem but
to represent it so that it becomes a general expression which can be used to find specific terms.
Section 7.3:
Number Patterns
In primary school, students have attempted questions involving number patterns. In this section, teachers can
ask the students to apply what they have learnt for number sequences on number patterns.
Teachers can get students to work in pairs to find a formula for the general term and hence find a specific term
for different number patterns (see Class Discussion: The Triangular Number Sequence). Through this class
discussion, students should learn that they need not use a large number of coins to find the total number of coins
needed to form a triangle with a base that has 100 coins. They need only to find the formula for the general term
and they are able to find the total number of coins by substituting n = 100 into the formula. They should also
learn that with the formula, they can find Tn easily for any n.
Section 7.4:
Number Patterns in Real-World Contexts
Teachers may get students to discover number patterns in real-world contexts (e.g. shells, pine cones, rocks,
wallpaper, floor tiles) and ask them to represent that number pattern into a general expression.
Through Worked Example 5, students will learn that in the real world, which in this case in Chemistry, the general
term of a number sequence is important and advantageous in finding specific terms. In this worked example,
finding the general term of the number of hydrogen atoms allowed one to find the member number, number of
carbon atom(s) and number of hydrogen atoms easily without going through tedious workings, especially if the
value of the specific term is large. For other figures, students should consider drawing the next figure in the
sequence so as to identify the pattern.
111
1
Challenge Yourself
Some of the questions (e.g. Questions 3, 4 and 5) are not easy for average students while others (Questions 1
and 2) should be manageable if teachers guide them as follows:
Questions 1 and 2: Teachers can get the students to draw a table and write down the first 6 terms. The students
have to observe carefully how each term in the sequence can be obtained and find a formula for the general term
to get the final answer to the question.
Questions 3, 4 and 5: Teachers have to get the students to think beyond just the four operations. The students
have to consider more ways and observe carefully how each term in the sequence is obtained. Once they have
figured this out, they are able to search on the Internet to find out the names for the sequences.
1
112
WORKED SOLUTIONS
Figure
Number,
n
Number of Coins
at the Base of the
Triangle, n
1
1
1
=1 =
1× 2
2
2
2
1+2
=3 =
2×3
2
3
3
1+2+3
=6 =
3× 4
2
4
4
1+2+3+4
= 10 =
4×5
2
5
5
1+2+3+4+5
= 15 =
5×6
2
Start with 1, then
multiply each term
by 2 to get the next
term .
6
6
1 + 2 + 3 + 4 + 5 + 6 = 21 =
6×7
2
:
:
1, 3, 9, 27, 81, 243, 729, … Start with 1, then
multiply each term
×3 ×3 ×3 ×3 ×3 ×3
by 3 to get the next
term .
n
n
2.
Class Discussion (Number Sequences)
1.
Sequence
Positive
even
numbers
2, 4, 6, 8, 10, 12, 14, …
Positive
odd
numbers
1, 3, 5, 7, 9, 11, 13, …
+2 +2 +2 +2 +2 +2
+2 +2 +2 +2 +2 +2
Multiples
of 3
3, 6, 9, 12, 15, 18, 21, …
Powers of 2
1, 2, 4, 8, 16, 32, 64, …
+3 +3 +3 +3 +3 +3
×2 ×2 ×2 ×2 ×2 ×2
Start with 2, then
add 2 to each term
to get the next
term .
Start with 1, then
add 2 to each term
to get the next
term .
Start with 3, then
add 3 to each term
to get the next
term .
Powers of 3
Rule
:
1 + 2 + 3 + 4 + ··· + n
=
1
n(n + 1)
2
Table 7.6
3. When n = 100,
Table 7.1
1
1
n(n + 1) =
× 100 × (100 + 1)
2
2
1
=
× 100 × 101
2
= 5050
Total number of coins needed to form a triangle with a base that has
100 coins = 5050
2. The sequence of positive odd numbers can be obtained by subtracting
1 from each term of the sequence 2, 4, 6, 8, 10, … .
Teachers may wish to note that there are other possible answers to
this question.
3. (a) Rule: Find the square of the position of each term . The next two
terms are 36 and 49 .
(b) Rule: Find the cube of the position of each term . The next two
terms are 216 and 343 .
Investigation (Fibonacci Sequence)
1.
2.
3.
4.
Class Discussion (Generalising Simple Sequences)
(a) Hence, Tn = 3n .
100th term, T100 = 3 × 100
= 300
2
(b) Hence, Tn = n .
100th term, T100 = 1002
= 10 000
(c) Hence, Tn = n3 .
100th term, T100 = 1003
= 1 000 000
1; 5; 13; 21
3, 5, 8, 13, 21, 34
Michaelmas Daisy has 55 petals .
4, 6; 7, 10
Journal Writing (Page 169)
Pascal’s Triangle was developed by the French Mathematician Blaise
Pascal . It is formed by starting with the number 1 . Each number in the
subsequent rows is obtained by finding the sum of the number which is
diagonally above it to the left and that which is diagonally above it to
the right . 0 is used as a substitute in the absence of a number in either
of the two positions .
Class Discussion (The Triangular Number Sequence)
1
1
1.
1
1
Figure 5
Total Number of Coins, Tn
Figure 6
1
1
1
1
113
5
7
2
21
10
5
8
13
21
1
4
10
20
35
3
1
3
6
15
2
1
3
4
6
1
1
5
15
35
1
6
21
1
7
1
1
(d) Since the common difference is 3, Tn = 3n + ? .
The term before T1 is c = T0
=1–3
= –2 .
\ General term of sequence, Tn = 3n – 2
2. (i) 23, 27
(ii) Since the common difference is 4, Tn = 4n + ? .
The term before T1 is c = T0
=3–4
= –1 .
\ General term of sequence, Tn = 4n – 1
(iii) T50 = 4(50) – 1
= 200 – 1
= 199
The Fibonacci sequence is a set of numbers that begins with 1 and
1, and each subsequent term is the sum of the previous two terms,
i .e . 1, 1, 2, 3, 5, 8, 13, 21, … The sums of the numbers on the diagonals
of Pascal’s Triangle form the Fibonacci sequence, as illustrated .
Teachers may wish to get students to describe the symmetry in Pascal’s
Triangle and to identify other patterns that can be observed from the
triangle.
Practise Now 1
1. (a) Rule: Add 5 to each term to get the next term . The next two
terms are 28 and 33 .
(b) Rule: Subtract 6 from each term to get the next term . The next
two terms are –50 and –56 .
(c) Rule: Multiply each term by 3 to get the next term . The next
two terms are 1215 and 3645 .
(d) Rule: Divide each term by –3 to get the next term . The next two
terms are –18 and 6 .
2. (a) 22, 29
(b) 15, 11
Practise Now 4
1. (i)
Figure 5
Figure 6
Practise Now 2
(ii)
(i) T4 = 4(4) + 7
= 16 + 7
= 23
(ii) T7 = 4(7) + 7
= 28 + 7
= 35
Sum of 4th term and 7th term of sequence = T4 + T7
= 23 + 35
= 58
Practise Now 3
Number of Dots
1
2+1×4=6
2
2 + 2 × 4 = 10
3
2 + 3 × 4 = 14
4
2 + 4 × 4 = 18
5
2 + 5 × 4 = 22
6
2 + 6 × 4 = 26
:
:
n
2 + n × 4 = 4n + 2
(iii) When n = 2013,
4n + 2 = 4(2013) + 2
= 8054
Number of dots in 2013th figure = 8054
2. (i) 8th line: 72 = 8 × 9
(ii) Since 110 = 10 × 11 = 10(10 + 1),
k = 10 .
1. (a) Since the common difference is 4, Tn = 4n + ? .
The term before T1 is c = T0
=5–4
= 1 .
\ General term of sequence, Tn = 4n + 1
(b) Since the common difference is 5, Tn = 5n + ? .
The term before T1 is c = T0
=7–5
= 2 .
\ General term of sequence, Tn = 5n + 2
(c) Since the common difference is 6, Tn = 6n + ? .
The term before T1 is c = T0
=2–6
= – 4 .
\ General term of sequence, Tn = 6n – 4
1
Figure Number
Practise Now 5
(i)
114
Member
Number
Number of carbon
atoms
Number of
hydrogen atoms
1
2
4
2
3
6
3
4
8
4
5
10
5
6
12
6
7
14
:
:
:
n
n+1
2n + 2
(ii) Let h + 1 = 55 .
h = 55 – 1
= 54
When n = h = 54,
2n + 2 = 2(54) + 2
= 110
Number of hydrogen atoms the member has = 110
(iii) Let 2k + 2 = 120 .
2k = 120 – 2
= 118
k = 59
When n = k = 59,
n + 1 = 59 + 1
= 60
Number of carbon atoms the member has = 60
Exercise 7B
1. (a) Since the common difference is 6, Tn = 6n + ? .
The term before T1 is c = T0
=7–6
= 1 .
\ General term of sequence, Tn = 6n + 1
(b) Since the common difference is 3, Tn = 3n + ? .
The term before T1 is c = T0
= –4 – 3
= –7 .
\ General term of sequence, Tn = 3n – 7
(c) Since the common difference is 7, Tn = 7n + ? .
The term before T1 is c = T0
= 60 – 7
= 53 .
\ General term of sequence, Tn = 7n + 53
(d) Since the common difference is –3, Tn = –3n + ? .
The term before T1 is c = T0
= 14 + 3
= 17 .
\ General term of sequence, Tn = –3n + 17
2. (i) T5 = 2(5) + 5
= 10 + 5
= 15
(ii) T8 = 2(8) + 5
= 16 + 5
= 21
(iii) 15 = 3 × 5
21 = 3 × 7
LCM of 5th term and 8th term of sequence = 3 × 5 × 7
= 105
3. (i) 18, 21
(ii) Since the common difference is 3, Tn = 3n + ? .
The term before T1 is c = T0
=3–3
= 0 .
\ General term of sequence, Tn = 3n
(iii) T105 = 3(105)
= 315
4. (i) 30, 34
(ii) Since the common difference is 4, Tn = 4n + ? .
The term before T1 is c = T0
= 10 – 4
= 6 .
\ General term of sequence, Tn = 4n + 6
(iii) T200 = 4(200) + 6
= 800 + 6
= 806
Exercise 7A
1. (a) Rule: Add 5 to each term to get the next term . The next two
terms are 39 and 44 .
(b) Rule: Subtract 8 from each term to get the next term . The next
two terms are 40 and 32 .
(c) Rule: Multiply each term by 2 to get the next term . The next
two terms are 384 and 768 .
(d) Rule: Divide each term by 2 to get the next term . The next two
terms are 50 and 25 .
(e) Rule: Divide each term by – 4 to get the next term . The next two
terms are 16 and – 4 .
(f) Rule: Multiply each term by –2 to get the next term . The next
two terms are –288 and 576 .
(g) Rule: Subtract 7 from each term to get the next term . The next
two terms are –87 and –94 .
(h) Rule: Add 10 to each term to get the next term . The next two
terms are –50 and – 40 .
(i) Rule: Add 10 to each term to get the next term . The next two
terms are 50 and 60 .
(j) Rule: Add 7 to each term to get the next term . The next two
terms are 80 and 87 .
(k) Rule: Multiply each term by 3 to get the next term . The next
two terms are 324 and 972 .
2. (a) 9, 15
(b) 12, 8
(c) –33, –32
(d) 88, 85
(e) 21, 28
3. (a) –67, –131
(b) 8, 13
(c) 144, 196
(d) –216, 343
(e) 81, 243
115
1
(b) (i) General term of sequence, Tn = 2n2 + 1 – 2
= 2n2 – 1
2
(ii) T388 = 2(388) – 1
= 301 088 – 1
= 301 087
8. (i)
5. (i)
Number of
points
1
2
3
4
5
6
Number of
segments
1+1
=2
2+1
=3
3+1
=4
4+1
=5
5+1
=6
6+1
=7
(ii) Let the number of points be n .
Number of segments = n + 1 .
When n = 49, number of segments = 49 + 1
= 50
(iii) 101 = n + 1
\ n = 101 – 1
= 100
Figure 5
Figure 6
6. (i)
(ii)
Figure 5
(ii)
Figure
Number
Number of Intersection(s)
between the Circles
1
0
2
1
3
2
4
3
5
4
6
5
:
:
n
n–1
Number of Small Triangles
1
4
2
9
3
16
4
25
5
36
6
49
:
:
n
(n + 1)2
(iii) When n = 20,
(n + 1)2 = (20 + 1)2
= 212
= 441
Number of triangles in 20th figure = 441
(iv) Let (n + 1)2 = 121 .
n + 1 = 11
or
n + 1 = –11
n = 11 – 1 or
n = –11 – 1
= 10
or
= –12 (N .A . since n > 0)
9. (i) 6th line: 54 = 6 × 9
(ii) Since 208 = 13 × 16 = 13(13 + 3),
k = 13 .
10. (i) 5th line: 1 + 3 + 5 + 7 + 9 + 11 = 36 = 62 = (5 + 1)2
(iii) Let n – 1 = 28 .
n = 28 + 1
= 29
7. (a) When n = 1,
2n2 + 1 = 2(1)2 + 1
=2+1
=3
When n = 2,
2n2 + 1 = 2(2)2 + 1
=8+1
=9
When n = 3,
2n2 + 1 = 2(3)2 + 1
= 18 + 1
= 19
When n = 4,
2n2 + 1 = 2(4)2 + 1
= 32 + 1
= 33
The first four terms of the sequence are 3, 9, 19 and 33.
1
Figure Number
Figure 6
(ii) c =
169
= 13
d + 1 = 13
d = 13 – 1
= 12
a = 13 + 12
= 25
116
11 . (a) (i)
(ii)
Number of
people
4
6
8
10
12
14
Number of
tables
4–2
=1
2
6–2
=2
2
8–2
=3
2
10 – 2
=4
2
12 – 2
=5
2
14 – 2
=6
2
Number
of tables
1
2
3
4
5
6
Number
of people
2(1) + 2 = 4
2(2) + 2 = 6
2(3) + 2 = 8
2(4) + 2 = 10
2(5) + 2 = 12
2(6) + 2 = 14
(b) (i) From (a)(i): When n = 20,
n – 2 20 – 2
=
2
2
=9
\ 9 tables will be needed to seat 20 people .
(ii) When n = 30,
n–2
30 – 2
=
2
2
= 14
\ 14 tables will be needed to seat 30 people .
(c) (i) From (a)(ii): When n = 22,
2(22) + 2 = 46
\ 46 people can be seated at 22 tables .
(ii) When n = 36,
2(36) + 2 = 74
\ 74 people can be seated at 36 tables .
12 . (i)
Number of
points on the
line segments
AB (including
the points A
and B)
Number of
possible line
segments
2
3
4
5
6
7
2 × (2 – 1)
2
3 × (3 – 1)
2
4 × (4 – 1)
2
5 × (5 – 1)
2
6 × (6 – 1)
2
7 × (7 – 1)
2
= 10
= 15
=1
=3
=6
= 21
(ii) Number of points including AB = 18 + 2
= 20
20 × (20 – 1)
2
= 190
Number of possible line segments =
117
1
13. (i) 1
(ii)
5
Row
10
10
5
1
Sum
1
1 = 1 = 20
2
1 + 1 = 2 = 21
3
1 + 2 + 1 = 4 = 22
4
1 + 3 + 3 + 1 = 8 = 23
5
1 + 4 + 6 + 4 + 1 = 16 = 24
6
1 + 5 + 10 + 10 + 5 + 1 = 32 = 25
:
n
14. (a)
:
1 + (n – 1) + ··· + (n – 1) + 1 = 2n – 1
Figure
1
Number of black
1
squares (b)
Number of white
1×2+1=3
squares (w)
Area of whole
4
figure (b + w)
Perimeter of
2(1 + 4) = 10
whole figure (cm)
2
3
4
5
6
2
3
4
5
6
2×2+1=5
3×2+1=7
4×2+1=9
7
10
13
16
19
2(2 + 4) = 12
2(3 + 4) = 14
2(4 + 4) = 16
2(5 + 4) = 18
2(6 + 4) = 20
(b) (i) Number of white squares in Figure 9 = 9 × 2 + 1
= 19
(ii) Perimeter of Figure 9 = 2(9 + 4)
= 26 cm
(iii) Number of white squares in Figure n = n (2 + 1)
= 2n + 1
(iv) Perimeter of Figure n = 2(n + 4)
= (2n + 8) cm
15. (i) 8th line:
16. (a) (i) 11, 13
(ii) 24, 28
(iii) 84, 112
(iv) 85, 113
(b) 6th line: 132 + 842 = 852
7th line: 152 + 1122 = 1132
17. (i)
1
2
1
2
=
–
+
8
9
10
8 × 9 × 10
(ii) Based on the pattern, nth line:
2
1
2
1
–
+
=
n ( n + 1)( n + 2)
n
n +1
n+2
1
2
1
2
\
–
+
=
10
11
12
10 × 11 × 12
2
=
1320
1
=
660
2
1
2
1
(iii)
=
–
+
p +1
p+2
7980
p
2
2
1
1
–
+
=
p ( p + 1)( p + 2)
p +1
p+2
p
Member
Number
Number of carbon
atoms
Number of
hydrogen atoms
1
3
4
2
4
6
3
5
8
4
6
10
5
7
12
6
8
14
:
:
:
n
n+2
2n + 2
(ii) Let h + 2 = 25 .
h = 25 – 2
= 23
When n = h = 23,
2n + 2 = 2(23) + 2
= 48
Number of hydrogen atoms the member has = 48
\ p(p + 1)(p + 2) = 7980
(p2 + p)(p + 2) = 7980
p3 + 2p2 + p2 + 2p – 7980 = 0
p3 + 3p2 + 2p – 7980 = 0
\ p = 19 or
p = –11 + 17 .292 (5 s .f .) (reject, p is a whole number) or
p = –11 – 17 .292 (5 s .f .) (reject, p > 1)
1
5 × 2 + 1 = 11 6 × 2 + 1 = 13
118
(iii) Let 2k + 2 = 64 .
2k = 64 – 2
= 62
k = 31
When n = k = 31,
n + 2 = 31 + 2
= 33
Number of carbon atoms the member has = 33
18. (i)
3. (i)
Figure 5
(ii)
M
1st Generation Ancestor
F
M
2nd Generation Ancestors
F
F
M
M
F
F
M
F
4rd Generation Ancestors
Number of 4th generation ancestors a male bee has = 5
(ii) The number of nth generation ancestors forms a sequence:
1, 2, 3, 5, ... The first two numbers of the sequence are 1 and 2,
and each subsequent term is the sum of the previous two terms .
(iii) Number of 5th generation ancestors a male bee has = 3 + 5
=8
M
F
1
5×1+1=6
2
5 × 2 + 1 = 11
3
5 × 3 + 1 = 16
4
5 × 4 + 1 = 21
5
5 × 5 + 1 = 26
:
:
n
5 × n + 1 = 5n + 1
1st Generation Ancestor
M
F
F
Number of Buttons
(iii) When n = 56,
5n + 1 = 5(56) + 1
= 281
Number of buttons in 56th figure = 281
(iv) Let 5n + 1 = 583 .
5n = 583 – 1
= 582
2
n = 116
5
2
Since n = 116 ∉ Z+, it is not possible for a figure in the
5
sequence to be made up of 583 buttons .
4. (i)
3rd Generation Ancestors
F
Figure Number
M
2nd Generation Ancestors
F
3rd Generation Ancestors
Figure 5
F
M
F
M
F
M
F
M
F
F
F
M
(ii)
4rd Generation Ancestors
F 5th Generation Ancestors
(iv) The sequence for the number of nth generation ancestors is
1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
Number of 10th generation ancestors a male bee has = 34 + 55
= 89
Review Exercise 7
1. (a) 53, 44
(b) 28, 40
1
1
,
27 81
(d) 121, 169
2. (i) 64, 81
(ii) General term of sequence, Tn = (n + 2)2
(iii) T25 = (25 + 2)2
= 272
= 729
(c)
Figure Number
Number of Triangles
1
1=
1× 2
2
2
3=
2×3
2
3
6=
3× 4
2
4
10 =
4×5
2
5
15 =
5×6
2
:
:
n
1
n(n + 1)
2
(iii) When n = 77,
1
1
n(n + 1) =
× 77 × (77 + 1)
2
2
1
=
× 77 × 78
2
= 3003
Number of triangles in 77th figure = 3003
119
1
1
n(n + 1) = 66 .
2
n(n + 1) = 132
Since 132 = 11 × 12 = 11(11 + 1),
n = 11 .
5. (i) 7th line: 13 + 23 + 33 + 43 + 53 + 63 + 73
= 784 = (1 + 2 + 3 + 4 + 5 + 6 + 7)2
3
3
(ii) 1 + 2 + 33 + 43 + ··· + 153 = (1 + 2 + 3 + 4 + ··· + 15)2
= 1202
= 14 400
3
3
3
3
(iii) Since 1 + 2 + 3 + 4 + ··· + k3 = 1296 = 362 =
(1 + 2 + 3 + 4 + ··· + 8)2,
k = 8 .
(iv) Let
12 + 35
2
6. (i) a =
=
2.
Number of Handshakes
2
1=
2 ×1
2
3
3=
3× 2
2
4
6=
4×3
2
5
10 =
5×4
2
6
15 =
6×5
2
:
:
n
1
n(n – 1)
2
2
1369
Number of handshakes that will take place =
= 37
(ii) Since the common difference is 2, TnA = 2n + ? .
The term before T1A is c = T0A
=4–2
= 2 .
\ General term of sequence A, TnA = 2n + 2
(iii) General term of sequence C, TnC =
=
(2 n + 2)2 + ( n 2 + 2 n )2
2 (18 ) + 2 + 18 2 + 2 (18 )
=
38 2 + 360 2
=
131 044
2
= 362
Challenge Yourself
Value
31
32
33
34
35
36
···
Last Digit
3
9
7
1
3
9
···
2015 ÷ 4 = 503 R 3
\ Last digit of 32015 = 7
1
1
n(n – 1)
2
3. (i) 4, 9
(ii) The general term, Tn, of the sequence is obtained by continuously
finding the sum of the digits of n2 until a single-digit number is
left, e .g . to obtain T7,
72 = 49 → 4 + 9 = 13 → 1 + 3 = 4, \ T7 = 4 .
4. (i) 11, 18
(ii) For n 3, Tn = Tn – 1 + Tn – 2 .
(iii) Lucas Numbers (which is different from Lucas Sequence)
5. (i) 10, 12
(ii) For n 4, Tn = Tn – 2 + Tn – 3 .
(iii) Perrin Numbers (or Perrin Sequence)
Tn2A + Tn2B
2
T18C =
1.
Number of People
120
5. Let the time the motorist spends on the expressway be x hours .
Then the time he spends on the stretch of road is 2x hours .
95x + 65 × 2x = 375
95x + 130x = 375
225x = 375
Revision Exercise B1
1. (a) 0 .15x + 2 .35(x – 2) = 1 .3
0 .15x + 2 .35x – 4 .7 = 1 .3
2 .5x – 4 .7 = 1 .3
2 .5x = 1 .3 + 4 .7
2 .5x = 6
\ x = 2 .4
7
5
(b)
–
=4
2 – 2y
1– y
5
7
=4
–
1– y
2(1 – y )
10 – 7
=4
2(1 – y )
3
=4
2(1 – y )
3 = 8(1 – y)
3 = 8 – 8y
3 – 8 = –8y
–5 = –8y
\y =
2
3
\ Total time taken = x + 2x
= 3x
2
= 3 1
3
= 5 hours
x=1
6.
x
–4
0
6
y= 1 x+3
2
1
3
6
y = –x + 6
10
6
0
y
y = –x + 6
10
5
8
9
2. (a) 12x > 60
\x > 5
(b) 15y < –24
8
24
y<–
15
3
\ y < –1
5
x – 4y
3
3.
=
5x + y
5
5(x – 4y) = 3(5x + y)
5x – 20y = 15x + 3y
5x – 15x = 3y + 20y
–10x = 23y
5
1
y= x+3
2
7
6
4
3
2
1
– 4 –3 –2 –1 0
x
1
2
3
4
5
6
7. (i) Since the common difference is 9, Tn = 9n + ? .
The term before T1 is c = T0
=6–9
= –3 .
\ General term of sequence, Tn = 9n – 3
(ii) Let 9k – 3 = 159 .
9k = 159 + 3
= 162
\ k = 18
8. (i) 8th line: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 82
(ii) Since 144 = 122,
k = 12 .
x
23
=–
y
10
x
23
\
=–
3y
30
4. Let the number of 20-cent coins in the box be x .
Then the number of 50-cent coins in the box is 54 – x .
0 .2x + 0 .5(54 – x) = 20 .7
0 .2x + 27 – 0 .5x = 20 .7
– 0 .3x + 27 = 20 .7
– 0 .3x = 20 .7 – 27
– 0 .3x = – 6 .3
x = 21
There are twenty-one 20-cent coins in the box .
121
1
y
5. (i)
Revision Exercise B2
C(2, 2)
1
(x – 3) – x + 5 = 3(x – 1)
3
1
x – 1 – x + 5 = 3x – 3
3
2
– x + 4 = 3x – 3
3
2
– x – 3x = –3 – 4
3
11
– x = –7
3
10
\x=1
11
2
3
(b)
–
+1=3
y
y
1
– +1 =3
y
1
– =3–1
y
1
– =2
y
1. (a)
2
D(–2, 0) 1
–2 –1 0
–1
2
3
B(4, –2)
–3
–4
A(0, – 4)
(ii) Coordinates of D = (–2, 0)
6. (a)
x
–5
0
5
y=x+2
y=x–3
–3
2
7
–8
–3
2
y
y=x+2
7
6
5
4
1
2
3
2. (a) 14x > –110
2
6
7
1
\ x > –7
– 5 – 4 –3 –2 –1 0
–1
(b) –18 < 3y
18
<y
3
\ y > –6
3. Let the smallest even number be x .
Then the next 6 even numbers are x + 2, x + 4, x + 6, x + 8, x + 10
and x + 12 .
x + x + 2 + x + 4 + x + 6 + x + 8 + x + 10 + x + 12 = 336
7x + 42 = 336
7x = 336 – 42
7x = 294
\ x = 42
The smallest of the 7 numbers is 42 .
4. 8 .5x + 3 .6(2x + 5) = 206 .4
8 .5x + 7 .2x + 18 = 206 .4
15 .7x + 18 = 206 .4
15 .7x = 206 .4 – 18
15 .7x = 188 .4
\ x = 12
–
y=2
x
1
2
3
4
5
–2
–3
y = –3
–4
–5
–6
–7
y=x–3
–8
1
1
×5×5+
×5×5
2
2
1
1
= 12 + 12
2
2
= 25 units2
Since the common difference is –3, Tn = –3n + ? .
The term before T1 is c = T0
= 44 + 3
= 47 .
\ General term of sequence, Tn = –3n + 47
Let –3k + 47 = –13 .
–3k = –13 – 47
= –60
\ k = 20
6th line: 62 – 2 × 6 = 24
Since 92 – 2 × 9 = 63,
k = 9 .
(b) Area enclosed by the four lines =
7. (i)
(ii)
8. (i)
(ii)
1
4
–2
1
–y =
2
\y =–
x
1
122
Chapter 8 Percentage
TEACHING NOTES
Suggested Approach
Although students have learnt percentage in primary school (i.e. how to express a part of a whole as a percentage, write fractions
and decimals as percentages, and vice versa, find a percentage part of a whole and solve up to 2-step word problems involving
percentage), many may still struggle with percentage. Teachers can introduce percentage as fractions by going right back to the
fundamentals. Teachers can give students practical applications of percentages and show the changes in fractions and proportions
through the examples to give them a better understanding of the concept.
Section 8.1:
Introduction to Percentage
Teachers can get students to work in pairs to find an advertisement/article in which percentage(s) can be found
and discuss about it together (see Class Discussion: Percentages in Real Life). After the discussion, students
should understand the meaning of percentage(s) better and interpret information more accurately. Students need
to be able to comment critically on the usefulness of percentages before they can have a confident grasp of the
topic.
Teachers can then build upon what students have learnt about percentage in primary school. Students may be
able to learn how to accurately calculate a percentage but they might struggle to explain the meaning behind it.
Teachers should emphasise on the basics of fractions and proportions before getting the students to calculate and
interpret percentages.
In Worked Example 5, students should learn that it is easy to see that more people passed the entrance test in
2011 but it is not easy to see which year had a higher proportion of people passing the entrance test. Teachers can
highlight to the students that two quantities can be easily compared using percentages because the proportions
are converted to the same base i.e. 100.
Section 8.2:
Percentage Change and Reverse Percentage
Teachers should guide students on how to use algebra in percentage change and reverse percentage. Students
may draw models, wherever applicable, to help them understand the problem.
Through the worked examples in this section, students should be able to tackle percentage change and reverse
percentage problems involving algebra. They should also learn how to identify whether the problem is a reverse
percentage or a percentage change problem. Teachers can highlight to the students that percentage change is
when they are given both the new value and the original value while a reverse percentage is when they need to
find the original value given a quantity after a percentage increase or decrease.
123
1
The term ‘up to’ in the phrase ‘Discount up to 80% on All Items’
suggests that the greatest percentage discount given on the items in
the shop is 80%. This means that some items in the shop may be
sold at a discount that is less than 80%. Hence, the shopkeeper is
not dishonest.
Most of the time, such advertisements come with terms and
conditions that may state the items which are not subjected to a
discount, such as ‘New Arrivals’. These terms and conditions are
normally shown in fine print.
As some items may be sold at a discount of less than 80% and some
items may not be subjected to a discount, the prices of the items in
the shop may not be low. In addition, the original prices of some
items may be very high, such that their prices are still high even
after a discount.
WORKED SOLUTIONS
Class Discussion (Percentage in Real Life)
1. Guiding Questions:
• What is the advertisement/article about?
• Are the percentages found in the advertisement/article expressed
using the percentage symbol or in words?
• What do the percentages mean in the context of the
advertisement/article?
Teachers may use this to assess students’ prior knowledge of
percentage, e.g. whether students are able to relate percentages
to fractions and to perform relevant calculations using the given
percentages to illustrate the meaning of the percentages in the
context of the advertisement/article. Teachers may also use this as
a trigger to show students the need to learn percentage, and link
back to the different scenarios in the advertisement/article.
Alternatively, teachers may wish to use the article titled ‘A smaller
and cheaper iPad’ (Today, 5 July 2012) and/or the apparel
advertisement (Page 3, Today, 5 July 2012) for this question.
Teachers may wish to ask students to list other instances where such
phrases are used. They may also want to take this opportunity to
highlight to students the importance of being informed consumers.
Students should not take information at face value. Instead, they
should learn how to interpret information accurately.
3. Guiding Questions:
The following shows examples of statements with percentages more
than 100%:
• In Singapore, the number of employers hiring ex-offenders has
increased by more than 100% from 2004 to 2011.
• The total number of registrants for a school during Phase 2B of
the Primary 1 registration is 120% of the number of vacancies
available.
• The number of mobile subscriptions in Singapore in 2011 is
about 150% of her population.
• The population of Singapore in 2010 is about 240% of that in
1970.
• The total fertility rate in Singapore in 1970 is about 270% of
that in 2010.
The phrase ‘this year’s sales is 200% of last year’s sales’ means that
the sales this year is 2 times of that of last year.
The article ‘A smaller and cheaper iPad?’ is about Apple’s apparent
intention to launch an iPad which is smaller and less expensive. This
is in order to counter its rivals’ new products so as to maintain its
stronghold in the tablet market. The percentage found in the article
is 61 per cent, which is expressed in words and in the context of
the article, it means that Apple has 61% of the tablet market share.
A conclusion that can be drawn is that the iPad is the most popular
tablet in the market as the total tablet market share is the base, i.e.
100%. The other brands have a total of 100% – 61% = 39% of the
tablet market share.
The advertisement shows that a particular brand of apparel is
holding a storewide end-of-season sale. The percentages found in
the advertisement are 70% and 10%, which are expressed using the
percentage symbol. In the context of the advertisement, it means
that shoppers can enjoy a discount of up to 70% on the items and
those with UOB cards are entitled to an additional 10% off if they
purchase a minimum of 3 items.
Teachers may ask students to refer to page 197 of the textbook for
an example of how this phrase may be used. Teachers should also
highlight to students that the use of percentages can be misleading,
e.g. a salesman who sold a car in January and two cars in February
can say that his sales in February is 200% of that in January.
Teachers may wish to ask students how the additional 10% off for
UOB card members is calculated, i.e. whether a shopper with a UOB
card gets to enjoy a maximum discount of 70% + 10% = 80% on
selected items. Teachers may also wish to get students to use any
amount,
e.g. $100, to illustrate the meaning of the percentages, i.e. 70% and
10%, in the context of the advertisement.
2. Guiding Questions:
• What is the meaning of the term ‘up to’?
• An advertisement with the phrase ‘Discount up to 80% on
All Items’ is displayed at the entrance of a shop. If an item in
the shop is sold at a discount of 10%, does this mean that the
shopkeeper is dishonest?
• Is it true that all the items in the shop are sold at a discount?
Could there be any exceptions?
• Does it mean that the prices of all the items in the shop are very
low?
Class Discussion (Expressing Two Quantities in Equivalent
Forms)
40
× 100%
50
= 80%
There are 80% as many male teachers as female
teachers.
The number of male teachers is 80% of the number of
female teachers.
4
The number of male teachers is
of the number of
5
female teachers.
1. (a) (i) Required percentage =
•
•
•
124
2. Mr Lee’s monthly salary in 2011 = 110% × $x
110
=
× $x
100
= $1.1x
Mr Lee’s monthly salary in 2012 = 90% × $1.1x
50
× 100%
40
= 125%
There are 125% as many female teachers as male
teachers.
The number of female teachers is 125% of the number
of male teachers.
(ii) Required percentage =
•
•
•
(b)
=
90
× $1.1x
100
= $0.99x
Hence, it is not correct to say that Mr Lee’s monthly salary in 2012
was $x.
5
The number of female teachers is
of the number of
4
male teachers.
In words
A is 80% of B.
B is 125% of A.
Percentage
A = 80% × B
B = 125% × A
Fraction
4
A = (fraction) × B
5
5
B = (fraction) × A
4
Decimal
A = 0.8 × B
B = 1.25 × A
Practise Now 1
(a) (i) 45% =
45
100
9
=
20
305
(ii) 305% =
100
61
=
20
1
=
3
20
5.5
(iii) 5.5% =
100
55
=
1000
11
=
200
5
61
(iv) 8 % =
%
7
7
61
= ÷ 100
7
61
1
= ×
7
100
61
=
700
17
17
(b) (i)
=
× 100%
20
20
1700
=
20
=
85%
Table 8.1
2. (i)
In words
P is 20% of Q.
Percentage P = 20% × Q
1
(fraction) × Q
5
Fraction
P=
Decimal
P = 0.2 × Q
R is 50% of S.
T is 125% of U.
R = 50% × S
T = 125% × U
R=
1
5
(fraction) × S T = (fraction) × U
2
4
R = 0.5 × S
T = 1.25 × Q
Table 8.2
(ii) The relationship between P and Q can be illustrated as follows:
P
Q
The relationship between R and S can be illustrated as follows:
R
S
The relationship between T and U can be illustrated as follows:
T
U
(ii) 23
Thinking Time (Page 198)
20% + 80%
, i.e. 50% of the total
2
number of students in the two groups had done the survey. This
is because there may be a different number of students in each
of the two groups, e.g. if Ethan conducted the survey on 20%
of a group of 100 students and on 80% of another group of
1. No, it is not correct to say that
1
116
=
5
5
116
= × 100%
5
11 600
=
%
5
=
2320%
20% × 100 + 80% × 200
× 100% = 60% of the
100 + 200
total number of students in the two groups had done the survey.
200 students, then
125
1
Practise Now 2
Practise Now (Page 190)
1. (a) 20% of $13.25 = 20% × $13.25
(a) (i) 12% =
12
100
=
0.12
20
= × $13.25
100
= $2.65
(ii) 413% =
413
100
=
4.13
(b) 15
3
3
% of $640 = 15 % × $640
4
4
3
15
4
=
× $640
100
= $100.80
2. 2500% of $4.60 = 2500% × $4.60
(iii) 23.6% =
23.6
100
=
0.236
(iv) 6
1
25
%=
%
4
4
25
= ÷ 100
4
25
1
= ×
4
100
25
=
400
=
0.0625
Alternatively,
2500
=
× $4.60
100
= $115
Practise Now 4
1. Method 1:
Number of students who were late for school = 3% × 1500
=
3
× 1500
100
= 45
Number of students who were punctual for school = 1500 – 45
= 1455
Method 2:
Percentage of students who were punctual for school = 100% – 3%
= 97%
Number of students who were punctual for school = 97% × 1500
6
1
% = 6.25%
4
6.25
=
100
=
0.0625
(b) (i) 0.76 = 0.76 × 100%
=
76%
(ii) 2.789 = 2.789 × 100%
=
278.9%
=
97
× 1500
100
= 1455
Practise Now 3
1. (i) Total number of teachers in the school = 45 + 75
= 120
2. Percentage of children who attended the dinner
=
100% – 35.5% – 40%
=
24.5%
Number of children who attended the dinner = 24.5% × 1800
Percentage of male teachers in the school =
45
× 100%
120
= 37.5%
=
24.5
× 1800
100
= 441
(ii) Method 1:
Percentage of female teachers in the school =
75
× 100%
120
= 62.5%
Practise Now 5
Method 2:
Percentage of female teachers in the school = 100% – 37.5%
= 62.5%
1400 ml
2. Required percentage =
× 100%
2.1 l
1400 ml
=
× 100%
2100 ml
Percentage of people who attended the New Year party in Village A
4000
=
× 100%
30 000
= 13
1
%
3
Percentage of people who attended the New Year party in Village B
2800
=
× 100%
25 000
=
2
× 100%
3
2
= 66 %
3
1
= 11.2%
\ Village A had a higher percentage of people who attended its New
Year party.
126
Practise Now 6
Practise Now 8
1. (a) Value of award for a Secondary 1 student in 2009
=
140% × $250
70% of the books = 35
35
1% of the books =
70
35
100% of the books =
× 100
70
= 50
There are 50 books on the bookshelf.
140
= × $250
100
=
$350
(b) (i) Percentage increase in value of award from 2008 to 2009
for a Primary 1 student
$250 – $150
=
× 100%
$150
$100
=
× 100%
$150
Practise Now 9
1. Method 1:
109% of original cost = $654
2
=
66 %
3
(ii) Percentage increase in value of award from 2008 to 2009
for a Primary 6 student
$300 – $200
=
× 100%
$200
$100
=
× 100%
$200
=
50%
2. (a) Required result = 75% × 32
1% of original cost =
$654
109
$654
100% of original cost =
× 100
109
= $600
The original cost of the article is $600.
Method 2:
Let the original cost of the article be $x.
0%
109%
x
75
= × 32
100
=
24
(b) Percentage decrease in value of car
$127 000 – $119 380
=
× 100%
$127 000
$7620
× 100%
=
$127 000
654
From the model, we form the equation:
109% × x = 654
1.09x = 654
x = 600
The original cost of the article is $600.
2. 120% of value in 2011 = $180 000
=
6%
Practise Now 7
$180 000
120
$180 000
100% of value in 2011 =
× 100
120
= $150 000
The value of the vase was $150 000 in 2011.
120% of value in 2010 = $150 000
$150 000
1% of value in 2010 =
120
$150 000
100% of value in 2010 =
× 100
120
= $125 000
The value of the vase was $125 000 in 2010.
1% of value in 2011 =
Original
Cost
Percentage
Change
New Cost
Rental
$2400
–5%
95
× $2400 = $2280
100
Wages
$1800
–6%
94
× $1800 = $1692
100
Utilities
$480
+7%
107
× $480 = $513.60
100
Business
$4680
$4485.60
Percentage decrease in monthly cost of running business
$4680 – $4485.60
× 100%
=
$4680
$194.40
=
× 100%
$4680
=4
100%
2
%
13
127
1
(d) 6
3
33
% =
%
5
5
33
= ÷ 100
5
33
1
= ×
5
100
33
=
500
4
2. (a) 4% =
100
=
0.04
Practise Now 10
1. Method 1:
97% of original monthly salary = $3346.50
1% of original monthly salary =
$3346.50
97
$3346.50
100% of original monthly salary =
× 100
97
= $3450
Devi’s original monthly salary is $3450.
Method 2:
Let Devi’s original monthly salary be $x.
0%
97%
(b) 633% =
633
100
=
6.33
100%
x
(c) 0.02% =
0.02
100
=
0.0002
0.97x
3346.50
(d) 33
From the model, we form the equation:
97% × x = 3346.50
0.97x = 3346.50
x = 3450
Devi’s original monthly salary is $3450.
2.
85% of value in 2011 = $86 700
$86 700
85
$86 700
100% of value in 2011 =
× 100
85
= $102 000
The value of the car was $102 000 in 2011.
85% of value in 2010 = $102 000
1% of value in 2011 =
3. (a)
(b)
$102 000
85
$102 000
100% of value in 2010 =
× 100
85
= $120 000
The value of the car was $120 000 in 2010.
1% of value in 2010 =
(c)
(d)
(e)
Exercise 8A
1. (a) 28% =
28
100
7
=
25
158
(b) 158% =
100
79
=
50
29
=
1
50
12.4
(c) 12.4% =
100
124
=
1000
31
=
250
1
3
3
=
× 100%
5
5
=
60%
9
9
=
× 100%
10
10
=
90%
6
6
=
× 100%
125
125
=
4.8%
6
6
=
× 100%
5
5
=
120%
12
12
=
× 100%
25
25
=
48%
6
31
=
× 100%
25
25
=
124%
0.78 = 0.78 × 100%
=
78%
0.25 = 0.25 × 100%
=
25%
0.07 = 0.07 × 100%
=
7%
0.095 = 0.095 × 100%
=
9.5%
1.35 = 1.35 × 100%
=
135%
(f) 1
4. (a)
(b)
(c)
(d)
(e)
128
2
101
%=
%
3
3
101
= ÷ 100
3
101
1
= ×
3
100
101
=
300
=
0.337 (to 3 s.f.)
(f) 2.6 = 2.6 × 100%
=
260%
5. (a) 50% of $70 = 50% × $70
45 minutes
× 100%
1 hour
45 minutes
=
× 100%
60 minutes
(b) Required percentage =
50
= × $70
100
=
$35
(b) 80% of 4.5 m = 80% × 4.5 m
3
× 100%
4
= 75%
1 year
(c) Required percentage =
× 100%
4 months
12 months
=
× 100%
4 months
=
80
= × 4.5 m
100
= 3.6 m
6. (i) Total number of students in the class = 20 + 18
= 38
= 3 × 100%
= 300%
15 mm
(d) Required percentage =
× 100%
1m
15 mm
=
× 100%
1000 mm
3
× 100%
=
200
= 1.5%
335 cm
(e) Required percentage =
× 100%
5m
335 cm
=
× 100%
500 cm
Percentage of boys in the class =
20
× 100%
38
12
= 52
%
19
12
(ii) Percentage of girls in the class = 100% – 52
%
19
7
=47
%
19
7. Percentage of cars which are not blue = 100% – 30%
= 70%
Number of cars which are not blue = 70% × 120
=
70
× 120
100
= 84
8. Percentage of annual income Jun Wei donated to charitable
organisations
$1200
=
× 100%
12 × $1600
$1200
=
× 100%
$19 200
67
× 100%
100
= 67%
1 kg
(f) Required percentage =
× 100%
800 g
1000 g
=
× 100%
800 g
=
5
× 100%
4
= 125%
=
=
6.25%
Percentage of annual income Lixin donated to charitable
organisations
$4500
=
× 100%
12 × $6800
$4500
=
× 100%
$81 600
60°
× 100%
360°
1
× 100%
=
6
2
= 16 %
3
63 cents
(h) Required percentage =
× 100%
$2.10
63 cents
=
× 100%
210 cents
(g) Required percentage =
35
=
5
%
68
\ Jun Wei donated a higher percentage of his annual income to
charitable organisations.
25 seconds
9. (a) Required percentage =
× 100%
3.5 minutes
25 seconds
=
× 100%
210 seconds
=
3
× 100%
10
= 30%
10. (a) 6
5
× 100%
42
19
= 11
%
21
=
129
1
1
% of 1.35 ml = 6 % × 1.35 ml
5
5
1
6
5
=
× 1.35 ml
100
837
=
ml
10 000
1
14. Number of remaining pages after Friday = 600 – 150
= 450
Number of pages that remains to be read = (100% – 40%) × 450
= 60% × 450
(b) 56
7
7
% of 810 m = 56 % × 810 m
8
8
7
56
8 × 810 m
=
100
11
= 460
m
16
(c) 0.56% of 15 000 l = 0.56% × 15 000 l
=
60
× 450
100
= 270
Required percentage =
270
× 100%
600
= 45%
=
0.56
× 15 000 l
100
= 84 l
(d) 2000% of 5¢ = 2000% × 5¢
Exercise 8B
2000
=
× 5¢
100
= 100¢
= $1
1. (a) Required value = 135% × 60
135
= × 60
100
= 81
(b) Required value = 225% × 28
11. Percentage of marks Kate obtains =
\ Kate gets a bronze award.
40
× 100%
60
2
= 66 %
3
225
= × 28
100
= 63
(c) Required value = 55% × 120
Percentage of marks Priya obtains =
\ Priya gets a silver award.
46
× 100%
60
2
= 76 %
3
55
= × 120
100
= 66
(d) Required value = 62
Percentage of marks Nora obtains =
49
× 100%
60
2
= 81 %
3
\ Nora gets a gold award.
12. Percentage of employees who were unaffected by the financial crisis
=
100% – 2.5% – 50.75%
=
46.75%
Number of employees who were unaffected by the financial crisis
= 46.75% × 12 000
2. (a)
1% of number =
17
20
17
100% of number =
× 100
20
= 85
The number is 85.
(b) 175% of number = 49
46.75
=
× 12 000
100
=
5610
13. Amount Ethan spent on room rental = 20.5% × $1850
1% of number =
49
175
49
100% of number =
× 100
175
= 28
The number is 28.
(c) 115% of number = 161
=
20.5
× $1850
100
= $379.25
Amount Ethan overspent = $379.25 + $690 + $940 – $1850
= $159.25
$159.25
Required percentage =
× 100%
$1850
= 8.61% (to 2 d.p.)
1
1
% × 216
2
1
62
= 2 × 216
100
=
135
20% of number = 17
1% of number =
161
115
161
100% of number =
× 100
115
= 140
The number is 140.
130
(d)
80% of number = 192
9.
$58.50
90
$58.50
100% of original bill =
× 100
90
= $65
The original bill is $65.
10. Value obtained after initial increase = 130% × 2400
3. Percentage increase in length of elastic band =
90 – 72
× 100%
72
18
=
× 100%
72
= 25%
4. (i) Value of award for a Secondary 1 student in the top 5% in 2009
=
130% × $500
=
Final number = 80% × 3120
=
108
×x
100
= 1.08x
Number of train passengers in 2012 = 108% × 1.08x
=
108
× 1.08x
100
= 1.1664x
Percentage increase in number of train passengers from 2010 to 2012
1.1664 x – x
=
× 100%
x
0.1664 x
=
× 100%
x
=
0.1664 × 100%
=
16.64%
=
80
× $120 000
100
= $96 000
Value of car at the end of 2011 = 90% × $96 000
Original
Cost
Percentage Change
New Cost
Raw materials
$100
+11%
111
× $100 = $111
100
Overheads
$80
+20%
120
× $80 = $96
100
Wages
$120
–15%
85
× $120 = $102
100
Printer
$300
12.
=
45% of the students = 135
130
× 2400
100
= 3120
80
= × 3120
100
= 2496
11. Let the number of train passengers in 2010 be x.
Number of train passengers in 2011 = 108% × x
130
= × $500
100
=
$650
(ii) Percentage increase in value of award from 2008 to 2009 for a
Primary 4 student in the next 5%
$350 – $250
=
× 100%
$250
$100
=
× 100%
$250
=
40%
5. Percentage decrease in price of desktop computer
$1360 – $1020
=
× 100%
$1360
$340
=
× 100%
$1360
=
25%
6. Value of car at the end of 2010 = 80% × $120 000
7.
90% of original bill = $58.50
1% of original bill =
192
1% of number =
80
192
100% of number =
× 100
80
= 240
The number is 240.
90
× $96 000
100
= $86 400
1% of the students =
135
45
135
100% of the students =
× 100
45
= 300
There are 300 students who take part in the competition.
8. 136% of original cost = $333 200
$309
Percentage increase in production cost of printer
$309 – $300
=
× 100%
$300
$9
=
× 100%
$300
=
3%
$333 200
136
$333 200
100% of original cost =
× 100
136
= $245 000
The cost of the house when it was built is $245 000.
1% of original cost =
131
1
16. Let Amirah’s height be x m.
108% of Huixian’s height = x m
13. 115% of value in 2011 = $899 300
$899 300
1% of value in 2011 =
115
$899 300
100% of value in 2011 =
× 100
115
= $782 000
The value of the condominium was $782 000 in 2011.
115% of value in 2010 = $782 000
1% of Huixian’s height =
x
m
108
x
100% of Huixian’s height =
× 100
108
25
=
xm
27
25
Huixian’s height is
x m.
27
25
Priya’s height = 90% ×
x
27
90
25
= ×
x
100
27
5
=x m
6
x
Required percentage =
× 100%
5
x
6
1
=
× 100%
5
6
$782 000
115
$782 000
100% of value in 2010 =
× 100
115
= $680 000
The value of the condominium was $680 000 in 2010.
14. 75% of value in 2011 = $11 250
1% of value in 2010 =
$11 250
75
$11 250
100% of value in 2011 =
× 100
75
= $15 000
The value of the surveying machine was $15 000 in 2011.
75% of value in 2010 = $15 000
1% of value in 2011 =
= 120%
$15 000
75
$15 000
100% of value in 2010 =
× 100
75
= $20 000
The value of the surveying machine was $20 000 in 2010.
15. 105% of value at the end of 2010 = $61 824
1% of value in 2010 =
Review Exercise 8
1m
× 100%
56 mm
1000 mm
=
× 100%
56 mm
1. Required percentage =
=
125
× 100%
7
5
= 1785 %
7
2. (i) Pocket money Michael receives in a year = 52 × $28
= $1456
$61 824
105
$61 824
100% of value at the end of 2010 =
× 100
105
= $58 880
The value of the investment portfolio was $58 880 at the end of
2010.
92% of original value = $58 880
1% of value at the end of 2010 =
Savings in a year =
20
× $1456
100
= $291.20
(ii) Spending in a year = $1456 – $291.20
= $1164.80
30
×b
a
3.
= 100
4b
4b
30
100
=
4
3
=
40
$58 880
92
$58 880
100% of original value =
× 100
92
= $64 000
The original value of the investment portfolio was $64 000.
1% of original value =
1
132
68
× 100%
80
= 85%
4. Huixian’s percentage score =
Challenge Yourself
1. Let the number of red jellybeans Amirah moves from Bottle A to
Bottle B be x, the number of yellow jellybeans Amirah moves from
Bottle A to Bottle B be y.
Priya’s percentage score =
86
× 100%
120
2
= 71 %
3
120
Rui Feng’s percentage score =
× 100%
150
= 80%
\ Huixian performs the best in her Science test.
120
× 120
100
= 144
Red
300
150
Yellow
100
150
Red
300 – x
150 + x
Yellow
100 – y
150 + y
After
300 –
100 –
300 –
100 –
x
80
=
y
20
x
4
=
y
1
300 – x = 4(100 – y)
300 – x = 400 – 4y
4y – x = 100 — (1)
150 + x
60
\
=
150 + y
40
150 + x
3
=
150 + y
2
2(150 + x) = 3(150 + y)
300 + 2x = 450 + 3y
2x – 3y = 150 — (2)
2 × (1): 8y – 2x = 200 — (3)
(2) + (3): 5y = 350
y = 70
Substitute y = 70 into (1): 4(70) – x = 100
280 – x = 100
x = 180
Number of jellybeans Amirah moves from Bottle A to Bottle B
=x+y
=
180 + 70
=
250
2. Percentage of water which is poured from Cup B into Cup A
\
120
1% of number of pears =
60
120
100% of number of pears =
× 100
60
= 200
Number of pears the vendor has = 200
Total number of fruits the vendor has = 120 + 144 + 200
= 464
6. 120% of number of pages Kate reads on the second day = 60
1% of number of pages Kate reads on the second day =
Bottle B
Before
5. Number of apples the vendor has =
60% of number of pears = 120
Bottle A
60
120
100% of number of pages Kate reads on the second day
60
= × 100
120
=
50
Number of pages Kate reads on the second day = 50
Number of pages in the book = 6 × 50
= 300
7. Percentage of goats left =
94
× 86%
100
= 80.84%
80.84% of original number of goats = 8084
1% of original number of goats =
8084
80.84
8084
100% of original number of goats =
× 100
80.84
= 10 000
The original number of goats in the village is 10 000.
8. Let Mr Neo’s original salary be $x.
60
= × 70%
100
=
42%
Percentage of water in Cup A before 60% of solution in Cup A is
poured into Cup B
= 40% + 42%
=
82%
Percentage of water in Cup A after 60% of solution in Cup A is
poured into Cup B
Mr Neo’s reduced salary =
85
× $x
100
= $0.85x
$ x – $0.85 x
Required percentage =
× 100%
$0.85 x
$0.15 x
=
× 100%
$0.85 x
=
40
× 82%
100
=
32.8%
=
0.15
× 100%
0.85
11
= 17
%
17
133
1
Chapter 9 Ratio, Rate, Time and Speed
TEACHING NOTES
Suggested Approach
Students have learnt how to solve problems involving ratios and speed in primary school. Teachers can bring in real-life examples
for ratio, rate, time and speed to arouse students’ interest in this topic. Students will also learn how to solve problems involving
ratio, rate, time and speed through worked examples that involve situations in real-world contexts.
Section 9.1:
Ratio
Teachers can build upon what students have learnt about ratio in primary school and introduce equivalent ratios
through a recap of equivalent fractions. Teachers should emphasise that ratio does not indicate the actual size of
quantities involved. Practical examples can be given to the students to let them recognise what equivalent ratios
are (e.g. using 2 different kinds of fruits).
Teachers should highlight some common errors in ratio (i.e. the ratio of a part of a whole with the ratio of two
parts, incorrect order of numbers expressed when writing ratio and incorrect numerator expressed when writing
ratio as a fraction).
To make learning interesting, students can explore more about the Golden Ratio (see chapter opener and
Investigation: Golden Ratio). Teachers can also get the students to find out what other man-made structures or
natural occurrences have in common with the Golden Ratio (see Performance Task at page 210 of the textbook).
Section 9.2:
Rate
Teachers should explain that rate is a relationship between two quantities with different units of measure (which
is different from ratio). Teachers can give real life examples (e.g. rate of flow, consumption) for students to
understand the concept of rate. Teachers can also get students to interpret using tables which show different
kinds of rates (e.g. interest rate, postage rate, parking rate etc.).
Students can get more practice by learning to calculate rates they are familiar with (see Investigation: Average
Pulse Rate). Teachers should impress upon them to distinguish between constant and average rates.
Section 9.3:
1
Time
Teachers should emphasise that the addition and subtraction of times are not simply the same as adding and
subtracting the numbers. For example, teachers can ask students why 30 + 40 = 70 = 110 (where 110 refers to 1
h 10 min). To prevent students from making careless mistakes, teachers should help students understand that:
6 hours 45 minutes is not the same as 6.45 hours, 1 hour is not the same as 100 minutes, 1 minute is not the same
as 100 seconds.
Another important learning point would be dealing with time during the period before midnight and early
morning. Teachers may also compare the time displayed on a digital clock with that on an analogue clock, and
show students how the time is read.
134
Section 9.4:
Speed
Teachers should inform students that speed is a special type of rate, i.e. speed is the distance covered per unit
time. Teachers can get students to match appropriate speed to examples given (e.g. speed of a moving bicycle,
lorry, car and aeroplane) to bring across the notion of speed.
Teachers can build upon what students have learnt about distance, time and speed in primary school. Students
need to know that average speed is defined as the total distance travelled by the object per unit time and not the
average of the speeds of the object. Teachers should also impress upon students that there are differences between
average speed and constant speed.
Teachers should teach students the conversion of units and highlight to them to use appropriate units when solving
problems.
Challenge Yourself
Questions 1 and 2: Teachers can guide the students by getting them to use appropriate algebraic variables to
represent the rates involved in the question. Students have to read the question carefully and form the linear
equations which then can be solved to get the answers.
135
1
WORKED SOLUTIONS
Journal Writing (Page 208)
Class Discussion (Making Sense of the Relationship
between Ratios and Fractions)
1. Aspect ratio is used to describe the relationship between the width
and height of an image . It does not represent the actual length and
height, but instead represents the proportion of its width and height .
This is usually represented by two numbers separated by a colon,
for example, 4 : 3 and 16 : 9 .
The standard size of televisions has an aspect ratio of 4 : 3 which
means the image is 4 units wide for every 3 units of height .
Meanwhile, the latest size of televisions for the aspect ratio is
16 : 9 which is 16 units of width for every 9 units of height .
The following are some examples of aspect ratio used in our daily
lives:
• 16 : 10 is used mainly in widescreen computer monitors.
• 16 : 9 is the aspect ratio used in cinema halls as well as High
Definition TV.
• 14 : 9 is a compromise aspect ratio used to create an image that
is viewable to both 4 : 3 and 16 : 9 televisions .
• 5 : 4 is a computer monitor resolution and also in mobile phones.
• 4 : 3 is used in the older TVs (mainly non-widescreen) and
computer monitors .
• 1 : 1 is an uncommon aspect ratio that is used mainly in
photography .
2. Example 1:
Scale drawings of maps and buildings are often represented by
ratios . This is because it is impossible for a map to be exactly of the
same size as the area it represents . Therefore, the measurements are
scaled down in a fixed proportion so that the map can be used easily.
Similarly, a scale drawing of a building will have the same shape as
the actual building except that is scaled down .
Example 2:
In Chemistry and Biology, ratios are used for simple dilution
of chemicals. A fixed unit volume of a chemical is added to an
appropriate volume of solvent in order to dilute the chemical . For
example, a 1 : 5 dilution (verbalize as “1 to 5” dilution) entails
combining 1 unit volume of solute (the material to be diluted) + 4
unit volumes (approximately) of the solvent to give 5 units of the
total volume .
There are 40 green balls and 60 red balls in a bag .
Let A and B represent the number of green balls and red balls respectively .
1. Find the ratio of A to B .
A : B = 40 : 60
= 2:3
We can conclude that:
The ratio of A to B is 2 : 3 .
The following statement is equivalent to the above statement .
2
2
A
(fraction) of B, i .e .
=
(fraction) .
3
3
B
2. Find the ratio of B to A .
B : A = 60 : 40
= 3:2
We can conclude that:
The ratio of B to A is 3 : 2 .
The following statement is equivalent to the above statement .
A is
3.
B is
3
B
3
(fraction) of A, i .e .
=
(fraction) .
2
A
2
A
20
20
B
20
20
20
4. Example:
There are 30 girls and 10 boys in a class .
Let G and B represent the number of girls and boys respectively .
G : B = 30 : 10
= 3:1
We can conclude that:
The ratio of G to B is 3 : 1 .
The following statement is equivalent to the above statement .
G is
3
G
3
(fraction) of B, i .e .
=
(fraction) .
1
B
1
OR
B : G = 10 : 30
= 1:3
We can conclude that:
The ratio of B to G is 1 : 3 .
The following statement is equivalent to the above statement .
B is
1
B
1
(fraction) of G, i .e .
=
(fraction) .
3
G
3
G
10
B
10
10
Investigation (Golden Ratio)
1.
10
2.
3.
1
136
AB = 1 .7 cm
BC = 1.05 cm
AC
AB
AB
BC
XY
YZ
2.75
= 1 .62 (to 2 s .f .)
1.7
1.7
=
= 1 .62 (to 2 s .f .)
1.05
= 2.75 cm
= 1 .7 cm
=
XY
2.75
=
= 1 .62 (to 2 s .f .)
YZ
1.7
1+ 5
= 1 .62 (to 2 s .f .)
2
4. All the values in the previous questions are all equal .
5. –
6. (a)
(b)
Thinking Time (Page 224)
Do a recap on the definition of average speed which is defined as the
total distance travelled by the total time taken . Average speed is different
from the general meaning of ‘average’ in statistics . The word ‘average’
here does not refer to the sum of all individual speeds divided by the
total number of individual speeds .
3+ 5
2
3+ 5
ϕ+1=
2
Both answers are the same .
ϕ2 =
1 = –1 + 5
2
ϕ
1
= ϕ – ____
ϕ
1+ 5
–1 + 5
ϕ– 1 =
–
2
2
ϕ
=1
It is equal to 1 .
Performance Task (Page 225)
1. (a) Teachers may wish to assign this activity as a pair work for the
students to do.
Students can help to record each other’s walking speed.
Average walking speed of a human
=
5 km/h
(b) Minimum average speed
2.4
=
11
×
60 + 30
3600
2.4
=
690
3600
Performance Task (Page 210)
Teachers may wish to give some examples of
• man-made structures such as the
a) Acropolis of Athens (468–430 BC), including the Parthenon;
b) Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D);
c) Cathedral of Chartres (begun in the 12th century), Notre-Dame
of Laon (1157–1205), and Notre Dame de Paris (1160);
d) Mexico City Metropolitan Cathedral (1667–1813) .
• natural occurrences
a) spiral growth of sea shells;
b) spiral of a pinecone;
c) petals of sunflower;
d) horns of antelopes, goats and rams;
e) tusks of elephants;
f) body dimensions of penguins .
=
12.5 km/h (to 3.s.f.)
(c) Average speed of a bicycle
=
22.5 km/h
Average speed of a sports car
= 280 km/h
(d) Average speed of an MRT train
=
45 km/h
(e) Average speed of an aeroplane
=
805 km/h
(f) Average speed of the spaceship
=
28 000 km/h
Investigation (Average Pulse Rate)
Average speed (km/h)
(a)
Walking
5 km/h
(b)
Running
12.5 km/h
(c)
Bicycle
22.5 km/h
Thinking Time (Page 216)
(d)
Sports car
280 km/h
1. The parking charges per minute are $0 .40 is a constant rate as the
rate of charges per minute is the same throughout .
The rate of petrol consumption is 13.5 km per litre is an average rate
as the rate of consumption is not the same per minute .
2. The following are 3 examples of average rate that can be found in
daily life:
• Average speed
• Downloading rate of a file
• Average daily population growth
The following are 3 example of constant rate that can be found in
daily life:
• Simple interest rate
• Income Tax rate
• Currency exchange rate
(e)
MRT train
45 km/h
(f)
Aeroplane
805 km/h
(g)
Spaceship
28 000 km/h
First
reading
Second
reading
Third
reading
Pulse rate (per minute)
2. Priya cycles to school while Devi walks to school .
3. Number of times a spaceship is as fast as an aeroplane
=
28 000
805
= 34 .8
4. Other examples of speeds which can be encountered in real life:
• Speed of a bus
• Speed of a cheetah
• Speed of the Singapore Flyer capsule
137
1
5. Teachers may wish to ask the students to present their findings to
the class .
2. Let the amount of money Kate had initially be $3x .
Then the amount of money Nora had initially is $5x .
Kate
Nora
Before
$3x
$5x
After
$(3x + 150)
$(5x – 150)
Practise Now 1
(i) Ratio of the number of lemons to the number of pears
= 33 : 20
(ii) Ratio of the number of pears to the number of fruits in the basket
= 20 : (33 + 20)
=
20 : 53
3 x + 150
7
=
5 x – 150
9
9(3x + 150) = 7(5x – 150)
27x + 1350 = 35x – 1050
27x – 35x = –1050 – 1350
–8x = –2400
x = 300
\ Amount of money Kate had initially
= $[3(300)]
= $900
\
Practise Now 2
(a) 240 g : 1 .8 kg = 240 g : 1800 g
= 2 : 15
Alternatively,
2.40 g
240 g
=
1.8 kg
1800 g
2
15
\ 240 g : 1.8 kg = 2 : 15
=
3 8
3
8
:
=
× 45 :
5 9
5
9
=
27 : 40
(c) 0 .36 : 1 .2 = 0 .36 × 100
=
36
=
3
(b)
Practise Now 5
x:y = 5:6
↓ ×2
= 10 : 12
(i) x : y : z = 10 : 12 : 27
(ii) x : z = 10 : 27
× 45
: 1 .2 × 100
: 120
: 10
y:z = 4:9
↓ ×3
= 12 : 27
Practise Now 6
Practise Now 3
Let the amount of money Khairul had initially be $6x .
Then the amount of money Michael and Ethan had initially is $4x and
$5x respectively .
3a : 7 = 8 : 5
3a 8
=
7
5
15a = 56
a =3
11
15
Practise Now 4
x
Non-fiction
From the model, we form the equation:
5x + 2x = 1421
7x = 1421
x = 203
There are 3 × 203 = 609 more fiction than non-fiction books in the
library .
1
Michael
Ethan
Before
$6x
$4x
$5x
After
$(6x – 45)
$(4x + 30)
$(5x + 15)
7
6 x – 45
=
4 x + 30
6
6(6x – 45) = 7(4x + 30)
36x – 270 = 28x + 210
36x – 28x = 210 + 270
8x = 480
x = 60
\ Amount of money Khairul had initially
= $[6(60)]
= $360
\
1. Let the number of fiction books = 5x .
Then the number of non-fiction book = 2x .
Fiction
Khairul
138
Practise Now 7
Number of words per minute that Amirah can type
Practise Now 10
10 45 11 00
720
16
= 45
Number of words per minute that Lixin can type
23 00 23 11
=
15 min
Practise Now 11
1. (i) 25 minutes =
25
hours
60
16.8
Speed of the train =
= 40.32 km/h
25
60
798
19
= 42
Thus, Lixin is the fastest typist .
=
(ii) 16 .8 km = 16 800 m
25 minutes = 25 × 60 = 1500 seconds
Practise Now 8
1. (a) Amount each child have to pay
16 800
= 11.2 m/s
1500
55 km
(55 × 1000) m
5
2. 55 km/h =
=
= 15
m/s
1h
3600s
18
12 minutes 30 seconds = (12 × 60) + 30 = 750 seconds
Speed of the train =
$2.70 × 32.5
=
36
= $2 .44
(b) (i) Distance travelled on 1 litre of petrol
Distance travelled = 15
5
1
× 750 = 11 458 m
18
3
3. Let the speed of the bus be x km/h.
265
25
265
=
25
= 10 .6 km
Distance travelled on 58 litres of petrol
= 10 .6 × 58
= 614 .8 km
(ii) Amount of petrol required to travel a distance of 1007 km
=
3 hours
13 20 hours
16 20 hours
Distance the car travelled in 3 hours = 90 × 3 = 270 km
270 + (3 × x) = 510
3x = 510 – 270
3x = 240
x = 80
The speed of the bus is 80 km/h.
1007
10.6
=
95 litres
Amount that the car owner has to pay
=
95 × $1.95
=
$185.25
2. In 1 minute, 5 people can finish
=
Practise Now 12
1. (i) Speed of the train
=
48.6 km/h
48.6 km
=
1h
48 600 m
=
(convert 48 .6 km to m and 1 h into s)
3600 s
20
60
= 6 buns
In 5 minutes, 5 people can finish
=6×5
= 30 buns
In 5 minutes, 10 people can finish
= 30 × 2
= 60 buns
=
13.5 m/s
(ii) Speed of the train
=
48.6 km/h
48.6 km
=
1h
4 860 000 cm
=
(convert 48 .6 km to cm and 1 h into min)
60 min
=
81 000 cm/min
Practice Now 9
7
1
h = 7 h 15 min
4
22 45
+7h
11 min
15 min + 11 min = 26 min
\ The bus journey was 12 h 26 min long .
828
=
18
= 46
Number of words per minute that Shirley can type
= 20 ÷ 3
12 h
+ 15 min
29 45
06 00
(05 45)
\ The ship arrived at Port Y at 06 00 or 6 a .m . on Saturday .
139
1
2. Speed of the fastest human sprinter
100 m
=
9.58 s
(100 ÷ 1000) km
=
(9.58 ÷ 3600) h
Practise Now 14
Let the distance for the car to travel from Town A to Town B to meet
the truck = x km .
Then the time taken for the car to travel from Town A to Town B to meet
the truck at an average speed of 72 km/h
277
km/h
479
No . of times a cheetah is as fast as the fastest human sprinter
110
=
277
37
479
x
hour, and
72
the time taken for the truck to travel from Town B to Town A to meet
the car at an average speed of 38 km/h
= 37
=
550 – x
hour
38
x
550 – x
\
=
72
38
38x = 39 600 – 72x
38x + 72x = 39 600
110x = 39 600
x = 360 km
Hence, the time taken for the two vehicles to meet
=
= 2 .93
Practise Now 13
Time taken for Farhan to swim a distance of 1.5 km
1.5
h
2.5
3
=
h
5
Total time taken
=
360
72
= 5 hours
=
3
1
1
+1 +1
5
2
9
3
3
10
=
+
+
5
2
9
54
135
100
=
+
+
90
90
90
289
=
hours
90
Distance that Farhan runs
=
Practise Now 15
Radius of the wheel of the car
0.75
2
= 0.375 m
Circumference of the wheel of the car
= 2 × p × 0.375
= 2 × 3 .142 × 0.375
= 2.3565 m
Distance travelled by a car in 1 minute
= 14 × 60
= 840 m
Number of revolutions made by the wheel per minute
=
1
9
10
=9×
9
= 10 km
Total distance travelled
= 1.5 + 40 + 10
= 51.5 km
Average speed for the entire competition
=9×1
=
840
2.3565
= 356 (to the nearest whole number)
=
Total distance travelled
Total time taken
Exercise 9A
51.5
289
90
11
km/h
= 16
289
=
1. (a) 1.5 kg : 350 g = 1500 g : 350 g
=
30 : 7
Alternatively,
1.5 kg
1500 g
=
350 g
350 g
30
7
\ 1.5 kg : 350 g = 30 : 7
=
(b)
1
140
15 9
15
9
:
=
× 168 :
× 168
24 7
24
7
= 105 : 216
=
35 : 72
(c) 0.45 : 0.85 = 0.45 × 100 : 0.85 × 100
=
45 : 85
=
9 : 17
(d) 580 ml : 1 .12 l : 104 ml =580 : 1120 : 104
= 145 : 280 : 26
From the model, we form the equation:
9x – 5x = 44
4x = 44
x = 11
Total amount of money that is shared between the two boys
=
(5 + 9) × 11
=
$154
2 3 5
2
3
5
:
: =
× 24 :
× 24 :
× 24
3 2 8
3
2
8
=16 : 36 : 15
(f) 0 .33 : 0 .63 : 1 .8 = 0 .33 × 100 : 0 .63 × 100 : 1 .8 × 100
=
33
:
63
: 180
=
11
:
21
: 60
2. (a) a : 400 = 6 : 25
a
6
=
400
25
25a = 2400
a = 96
(b) 5b : 8 = 2 : 5
(e)
8. (i) Number of toys Huixian makes =
1530
× 16
12 + 16 + 17
1530
=
× 16
45
=
544
1530
× 17
12 + 16 + 17
1530
=
× 17
45
= 578
Amount of money Priya earns = 578 × $1.65
= $953.70
(ii) Number of toys Priya makes =
5b
2
=
8
5
25b = 16
16
b=
25
3.
4.
5.
6.
7.
9. (a) 4
(b) 0.75 : 3
2x
3y
=
5
8
16x = 15y
x
15
=
y
16
x : y = 15 : 16
a : b : c = 75 : 120 : 132
(i) a : b : c = 25 : 40 : 44
(ii) b : a = 40 : 25
=
8 : 5
(iii) b : c = 40 : 44
= 10 : 11
(i) Ratio of the number of boys to the number of girls = 14 : 25
(ii) Ratio of the number of girls to the total number of players in
the team = 25 : 39
(i) Ratio of the number of athletes to the number of volunteers
= 3600 : 20 000
= 9 : 50
(ii) Ratio of the number of media representatives to the number of
athletes to the number of spectators
= 1200 : 3600 : 370 000
=3 : 9 : 925
Let the amount of money that Rui Feng gets = 5x .
Then the amount of money that Vishal gets = 9x .
Rui Feng
1
kg : 630 g = 4200 g : 630 g
5
= 20 : 3
(c) 0 .6 kg :
(d)
75
5
: 3
100
16
3
53
=
:
4
16
3
53
=
× 16 :
× 16
4
16
=12 :
53
3
kg : 400 g = 600 g : 750 g : 400 g
4
= 12 : 15 : 8
1
3
1
15
: 2.5 : 3
=
× 12 : 2.5 × 12 :
× 12
3
4
3
4
=
4
: 30 :
45
(e) 1 .2 : 3
10. (a)
5
=
16
2
1
4
9
4
3
33
: 5.5 = 1.2 × 10 :
× 10 : 5.5 × 10
10
10
= 12 :
33 : 55
:
6
=
m
: 1
:
6
=
m
:
1
5
6
5
9
6
× 20 : 6 × 20 = m × 20 :
× 20
4
5
45 : 120 = 20m : 24
9
: 24 = 20m : 24
9 = 20m
20m = 9
9
m =
20
x
$44
Vishal
141
1
(b)
x
: 3
:
9
2
=
15
:
4
4
Amount of profit Farhan received
1
: y
2
11
× $494 500
7 + 11 + 5
=
$236 500
Amount of profit Michael received
5
=
× $494 500
7 + 11 + 5
=
9
9
15
x×4:3×4:
× 4=
×4:
×4:y×4
4
2
2
4x : 12 : 18 =
15 : 18 : 4y
18
4x
15
12
=
=
4y
12
18
18
9
x
5
2
=
=
2y
3
6
3
6x = 15
4y = 27
=
$107 500
14. Let the number that must be added be x .
3+ x
2
=
8+x
3
3(3 + x) = 2(8 + x)
9 + 3x = 16 + 2x
3x – 2x = 16 – 9
x =7
The number is 7 .
15. Let the amount of money Ethan had initially be $5x .
Then the amount of money Jun Wei and Raj had initially is $6x and
$9x respectively .
15
27
y =
6
4
3
5
x =
y =6
4
2
1
x =2
2
3
1 1
11. p : q =
:2
p:r =
:
4
3 2
↓ ×4
↓ ×6
= 3 :8
= 2 : 3
↓
↓
= 6 : 16
= 6 : 9
(i) p : q : r = 6 : 16 : 9
(ii) q : r = 16 : 9
12. (i) Let the initial number of teachers in the school be x .
Then the number of students in the school is 15x .
15x = 1200
x = 80
The initial number of teachers in the school is 80 .
(ii) Let the number of teachers who join the school be y .
x =
Before
Jun Wei
Raj
$5x
$6x
$9x
$6x
$9x
$(5x – 50)
After
5 x – 50
3
=
6x
4
4(5x – 50) = 3(6x)
20x – 200 = 18x
20x – 18x = 200
2x = 200
x = 100
\ Amount of money Ethan has after giving $50 to his mother
=
$[5(100) – 50]
=
$450
x
y 5
3
16.
=
=
y
z 8
4
4x = 3y
8y = 5z
\
80 + y
3
=
1200
40
40(80 + y) = 3(1200)
3200 + 40y = 3600
40y = 3600 – 3200
40y = 400
y = 10
The number of teachers who join the school is 10 .
13. Ratio of Ethan, Farhan and Michael’s property investment
=
$427 000 : $671 000 : $305 000
= 427 :
671 : 305
= 7
:
11 :
5
Total amount of profit earned
=
$1 897 500 – ($427 000 + $671 00 + $305 000)
=
$494 500
Amount of profit Ethan received
3
y
4
2y
=
3x – y + 2 z
x =
=
=
7
× $494 500
7 + 11 + 5
=
$150 000
=
=
=
1
Ethan
142
z=
2y
3
8
3 y – y + 2 y
4
5
2y
9
16
y–y+
y
4
5
2y
45
64
20
y–
y+
y
20
20
20
2y
89
y
20
40
89
8
y
5
(ii) Amount of petrol required to travel a distance of 2013 .2 km
Exercise 9B
2013.2
11.8
36
= 170
litres
59
Amount that the car owner has to pay
=
1. (a) Number of words that she can type per minute
1800
60
(1 hour = 60 minutes)
= 30
(b) Cost of one unit of electricity
120.99
=$
654
= $0 .19
(c) His monthly rental rate
4800
=$
3
= $1600
(d) Its mass per metre
15
=
3.25
8
=4
kg/m
13
2. Time taken for Ethan to blow 1 balloon
=
36
× $1 .99
59
=
$339.51
6. (i) Amount of fertiliser needed for a plot of land that has an area
of 1 m2
= 170
200
8
=
25 g
Amount of fertiliser needed for a plot of land that has an area
of 14 m2
=
25 × 14
=
350 g
(ii) Area of land that can be fertilised by 450 g of fertiliser
=
450
25
= 18 m2
7. (i) Temperature of the metal after 9 minutes
= 428 °C – [(23 °C × 3) + (15 °C × 6)]
= 269 °C
(ii) Temperature of the metal after 18 minutes
= 428 °C – [(23 °C × 3) + (15 °C × 15)]
= 134 °C
Amount of temperature needed for the metal to fall so that it
will reach a temperature of 25 °C
=
134 °C – 25 °C
= 109 °C
Time needed for the metal to reach a temperature of 25 °C
=
20
15
= 1.3 minutes
Time taken for Jun Wei to blow 1 balloon
=
=
25
18
= 1.38 minutes
Time taken for Vishal to blow 1 balloon
21
16
=
1.3125 minutes
Thus, Vishal can blow balloons at the fastest rate.
3. 3 hours = 180 minutes
Number of ornaments made in 3 hours
=
109
8
5
= 13 minutes
8
8. 4 weeks ⇒ fifteen 2-litre bottles of cooking oil
=
180
×4
15
= 48
Amount earned by the worker
= 48 × $1.15
=
$55.20
4. (i) Amount he is charged for each minute of outgoing calls
=
15 × 2
= 7.5 litres of cooking oil
4
10 weeks ⇒ 10 × 7.5 = 75 litres of cooking oil
Number of 5-litre tins of cooking oil needed for a 10-week period
1 week ⇒
39
650
= $0 .06
(ii) Amount he has to pay
= $0 .06 × 460
= $27 .60
5. (i) Distance travelled on 1 litre of petrol
=$
75
5
=
15
9. (i) Total amount to be paid to the man
= 224 × $7.50
= $1680
=
259.6
22
= 11 .8 km
Distance travelled on 63 litres of petrol
= 11 .8 × 63
= 743 .4 km
=
143
1
(ii) Number of normal working hours from 9 a .m . to 6 p .m . excluding
lunch time
= 8 hours
Let the number of overtime hours needed to complete the project
in 4 days by each worker be x .
4[4(8 + x)] = 224
16(8 + x) = 224
128 + 16x = 224
16x = 224 – 128
16x = 96
x =6
Overtime hourly rate
=
1.5 × $7.5
=
$11.25
Total amount to be paid to the 4 men if the project is to be
completed in 4 days
= 4{4[(8 × $7.5) + (6 × $11.25)]}
= $2040
10. 10 chefs can prepare a meal for 536 people in 8 hours and so
1 chef can prepare a meal for 536 people in 8 × 10 = 80 hours .
Hence, 22 chefs can prepare a meal for 536 × 22 = 11 792 people in
536 × 22
80 hours and so 22 chefs can prepare a meal for
people
80
80
in
= 1 hour .
80
536 × 22
Thus 22 chefs can prepare a meal for
× 5 people in
80
1 × 5 = 5 hours.
3.
Journey Time
Arrival Time
(a)
02 40
55 minutes
03 35
(b)
22 35
8 hours
06 35
(next day)
(c)
15 45
2
(d)
09 48
(e)
20 35
(Tuesday)
(f)
22 35
1
h or 2 h 15 min
4
17 50
12
7
h or 12 h 28 min
15
22 16
10
2
h or 10 h 40 min
3
07 15
(Wednesday)
1
1
h
4
23 50
4. 8.35 a.m. ⇒ 08 35
3 .12 p .m . ⇒ 15 12
08 35
15 00 15 12
09 00
25 min
7h
12 min
25 min + 12 min = 37 min
\ The journey took 6 h 37 min .
5. 21 55
+9h
6.
+4h
+ 5 min
+ 13 min
30 55
07 00
07 13
(06 55)
\ The train arrived at its destination at 07 13 or 7 .13 a .m . on Tuesday .
536 × 22
×5
80
= 737
Exercise 9C
1. (a)
(b)
(c)
(d)
2. (a)
(b)
(c)
(d)
Departure
Time
08 00
21 42
00 00
02 42
3 .30 a .m .
11 .12 p .m .
7.15 p.m.
12 .00 a .m .
?
10 51
–4h
?
14 51
+ 15 min
– 9 min
15 00
– 6 min
15 06
15 06
Working backwards, the car started the journey at 10 51.
7. (i)
22 55 23 00 00 00
5 min
06 00 06 05
5 min + 5 min = 10 min
\The journey took 7 h 10 min .
(ii) 35 min = 30 min + 5 min
5 min before 06 05 is 06 00
30 min before 06 00 is 05 30
\ The coach reached its destination at 05 30.
8. Assume time taken includes breaks in between stations .
(a) Depart from A: 21 30; Arrive at C: 02 25
21 30
22 00
30 min
00 00
4h
30 min + 25 min = 55 min
\ The time taken from A to C was 4 h 55 min.
1
5 min
7h
144
02 00 02 25
25 min
(b) Depart from B: 22 30; Arrive at E: 07 50
22 30
23 00 00 00
30 min
3. (a) 8.4 km/min
8.4 km
=
1 min
8.4 km
=
1
60 h
=
504 km/h
(b) 315 m/s
315 m
=
1s
07 50
07 00
50 min
8h
30 min + 50 min = 80 min
= 1 h 20 min
\ The time taken from B to E was 9 h 20 min .
(c) Depart from C: 02 30; Arrive at F: 09 20
02 30
03 00
09 00 09 20
30 min
6h
20 min
30 min + 20 min = 50 min
\ The time taken from C to F was 6 h 50 min.
(d) Depart from D: 04 20; Arrive at G: 10 45
04 20
05 00
10 00
5h
40 min
22 00
10 00
30 min
1
3600 h
=
1134 km/h
(c) 242 m/min
242 m
=
1 min
10 45
45 min
40 min + 45 min = 85 min
= 1 h 25 min
\ The time taken from D to G was 6 h 25 min.
(e) Depart from A: 21 30; Arrive at G: 10 45
21 30
=
12 h
242
1000 km
=
1
60 h
13
km/h
25
(d) 125 cm/s
125 cm
=
1s
= 14
10 45
45 min
30 min + 45 min = 75 min
= 1 h 15 min
\ The time taken from A to G was 13 h 15 min.
125
100 000 km
=
1
3600 h
=
4.5 km/h
4. (a) 65 cm/s
65 cm
=
1s
Exercise 9D
30
1
=
hour
60
2
Speed of the particle
24.6 km
=
1
2 hour
1. (i) 30 minutes =
65
100 km
=
1s
13
m/s
=
20
(b) 367 km/h
367 km
=
1h
367 × 1000 m
=
3600 s
367 000 m
=
3600 s
=
49.2 km/h
(ii) 24 .6 km = 24 .6 × 1000 = 24 600 m
30 minutes = 30 × 60 = 1800 s
Speed of the particle
24 600 m
=
1800 s
= 13
2
m/s
3
2. 12 24 hours
1 hour 48 minutes
14 12 hours
= 107
48
4
1 hour 48 minutes = 1
= 1 hours
60
5
Distance between the two stations
= 200 × 1
315
km
1000
17
m/s
18
4
5
= 360 km
= 360 × 1000 m
= 360 000 m
145
1
(c) 1000 cm/min
1000 cm
=
1 min
7.
1000
100 m
=
60 s
Time taken = 12 s
X
60
15
=4s
(ii) Average speed of the object for its entire journey from X to Y
Total distance travelled
=
Total time taken
=
1
m/s
6
(d) 86 km/min
86 km
=
1 min
86 × 1000 m
=
60 s
120
12 + 4
120
=
16
1
= 7 m/s
2
=
1
m/s
3
5. Speed of the fastest Singaporean sprinter
100 m
=
10.37 s
= 1433
8.
L
100
1000 km
=
10.37
3600 h
742
km/h
1037
Number of times the bullet train is as fast as the fastest Singaporean
sprinter
365
742
34 1037
19
57
1
=
h
3
Time taken to travel the remaining part of the journey
=
160
6+4
160
=
10
=
16 m/s
9. Time taken to travel the first 50 km of its journey
50 × 1000 m
=
25 m/s
=
55
110
1
=
h
2
Average speed of the car for its entire journey
Total distance travelled
=
Total time taken
=
= 2000 s
2000
h
3600
5
=
h
9
Time taken to travel the next 120 km of its journey
=
19 + 55
1 1
+
3 2
74
5
6
120
80
1
=1 h
2
=
4
km/h
5
1
N
100
25
=4s
Average speed of the object for its entire journey from L to N
Total distance travelled
=
Total time taken
3701
7200
6. Time taken to travel the first part of the journey
= 88
Speed = 25 m/s
M
160 m
=
= 10
=
Speed = 10 m/s
Time taken = 6 s
Distance travelled from L to M
= 10 × 6
= 60 m
Thus, distance travelled from M to N
= 160 – 60
= 100 m
Time taken to travel from M to N
= 34
=
Y
(i) Time taken to travel from M to Y
=
=
Speed = 15 m/s
M
120 m
146
Then the time taken for Nora to meet Lixin will be (x + 6) min .
Distance between Town A and Town B
= 100 × (x + 6) + 75 × (x + 6)
= 100x + 600 + 75x + 450
=
(175x + 1050) m
Thus,
180x = 175x + 1050
10x – 175x = 1050
5x = 1050
x = 210
Distance between Town A and Town B
= 180 × 210
= 37 800 m
Distance travelled for the last part of its journey
35
= 90 ×
60
=
52.5 km
Average speed of the object for its entire journey
Total distance travelled
=
Total time taken
50 + 120 + 52.5
1 35
5
+1 +
9
2 60
6
= 84
km/h
19
10. Radius of the wheel of the car
=
60
2
= 30 cm
= 0 .3 m
Circumference of the wheel of the car
= 2 × p × 0 .3
= 2 × 3 .142 × 0 .3
=
1.8852 m
Distance travelled by the car in 1 hour
= 13 .2 × 60 × 60
=
47 520 m
Number of revolutions made by the wheel per minute
=
Review Exercise 9
1 1
:
b:c=3:4
2 3
↓ ×2
↓ ×6
= 3: 2
=6:8
↓ ×3
= 9:6
\ a : c = 9 : 8 .
2. (i) Let the mass of type A coffee beans in the mixture be 3x kg .
Then the mass of type B and C coffee beans in the mixture be
5x kg and 7x kg respectively .
\ 3x + 5x + 7x =35
15x =45
x =3
Mass of type A coffee beans in the mixture = 3 × 3 = 9 kg
Mass of type B coffee beans in the mixture = 5 × 3 = 15 kg
Mass of type C coffee beans in the mixture = 7 × 3 = 21 kg
(ii) Cost of the mixture per kg
1. a : b =
47 520
1.8852
=
25 207 (to the nearest whole number)
11. Length of the goods train
=
8
8
= 72 ×
+ 54 ×
3600
3600
7
km
25
7
=
× 1000
25
= 280 m
=
12.
(9 × $7) + (15 × $10) + (21 × $13)
45
= $10 .80
3. (i) Let the number of books in the box be 4x .
Then the initial number of toys in the box be 5x .
\ 4x = 36
x =9
So the initial number of toys in the box is 5 × 9 = 45.
(ii) Let the number of toys that are given away be y .
36
12
\
=
45 – y
11
11(36) = 12(45 – y)
396 = 540 – 12y
12y =540 – 396
12y = 144
y = 12
The number of toys that are given away is 12 .
=
Nora: 100 m/min
Town A
Town B
Kate: 80 m/min
Let the time taken for Nora to meet Kate be x min .
So distance between Town A and Town B
= 100 × x + 80 × x
= 180x m
Nora: 100 m/min
Town A
Town B
Kate: 80 m/min
6 min
Lixin: 75 m/min
147
1
4. (i) Total cost of placing an advertisement containing 22 words
=
350 + (22 × 25)
= 900 cents
= $9
(ii) Let the number of words he can use be x .
Then 350 + 25x 1500
25x 1500 – 350
25x 1150
x 46
The greatest number of words he can use is 46 .
+2h
+ 15 min
5. (i)
?
?
12 06
09 51
11 51
12 00
12 06
–2h
– 9 min
Time taken for the athlete to run
5 × 1000
3
2
= 1666 s
3
2
= 1666 ÷ 3600
3
25
=
h
54
His average speed for the entire competition
Total distance travelled
=
Total time taken
750
+ 20 + 5
= 1000
15 30 25
+
+
60 60 54
30
= 21
km/h
131
2
8. Let the distance from A to C be x m .
5
3
Then the distance from C to B be x m .
5
=
– 6 min
Working backwards, the time the journey started was 09 51.
(ii) Total distance = 198 km
1
9
Total time = 2 h 15 min = 2 h =
h
4
4
198
Average speed =
9
4
= 88 km/h
195
6. (i) Time taken =
52
3
=3
h
4
=
3 h 45 min
+3h
+ 15 min
Time Taken
= 30 s
A
2xm
5
+ 30 min
5 min
3h
3
x ÷ 30
5
3
1
= x×
5
30
1
=
xs
50
(ii) Average speed of the object for its entire journey from A to B
xm
=
1
30 + 50 x s
18 15
15 min
5 min + 15 min = 20 min
\ The time taken was 3 h 20 min .
1
3 h 20 min = 3 h
3
10
=
h
3
195
Average speed =
10
3
= 58.5 km/h
7. Distance that the athlete cycles
50 x
m/s
1500 + x
9. Let the first part of the journey be x km .
Then the remaining part of the journey be (150 – x) km .
Time taken for the entire journey = 4.5 h
=
x
150 – x
+
35
5
x
150 – x
35 ×
+ 35 ×
35
5
x + 7(150 – x)
x + 1050 – 7x
x – 7x
– 6x
x
\ Distance = 148.75 km
\
30
= 40 ×
60
= 20 km
1
B
3xm
5
=
(ii)
18 00
C
(i) Time taken for the object to travel from C to B
3
x
= 5
30
08 45
11 45
12 00
12 30
\ The time at which the lorry arrives at its destination is 12 30 .
14 55 15 00
Average speed
= 30 m/s
148
= 4.5
= 35 × 4.5
= 157.5
=157.5
=157.5 – 1050
=–892.5
=148.75
For the second race,
Let the time for the first person to pass the end point be t s .
Time taken for Vishal to finish the 100 m race
10. Radius of the wheel of the car
48
=
2
= 24 cm
Circumference of the wheel of the car
= 2 × p × 24
= 2 × 3 .142 × 24
=
150.816 cm
Distance travelled by a car in 1 minute
=
3.5 ÷ 60
110
x
110
100 y
=
)
(Substitute x =
90
100 y
90
=
99
s
y
Time taken for Jun Wei to finish the 100 m race
100
=
s
y
At time t s, distance that Vishal covered
99
t=
y
=
7
km
120
7
=
× 100 000 cm
120
1
=
5833 cm
3
Number of revolutions made by the wheel per minute
1
5833
3
=
150.816
= 39 (to the nearest whole number)
=
ty = 99 m
At time t s, distance that Jun Wei covered
100
t=
y
ty = 100 m
\ Vishal win the race by 100 – 99 = 1 m.
Challenge Yourself
1. Let the rate of the moving escalator be x steps per second .
When she is walking down at a rate of 2 steps per second, then the
total steps (including the steps covered by the moving escalator)
covered in 1 second is (x + 2) . Since she use 18 steps to reach the
bottom from the top, therefore, the time taken is (18 ÷ 2) = 9 seconds .
When she is exhausted, then the total steps (including the steps
covered by the moving escalator) covered in 1 second is (x + 1) .
Since she use 12 steps to reach the bottom from the top, therefore,
the time taken is (12 ÷ 1) = 12 seconds .
Hence,
9(x + 2) = 12(x + 1)
9x + 18 = 12x + 12
9x – 12x = 12 – 18
–3x = –6
x =2
\ Total steps covered by the moving escalator = 9(2 + 2) = 36 .
Hence the time taken for her to reach the bottom from the top if she
stands on the escalator
36
2
= 18 s
2. Let Vishal’s speed be x m/s and Jun Wei’s speed be y m/s.
Then in the first race, when Vishal ran pass the end point 100 m,
Jun Wei is only at 90 m of the race. Hence, at the same time,
90
100
=
y
x
100y = 90x
100 y
x=
90
=
149
1
Chapter 10 Basic Geometry
TEACHING NOTES
Suggested Approach
Students have learnt angle measurement in primary school. They have learnt the properties, namely, angles on a straight line,
angles at a point and vertically opposite angles. However, students are unfamiliar with the types of angles and using algebraic
terms in basic geometry. There is a need to guide students to apply basic algebra and linear equations in this topic. Students
will learn how to do this through the worked examples in this topic. Teachers can introduce basic geometry by showing real-life
applications (see chapter opener on page 231).
Section 10.1: Points, Lines and Planes
Teachers should illustrate what a point, a line, intersecting lines and planes look like. Teachers can impress
upon the students that there is a difference between a line and a ray. A ray has a direction while a line has no
direction. Teachers can highlight to the students that for a ray, the arrowhead indicates the direction in which
the ray extends while for a line, its arrowhead is to indicate that the line continues indefinitely.
The thinking time on page 234 of the textbook requires students to think and determine whether each of the
statements is true or false. Teachers should make use of this opportunity to highlight and clear some common
misconceptions about points, lines and planes.
Section 10.2: Angles
Teachers can build upon prerequisites, namely angle measurement, to introduce the types of angles by classifying
angle measurements according to their sizes.
To make practice more interesting, teachers can get the students to work in groups to measure and classify the
various types of angles of different objects (i.e. scissors, set square, compass and the hands of a clock).
Teachers should recap with students on what they have learnt in primary school, i.e. angles on a straight line,
angles at a point and vertically opposite angles. After going through Worked Examples 1 to 4, students should
be able to identify the properties of angles and use algebraic terms to form and solve a linear equation to find
the value of the unknowns. Students are expected to state reasons in their working.
Section 10.3: Angles Formed by Two Parallel Lines and a Transversal
Teachers can get students to discuss examples where they encounter parallel lines in their daily lives and ask
them what happens when a line or multiple lines cut the parallel lines.
To make learning more interactive, students are given the opportunity to explore the three angle properties observed
when a pair of parallel lines is cut by a transversal (see Investigation: Corresponding Angles, Alternate Angles
and Interior Angles). Through this investigation, students should be able to observe the properties of angles
associated with parallel lines. The investigation also helps students to learn how to solve problems involving
angles formed by two parallel lines and a transversal. Students are expected to use appropriate algebraic variables
to form and solve linear equations to find the value of the unknowns. Teachers should emphasise the importance
of stating the properties when the students are solving questions on basic geometry.
Challenge Yourself
Question 1: Teachers can guide the students by hinting to them that this question is similar to a problem involving
number patterns. Students have to draw a table and write down the first few numbers of rays between OA and
OB, and their respective number of different angles. The students will then have to observe carefully and find
an expression that represents rays between OA and OB.
Question 2: Teachers can guide the students by telling them to find the different angles that both the hour hand
and minute hand makes from one specific position to another.
Question 3: Teachers can guide the students by telling them to find the number of times the bell will sound
between certain times of the day.
1
150
2. 2c° + 100° + 3c°
2c° + 3c°
5c°
c°
\c
WORKED SOLUTIONS
Thinking Time (Page 234)
(a) False. There are an infinite number of points lying on a line segment.
(b) False. There is exactly one line that passes through any three distinct
points which are collinear; there is no line that passes through any
three distinct points which are non-collinear.
(c) False. There is exactly one line that passes through any two distinct
points.
(d) False. Two distinct lines intersect at one point; two coincident lines
intersect at an infinite number of points; two parallel lines do not
intersect at any point.
(e) True.
Practise Now 2
1. 58° + 148° + 7a° = 360° (/s at a point)
7a° = 360° – 58° – 148°
= 154°
a° = 22°
\ a = 22
2. b° + 90° + b° + 4b° = 360° (/s at a point)
b° + b° + 4b° = 360° – 90°
6b° = 270°
b° = 45°
\ b = 45
Investigation (Corresponding Angles, Alternate Angles and
Interior Angles)
/a = /b
/a = /c
/a + /d = 180°
/a = /b (corr. /s)
/a = /c (alt. /s)
/a + /d = 180° (int. /s)
/b = /a (corr. /s)
/c = /b (vert. opp. /s)
= /a
\ /a = /c (proven)
6. Method 1:
/b = /a (corr. /s)
/b + /d = 180° (adj. /s on a str. line)
\ /a + /d = 180° (proven)
Method 2:
/c = /a (alt. /s)
/c + /d = 180° (adj. /s on a str. line)
\ /a + /d = 180° (proven)
1.
2.
3.
(a)
(b)
(c)
5.
Practise Now 3
(i) AOC + 90° + 53° = 180° (adj. /s on a str. line)
AOC = 180° – 90° – 53°
= 37°
(ii) BOD = AOC
= 37° (vert. opp. /s)
Practise Now 4
3a° + 40° = a° + 60° (vert. opp. /s)
3a° – a° = 60° – 40°
2a° = 20°
a° = 10°
\ a = 10
a° + 60° + 4b° + 10° = 180° (adj. /s on a str. line)
10° + 60° + 4b° + 10° = 180°
4b° = 180° – 10° – 60° – 10°
= 100°
b° = 25°
\ b = 25
Practise Now (Page 236)
(a)
(b)
(c)
(d)
(e)
(f)
= 180° (adj. /s on a str. line)
= 180° – 100°
= 80°
= 16°
= 16
Acute
Reflex
Obtuse
Obtuse
Reflex
Acute
Practise Now (Page 246)
(a) (i) /a and /m, /b and /n, /c and /o, /d and /p, /e and /i,
/f and /j, /g and /k, /h and /l
(ii) /c and /m, /d and /n, /g and /i, /h and /j
(iii) /c and /n, /d and /m, /g and /j, /h and /i
(b) No, /c ≠ /g as PQ is not parallel to RS .
Practise Now 1
1. (a) 122° + a° = 180° (adj. /s on a str. line)
a° = 180° – 122°
= 58°
\ a = 58
(b) 95° + 65° + b° = 180° (adj. /s on a str. line)
b° = 180° – 95° – 65°
= 20°
\ b = 20
151
1
DCQ = CDR = 35° (alt. /s, PQ // RS)
b° + ACQ + DCQ = 360° (/s at a point)
b° + 48° + 35° = 360°
b° = 360° – 35° – 48°
= 277°
\ b = 277
Practise Now 5
a° = 54° (corr. /s, AB // CD)
\ a = 54
c° + 106° = 180° (int. /s, AB // CD)
c° = 180° – 106°
= 74°
\ c = 74
b° = c° (vert. opp. /s)
= 74°
\ b = 74
d° = c° (corr. /s, AB // CD)
= 74°
\ d = 74
2. 2e° + 30° = 69° (corr. /s, AB // CD)
2e° = 69° – 30°
= 39°
e° = 19 .5°
\ e = 19 .5
f ° = 2e° (corr. /s, AB // CD)
= 39°
\ f = 39
1.
Practise Now 7
C
19°
X
A
C 2a°
20°
P
D
Q
E
XCQ = 44° (alt. /s, AB // PQ)
ECQ = 20° (alt. /s, PQ // DE)
2a° = XCQ + ECQ
= 44° + 20°
= 64°
a° = 32°
\ a = 32
2.
228°
X
b°
C
E
32°
Since BWQ = DYQ (= 122°), then AB // CD (converse of corr. /s) .
\ BXS = CZR = 65° (alt. /s, AB // CD)
Q
D
R
293°
S
Exercise 10A
1. (a)
(b)
(c)
(d)
2. (a)
(b)
(c)
(d)
(e)
(f)
F
XAC = 228° – 180°
= 48°
ACQ = XAC (alt. /s, XB // PQ)
= 48°
EDR = 32° (alt. /s, RS // EF)
EDR + CDR + 293° = 360° (/s at a point)
32° + CDR + 293° = 360°
CDR = 360° – 32° – 293°
= 35°
1
E
245°
Practise Now 8
B
A
P
D
P
B
44°
2b°
B
DEF + 245° = 360° (/s at a point)
DEF = 360° – 245°
= 115°
5a° = 115° (corr. /s, BC // EF)
a° = 23°
\ a = 23
CDQ = 19° (alt. /s, BC // PQ)
EDQ + DEF = 180° (int. /s, PQ // EF)
EDQ + 115° = 180°
EDQ = 180° – 115°
= 65°
2b° = CDQ + EDQ
= 19° + 65°
= 84°
b° = 42°
\ b = 42
Practise Now 6
1.
Q
5a°
A
F
152
a = 79, b = 106, c = 98
d = 50, e = 228
f = 117, g = 45
h = 243, i = 94, j = 56
Obtuse
Reflex
Acute
Reflex
Acute
Obtuse
3. (a) Complementary angle of 18° = 90° – 18°
= 72°
(b) Complementary angle of 46° = 90° – 46°
= 44°
(c) Complementary angle of 53° = 90° – 53°
= 37°
(d) Complementary angle of 64° = 90° – 64°
= 26°
4. (a) Supplementary angle of 36° = 180° – 36°
= 144°
(b) Supplementary angle of 12° = 180° – 12°
= 168°
(c) Supplementary angle of 102° = 180° – 102°
= 78°
(d) Supplementary angle of 171° = 180° – 171°
= 9°
5. (a) a° + 33° = 180° (adj. /s on a str. line)
a° = 180° – 33°
= 147°
\ a = 147
(b) b° + 42° + 73° = 180° (adj. /s on a str. line)
b° = 180° – 42° – 73°
= 65°
\ b = 65
(c) 4c° + 80° + c° = 180° (adj. /s on a str. line)
4c° + c° = 180° – 80°
5c° = 100°
c° = 20°
\ c = 20
(d) 4d° + 16° + 2d° + 14° = 180° (adj. /s on a str. line)
4d° + 2d° = 180° – 16° – 14°
6d° = 150°
d° = 25°
\ d = 25
6. (a) x° + y° + z° = 180° (adj. /s on a str. line)
When y° = 45°, z° = 86°,
x° + 45° + 86° = 180°
x° = 180° – 45° – 86°
= 49°
\ x = 49
(b) x° + y° + z° = 180° (adj. /s on a str. line)
When x° = 2y°, z° = 3y°,
2y° + y° + 3y° = 180°
6y° = 180°
y° = 30°
\ y = 30
7. (a) a° + 67° + 52° + 135° = 360° (/s at a point)
a° = 360° – 67° – 52° – 135°
= 106°
\ a = 106
(b) 5b° + 4b° + 3b° = 360° (/s at a point)
12b° = 360°
b° = 30°
\ b = 30
8.
9.
10.
11.
153
(c) 16c° + 4c° + 90° + 4c° = 360° (/s at a point)
16c° + 4c° + 4c° = 360° – 90°
24c° = 270°
c° = 11 .25°
\ c = 11 .25
(d) (7d + 23)° + 6d° + 139° + 5d° = 360° (/s at a point)
7d° + 23° + 6d° + 139° + 5d° = 360°
7d° + 6d° + 5d° = 360° – 23° – 139°
18d° = 198°
d° = 11°
\ d = 11
(i) AOC = 48° (vert. opp. /s)
(ii) 90° + DOE + 48° = 180° (adj. /s on a str. line)
DOE = 180° – 90° – 48°
= 42°
(a) 40° + 30° + a° = 117° (vert. opp. /s)
a° = 117° – 40° – 30°
= 47°
\ a = 47
(b) 7b° + 3b° = 180° (adj. /s on a str. line)
10b° = 180°
b° = 18°
\ b = 18
c° = 7b° (vert. opp. /s)
= 7(18°)
= 126°
\ c = 126
(a) x° + y° + z° = 180° (adj. /s on a str. line)
y° + x° + z° = 180°
When y° = x° + z°,
y° + y° = 180°
2y° = 180°
y° = 90°
\ y = 90
(b) x° + y° + z° = 180° (adj. /s on a str. line)
When x° = y° = z°,
z° + z° + z° = 180°
3z° = 180°
z° = 60°
\ z = 60
AOB + DOA = 180° (adj. /s on a str. line)
AOB + 5AOB = 180°
6AOB = 180°
\ AOB = 30°
BOC = 2AOB
= 2 × 30°
= 60°
COD = 4AOB
= 4 × 30°
= 120°
DOA = 5AOB
= 5 × 30°
= 150°
1
= 180° (adj. /s on a str. line)
= 180°
= 180°
= 180° – 132°
= 48°
y° = 16°
\ y = 16
(ii) Obtuse AOD = (186 – 4x)° + 34°
= [186 – 4(22)]° + 34°
= 98° + 34°
= 132°
Reflex COE = 180° + (186 – 4x)°
= 180° + 98°
= 278°
12. (a) 7a° + 103° = 180° (adj. /s on a str. line)
7a° = 180° – 103°
= 77°
a° = 11°
\ a = –11
2b° + 13° = 103° (vert. opp. /s)
2b° = 103° – 13°
= 90°
b° = 45°
\ b = 45
(b) 62° + 49° + 3c° = 180° (adj. /s on a str. line)
3c° = 180° – 62° – 49°
= 69°
c° = 23°
\ c = 23
d° = 3c° (vert. opp. /s)
= 69°
\ d = 69
e° = 62° + 49° (vert. opp. /s)
= 111°
\ e = 111
(c) 7f ° + 5° = 2f ° + 35° (vert. opp. /s)
7f ° – 2f ° = 35° – 5°
5f ° = 30°
f ° = 6°
\f =6
2f ° + 35° + 5g° + 18° = 180° (adj. /s on a str. line)
2(6°) + 35° + 5g° + 18° = 180°
12° + 35° + 5g° + 18° = 180°
5g° = 180° – 12° – 35° – 18°
= 115°
g° = 23°
\ g = 23
(d) 24° + 90° + h° = 104° + 32° (vert. opp. /s)
h° = 104° + 32° – 24° – 90°
= 22°
\ h = 22
24° + 90° + h° + 2i° = 180° (adj. /s on a str. line)
24° + 90° + 22° + 2i° = 180°
2i° = 180° – 24° – 90° – 22°
= 44°
i° = 22°
\ i = 22
j° = 2i° (vert. opp. /s)
= 44°
\ j = 44
13. (i) (186 – 4x)° + 34° = 6x° (vert. opp. /s)
186° – 4x° + 34° = 6x°
6x° + 4x° = 186° + 34°
10x° = 220°
x° = 22°
\ x = 22
1
6x° + 3y°
6(22°) + 3y°
132° + 3y°
3y°
Exercise 10B
1. (a) (i) BXR and DZR, AXR and CZR, AXS and CZS, BXS and DZS,
BWP and DYP, AWP and CYP, AWQ and CYQ,
BWQ and DYQ
(ii) AXS and DZR, BXS and CZR, AWQ and DYP,
BWQ and DYP
(iii) AXS and CZR, BXS and DZR, AWQ and CYP,
BWQ and DYP
(b) No, BWQ ≠ AXR as PQ is not parallel to RS .
(c) No, the sum of DYP and CZR is not equal to 180° as PQ is not
parallel to RS .
2. (a) a° = 117° (vert. opp. /s)
\ a = 117
b° = 117° (corr. /s, AB // CD)
\ b = 117
c° + a° = 180° (int. /s, AB // CD)
c° + 117° = 180°
c° = 180° – 117°
= 63°
\ c = 63
d° = 78° (corr. /s, AB // CD)
\ d = 78
(b) e° = 31° (alt. /s, AB // CD)
\ e = 31
f ° = 35° + 31° (alt. /s, AB // CD)
= 66°
\ f = 66
(c) g° = 83° (alt. /s, AB // CD)
\ g = 83
h° = 69° (corr. /s, AB // CD)
\ h = 69
(d) i° + 75° + 60° = 180° (int. /s, AB // CD)
i° = 180° – 75° – 60°
= 45°
\ i = 45
j° = 60° (alt. /s, AB // CD)
\ j = 60
154
a° = 38° (corr. /s, AB // CD)
\ a = 38
a° + 30° = 2b° (corr. /s, AB // CD)
38° + 30° = 2b°
2b° = 68°
b° = 34°
\ b = 34
(b) 7c° = 140° (corr. /s, AB // CD)
c° = 20°
\ c = 20
2d° = 7c° (vert. opp. /s)
= 140°
d° = 70°
\ d = 70
(c) 7e° + 3e° = 180° (int. /s, AB // CD)
10e° = 180°
e° = 18°
\ e = 18
(d) (2f + 6)° = (3f – 23)° (alt. /s, AB // CD)
2f ° + 6° = 3f ° – 23°
3f ° – 2f ° = 6° + 23°
f ° = 29°
\ f = 29
4. (a)
A
5. (a)
3. (a)
D
C
CEQ + 128° = 180° (int. /s, PQ // CD)
CEQ = 180° – 128°
= 52°
AEQ = 92° – CEQ
= 92° – 52°
= 40°
a° + AEQ = 180° (int. /s, AB // PQ)
a° + 40° = 180°
a° = 180° – 40°
= 140°
\ a = 140
P
(b)
X
A
B
(4b – 10)°
B
Y
D
(2b – 2)°
Q
Q
114°
CYP = (2b – 2)° (vert. opp. /s)
CYP + (4b – 10)° = 180° (int. /s, AB // CD)
(2b – 2)° + (4b – 10)° = 180°
2b° – 2° + 4b° – 10° = 180°
2b° + 4b° = 180° + 2° + 10°
6b° = 192°
b° = 32
\ b = 32
D
AEQ + 142° = 180° (int. /s, AB // PQ)
AEQ = 180° – 142°
= 38°
CEQ + 114° = 180° (int. /s, PQ // CD)
CEQ = 180° – 114°
= 66°
a° = AEQ + CEQ
= 38° + 66°
= 104°
\ a = 104
(c)
R
C
69°
E
37°
A
P
B
A
C
Q
C
C
b°
92°
128°
a°
P
E
P
E
(b)
B
a°
142°
P
A
B
28°
E
Q
94°
F
c°
S
19°
D
AEP = 28° (alt. /s, AB // PQ)
FEP = 94° – AEP
= 94° – 28°
= 66°
EFS = FEP
= 66° (alt. /s, PQ // RS)
DFS = 19° (alt. /s, RS // CD)
c° + EFS + DFS = 360° (/s at a point)
c° + 66° + 19° = 360°
c° = 360° – 66° – 19°
= 275°
\ c = 275
Q
D
AEP = 69° (alt. /s, AB // PQ)
CEP = 37° (alt. /s, PQ // CD)
b° = AEP + CEP
= 69° + 37°
= 106°
\ b = 106
155
1
BAC + DCA = 180° (int. /s, AB // CE)
BAC + 122° = 180°
BAC = 180° – 122°
= 58°
7y° + BAC = 360°
7y° + 58° = 360°
7y° = 360° – 58°
= 302°
y° = 43.1° (to 1 d.p.)
\ y = 43 .1
6. (i) CDF = 86° (alt. /s, CE // FG)
(ii) HDE = 86° (vert. opp. /s)
EDA = HDE – 47°
= 86° – 47°
= 39°
BAD + EDA = 180° (int. /s, AB // CE)
BAD + 39° = 180°
BAD = 180° – 39°
= 141°
7. (i) AEB = 68° (alt. /s, BF // AD)
(ii) EAB = 58° (alt. /s, AB // CD)
FBA + EAB = 180° (int. /s, BF // AD)
FBA + 58° = 180°
FBA = 180° – 58°
= 122°
ABE = FBA – 68°
= 122° – 68°
= 54°
8.
B
C
A
P
46°
72°
D
E
52°
F
10.
A
Q
x°
B
D
P
5y°
147°
E
x° = 147° (corr. /s, BC // EF)
\ x = 147
CDQ = 32° (alt. /s, BC // PQ)
QDE + 147° = 180° (int. /s, PQ // EF)
QDE = 180° – 147°
= 33°
5y° + CDQ + QDE = 360° (/s at a point)
5y° + 32° + 33° = 360°
5y° = 360° – 32° – 33°
= 295°
y° = 59°
\ y = 59
11. Since AXS + CZR = 104° + 76° = 180°,
then AB // CD (converse of int. /s) .
\ BWP = DYP = 46° (corr. /s, AB // CD)
A
12.
Q
G
(i) CGF = 52° (corr. /s, FD // GC)
(ii) BCG + CGF = 180° (int. /s, AC // EG)
BCG + 52° = 180°
BCG = 180° – 52°
= 128°
BCF = BCG – 72°
= 128° – 72°
= 56°
(iii) BDQ = 46° (alt. /s, AC // PQ)
FDQ = 52° (alt. /s, PQ // EG)
Reflex BDF + BDQ + FDQ = 360° (/s at a point)
Reflex BDF + 46° + 52° = 360°
Reflex BDF = 360° – 46° – 52°
= 262°
9. FDC + 58° = 180° (adj. /s on a str. line)
FDC = 180° – 58°
= 122°
DCA = FDC (alt. /s, DF // AC)
= 122°
DCA + 4x° = 360° (/s at a point)
122° + 4x° = 360°
4x° = 360° – 122°
= 238°
x° = 59 .5°
\ x = 59 .5
1
C
32°
B
w°
C
P
R
x°
y°
Q
D
S
z°
E
F
G
QCA + w° = 180° (int. /s, AB // PQ)
QCA = 180° – w°
QCD = x° – QCA
= x° – (180° – w°)
= x° – 180° + w°
CDR = QCD (alt. /s, PQ // RS)
= x° – 180° + w°
FDR = y° – CDR
= y° – (x° – 180° + w°)
= y° – x° + 180° – w°
FDR + z° = 180° (int. /s, RS // EG)
y° – x° + 180° – w° + z° = 180°
w° + x° = y° + z°
\w+x=y+z
156
F
4. (a) A
Review Exercise 10
B
a°
1. (a) 32° + 4a° + 84° = 180° (adj. /s on a str. line)
4a° = 180° – 32° – 84°
= 64°
a° = 16°
\ a = 16
84° + 2b° = 180° (adj. /s on a str. line)
2b° = 180° – 84°
= 96°
b° = 48°
\ b = 48
(b) c° + 68° = 180° (adj. /s on a str. line)
c° = 180° – 68°
= 112°
\ c = 112
68° + 3d° – 5° + 30° = 180° (adj. /s on a str. line)
3d° = 180° – 68° + 5° – 30°
= 87°
d° = 29°
\ d = 29
2. (a) 4a° + 2a° + a° + a° + 2a° = 360° (/s at a point)
10a° = 360°
a° = 36°
\ a = 36
(b) (3b – 14)° + (4b – 21)° + (2b + 1)° + (b + 34)°
= 360° (/s at a point)
3b° – 14° + 4b° – 21° + 2b° + 1° + b° + 34° = 360°
3b° + 4b° + 2b° + b° = 360° + 14° + 21° – 1° – 34°
10b° = 360°
b° = 36°
\ b = 36
3. (a) COF = 4a° – 17° (vert. opp. /s)
2a° + COF + 3a° – 10° = 180° (adj. /s on a str. line)
2a° + 4a° – 17° + 3a° – 10° = 180°
2a° + 4a° + 3a° = 180° + 17° + 10°
9a° = 207°
a° = 23°
\ a = 23
(b) COF = 2b° + 15° (vert. opp. /s)
2b° + 2b° + 15° + b° = 180° (adj. /s on a str. line)
2b° + 2b° + b° = 180° – 15°
5b° = 165°
b° = 33°
\ b = 33
b° + 3c° + 2b° + 15° = 180° (adj. /s on a str. line)
33° + 3c° + 2(33°) + 15° = 180°
33 + 3c° + 66° + 15° = 180°
3c° = 180° – 33° – 66° – 15°
= 66°
c° = 22°
\ c = 22
E
P
250°
Q
126°
C
D
DEP + 126° = 180° (int. /s, PQ // CD)
DEP = 180° – 126°
= 54°
BEP + DEP + 250° = 360° (/s at a point)
BEP + 54° + 250° = 360°
BEP = 360° – 54° – 250°
= 56°
a° + BEP = 180° (int. /s, AB // PQ)
a° + 56° = 180°
a° = 180° – 56°
= 124°
\ a = 124
(b) (6b – 21)° + (5b – 52)° = 180° (int. /s, AB // CD)
6b° – 21° + 5b° – 52° = 180°
6b° + 5b° = 180° + 21° + 52°
11b° = 253°
b° = 23°
\ b = 23
3c° = (6b – 21)°
= [6(23) – 21]°
= 117°
c° = 39°
\ c = 39
(c)
A
B
276°
P
(5d – 13)°
E
Q
(4d + 28)°
C
D
QEA + (5d – 13)° = 180° (int. /s, AB // PQ)
QEA + 5d° – 13° = 180°
QEA = 180° – 5d° + 13°
= 193° – 5d°
QEC + (4d + 28)° = 180° (int. /s, PQ // CD)
QEC + 4d° + 28° = 180°
QEC = 180° – 4d° – 28°
= 152° – 4d°
276° + QEA + QEC = 360° (/s at a point)
276° + 193° – 5d° + 152° – 4d° = 360°
5d° + 4d° = 276° + 193° + 152° – 360°
9d° = 261°
d° = 29°
\ d = 29
157
1
A
(d)
F
R
285°
E
P
Q
e°
S
18°
C
5. (i) CDF = 148° (alt. /s, GC // DF)
CDE + 84° + CDF = 360° (/s at a point)
CDE + 84° + 148° = 360°
CDE = 360° – 84° – 148°
= 128°
(ii) ABC = CDE (alt. /s, AB // DE)
= 128°
ABH = ABC – 74°
= 128° – 74°
= 54°
6. (i) DEH + 26° = 180° (adj. /s on a str. line)
DEH = 180° – 26°
= 154°
(ii) BEH = 62° (alt. /s, EC // GH)
DEB + BEH + DEH = 360° (/s at a point)
DEB + 62° + 154° = 360°
DEB = 360° – 62° – 154°
= 144°
ABC = DEB (corr. /s, AB // DF)
= 144°
B
7.
A
C
B
37°
D
AEP = 37° (alt. /s, AB // PQ)
AEP + 285° + FEP = 360° (/s at a point)
37° + 285° + FEP = 360°
FEP = 360° – 37° – 285°
= 38°
EFS = 38° (alt. /s, PQ // RS)
DFS = 18° (alt. /s, RS // CD)
e° = EFS + DFS
= 38° + 18°
= 56°
\ e = 56
(e)
C
B
A
123°
X
D
122°
f°
316°
E
XEC = 122° (alt. /s, CD // XE)
XEB + 123° = 180° (int. /s, AB // XE)
XEB = 180° – 123°
= 57°
f ° = XEC – XEB
= 122° – 57°
= 65°
\ f = 65
(f)
Q
X
D
C
238°
5g°
E
A
P
X
BEQ = 37° (alt. /s, AB // PQ)
ECX = 238° – 180°
= 58°
CEQ = ECX (alt. /s, PQ // XD)
= 58°
5g° = BEQ + CEQ
= 37° + 58°
= 95°
g° = 19°
\ g = 19
1
58°
Y
F
G
(i) DEF + reflex DEF = 360° (/s at a point)
DEF + 316° = 360°
DEF = 360° – 316°
= 44°
BDE = DEF (alt. /s, DB // FE)
= 44°
(ii) DYF = 58° (corr. /s, YB // FE)
ABD = DYF (alt. /s, AC // XG)
= 58°
8. Since BWQ + DYP = 123° + 57° = 180°,
then AB // CD (converse of int. /s) .
\ DZR = AXS = 118° (alt. /s, AB // CD)
B
37°
E
D
158
9. A
P
D
1
× 360°
12
= 30°
Angle hour hand moves from 12 noon to 7 p.m. = 7 × 30°
= 210°
Q
x°
R
2. Angle hour hand moves in 1 hour =
B
w°
C
y°
S
z°
F
Angle hour hand moves from 7 p.m. to 7.20 p.m. =
20
× 30°
60
= 10°
Angle hour hand moves from 12 noon to 7.20 p.m. = 210° + 10°
= 220°
E
BCQ = w° (alt. /s, AB // PQ)
DCQ = x° – BCQ
= x° – w°
CDS + DCQ = 180° (int. /s, PQ // RS)
CDS + x° – w° = 180°
CDS = 180° – x° + w°
EDS = y° – (180° – x° + w°)
= y° – 180° + x° – w°
EDS = z° (alt. /s, RS // FE)
y° – 180° + x° – w° = z°
w° + z° + 180° = x° + y°
\ w + z + 180 = x + y
Angle minute hand moves from 7 p.m. to 7.20 p.m. =
20
× 360°
60
= 120°
Smaller angle between minute hand and hour hand at 7.20 p.m.
= 220° – 120°
= 100°
Larger angle between minute hand and hour hand at 7.20 p.m.
= 360° – 100° (/s at a point)
= 260°
3. From 9 a.m. to before 9 p.m. on any particular day, the bell will
sound twice every hour, except for the hour from 1 p.m. to before
2 p.m. and the hour from 2 p.m. to before 3 p.m., when it only sounds
once during each hour.
Likewise, from 9 p.m. on any particular day to before 9 a.m. the
next day, the bell will sound twice every hour, except for the hour
from 1 a.m. to before 2 a.m. and the hour from 2 a.m. to before
3 a.m, when it only sounds once during each hour.
\ Number of times bell will sound from 9 a.m. on a particular day
to before 9 p.m. the next day
= 2 × 30 + 1 × 6
= 60 + 6
= 66
Since the bell will sound at 9 p.m. the next day,
Number of times bell will sound from 9 a.m. on a particular day to
9 p.m. the next day
= 66 + 1
= 67
Challenge Yourself
1.
Number of Rays between
OA and OB
Number of Different Angles
0
1=
1
×1×2
2
1
3=
1
×2×3
2
2
6=
1
×3×4
2
3
10 =
1
×4×5
2
4
15 =
1
×5×6
2
:
n
:
1
(n + 1)(n + 2)
2
Number of different angles in the figure =
1
(n + 1)(n + 2)
2
159
1
Chapter 11 Triangles, Quadrilaterals and Polygons
TEACHING NOTES
Suggested Approach
Students have learnt about triangles, and quadrilaterals such as parallelograms, rhombuses and trapeziums in primary school.
They would have learnt the properties and finding unknown angles involving these figures. In this chapter, students begin from
3-sided triangles, to 4-sided quadrilaterals and finally n-sided polygons. The incremental approach is to ensure that students
have a good understanding before they move on to a higher level. Teachers may want to dedicate more time and attention to
the section on polygons in the last section of this chapter.
Section 11.1: Triangles
Students have learnt about isosceles triangles, equilateral triangles and right-angled triangles in primary
school. In this chapter, students should be aware that triangles can be classified by the number of equal sides
or the types of angles. Teachers may want to check students’ understanding on the classification of triangles
(see Thinking Time on page 260). Teachers should highlight to the students that equilateral triangles are a
special type of isosceles triangles while scalene triangles are triangles that are not isosceles, and are definitely
not equilateral triangles.
Students should explore and discover that the longest side of a triangle is opposite the largest angle, and the sum
of two sides is always larger than the third side (see Investigation: Basic Properties of a Triangle).
Teachers should ensure students are clear what exterior angles are before stating the relation between exterior
angles and its interior opposite angles. Some may think that the exterior angle of a triangle is the same as the
reflex angle at a vertex of a triangle.
Section 11.2: Quadrilaterals
Teachers may want to first recap students’ knowledge of parallelograms, rhombuses and trapeziums based
on what they have learnt in primary school. Teachers can use what students have learnt in Chapter 10,
reintroduce and build up their understanding of the different types of quadrilaterals and their properties
(see Investigation: Properties of Special Quadrilaterals and Investigation: Symmetric Properties of Special
Quadrilaterals). For further understanding, teachers may wish to show the taxonomy of quadrilaterals to
demonstrate their relations.
Before proceeding onto the next section, teachers may want to go through with the students the angle properties
of triangles and quadrilaterals. This reinforces the students’ knowledge as well as prepares them for the section
on polygons.
Section 11.3: Polygons
Teachers should emphasise to the students that triangles and quadrilaterals are polygons so that they are aware
that all the concepts which they have learnt so far remains applicable in this topic. Students should learn the
different terms with regards to polygons. In this section, most polygons studied will be simple, convex polygons.
Students need to know the names of polygons with 10 sides or less and the general naming convention of polygons
(see Class Discussion: Naming of Polygons). Through the class discussion, students should be able to develop a
good understanding on polygons and be able to name them. They should also know and appreciate the properties
of regular polygons (see Investigation: Properties of a Regular Polygon and Investigation: Symmetric Properties
of Regular Polygons).
Teachers can ask students to recall the properties of triangles and quadrilaterals during the investigation of the
sum of interior angles and sum of exterior angles of a polygon. Students should see a pattern in how the sum of
interior angles differs as the number of sides increases and understand its formula, (see Investigation: Sum of
Interior Angles of a Polygon) as well as discover that the sum of exterior angles is always equal to 360° regardless
of the number of sides of the polygon (see Investigation: Sum of Exterior Angles of a Pentagon).
1
160
Challenge Yourself
Some of the questions (e.g. Questions 1 and 2) may be challenging for most students while the rest of the questions
can be done with guidance from teachers.
Question 1: Two new points need to be added. The first point (say, E) is the midpoint of BC and the second point
(say, F) lies on the line AE such that nBCF is equilateral. Draw the lines AE, CF and DF. Begin by finding ABC
and continue from there.
Question 2: Draw DG such that BC // DG, and mark E at the point where DG cuts CD. Join E and F. Begin by
finding ACB and continue from there.
161
1
6. (a) Square
: All sides are equal in length .
Parallelogram : Opposite sides are equal in length .
Rhombus
: All sides are equal in length .
Trapezium
: All sides are not equal in length .
Kite
: There are two pairs of equal adjacent sides .
(b) Square
: All four interior angles are right angles .
Parallelogram : Opposite interior angles are equal .
Rhombus
: Opposite interior angles are equal .
Trapezium
: All four interior angles are not equal .
Kite
: One pair of opposite interior angles is equal .
(c) Square
: The two diagonals are equal in length .
Parallelogram : The two diagonals are not equal in length .
Rhombus
: The two diagonals are not equal in length .
Trapezium
: The two diagonals are not equal in length .
Kite
: The two diagonals are not equal in length .
(d) Square
: The diagonals bisect each other .
Parallelogram : The diagonals bisect each other .
Rhombus
: The diagonals bisect each other .
Trapezium
: The diagonals do not bisect each other .
Kite
: The diagonals do not bisect each other .
(e) Square
: The diagonals are perpendicular to each other .
Parallelogram : The diagonals are not perpendicular to each
other .
Rhombus
: The diagonals are perpendicular to each other .
Trapezium
: The diagonals are not perpendicular to each
other .
Kite
: The diagonals are perpendicular to each other .
(f) Square
: The diagonals bisect the interior angles .
Parallelogram : The diagonals do not bisect the interior angles .
Rhombus
: The diagonals bisect the interior angles .
Trapezium
: The diagonals do not bisect the interior angles .
Kite
: One diagonal bisects the interior angles .
WORKED SOLUTIONS
Thinking Time (Page 260)
A represents isosceles triangles .
B represents scalene triangles .
C represents acute-angled triangles .
D represents right-angled triangles .
Investigation (Basic Properties of a Triangle)
1. The side opposite /B is b and the side opposite /C is c .
2. The largest angle is /C and the smallest angle is /B .
The side opposite the largest angle, /C is the longest side and the
side opposite the smallest angle, /B is the shortest side .
3. The bigger the angle, the longer the side opposite it . The angle
opposite the side shortest in length will be the smallest angle. This
applies to the longest side as well i.e. the longest side is always
opposite the largest angle .
4. The sum of the lengths of the two shorter sides of a triangle is always
longer than the length of the longest side .
5. Yes, since the sum of the angles facing the two shorter sides are
greater than the largest angle facing the longest side, hence, the sum
of the lengths of the two shorter sides of a triangle is always longer
than the length of the longest side .
6. No, it is not possible to form a triangle .
7. a + b = c . It is still not possible to form a triangle .
8. The sum of the lengths of any two line segments has to be greater
than the length of the third line segment
From the investigation, two basic properties of a triangle are:
• The largest angle of a triangle is opposite the longest side, and the
smallest angle is opposite the shortest side .
• The sum of the lengths of any two sides of a triangle must be greater
than the length of the third side .
Thinking Time (Page 271)
Investigation (Properties of Special Quadrilaterals)
(a) Yes
(b) Yes
(d) Yes
(e) Yes
A represents kites .
B represents parallelograms .
C represents rhombus .
D represents squares .
1. AB = 2 .8 cm, BC = 1 .8 cm, DC = 2 .8 cm, AD = 1 .8 cm
AB = DC and BC = AD (Opposite sides are equal in length .)
2. BAD = 90°, ABC = 90°, BCD = 90°, ADC = 90°
BAD = ABC = BCD = ADC = 90° (All four interior angles are right
angles .)
3. AE = 1 .7 cm, BE = 1 .7 cm, CE = 1 .7 cm, DE = 1 .7 cm
AE = BE = CE = DE = 1 .7 cm (Diagonals bisect each other .)
4. AE + CE = 1 .7 + 1 .7 = 3 .4 cm,
BE + DE = 1 .7 + 1 .7 = 3 .4 cm
Both of the sums are equal . (The two diagonals are equal in length .)
5. The following properties hold:
• Opposite sides are equal in length.
• All four interior angles are right angles.
• Diagonals bisect each other.
• The two diagonals are equal in length.
1
162
(c) Yes
Class Discussion (Naming of Polygons)
Journal Writing (Page 278)
Triangle (3-sided)
Quadrilateral (4-sided)
Pentagon (5-sided)
Hexagon (6-sided)
Heptagon (7-sided)
Octagon (8-sided)
Nonagon (9-sided)
Decagon (10-sided)
Since a regular polygon is a polygon with all sides equal and all angles
equal, the statement made by Devi is correct as she stated one of the two
properties of a regular polygon .
On the other hand, the statement made by Michael is wrong as he stated
an incomplete definition of a regular polygon, i .e . the conditions of a
regular polygon . A polygon with all sides equal may not be regular,
e .g . a square is a regular polygon (see Fig . (a)) but a rhombus is not a
regular polygon (see Fig . (b)) . This is because even though a rhombus is
a polygon with all sides equal, not all its angles are equal . The hexagon
shown in Fig . (d) is a regular polygon but the hexagon shown in Fig . (e)
is not a regular polygon because even though all its sides are equal, not
all its angles are equal . Hence, it does not mean that a polygon with all
sides equal is regular .
Thinking Time (Page 277)
Fig. (a)
Fig. (b)
Fig. (c)
Fig. (d)
Fig. (e)
Fig. (f)
The name of a regular triangle is an equilateral triangle and the name of
a regular quadrilateral is a square .
Investigation (Properties of a Regular Polygon)
1. Yes .
(a) Rhombus
(b)
In addition, a polygon with all angles equal may not be regular . For
example, a rectangle is a polygon (see Fig . (c)) but it is not regular
because not all its sides are equal although all its angles are equal . Another
example is the hexagon as shown in Fig . (f) . It is not a regular polygon
because even though all its angles are equal, not all its sides are equal .
Hence, it does not mean that a polygon with all angles equal is regular .
2. Yes .
(a) Square and Rectangle
(b)
In conclusion, a regular polygon is a polygon with all sides equal and
all angles equal .
163
1
Investigation (Sum of Interior Angles of a Polygon)
Investigation (Sum of Exterior Angles of a Pentagon)
1.
1. –
2. The sum of exterior angles of a pentagon is 360° as all the exterior
angles will meet at a vertex .
From the investigation, we observe that the sum of exterior angles
of a pentagon is 360° .
A proof of the above result is given as follows:
Consider the pentagon in Fig . 11 .24 .
We have /a + /p = 180°, /b + /q = 180°,
/c + /r = 180°, /d + /s = 180° and /e + /t = 180° .
\ /a + /p + /b + /q + /c + /r + /d + /s + /e + /t
= 5 × 180°
(/a + /b + /c + /d + /e) + (/p + /q + /r + /s + /t) = 900°
Since the sum of interior angles of a pentagon
= /a + /b + /c + /d + /e
= (5 – 2) × 180° = 540°,
540° + (/p + /q + /r + /s + /t) = 900° .
\ /p + /q + /r + /s + /t = 900° – 540° = 360°
By using this method, we can show that the sum of exterior angles
of a hexagon, of a heptagon and of an octagon is also 360° .
Polygon
Number
of sides
Number of
Triangle(s)
formed
Sum of Interior Angles
3
1
1 × 180° = (3 – 2) × 180°
4
2
2 × 180° = (4 – 2) × 180°
5
3
3 × 180° = (5 – 2) × 180°
Triangle
Quadrilateral
Pentagon
Thinking Time (Page 285)
6
4
4 × 180° = (6 – 2) × 180°
7
5
5 × 180° = (7 – 2) × 180°
8
6
6 × 180° = (7 – 2) × 180°
n
(n – 2)
(n – 2) × 180°
1. (i) No . Since 70° is not an exact divisor of 360°, hence a regular
polygon to have an exterior angle of 70° is not possible .
(ii) Since
360° = 3 × 120°,
360° = 4 × 90°,
360° = 6 × 60°,
360° = 8 × 45°,
360° = 9 × 40°,
360° = 10 × 36°,
360° = 12 × 30°,
360° = 15 × 24°,
360° = 18 × 20°,
360° = 20 × 18°,
360° = 25 × 15°,
360° = 30 × 12°,
360° = 40 × 9°,
360° = 45 × 8°,
360° = 60 × 6°,
360° = 90 × 4°,
360° = 120 × 3°,
360° = 180 × 2°,
All the possible values of the angle are 2°, 3°, 4°, 6°, 8°, 9°, 12°,
15°, 18°, 20°, 24°, 30°, 36°, 40°, 45°, 60°, 90° and 120° .
2. No, it is not possible as a concave polygon has one or more interior
angles that are greater than 180° while as a regular polygons has all
interior angles that are less than 180° .
Hexagon
Heptagon
Octagon
n-gon
2. If a polygon has n sides, then it will form (n – 2) triangles .
Investigation (Tesellation)
1. The only regular polygons that tessellate on their own are equilateral
triangles, squares and regular hexagons . Combinations of other
regular polygons such as a square and a regular octagon can produce
tessellations .
2. See Fig . 11 .17 in the textbook for an example .
3. The sum of the corner angles will add up to 360° .
1
164
54°
2
= 27°
\ CDE = 27°
Practise Now 1
x° =
1. 90° + 65° + a° = 180° (/ sum of n)
a° = 180° – 90° – 65°
= 25°
\ a = 25
2. Since AC = BC, \ CAB = CBA = b°
b° + 52° + b° = 180° (/ sum of n)
2b° = 180° – 52°
= 128°
Practise Now 4
1. (i) ABC = 108° (opp . /s of // gram)
9x° = 108°
108°
9
= 12°
\ x = 12
(ii) (DCE + 38°) + 108° = 180° (int . /s, AD // BC)
DCE = 180° – 38° – 108°
= 34°
2. (5x + 6)° + (2x + 13)° = 180° (int . /s, AB // DC)
7x° + 19° = 180°
7x° = 180° – 19°
= 161°
x=
128°
2
= 64°
\ b = 64
b° =
Practise Now 2
a° = 53° + 48° (ext . / of n)
= 101°
\ a = 101
(b)
FDE = 93° (vert . opp . /s)
b° + 33° + 93° = 180° (/ sum of n)
b° = 180° – 33° – 93°
= 54°
\ b = 54
c° = 41° + 93° (ext . / of nABD)
= 134°
\ c = 134
(a)
161°
7
= 23°
\ x = 23
[5(23) + 6]° + (y + 17°) = 180° (int . /s, AB // DC)
y° = 180° – 121° – 17°
= 42°
\ y = 42
x° =
Practise Now 3
Practise Now 5
1. (i) DAE = 90° (right angle)
51° + 90° + AED = 180° (/ sum of nAED)
AED = 180° – 51° – 90°
= 39°
(ii) CDE + 51° = 90° (/ADC is a right angle)
CDE = 90° – 51°
= 39°
68° + 39° + CED = 180° (/ sum of nCDE)
CED = 180° – 68° – 39°
= 73°
2. (i) Since EB = EC (diagonals bisect each other), \ EBC = 63°
63° + BEC + 63° = 180° (/ sum of nBEC)
BEC = 180° – 63° – 63°
= 54°
(ii) DEC + 54° = 180° (adj . /s on a str . line)
DEC = 180° – 54°
= 126°
Since ED = EC (diagonals bisect each other),
\ CDE = DCE = x° .
x° + 126° + x° = 180° (/ sum of nCDE)
2x° = 180° – 126°
= 54°
1. (i) CAB = 32° (alt . /s, AB // DC)
Since BA = BC, \ ACB = CAB = 32°
32° + ABC + 32° = 180° (/sum of nABC)
ABC = 180° – 32° – 32°
= 116°
(ii) Since AC = CE, \ CEA = CAE = 32°
32° + (32° + BCE) + 32° = 180° (/ sum of nABC)
BCE = 180° – 32° – 32° – 32°
= 84°
2. BDC = (3x + 13)° (diagonals bisect interior angles of a rhombus)
DAC = (x + 45)° (diagonals bisect interior angles of a rhombus)
2(3x + 13)° + 2(x + 45)° = 180° (int . / s, AB // DC)
6x° + 26° + 2x° + 90° = 180°
8x° = 180° – 26° – 90°
8x° = 64°
64°
8
= 8°
\x =8
x° =
165
1
2. The sum of exterior angles of the regular decagon is 360° .
\ Size of each exterior angle of the regular decagon
Practise Now 6
1. Sum of interior angles of a pentagon
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
a° + 121° + a° + a° + 107° = 540°
3a° = 540° – 121° – 107°
3a° = 312°
360°
10
= 36°
\ Size of each interior angle of the regular decagon
= 180° – 36°
= 144°
3. The sum of exterior angles of an n-sided polygon is 360° .
25° + 26° + 3(180° – 161°) + (n – 5)(180° – 159°) = 360°
25° + 26° + 3(19°) + (n – 5)(21°) = 360°
25° + 26° + 57° + n(21°) – 105° = 360°
n(21°) = 360° – 25° – 26° – 57° + 105°
= 357°
=
312°
3
= 104°
\ a = 104
2. Sum of interior angles of a hexagon
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
3b° + 4b° + 104° + 114° + 128° + 122° = 720°
7b° = 720° – 104° – 114°
– 128° – 122°
7b° = 252°
a° =
357°
21°
= 17
n=
Practise Now 9
E
252°
7
= 36°
\ b = 36
D
b° =
F
Practise Now 7
C
A
(i) Sum of interior angles of a regular polygon with 24 sides
= (n – 2) × 180°
= (24 – 2) × 180°
= 3960°
(ii) Size of each interior angle of a regular polygon with 24 sides
B
G
Size of each exterior angle of the hexagon
360°
6
= 60°
CBG = BCG = 60°
BGC + 60° + 60° = 180° (/ sum of nBCG)
BGC = 180° – 60° – 60°
= 60°
=
3960°
24
= 165°
=
Practise Now 8
Practise Now 10
1. (a) The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
(i) Sum of interior angles of a pentagon
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
Since PBC is an interior angle of a pentagon,
360°
40°
=9
(b) Size of each exterior angle of a regular polygon
= 180° – 178°
= 2°
The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
=
540°
= 108° .
5
(ii) Since CRQ is an interior angle of a pentagon,
\ CRQ = 108° .
Let QCR = CQR = x° (base /s of isos . nCQR)
x° + x° + 108° = 180° (/ sum of nCQR)
2x° = 180° – 108°
2x° = 72°
\ PBC =
360°
2°
= 180
=
72°
2
= 36°
\ QCR = 36°
x° =
1
166
A
18°
y° =
2
= 9°
\ BDC = 9°
(v) Let the exterior angle of the n-sided polygon be a° .
a° + 162° = 180° (adj /s on a str . line)
a° = 180° – 162°
= 18°
Since the sum of the exterior angles of the n-sided polygon is 360°,
2. (a)
(b)
\n =
(c)
Exercise 11A
(d)
360°
18°
= 20
C
1. (a)
3. (a)
60°
20°
A
B
/C = 180° – 20° – 60° (/ sum of n)
= 100°
It is a scalene triangle and an obtuse-angled triangle .
(b)
C
(d)
(iii) BCD + 108° + 90° = 360° (/s at a point)
BCD = 360° – 108° – 90°
= 162°
(iv) Let BDC = BCD = y° (base /s of isos . nBCD)
y° + y° + 162° = 180° (/ sum of nBCD)
2y° = 180° – 162°
2y° = 18°
(b)
C
42°
48°
B
/C = 180° – 42° – 48° (/ sum of n)
= 90°
It is a scalene triangle and right-angled triangle .
Third angle of the triangle
= 180° – 40° – 40° (/ sum of n)
= 100°
Third angle of the triangle
= 180° – 87° – 87° (/ sum of n)
= 6°
Third angle of the triangle
= 180° – 15° – 15° (/ sum of n)
= 150°
Third angle of the triangle
= 180° – 79° – 79° (/ sum of n)
= 22°
39° + 90° + a° = 180° (/ sum of n)
a° = 180° – 39° – 90°
= 51°
\ a = 51
68° + 2b° + 64° = 180° (/ sum of n)
2b° = 180° – 68° – 64°
= 48°
48°
2
= 24°
\ b = 24
(c) 4c° + 3c° + 40° = 180° (/ sum of n)
4c° + 3c° = 180° – 40°
7c° = 140°
b° =
A
70°
40°
B
/C = 180° – 70° – 40° (/ sum of n)
= 70°
It is an isosceles triangle and acute-angled triangle .
(c)
140°
7
= 20°
\ c = 20
(d) 3d° + 4d° + d° = 180° (/ sum of n)
8d° = 180°
c=
C
180°
8
= 22 .5°
\ d = 22 .5
(e) Since BA = BC, \ BCA = BAC = 62°
62° + e° + 62° = 180° (/ sum of n)
e° = 180° – 62° – 62°
= 56°
\ e = 56
d° =
A
60°
60°
B
/C = 180° – 60° – 60° (/ sum of n)
= 60°
It is an equilateral triangle and acute-angled triangle .
167
1
(f) Since AC = BC = AB, \ CAB = CBA = ACB = f °
f ° + f ° + f ° = 180° (/ sum of n)
3f ° = 180°
7. (a) a° + 90° = 115° (ext . / of nBCE)
a° = 115° – 90°
= 25°
\ a = 25
b° = 90° + 32° (ext . / of nEFG)
= 122°
\ b = 122
(b) ABE = ABD = 89° + 27° (ext . / of nBCD)
= 116°
c° = 116° + 22° (ext . / of nABE)
= 138°
\ c = 138
8. (a) 82° + 40° + a° = 180° (/ sum of n)
a° = 180° – 82° – 40°
= 58°
\ a = 58
ADB = 82° (vert . opp . /s)
b° = 45° + 82° (ext . / of nABD)
= 127°
\ b = 127
(b) EDF + 44° + 57° = 180° (/ sum of n)
EDF = 180° – 44° – 57°
= 79°
ADB = 79° (vert . opp . /s)
c° = 51° + 79° (ext . / of nABD)
= 130°
\ c = 130
9. (a) BAC + ACD = 180° (int . /s, AB // CD)
108° + (a° + 37°) = 180°
a° = 180° – 108° – 37°
= 35°
\ a = 35
b° = 71° + 37° (ext . / of nABD)
= 108°
\ b = 108
(b) AHF = 45° (vert . opp . /s)
AHI + CIH = 180° (int . /s, AB // CD)
(45° + 64°) + (32° + c°) = 180°
c° = 180° – 45° – 64° – 32°
= 39°
\ c = 39
d° + 39° + 64° = 180° (/ sum of n)
d° = 180° – 39° – 64°
= 77°
\ d = 77
180°
3
= 60°
\ f = 60
4. (a) a° = 47° + 55° (ext . / of n)
= 102°
\ a = 102
(b) 90° + b° + 50° = 180° (/ sum of n)
b° = 180° – 90° – 50°
= 40°
\ b = 40
90° + c° + 35° = 180° (/ sum of n)
c° = 180° – 90° – 35°
= 55°
\ c = 55
(c) d° + 110° = 180° (adj . /s on a str . line)
d° = 180° – 110°
= 70°
\ d = 70
2e° + 3e° = 110° (ext . / of n)
5e° = 110°
f° =
110°
5
= 22°
\ e = 22
5. 3x° + 4x° + 5x° = 180° (/ sum of n)
12x° = 180°
e° =
180°
12
= 15°
\ x = 15
Smallest angle of the triangle
= 3(15°)
= 45°
6. (i) Let ADB = BDC = x°
90° + 20° + 2x° = 180° (/ sum of n)
2x° = 180° – 90° – 20°
= 70°
x° =
70°
2
= 35°
\ BDC = 35°
(ii) CBD + 20° + 35° = 180° (/ sum of n)
CBD = 180° – 20° – 35°
= 125°
x° =
1
168
(c) Since EB = EC, \ ECB = EBC = 2e°
f ° = 2e° + 2e° (ext . / of nBCE)
= 4e°
e° + f ° = 120° (ext . / of nBEF)
e° + 4e° = 120°
5e° = 120°
12. (i) DCE + 61° + 41° = 180° (/ sum of n)
DCE = 180° – 61° – 41°
= 78°
ACB = 78° (vert . opp . /s)
(ii) ABC + 78° + 50° = 180° (/ sum of n)
ABC = 180° – 78° – 50°
= 52°
13. (i) DEC = BCE = 47° (alt . /s, AC // ED)
32° + 47° + DFE = 180° (/ sum of nDEF)
DFE = 180° – 32° – 47°
= 101°
(ii) CBD = BDE = 32° (alt /s, AC // ED)
106° + EBD + 32° = 180° (adj . /s on a str . line)
EBD = 180° – 106° – 32°
= 42°
BDC = EBD = 42° (alt . /s, BE // CD)
14. Let CBO be x° .
120°
5
= 24°
\ e = 24
f ° = 4(24°)
= 96°
\ f = 96
ABE = DEB (alt /s, AB // CD)
g° + 2(24°) = 96°
g° = 96° – 48°
= 48°
\ g = 48
(d) AFE = CGF = 68° (corr . /s, AB // CD)
68° + h° = 180° (adj . /s on a str . line)
h° = 180° – 68°
= 112°
\ h = 112
FJI = KJB = 65° (vert . opp . /s)
i° = 65° (corr . /s, AB // CD)
\ i = 65
IGH = CGF = 68° (vert . opp . /s)
68° + j° + 65° = 180° (/ sum of nGHI)
j° = 180° – 68° – 65°
= 47°
\ j = 47
°
1
10. (x – 35)° + (x – 25)° + x – 10 = 180° (/ sum of n)
2
5
x° – 70° = 180°
2
5
x° = 180° + 70°
2
5
x° = 250°
2
250°
x° =
5
2
= 100°
\ x = 100
11. (i) ABC + 50° + 26° = 180° (/ sum of n)
ABC = 180° – 50° – 26°
= 104°
(ii) CBD = 50° + 26° (ext . / of n)
= 76°
e° =
1
1
x° and BAO = 1 x° .
2
2
1
Since OA = OC, \ ACO = CAO = x° .
2
Since OB = OC, \ CBO = BCO = x° .
Then CAO =
Since OA = OB, \ BAO = ABO = 1
1
x° .
2
Hence,
CAB + ABC + BCA = 180° (/ sum of nABC)
1
1
1
1
2 x ° + 1 2 x ° + 1 2 x ° + x ° + 2 x ° + x ° = 180°
6x° = 180
180°
6
= 30°
x° =
1
(30°) = 15° .
2
15. Since AB = AC, then let ABC = ACB = x° .
DBE = 180° – x° (adj . /s on a str . line)
Since BD = BE, then
\ CAO =
180° – (180° – x ° )
x°
=
.
2
2
Since AF = DF, \ FAD = FDA
BDE = BED =
FAD = FDA = BDE =
x°
.
2
x°
+ x + x° = 180° (/ sum of nABC)
2
1
2 x° = 180°
2
180°
x=
1
2
2
= 72°
\ ABC = 72°
169
1
4. (a) Since ABCD is a square, \ DAC = BAC = 45° and hence
DAE = 45° .
(Diagonals bisect the interior angles)
AED + 82° = 180° (adj . /s on a str . line)
AED = 180° – 82°
= 98°
45° + 98° + a° = 180° (/ sum of nADE)
a° = 180° – 45° – 98°
= 37°
\ a = 37
Since ABCD is a square, \ BAC = DAC = 45° and hence
EAF = 45° .
(Diagonals bisect the interior angles)
AEF = 82° (vert . opp . /)
b° = 45° + 82° (ext . / of nAEF)
= 127°
\ b = 127
(b) Since ABCD is a square, \ BCA = DCA = 45° and hence
ECF = 45° .
(Diagonals bisect the interior angles)
c° + 45° + c° = 180° (/ sum of nCEF)
2c° = 180° – 45°
= 135°
Exercise 11B
1. (a) a° + 54° = 90° (BCD is a right angle)
a° = 90° – 54°
= 36°
\ a = 36
b° = 36° (alt . /s, AB // DC)
\ b = 36
(b) EBC = 90° (right angle)
90° + 39° + c° = 180° (/ sum of nBCE)
c° = 180° – 90° – 39°
= 51°
\ c = 51
DCE + 39° = 90° (BCD is a right angle)
DCE = 90° – 39°
= 51°
51° + d° + 78° = 180° (/ sum of nCDE)
d° = 180° – 51° – 78°
= 51°
\ d = 51
2. (a) a° = 106° (opp . /s of // gram)
\ a = 106
b° = 48° (alt . /s, AD // BC)
\ b = 48
(b) 4c° + 5c° = 180° (int . /s, AB // DC)
9c° = 180°
135°
2
= 67 .5°
\ c = 67 .5
Since ABCD is a square, \ CED = 90° .
(Diagonals bisect each other at right angles)
Hence,
d° + 67 .5° = 90°
d° = 90° – 67 .5°
= 22 .5°
\ d = 22 .5
5. (a) Since ABCD is a rhombus, \ ACB = ADC = 114° (Opposite
angles are equal) and hence a = 114 .
Since ABCD is a rhombus, \ AB = CB and hence
ACB = CAB = b° .
b° + 114° + b° = 180° (/ sum of nABC)
2b° = 180° – 114°
= 66°
c° =
180°
9
= 20°
\ c = 20
2d° = 4(20°) (opp . / s of // gram)
c° =
80°
2
= 40°
\ d = 40
3. (a) Since ABCD is a kite, \ AD = CD and so ACD = CAD = a°
a° + 100° + a° = 180° (/ sum of nACD)
2a° = 180° – 100°
= 80°
d° =
80°
2
= 40°
\ a = 40
Since ABCD is a kite, \ AB = CB and so CAB = ACB = 61° .
61° + b° + 61° = 180° (/sum of nABC)
b° = 180° – 61° – 61°
= 58°
\ b = 58
(b) Since ABCD is a kite, \ DAC = BAC = 40° .
(One diagonal bisects the interior angles)
40° + 26° + c° = 180° (/ sum of nACD)
c° = 180° – 40° – 26°
= 114°
\ c = 114
a° =
1
66°
2
= 33°
\ b = 33
(b) CBD = BDA = 38° (alt . /s, AD // BC)
c° = 38°
\ c = 38
Since ABCD is a rhombus, \ AB = AD and hence
BDA = DBA = 38° .
38° + d° + 38° = 180° (/ sum of nABD)
d° = 180° – 38° – 38°
= 104°
\ d = 104
b° =
170
9. ADB = (3x + 7)° (diagonals bisect interior angles of a rhombus)
DAC = (2x + 53)° (diagonals bisect interior angles of a rhombus)
2(3x + 7)° + 2(2x + 53)° = 180° (int . /s, AB // DC)
6x° + 14° + 4x° + 106° = 180°
10x° = 180° – 14° – 106°
10x° = 60°
60°
x° =
10
= 6°
\x =6
10. 5x° + x° = 180° (int . /s, AB // DC)
6x° = 180°
180°
x° =
6
= 30°
\ x = 30
2 .2(30°) + y° = 180° (int . /s, AB // DC)
y° = 180° – 66°
= 114°
\ y = 114
11. (i) Since ABCD is a kite, \ BAC = DAC = 25°
(One diagonal bisects the interior angles) and
since AB = AD, \ BDA = DBA = x°
x° + 2(25°) + x° = 180° (/ sum of nABD)
2x° = 180° – 50°
= 130°
130°
x° =
2
= 65°
\ ABD = 65°
(ii) Since ABCD is a kite, \ BCA = DCA = 44°
One diagonal bisects the interior angles) and
since CB = CD, \ BDC = DBC = y°
y° + 2(44°) + y° = 180° (/ sum of nBCD)
2y° = 180° – 88°
= 92°
92°
y° =
2
= 46°
\ CBD = 46°
12. D
C
(c) DCA = CAB = 42° (alt, /s, AB // DC)
e° = 42°
\ e = 42
Since ABCD is a rhombus, \ ADB = CDB = f ° .
(Diagonals bisect the interior angles)
Also, AD = CD and hence CAD = ACD = 42°
42° + 2f ° + 42° = 180° (/ sum of nACD)
2f ° = 180° – 42° – 42°
= 96°
96°
2
= 48°
\ f = 48
6. (i) AED = 52° (vert . opp . /s)
Since AE = DE, \ ADE = DAE = x° .
x° + 52° + x° = 180° (/ sum of nADE)
2x° = 180° – 52°
= 128°
f° =
128°
2
= 64°
\ ADB = ADE = 64°
ADC = 90° (right angle of a rectangle)
64° + 90° + ACD = 180° (/ sum of nACD)
ACD = 180° – 64° – 90°
= 26°
ADE + 65° = 180° (int . /s, AB // DC)
ADE = 180° – 65°
= 115°
BCD = 65° (opp . /s of // gram)
CBE + 65° = 125° (ext . / of nBCE)
CBE = 125° – 65°
= 60°
ABD = 46° (alt . /s, AB // DC)
Since ABCD is a rhombus, \ AB = AD and hence
BDA = DBA = 46° .
46° + BAD + 46° = 180° (/ sum of nABD)
BAD = 180° – 46° – 46°
= 88°
DBC = 46° (alt . /s, AD // BC)
Since BC = BE, \ BCE = BEC = x° .
x° + x° = 46° (ext . / of nBCE)
2x° = 46°
x° =
(ii)
7. (i)
(ii)
8. (i)
(ii)
118°
A
46°
x° =
2
= 23°
\ BCE = 23°
E
B
(i) Since E is the midpoint of AB, \ CE = DE and hence
CDE = DCE = x° .
x° + 118° + x° = 180° (/ sum of nCDE)
2x° = 180° – 118°
= 62°
62°
x =
2
= 31°
ADE + 31° = 90° (ADC is a right angle)
= 59°
(ii) From (i), DCE = x° = 31° .
171
1
13.
S
Since AB = AD, \ ABD = ADB = x°
x° + 118° + x° = 180° (/ sum of nABD)
2x° = 180° – 118°
= 62°
R
70°
62°
2
= 31°
\ ABD = 31°
(ii) ABC + 52° = 180° (int . /s, AB // DC)
ABC = 180° – 52°
= 128°
From (i), ABD = 31° .
36° + CBD = 128°
CBD = 128° – 31°
= 97°
x° =
P
42°
Q
(i) PQR + 70° = 180° (int . /s, PQ // SR)
PQR = 180° – 70°
= 110°
(ii) 42° + 110° + PRQ = 180° (/ sum of nPQR)
PRQ = 180° – 42° – 110°
= 28°
14.
Z
S
16.
Y
64°
P
108°
W
R
42°
X
(i) Since WXYZ is a rhombus, WZY = WXY = 108° (opp . /s of a //
gram) and XZY = XZW = x° (Diagonal bisect the interior angles),
hence WZY = 2x°
2x° = 108°
Q
(i) Since PS = RS, \ RPS = PRS = x° .
x° + 64° + x° = 180° (/ sum of nPRS)
2x° = 180° – 64°
= 116°
108°
2
= 54°
\ XZY = 54°
(ii) XYZ + 108° = 180° (int . /s, WX // ZY)
XYZ = 180° – 108°
= 72°
(iii) Since WXYZ is a rhombus, XWZ = XYZ = 72° (opp . /s of a
// gram) and XWY = ZWY = y° (Diagonals bisect the interior
angles), hence XWZ = 2y°
2y° = 72°
x° =
116°
2
= 58°
\ PRS = 58°
(ii) Since PQ = QR, \ QPR = QRP = 42° .
42° + PQR + 42° = 180° (/ sum of nPQR)
PQR = 180° – 42° – 42°
= 96°
x° =
Exercise 11C
72°
2
= 36°
\ XWY = 36°
y° =
15. D
1. (a) Sum of interior angles of a 11-gon
= (n – 2) × 180°
= (11 – 2) × 180°
= 1620°
(b) Sum of interior angles of a 12-gon
= (n – 2) × 180°
= (12 – 2) × 180°
= 1800°
(c) Sum of interior angles of a 15-gon
= (n – 2) × 180°
= (15 – 2) × 180°
= 2340°
C
52°
62°
A
B
(i) BAD + 62° = 180° (int . /s, AB // DC)
BAD = 180° – 62°
= 118°
1
172
(b) (i) Sum of interior angles of a regular polygon with 18 sides
= (n – 2) × 180°
= (18 – 2) × 180°
= 2880°
(ii) Hence, size of each interior angle of a regular polygon with
18 sides
(d) Sum of interior angles of a 20-gon
= (n – 2) × 180°
= (20 – 2) × 180°
= 3240°
2. (a) Sum of interior angles of a quadrilateral
= (n – 2) × 180°
= (4 – 2) × 180°
= 360°
78° + 62° + a° + 110° = 360°
a° = 360° – 78° – 62° – 110°
= 110°
\ a = 110
(b) Sum of interior angles of a quadrilateral
= (n – 2) × 180°
= (4 – 2) × 180°
= 360°
b° + 78° + 2b° + 84° = 360°
3b° = 360° – 78° – 84°
= 198°
2880°
18
= 160°
4. (a) The sum of exterior angles of the regular polygon is 360° .
\ Size of each exterior angle of the regular polygon
=
360°
24
= 15°
\ Size of each interior angle of a regular polygon with 24 sides
= 180° – 15°
= 165°
(b) The sum of exterior angles of the regular polygon is 360° .
\ Size of each exterior angle of the regular polygon
=
360°
36
= 10°
\ Size of each interior angle of a regular polygon with 36 sides
= 180° – 10°
= 170°
5. (a) The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
198°
3
= 66°
\ b = 66
(c) Sum of interior angles of a pentagon
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
c° + 152° + 38° + 2c° + 101° = 540°
3c° = 540° – 152° – 38° – 101°
3c° = 249°
=
b° =
360°
90°
=4
(b) The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
=
249°
3
= 83°
\ c = 83
(d) Sum of interior angles of a hexagon
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
102° + 5d° + 4d° + 4d° + 108° + 4d° = 720°
17d° = 720° – 102° – 108°
= 510°
c° =
360°
45°
=8
(c) The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
=
360°
12°
= 30
(d) The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
=
510°
17
= 30°
\ d = 30
3. (a) (i) Sum of interior angles of a hexagon
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
(ii) Hence, size of each interior angle of a hexagon
360°
4°
= 90
6. (a) Size of each interior angle of a regular polygon
= 180° – 140°
= 40°
The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
d° =
=
360°
40°
=9
720°
6
= 120°
=
=
173
1
(b) Size of each interior angle of a regular polygon
= 180° – 162°
= 18°
The sum of exterior angles of the regular polygon is 360°
\ Number of sides of the polygon
9. The sum of exterior angles of an n-sided polygon is 360° .
15° + 25° + 70° + (n – 3) × 50° = 360°
15° + 25° + 70° + n(50°) – 150° = 360°
n(50°) = 360° – 15° – 25° – 70° + 150°
n(50°) = 400°
360°
18°
= 20
(c) Size of each interior angle of a regular polygon
= 180° – 172°
= 8°
The sum of exterior angles of the regular polygon is 360° .
\ Number of sides of the polygon
400°
50°
=8
10. The sum of exterior angles of a n-sided polygon is 360° .
3(50°) + (180° – 127°) + (180° – 135°) + (n – 5)(180° – 173°)
= 360°
150° + 53° + 45° + (n – 5)(7°) = 360°
150° + 53° + 45° + n(7°) – 35° = 360°
n(7°) = 360° – 150° – 53° – 45° + 35°
= 147°
n=
=
360°
8°
= 45
(d) Size of each interior angle of a regular polygon
= 180° – 175°
= 5°
The sum of exterior angles of the regular polygon is 360°
\ Number of sides of the polygon
=
147°
7°
= 21
n =
11.
E
D
F
360°
5°
= 72
7. Sum of interior angles of a pentagon
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
2x° + 3x° + 4x° + 5x° + 6x° = 540°
20x° = 540°
=
C
G
H
B
A
Size of each exterior angle of the heptagon
360°
7
= 51 .43°
BHC + 51 .43° + 51 .43° = 180° (/ sum of nBCH)
BHC = 180° – 51 .43° – 51 .43°
= 77 .1° (to 1 d .p .)
=
540°
20
= 27°
Hence, the largest interior angle of the pentagon
= 6(27°)
= 162°
8. (i) The sum of exterior angles of the triangle is 360° .
3y° + 4y° + 5y° = 360°
12y° = 360°
x° =
D
B
A
360°
12
= 30°
\ y = 30
(ii) Smallest interior angle of the triangle
= 180° – 5(30°)
= 180° – 150°
= 30°
(i) Sum of interior angles of a regular polygon with 20 sides
= (n – 2) × 180°
= (20 – 2) × 180°
= 3240°
Hence, size of each interior angles of a regular polygon with
20 sides
y° =
1
C
12.
3240°
20
= 162°
\ ABC = 162°
=
174
(ii) Since size of each interior angle of a regular polygon with
20 sides = 162°,
\ BCD = 162°
Let CBD = CDB = x° (base /s of isos . nBCD)
x° + x° + 162° = 180° (/ sum of nBCD)
2x° = 180° – 162°
2x° = 18°
Hence,
ACD = BCD – BCA
= 108° – 36°
= 72°
(vi) Since size of each interior angle of a hexagon = 120°,
\ BAS = 120°
Since size of each interior angle of a pentagon = 108°,
\ BAE = 108°
120° + 108° + SAE = 360° (/s at a point)
SAE = 360° – 120° – 108°
= 132°
Let ASE = AES = x° (base / of isos . nAES)
x° + x° + 132° = 180° (/ sum of nAES)
2x° = 180° – 132°
2x° = 48°
18°
2
= 9°
\x =9
x° =
Hence,
ABD = ABC – CBD
= 162° – 9°
= 153°
13. (i) Sum of interior angles of a hexagon
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
\ Size of each interior angle of a hexagon
48°
2
= 24°
\ ASE = 24°
Let the interior angle be 5x° and the exterior angle be x° .
5x° + x° = 180° (adj . /s on a str . line)
6x° = 180°
x° =
14. (i)
720°
6
= 120°
Since ABP is an interior angle of a hexagon,
\ ABP = 120° .
(ii) Since PQR is an interior angle of a hexagon,
\ PQR = 120° .
=
180°
6
= 30°
Since sum of exterior angles of a n-sided polygon is 360°,
x° =
360°
= 12
30°
(ii) ABC = 5(30°) = 150° (int . / of a 12-sided polygon)
Let BAC = BCA = x° (base /s of isos . nABC)
x° + x° + 150° = 180° (/ sum of nABC)
2x° = 180° – 150°
2x° = 30°
30°
x° =
2
= 15°
Hence,
ACD = BCD – BCA
= 150° – 15°
= 135°
(iii) ABC = BCD = 150° (int . / of a 12-sided polygon)
BAD = ADC = y° (base /s of isos . quadrilateral, BA = CD)
y° + y° + 150° + 150° = 360° (/ sum of quadrilateral)
2y° = 360° – 150° – 150°
2y° = 60°
\n=
120°
PQX =
(QA is a line of symmetry)
2
= 60°
360°
(/s at a point)
6
= 60°
(iv) Sum of interior angles of a pentagon
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
\ Size of each interior angle of a pentagon
(iii) AXB =
540°
5
= 108°
Since ABC is an interior angle of a pentagon,
\ ABC = 108° .
(v) Since size of each interior angle of a pentagon = 108°,
\ BCD = 108°
Let BAC = BCA = x° (base /s of isos . nABC)
x° + x° + 108° = 180° (/ sum of nABC)
2x° = 180° – 108°
2x° = 72°
=
60°
2
= 30°
\ ADC = 30°
CDE = 150° (int . / of a 12-sided polygon)
Hence,
ADE = CDE – ADC
= 150° – 30°
= 120°
y° =
72°
2
= 36°
\ x = 36
x° =
175
1
15. (i) Since sum of exterior angles of a n-sided polygon is 360°,
17. Sum of interior angles of a pentagon = 540°
Let the exterior angle of the pentagon be x° .
5(180° – x°) = 540°
900° – 5x° = 540°
–5x° = 540° – 900°
–5x° = –360°
360°
\n=
= 10
36°
(ii) Size of an interior angle of the n-sided polygon
= 180° – 36° (adj . /s on a str . line)
= 144°
Let CBD = CDB = x° (base /s of isos . nBCD)
x° + x° + 144° = 180° (/ sum of nBCD)
2x° = 180° – 144°
2x° = 36°
360°
5
= 72°
/a + 72° + 72° = 180°
/a = 180° – 72° – 72°
= 36°
Hence, /a + /b + /c + /d + /e = 5 × 36° = 180°
18. a1 + x1 = 180° (adj . /s on a str . line)
a2 + x2 = 180° (adj . /s on a str . line)
a3 + x3 = 180° (adj . /s on a str . line)
a4 + x4 = 180° (adj . /s on a str . line)
an + xn = 180° (adj . /s on a str . line)
Hence,
a1 + x1 + a2 + x2 + a3 + x3 + a4 + a4 + ··· + an + xn = n × 180°
x° =
36°
2
= 18°
\ CDB = 18°
CDE = 144° (int . / of a 10-sided polygon)
Hence,
BDE = CDE – CDB
= 144° – 18°
= 126°
(iii) Let XCD = XDC = 18° (base /s of isos . nCDX, CX = DX)
18° + 18° + CXD = 180° (/ sum of nCDX)
CXD = 180° – 18° – 18°
= 144°
16. ACE = /a + /b (ext . / of nABC)
JCE + /a + /b = 180° (adj . /s on a str . line)
JCE = 180° – /a – /b
DEC = /c + /d (ext . / of nDEF)
GEC + /c + /d = 180° (adj . /s on a str . line)
GEC = 180° – /c – /d
HGJ = /e + /f (ext . / of nGHI)
EGJ + /e + /f = 180° (adj . /s on a str . line)
EGJ = 180° – /e – /f
CJG = /g + /h (ext . / of nJKL)
CGJ + /g + /h = 180° (adj . /s on a str . line)
CJG = 180° – /g – /h
Sum of interior angles of quadrilateral = (4 – 2) × 180° = 360°
\ JCE + GEC + EGJ + CJG = 360°
(180° – /a – /b) + (180° – /c – /d) + (180° – /e – /f ) +
(180° – /g – /h) = 360°
–/a – /b – /c – /d – /e – /f – /g – /h
= 360° – 180° – 180° – 180° – 180°
–/a – /b – /c – /d – /e – /f – /g – /h = –360°
Hence,
/a + /b + /c + /d + /e + /f + /g + /h = 360°
x° =
1
a1 + a2 + a3 + a4 + ··· + an + x1 + x2 + x3 + x4 + ··· + xn = n × 180°
(n – 2) × 180° + x1 + x2 + x3 + x4 + ··· + xn = n × 180°
x1 + x2 + x3 + x4 + ··· + xn = n × 180° – (n – 2) + 360°
x1 + x2 + x3 + x4 + ··· + xn = 180°n – 180°n + 360°
\ x1 + x2 + x3 + x4 + ··· + xn = 360°
19. (i) Two regular polygons are equilateral triangles and squares .
(ii) The interior angles of the polygons meeting at a vertex must
add to 360° .
(iii)
Shape
Interior Angle in degrees
Triangle
60
Square
90
Pentagon
108
Hexagon
120
More than six sides
More than 120 degrees
Since the interior angles of the polygon meeting at a vertex must
add to 360°, hence the interior angle must be an exact divisor of
360° . This will work only for triangles, squares and hexagons
as the interior angle are all divisor of 360° .
(iv) The reason is that the hexagon has the smallest perimeter for
a given area as compared to the square and the triangle . This
will allow the bees to make more honey using less wax and less
work .
176
Since DA = DC, \ DCA = DAC = 39° .
39° + ABC + 39° = 180° (/ sum of nACD)
ADC = 180° – 39° – 39°
= 102°
39° + d° = 102°
d° = 102° – 39°
= 63°
\ d = 63
(c) e° + 62° + 52° = 180° (/ sum of nBCD)
e° = 180° – 62° – 52°
= 66°
\ e = 66
48° + f ° + 66° = 180° (adj . /s on a str . line)
f ° = 180° – 48° – 66°
= 66°
\ f = 66
(d) 110° + DBC = 180° (adj . /s on a str . line)
DBC = 180° – 110°
= 70°
Since DB = DC, \ DCB = DBC = 70° .
Hence, g° = 70°
\ g = 70
70° + h° = 110° (ext . / of nBCD)
h° = 110° – 70°
= 40°
\ h = 40
(e) Since DB = DC, \ DBC = DCB = 3i° .
(5i + 4)° + 3i° = 180° (adj . /s on a str . line)
8i° = 180° – 4°
= 176°
Review Exercise 11
1. (a) Since AB = AC, \ ACB = ABC = 3a° .
3a° + 2a° + 3a° = 180° (/ sum of nABC)
8a° = 180°
180°
8
= 22 .5°
\ a = 22 .5
(b) Since DA = DB, \ DBA = DAB = 32° .
32° + ADB + 32° = 180° (/ sum of nABD)
ADB = 180° – 32° – 32°
= 116°
116° + b° = 360° (/s at a point)
b° = 360° – 116°
= 244°
\ b = 244
Since CA = CB, \ CAB = CBA = x° .
x° + 64° + x° = 180° (/ sum of nABC)
2x° = 180° – 64°
= 116°
a° =
116°
2
= 58°
c° + 32° = 58°
c° = 58° – 32°
= 26°
\ c = 26
2. (a) Since BA = BD, \ BDA = BAD = a° .
a° + 40° + a° = 180° (/ sum of nABD)
2a° = 180° – 40°
= 140°
x° =
176°
8
= 22°
\ i = 22
3(22°) + 2j° = [5(22) + 4]° (ext . / of nBCD)
2j° = 114° – 66°
= 48°
i° =
140°
2
= 70°
\ a = 70
CBD + 40° = 180° (adj . / s on a str . line)
CBD = 180° – 40°
= 140°
Since BC = BD, \ BCD = BDC = b° .
b° + 140° + b° = 180° (/ sum of nBCD)
2b° = 180° – 140°
= 40°
a° =
48°
2
= 24°
\ j = 24
(f) k° + 78° = 3k° (ext . / of nABD)
3k° – k° = 78°
2k° = 78°
j° =
40°
2
= 20°
\ b = 20
(b) Since BA = BD, \ BDA = BAD = c° .
c° + c° = 78° (ext . / of nABD)
2c° = 78°
b° =
78°
2
= 39°
\ k = 39
39° + l° + 78° = 180° (/ sum of nABD)
l° = 180° – 39° – 78°
= 63°
\ l = 63
k° =
78°
2
= 39°
\ c = 39
c° =
177
1
4. (a) 112° + ABC = 180° (adj . /s on a str . line)
ABC = 180° – 112°
= 68°
62° + HED = 180° (adj . /s on a str . line)
HED =180° – 62°
= 118°
a° + BCD = 180° (adj . /s on a str . line)
BCD = 180° – a°
Sum of the interior angles of a pentagon = (5 – 2) × 180° = 540° .
\ 114° + 68° + 180° – a° + 95° + 118° = 540°
–a° = 540° – 114° – 68° – 180° – 95° – 118°
= –35°
\ a = 35
(b) Sum of exterior angles of a hexagon = 360°
\ 2b° + 4b° + 3b° + b° + b° + b° = 360°
12b° = 360°
3. (a) Since AB = AC, ACB = ABC = a° .
a° + BAC + a° = 180° (/ sum of nABC)
BAC = 180° – 2a°
DCA = 180° – 2a° (alt . /s, AB // DC)
Since AC = AD = CD, DCA = CDA = CAD = 60°
180° – 2a° = 60°
–2a° = 60° – 180°
= –120°
–120°
–2
= 60°
\ a = 60
(b) b° + b° + 76° = 180° (int . /s, AB // DC)
2b° = 180° – 76°
= 104°
a° =
104°
2
= 52°
\ b = 52
c° + c° + 118° = 180° (int . /s, AB // DC)
2c° = 180° – 118°
= 62°
b° =
62°
2
= 31°
\ c = 31
52° + 31° + d° = 180° (/ sum of nABE)
d° = 180° – 52°– 31°
= 97°
\ d = 97
(c) Since EA = EB, EAB = EBA = 58° .
58° + e° = 180° (int . /s, AB // DC)
e° = 180° – 58°
= 122°
\ e = 122
f ° = 58° (corr . /s, AB // DC)
\ f = 58
Since ED = EC, EDC = ECD = 58° .
58° + g° + 58° = 180° (/ sum of nCDE) .
g° = 180° – 58° – 58°
= 64°
\ g = 64
c° =
1
360°
12
= 30°
\ b = 30
c° + 3(30°) = 180° (adj . /s on a str . line)
c° = 180° – 90°
= 90°
\ c = 90
ACD = 40° (alt . /s, AB // DC)
CAD + 108° + 40° = 180° (int . /s, AD// BC)
CAD = 180° – 108° – 40°
= 32°
Since AB = AD, \ ADB = ABD = 62° .
62° + BAD + 62° = 180° (/ sum of nABD)
BAD = 180° – 62° – 62°
= 56°
Since CB = CD, \ BDC = DBC = x° .
x° + 118° + x° = 180° (/ sum of nBCD)
2x° = 180° – 118°
= 62°
b° =
5. (i)
(ii)
6. (i)
(ii)
62°
2
= 31°
\ BDC = 31°
x° =
178
7. Since nABE is an equilateral triangle, AB = AE = BE and
EAB = EBA = AEB = 60° .
DAE + 60° = 90° (right angle of a square)
DAE = 90° – 60°
= 30°
Since AD = AB, \ AE = AD and AED = ADE = x° .
x° + 30° + x° = 180° (/ sum of nADE)
2x° = 180° – 30°
= 150°
10. Sum of interior angles of a pentagon
= (5 – 2) × 180°
= 3 × 180°
= 540°
Let the 5 interior angles be 3x°, 4x°, 5x°, 5x° and 7x° .
3x° + 4x° + 5x° + 5x° + 7x° = 540°
24x° = 540°
540°
24
= 22 .5°
(i) Largest interior angle = 7 × 22 .5°
= 157 .5°
(ii) Largest exterior angle = 180° – 3 × 22 .5°
= 112 .5°
11. Sum of exterior angles of a n-sided polygon = 360°
35° + 72° + (n – 2) × 23° = 360°
23°n = 360° – 35° – 72° + 46°
= 299°
x° =
150°
2
= 75°
CBE + 60° = 90° (right angle of a square)
CBE = 90° – 60°
= 30°
Since BC = AB, \ BE = BC and BEC = BCE = y° .
y° + 30° + y° = 180° (/ sum of nBEC)
2y° = 180° – 30°
= 150°
x° =
299°
23°
= 13
12. Let the interior angle be 13x° and the exterior angle be 2x° .
13x° + 2x° = 180° (adj . /s on a str . line)
15x° = 180°
n=
150°
2
= 75°
75° + 60° + 75° + CED = 360° (/s at a point)
CED = 360° – 75° – 60° – 75°
= 150°
8. Sum of interior angles of a (2n – 3)-sided polygon
= [(2n – 3) – 2] × 180°
Hence,
[(2n – 3) – 2] × 180° = 62 × 90°
(2n – 5) × 180° = 5580°
360°n – 900° = 5580°
360°n = 5580° + 900°
360°n = 6480°
y° =
180°
15
= 12°
Sum of exterior angles of a n-sided polygon = 360°
Hence,
360°
n=
2(12° )
x° =
= 15
13. Sum of the interior angles of a n-sided polygon = (n – 2) × 180°
Sum of the exterior angles of a n-sided polygon = 360°
(n – 2) × 180° = 4 × 360°
180°n = 1440° + 360°
= 1800°
6480°
360°
= 18
9. Sum of interior angles of a n-sided polygon
= (n – 2) × 180°
126° + (n – 1) × 162° = (n – 2) × 180°
126° + 162°n – 162° = 180°n – 360°
180°n – 162°n = 360° + 126° –162°
18°n = 324°
n =
1800°
180°
= 10
n=
324°
18°
= 18
n =
179
1
180° – 20°
(base /s of isos . nABC)
2
160°
=
2
= 80°
\ DCF = DCA
= ACB – 60°
= 80° – 60°
= 20°
BFC + FCB + 50° = 180° (/ sum of nBCF)
BFC + ACB + 50° = 180°
BFC + 80° + 50° = 180°
BFC = 180° – 80° – 50°
= 50°
Since CBF = BFC = 50°, i .e . CB = CF,
then nBCF is an isosceles triangle .
Challenge Yourself
1.
2. ACB =
180° – 20°
(base /s of isos . nABC)
2
160°
=
2
= 80°
ABC =
A
20°
D
G
F
A
B
C
E
20°
Draw E on BC such that AE ^ BC .
Draw F on AE such that nBCF is an equilateral triangle .
Then ABF = 80° – 60° = 20° and BF = BC = AD .
Consider the quadrilateral ABFD .
A
F
E
20°
B
D
50°
60°
C
Draw G on AG such that DG // BC .
Draw BG to cut CD at E .
Draw EF .
By symmetry, BE = CE, so nBCE is an isosceles triangle .
Since the base angle of nBCE is 60°,
then nBCE is an equilateral triangle,
i .e . BEC = 60° and EBF = 60° – 50° = 10° .
\ CE = CB (sides of equilateral nBCE)
= CF (sides of isosceles nBCF)
Since CE = CF,
then nCEF is an isosceles triangle .
ˆ
180° – ECF
CFE =
(base /s of isos . nCEF)
2
ˆ
180° – DCF
=
2
180° – 20°
=
2
160°
=
2
= 80°
\ BFE = CFE – BFC
= 80° – 50°
= 30°
G
F
20°
B
Since ABF = BAD = BAC = 20° and BF = AD,
then by symmetry, AB // DF and ABFD is an isosceles trapezium .
In the isosceles trapezium ABFD, by symmetry, AG = BG, so nABG
is an isosceles triangle .
20°
= 10° (AE bisects BAC),
2
then ABG = BAG = 10° (base /s of isos . nABG) .
\ ADB + ABD + BAD = 180° (/ sum of nABD)
ADB + ABG + 20° = 180°
ADB + 10° + 20° = 180°
ADB = 180° – 10° – 20°
= 150°
Teachers may wish to note the usefulness of the symmetric properties
of an isosceles trapezium. Otherwise, formal proofs using congruent
triangles are beyond the scope of Secondary 1 syllabus.
Since BAG = BAE =
1
G
D
180
4. (i) An exterior angle of a concave polygon has a negative measure
and is inside the polygon as shown in the diagram below .
E .g .
FEG = EBF + BFE (ext . / of nBEF)
= 10° + 30°
= 40°
DEG = BEC (vert . opp . /s)
= 60°
DGE = CBE (alt . /s, DG // BC)
= 60°
Since the base angle of nDEG is 60°,
then nDEG is an equilateral triangle,
i .e . EDG = 60° and DE = DG .
AGD = ACB (corr . /s, DG // BC)
= 80°
\ FGE + DGE + AGD = 180° (adj . /s on a str . line)
FGE + 60° + 80° = 180°
FGE = 180° – 60° – 80°
= 40°
Since FEG = FGE = 40°,
then nEFG is an isosceles triangle,
i .e . FE = FG .
Consider the quadrilateral DEFG .
e4°
i5°
i4°
e5°
i1°
e3°
e1°
i2°
e2°
(ii) Yes . Exterior angle of the vertex which is “pushed in” will
flip over into the inside of the polygon and becomes negative.
Adding all the exterior angles as before, they will still add to
360° .
E .g . i1° + (–e1°) + i2° + e2° + i3° + e3° + i4° + e4° + i5° + e5° +
= 5 × 180°
[i1° + i2° + i3° + i4° + i5°] + (–e1°) + e2° + e3° + e4° + e5°
= 900°
(–e1°) + e2° + e3° + e4° + e5°
= 900 – [i1° + i2° + i3° + i4° + i5°]
(–e1°) + e2° + e3° + e4° + e5° = 900° – (5 – 2) × 180°
(–e1°) + e2° + e3° + e4° + e5° = 900° – 540°
(–e1°) + e2° + e3° + e4° + e5° = 360°
The above proof holds for any n-sided polygon .
G
D
F
E
5. In a n-sided polygon, each diagonal connects one vertex to another
vertex which is not its next-door neighbour . Since there are n
vertices in an n-sided polygon, therefore there are n starting points
for the diagonals . For each diagonal, it (e .g . V1) can join to other
(n – 3) vertices since it cannot join itself (V1) or either of the two
neighbouring vertices (V2 and Vn) . So the total number of diagonals
formed is n × (n – 3) . However, in this way, each diagonal would
be formed twice (to and from each vertex), so the product n(n – 3)
must be divided by 2 . Hence the formula is n ( n – 3) .
2
E .g .
Since DE = DG and FE = FG,
then DEFG is a kite .
In the kite DEFG, the longer diagonal DF bisects EDG .
\ CDF = EDF
60°
2
= 30°
3. Yes . For any n-sided concave polygon, it can still form (n – 2)
triangles in the polygon .
Hence the sum of the interior angles is still the same .
E .g .
=
V1
Vn
V2
Vn – 1
V3
V4
181
1
Chapter 12 Geometrical Constructions
TEACHING NOTES
Suggested Approach
Students have learnt how to draw triangles and quadrilaterals using rulers, protractors and set squares in primary school. Teachers
need to reintroduce these construction tools and demonstrate the use of these if students are still unfamiliar with them. When
students are comfortable with the use of these construction tools and the compasses, teachers can proceed to the sections on
construction of triangles and quadrilaterals.
Section 12.1: Introduction to Geometrical Constructions
Teachers may wish to recap with students how rulers, protractors and set squares are used. More emphasis should
be placed on the use of protractors, such as the type of scale (inner or outer) to use, depending on the type of
angle (acute or obtuse). Teachers need to impress upon students to avoid parallax errors when reading the length
using a ruler, or an angle using a protractor.
Teachers should show and lead students on the use of compasses. Students are to know and be familiar with the
useful tips in using the construction tools.
Section 12.2: Perpendicular Bisectors and Angle Bisectors
Teachers should state and define perpendicular bisectors and angle bisectors. Stating what perpendicular and
bisect means individually will help students to remember their meanings.
For the worked examples in this section, teachers are encouraged to go through the construction steps one by
one with the students. Students should follow and construct the same figures as shown in the worked examples.
Teachers should allow students to use suitable geometry software to explore and discover the properties
of perpendicular bisectors and angle bisectors (see Investigation: Property of a Perpendicular Bisector and
Investigation: Property of an Angle Bisector), that is, their equidistance from end-points and sides of angles
respectively.
Section 12.3: Construction of Triangles
Students should be able to construct the following types of triangles at the end of this section:
• Given 2 sides and an included angle
• Given 3 sides
• Given 1 side and 2 angles
As a rule of thumb, students should draw the longest line as a horizontal line. Teachers are to remind their
students to mark all angles, vertices, lengths and other markings (same angles, same sides, right angles etc.)
clearly. Students should not erase any arcs that they draw in the midst of construction and check their figure at
the end.
Section 12.4: Construction of Quadrilaterals
Students should be able to construct parallelograms, rhombuses, trapeziums and other quadrilaterals at the end
of this section.
As a rule of thumb, students should draw the longest line as a horizontal line. Teachers are to remind their
students to mark all angles, vertices, lengths and other markings (same angles, same sides, right angles etc.)
clearly. Students should not erase any arcs they draw in the midst of construction and check their figure at the
end.
1
182
WORKED SOLUTIONS
Investigation (Property of a Perpendicular Bisector)
4. The length of AC is equal to the length of BC .
5. Any point on the perpendicular bisector of AB is equidistant from A and B .
6. Any point which is not on the perpendicular bisector of AB is not equidistant from A and B .
Investigation (Property of an Angle Bisector)
5. The length of PR is equal to the length of QR .
6. Any point on the angle bisector of BAC is equidistant from AB and AC .
7. Any point which is not on the angle bisector of BAC is not equidistant from AB and AC .
Practise Now 1
A
B
8 cm
Practise Now 2
C
78°
A
B
Practise Now 3
C
4 .8 cm
S
130°
A
7 .6 cm
B
(i) Length of AC = 11 .3 cm
(ii) Length of BS = 4 .0 cm
183
1
Practise Now 4
R
T
9 .8 cm
7 .2 cm
P
Q
8 .4 cm
(i) Required angle, QPR = 77°
(ii) Length of QT = 5 .3 cm
Practise Now 5
Z
U
X
56°
48°
8 cm
Y
(iii) The point U is equidistant from the points Y and Z, and equidistant from the lines XY and XZ .
1
184
Practise Now 6
D
1.
C
5 .5 cm
120°
A
8 .5 cm
B
Length of AC = 12 .2 cm
2. D
C
6 .5 cm
A
B
10 .5 cm
Length of AC = 12 .3 cm
185
1
Practicse Now 7
S
1.
6 cm
9 cm
9 cm
P
R
4 .5 cm
6 cm
Q
QRS = 71°
2.
S
P
R
12 cm
7 .5 cm
Q
QRS = 74°
1
186
Practise Now 8
R
9 .2 cm
95°
S
6 .2 cm
80°
P
Q
5 .6 cm
(i) Length of PS = 7 .0 cm
(ii) PSR = 54°
Exercise 12A
1.
A
B
9 .5 cm
2.
C
A
56°
B
187
1
C
3.
P
5.
6 .5 cm
10 cm
10 cm
80°
A
B
8 cm
Length of AC = 9 .4 cm
4.
C
Q
9 cm
QPR = 53°
6.
9 cm
A
5 cm
B
Length of AC = 7 .5 cm
9 .5 cm
1
188
R
7.
Z
60°
45°
X
Y
10 .2 cm
Length of XZ = 9 .1 cm
C
8.
S
A
6 .5 cm
88°
9 .8 cm
B
(i) Length of AC = 11 .6 cm
(ii) Length of BS = 5 .9 cm
189
1
C
9.
8 .8 cm
S
60°
A
B
9 .4 cm
(i) Required angle, BAC = 52°
(ii) Length of CS = 3 .9 cm
10.
P
T
9 .5 cm
Q
8 .5 cm
R
9 .8 cm
(i) Required angle, PQR = 52°
(ii) Length of QT = 8 .0 cm
1
190
11.
P
9 .2 cm
T
8 .8 cm
Q
R
10 .4 cm
(i) Required angle, QPR = 71°
(ii) PT = 4 .2 cm
12.
Z
13.
Z
U
55°
64°
X
U
8 cm
Y
74°
X
(i) Length of XZ = 7 .5 cm
(ii) Length of UY = 7 .2 cm
49°
8 cm
Y
(iii) The point U is equidistant from the points X and Y, and
equidistant from the lines XY and YZ .
191
1
14.
5 cm
C
11 cm
S
62°
T
A
B
10 .2 cm
(i) Length of BC = 10 .9 cm
(iii) Length of ST = 4 .7 cm
C
15.
4 .6 cm
54°
A
S
1
8 .5 cm
B
192
R
16.
7 .9 cm
9 .2 cm
T
P
8 .3 cm
17.
Q
P
O
Q
(ii) Diameter
193
1
Exercise 12B
A
1.
D
10 cm
80°
B
C
12 cm
Length of diagonal BD = 16 .9 cm
2. 96 mm = 9 .6 cm
84 mm = 8 .4 cm
84 mm
96 mm
Length of each of the two diagonals = 12 .8 cm
1
194
3.
D
C
115°
A
B
6 cm
Length of each of the two diagonals = 10 .1 cm, 6 .5 cm
4.
9 cm
S
P
9 cm
Q
7 .5 cm
R
12 cm
QPS = 133°
5. 60 mm = 6 cm
9 mm = 0 .9 cm
R
S
9 mm
P
60 mm
Q
QPS = 171°
195
1
6.
Q
75°
5 .3 cm
6 .3 cm
P
60°
R
S
6 .7 cm
(i) Length of PR = 7 .1 cm
(ii) RPS = 70°
7.
6 cm
W
Z
4 .5 cm
60°
X
Y
8 cm
Length of YZ = 3 .9 cm
Length of WY = 6 .9 cm
8. 56 mm = 5 .6 cm
112 mm = 11 .2 cm
Z
W
56 mm
80°
X
70°
Y
112 mm
Length of WY = 11 .6 cm
Length of XZ = 10 .7 cm
1
196
9.
T
D
C
6 cm
(i) Length of diagonal BD = 8 .4 cm
(ii) Length of AT = 7 .1 cm
115°
A
B
9 cm
R
10.
4 .8 cm
S
U
4 .8 cm
P
4 cm
(i) Length of QS = 5 .4 cm
(ii) Length of SU = 4 .5 cm
Q
11.
S
P
R
10 cm
(iii) Length of PQ = 7 .0 cm
Q
197
1
R
12.
S
6 cm
3 .5 cm
P
60°
45°
10 cm
T
Q
(i) QRS = 119°
(ii) Length of PT = 5 .4 cm
13.
U
S
R
7 cm
3 .1 cm
50°
P
Q
11 cm
(i) QRS = 109°
(ii) Length of RU = 4 .1 cm
1
198
14.
5 cm
X
S
W
120°
U
6 cm
Y
9 .8 cm
Z
T
(i) Length of WY = 8 .6 cm
(ii) Length of ST = 6 .5 cm
(iii) WUX = 105°
15. S
T
R
5 .8 cm
4 .6 cm
120°
105°
P
6 .5 cm
Q
Review Exercise 12
C
1.
6 cm
S
60°
A
4 .5 cm
B
(i) Length of AC = 5 .4 cm
(ii) Length of CS = 3 .3 cm
199
1
2.
R
8 .8 cm
10 .2 cm
P
T
Q
12 cm
3.
X
60°
Y
60°
U
Z
8 cm
(iii) Rhombus
V
1
200
(i) Required angle, QPR = 46°
(ii) RT = 7 .9 cm
4.
T
D
C
5 .5 cm
120°
A
S
8 cm
B
(i) Length of BD = 7 .1 cm
(ii) Length of ST = 6 .5 cm
5.
U
S
6 cm
R
2 cm
60°
P
8 cm
Q
(i) Required angle, QRS = 123°
(ii) Length of QU = 6 .5 cm
201
1
6.
B
S
A
C
10 cm
45°
D
(iii) AB = BC = AD = CD = 7 .1 cm
ABCD is a square .
(iv) Length of DS = 9 .3 cm
Challenge Yourself
1.
2.
R
5 cm
P
6 cm
6 cm
7 cm
P
8 cm
Q
Q
Incircle
Circumcircle
1
202
8 cm
R
(ii) ACD = 56° (alt. /s, AB // DC)
ACB = 70° – ACD
= 70° – 56°
= 14°
7. (i) (2x + 17)° + (3x – 25)° + (2x + 49)° + (x + 40)° + (4x – 17)°
+ (3x – 4)° = (6 – 2) × 180°
2x° + 17° + 3x° – 25° + 2x° + 49° + x° + 40° + 4x° – 17° + 3x° – 4°
= 4 × 180°
2x° + 3x° + 2x° + x° + 4x° + 3x°
= 720° – 17° + 25° – 49° – 40° + 17° + 4°
15x° = 660°
x° = 44°
\ x = 44
(ii) Smallest interior angle = (x + 40)°
= (44 + 40)°
= 84°
(iii) Smallest exterior angle
= 180° – largest interior angle (adj. /s on a str. line)
= 180° – (4x – 17)°
= 180° – [4(44) – 17]°
= 180° – (176 – 17)°
= 180° – 159°
= 21°
C
8. (i)
Revision Exercise C1
1. 35% of students = 140
140
1% of students =
35
140
100% of students =
× 100
35
= 400
\ The total number of students who take part in the competition is
400 .
2. Value first obtained = (100 – 15)% of 5600
= 85% × 5600
85
=
× 5600
100
= 4760
1100
110% of 4760 =
× 4760
100
= 5236
\ The final number is 5236.
3.
Height of hall
6
=
7
28
6
Height of hall =
× 28
7
= 24 m
Ratio of breadth of hall to height of hall = 21 : 24
= 7:8
4. 1035 hours
53 minutes
1128 hours
53
× 575
25
= 1219
Number of words in the report =
6 hours 25 minutes
X
6.5 cm
5. (i) 0845 hours
1510 hours
The train takes 6 hours 25 minutes to travel from Town A to
Town B .
7 cm
(ii) Distance between Town A and Town B = 108 × 6
25
60
5
= 108 × 6
12
= 693 km
6.
D
A
6 cm
B
(ii) Length of BX = 5.6 cm
C
70°
A
56°
B
(i) ABC + 70° = 180° (int. /s, AB // DC)
ABC = 180° – 70°
= 110°
203
1
195
20
3
60
195
=
1
3
3
1
= 58 km/h
2
Average speed of lorry on the return journey =
Revision Exercise C2
3780 – 3500
× 100%
3500
280
=
× 100%
3500
= 8%
1. Percentage increase in salary =
6. 26° + x° = 180° (adj. /s on a str. line)
x° = 180° – 26°
= 154°
\ x = 154
BAD = 62° (alt. /s, AB // CD)
BAQ = BAD – 26°
= 62° – 26°
= 36°
RSA = SAQ (alt. /s, PQ // RS)
= 36°
RSA + y° + y° = 180° (adj. /s on a str. line)
36° + y° + y° = 180°
y° + y° = 180° – 36°
2y° = 144°
y° = 72°
\ y = 72
7. 95° + (n – 1) × 169° = (n – 2) × 180°
95 + 169n – 169 = 180n – 360
169n – 180n = –360 – 95 + 169
–11n = –286
\ n = 26
2. Percentage of students who did not have to stay back = 100% – 25%
= 75%
75
75% of 40 =
× 40
100
= 30
\ 30 students did not have to stay back for detention.
3. X : Y = 8 : 15
= 56 : 105
Y : Z = 21 : 32
= 105 : 160
\ X : Z = 56 : 160
= 7 : 20
10
× 31
4
1
= 77
2
1230 hours
4. Number of times light can circle the world =
5. 0845 hours
3 hours 45 minutes
45
60
3
= 52 × 3
4
= 195 km
Distance between Town A and Town B = 52 × 3
1455 hours
3 hours 20 minutes
1815 hours
8. Length of BC = 7 .7 cm
Length of CD = 11 .7 cm
C
D
82°
5 .3 cm
105°
110°
A
1
7.6 cm
204
B
Chapter 13 Perimeter and Area of Plane Figures
TEACHING NOTES
Suggested Approach
In the previous chapter, students have learnt the construction of plane figures such as triangles and quadrilaterals. Here, they will
learn how to convert units of area, as well as find the perimeter and area of triangles and quadrilaterals. Students will revise what
they have learnt in primary school as well as learn the perimeter and area of parallelograms and trapeziums. Teachers should
place more focus on the second half of the chapter and ensure students are able to solve problems involving the perimeter and
area of parallelograms and trapeziums.
Section 13.1: Conversion of Units
Teachers may wish to recap with the students the conversion of unit lengths from one unit of measurement to
another (i.e. mm, cm, m and km) before moving onto the conversion of units for areas.
Teachers may ask students to remember simple calculations such as 1 cm2 = 1 cm × 1 cm to help them in their
calculations when they solve problems involving the conversion of units.
Section 13.2: Perimeter and Area of Basic Plane Figures
This section is a recap of what students have learnt in primary school. Students are reminded to be clear of the
difference in the units used for perimeter and area (e.g. cm and cm2).
Teachers can impress upon the students that the value of p in calculators is used when its value is not stated in
the question. Unless specified, all answers that are not exact should be rounded off to 3 significant figures.
Section 13.3: Perimeter and Area of Parallelograms
Teachers should illustrate the dimensions of a parallelogram to the students so that they are able to identify the
base and height of parallelograms. It is important to emphasise to the students that the height of a parallelogram
is with reference to the base and it must be perpendicular to the base chosen. Also, the height may lie within, or
outside of the parallelogram. Teachers can highlight to the students that identifying the height of a parallelogram
is similar to identifying the height of a triangle.
Teachers should guide students in finding the formula for the area of a parallelogram (see Investigation: Formula
for Area of a Parallelogram). Both possible methods should be shown to students (The second method involves
drawing the diagonal of the parallelogram and finding the area of the two triangles).
Section 13.4: Perimeter and Area of Trapeziums
Teachers should recap with students the properties of a trapezium. Unlike the parallelogram, the base of the
trapezium is not required and the height must be with reference to the two parallel sides of the trapezium. Thus,
the height lies either inside the trapezium, or it is one of its sides (this occurs in a right trapezium, where two
adjacent angles are right angles).
Teachers should guide students in finding the formula for the area of a trapezium (see Investigation: Formula for
Area of a Trapezium). Both possible methods should be shown to students (Again, the second method involves
drawing the diagonal of the trapezium and finding the area of the two triangles).
Teachers can enhance the students’ understanding and appreciation of the areas of parallelograms and trapeziums
by showing them the link between the area of a trapezium, a parallelogram and a triangle (see Thinking Time on
page 329).
205
1
Area of parallelogram ABCD = area of nABC + area of nADC
WORKED SOLUTIONS
1
1
× AB × CF +
× DC × DE
2
2
1
1
= bh + bh
2
2
= bh
=
Class Discussion (International System of Units)
1. The seven basic physical quantities and their base units are shown
in the following table:
Basic Physical Quantity
Base Unit
Length
metre (m)
Mass
kilogram (kg)
Time
second (s)
Electric current
ampere (A)
Thermodynamic Temperature
kelvin (K)
Amount of substance
mole (mol)
Luminous intensity
candela (cd)
Thinking Time (Page 325)
From the geometry software template ‘Area of Parallelogram’, we can
conclude that the formula for the area of parallelogram is also applicable
to oblique parallelograms .
Investigation (Formula for Area of a Trapezium)
1. The new quadrilateral AFGD is a parallelogram .
2. Length of AF = length of AB + length of EF
=b+a
=a+b
Scientists developed the International System of Units (SI units)
so that there is a common system of measures which can be used
worldwide .
2. Measurements of Lengths:
1 foot (ft) = 0 .3048 m
1 inch (in) = 0 .0254 m
1 yard (yd) = 0 .9144 m
1 mile
= 1609 .344 m
Measurement of Areas:
1 acre = 4046 .8564 m2
1
× area of parallelogram AFGD
2
1
=
× AF × h
2
1
= (a + b)h
2
3. Area of trapezium ABCD =
4. Method 1:
Divide the trapezium ABCD into two triangles ABD and DCB by
drawing the diagonal BD as shown below:
a
C
D
F
Investigation (Formula for Area of a Parallelogram)
1. The new quadrilateral CDEF is a rectangle .
2. Length of CF = length of DE = h
Length of EF = length of EB + length of BF
= length of EB + length of AE
=b
3. Area of parallelogram ABCD = area of rectangle CDEF
= EF × CF
= bh
4. Divide the parallelogram ABCD into two triangles ABC and ADC
by drawing the diagonal AC as shown below:
D
h
A
E
B
b
Length of FB = length of DE = h
Area of trapezium ABCD = area of nABD + area of nDCB
C
1
2
1
=
2
1
=
2
1
=
2
=
h
A
E
b
B
F
Length of CF = length of DE = h
1
206
× AB × DE +
×b×h+
(b + a)h
(a + b)h
1
× DC × FB
2
1
×a×h
2
Method 2:
Divide the trapezium ABCD into a parallelogram AFCD and a
triangle FBC by drawing a line FC // AD as shown below:
a
Practise Now 1
(a) 16 m2 = 16 × 10 000 cm2
= 160 000 cm2
(b) 357 cm2 = 357 × 0 .0001 m2
= 0 .0357 m2
C
D
Practise Now (Page 318)
(a)
h
h
b
A
E
F
b
G
B
(b)
Length of CG = length of DE = h
Length of AF = length of DC = a
\ Length of FB = length of AB – length of AF
=b–a
Area of trapezium ABCD
= area of parallelogram AFCD + area of nFBC
= AF × DE +
=a×h+
h
b
1
× FB × CG
2
1
× (b – a) × h
2
(c)
1
(2a + b – a)h
2
1
= (a + b)h
2
=
h
b
Teachers may wish to get higher-ability students to come up with
more methods to find a formula for the area of a trapezium.
(d)
b
Thinking Time (Page 329)
1. (i) The new figure is a parallelogram.
h
1
(ii) Area of trapezium = (a + b)h
2
When a = b,
1
1
(a + b)h = (b + b)h
2
2
1
= (2b)h
2
= bh
= area of parallelogram
2. (i) The new figure is a triangle.
(ii) Area of trapezium =
(e)
b
h
1
(a + b)h
2
When a = 0,
1
1
(a + b)h = (0 + b)h
2
2
1
= bh
2
= area of triangle
207
1
(f)
Practise Now (Page 324)
b
(a)
h
b
h
(b)
Practise Now 2
b
h
1. Length of each side of square field =
64
4
= 16 m
Area of field = 162
= 256 m2
Area of path = (16 + 3 .5 + 3 .5)2 – 256
= 232 – 256
= 529 – 256
= 273 m2
2. Area of shaded region
= area of rectangle ABCD – area of nARQ – area of nBRS
– area of nCPS – area of nDPQ
(c)
h
b
(d)
b
h
1
1
= 25 × 17 –
× (25 – 14) × 5 –
× 14 × 3
2
2
1
1
– × (25 – 8) × (17 – 3) –
× (17 – 5) × 8
2
2
1
1
1
= 425 –
× 11 × 5 – 21 –
× 17 × 14 –
× 12 × 8
2
2
2
1
= 425 – 27 – 21 – 119 – 48
2
1 2
= 209 m
2
(e)
h
b
Practise Now 3
(f)
3
× 2p(14) + 2(14)
4
= 21p + 28
= 94 .0 cm (to 3 s .f .)
(i) Perimeter of unshaded region =
h
3
× p(14)2
4
= 147p
= 462 cm2 (to 3 s .f .)
(iii) Area of shaded region = area of square – area of unshaded region
= (2 × 14)2 – 147p
= 282 – 147p
= 784 – 147p
= 322 cm2 (to 3 s .f .)
b
(ii) Area of unshaded region =
1
Practise Now 4
(i) Area of parallelogram = 24 × 7
= 168 m2
(ii) Perimeter of parallelogram = 2(30 + 7)
= 2(37)
= 74 m
208
Practise Now 7
Practise Now 5
Area of parallelogram = PQ × ST = 480 m2
20 × ST = 480
ST = 24
Length of ST = 24 m
1
× (5 + 13 .2) × 4
2
1
=
× 18 .2 × 4
2
= 36 .4 m2
(ii) Perimeter of trapezium = 5 + 6 + 13 .2 + 5 .5
= 29 .7 m
(i) Area of trapezium =
Practise Now 6
1. Total area of shaded regions
= area of parallelogram ABJK + area of parallelogram CDIJ
+ area of parallelogram DEGH
= 4 × 12 + (2 × 4) × 12 + 4 × 12
= 48 + 8 × 12 + 48
= 48 + 96 + 48
= 192 m2
2. Area of nCDF =
Practise Now 8
(i) Area of trapezium =
1
× DC × CF = 60 cm2
2
1
× DC × 3CG = 60
2
3
× DC × CG = 60
2
DC × CG = 40
Area of parallelogram ABCD = DC × CG
= 40 cm2
1
× (PQ + RS) × PS
2
1
× (14 + 10) × PS
2
1
× 24 × PS
2
12 × PS
PS
= 72 m2
= 72
= 72
= 72
=6
Length of PS = 6 m
(ii) Perimeter of trapezium = PQ + QR + RS + PS
14 + QR + 10 + 6
30 + QR
QR
Length of QR = 7 .2 m
= 37 .2 m
= 37 .2
= 37 .2
= 7 .2
Practise Now (Page 328)
Practise Now 9
(a)
Area of figure = area of trapezium + area of semicircle
1
1
1
× (48 + 16) × 20 + p
1424
2
2 2
1
1
=
× 64 × 20 + p × 356
2
2
= 640 + 178p
= 1200 m2 (to 3 s .f .)
2
=
h
(b)
Exercise 13A
h
(c)
1. (a) 40 m2 = 40 × 10 000 cm2
= 400 000 cm2
2
(b) 16 cm = 16 × 0 .0001 m2
= 0 .0016 m2
(c) 0 .03 m2 = 0 .03 × 10 000 cm2
= 300 cm2
(d) 28 000 cm2 = 28 000 × 0 .0001 m2
= 2 .8 m2
h
259
18.5
= 14 cm
(ii) Perimeter of rectangle = 2(18 .5 + 14)
= 2(32 .5)
= 65 cm
2. (i) Breadth of rectangle =
209
1
3. Area of figure = area of square – area of triangle
5. Let the diameter of the semicircle be x cm .
1
= 92 –
× 3 × 2 .5
2
= 81 – 3 .75
= 77 .25 m2
4. (a) Diameter of circle = 2 × 10
= 20 cm
Circumference of circle = 2p(10)
= 20p
= 62 .8 cm (to 3 s .f .)
Area of circle = p(10)2
= 100p
= 314 cm2 (to 3 s .f .)
1
× p × x + x = 144
2
1
22
×
× x + x = 144
2
7
11
x + x = 144
7
18
x = 144
7
x = 56
\ Diameter of semicircle = 56 cm
= 0 .56 m
21
6. (a) (i) Perimeter of figure = 2p + 2(36 – 21)
2
= 2p(10 .5) + 2(15)
= 21p + 30
= 96 .0 cm (to 3 s .f .)
(ii) Area of figure = area of two semicircles + area of rectangle
= p(10 .5)2 + 15 × 21
= 110 .25p + 315
= 661 cm2 (to 3 s .f .)
3.6
2
= 1 .8 m
Circumference of circle = 2p(1 .8)
= 3 .6p
= 11 .3 m (to 3 s .f .)
Area of circle = p(1 .8)2
= 3 .24p
= 10 .2 m2 (to 3 s .f .)
(b) Radius of circle =
(b) (i) Perimeter of figure =
1
× 2p(5) + 2(5) + 200
2
= 5p + 10 + 200
= 39 .9 cm (to 3 s .f .)
(ii) Area of figure = area of semicircle + area of triangle
176
2p
88
=
p
= 28 .0 mm (to 3 s .f .)
(c) Radius of circle =
1
1
× p(5)2 + × 10 × 10
2
2
25
=
p + 50
2
= 89 .3 cm2 (to 3 s .f .)
18
18
1
(c) (i) Perimeter of figure =
× 2p + 2p
4
2
2
=
88
p
Diameter of circe = 2 ×
176
p
= 56 .0 mm (to 3 s .f .)
=
88
Area of circle = p
π
2
1
× 2p(9) + 2p(4 .5)
2
= 9p + 9p
= 18p
= 56 .5 cm (to 3 s .f .)
=
7744
= p 2
π
7744
=
p
= 2460 mm2 (to 3 s .f .)
(ii) Area of figure
= area of big semicircle + area of two small semicircles
616
p
= 14 .0 cm (to 3 s .f .)
(d) Radius of circle =
1
× p(9)2 + p(4 .5)2
2
81
p + 20 .25p
=
2
= 60 .75p
= 191 cm2 (to 3 s .f .)
7. (i) Perimeter of figure = 2p(2) + 2(9 – 2 × 2) + 2(3)
= 4p + 2(5) + 6
= 4p + 10 + 6
= 4p + 16
= 28 .6 m (to 3 s .f .)
=
616
p
= 28 .0 cm (to 3 s .f .)
Diameter of circle = 2 ×
616
Circumference of circle = 2p
π
= 88 .0 cm (to 3 s .f .)
1
210
(ii) Area of figure = area of rectangle – area of four quadrants
= 9 × [2(2) + 3] – p(2)2
=9×7–4p
= 63 – 4p
= 50 .4 m2 (to 3 s .f .)
8. Let the breadth of the rectangular field be x m .
Then the length of the field is (x + 15) m .
2[(x + 15) + x] = 70
2(2x + 15) = 70
2x + 15 = 35
2x = 20
x = 10
\ Breadth of field = 10 m
Length of field = 10 + 15
= 25 m
Area of field = 25 × 10
= 250 m2
Area of path = (25 + 2 .5 + 2 .5) × (10 + 5 + 5) – 250
= 30 × 20 – 250
= 600 – 250
= 350 m2
9. Area of shaded region = area of quadrilateral PQRS
Cost incurred = 28p × $55
= $4838 .05 (to the nearest cent)
7
1
× 2p + 2(5 .7)
2
2
1
=
× 2p(3 .5) + 11 .4
2
= 3 .5p + 11 .4
= 22 .4 cm (to 3 s .f .)
(ii) Area of figure = area of semicircle BCD + area of nABD
13. (i) Perimeter of figure =
1
1
× p(3 .5)2 +
× 7 × (8 – 3 .5)
2
2
1
= 6 .125p +
× 7 × 4 .5
2
= 6 .125p + 15 .75
= 35 .0 cm2 (to 3 s .f .)
=
14. (i) Perimeter of shaded region
10
3
1
1
× 2p(10) + × 2p + × 2p(10 – 3) + 3 + (10 – 3)
2
4
2
4
1
1
= 15p +
× 2p(5) +
× 2p(7) + 3 + 7
2
4
7
= 15p + 5p + p + 10
2
47
=
p + 10
2
= 83 .8 cm (to 3 s .f .)
(ii) Area of shaded region = area of big semicircle
+ area of small semicircle
+ area of region ABCE
=
1
1
× AR × RP +
× RB × RP
2
2
1
=
× RP × (AR + RB)
2
1
=
× AD × AB
2
1
=
× 23 × (7 + 13 .5)
2
1
=
× 20 .5 × 23
2
= 235 .75 m2
10. Area of shaded region = area of nABC – area of nADE
=
1
1
× p(10)2 +
× p(5)2
2
2
1
+
× p(102 – 72)
4
25
1
= 50p +
p+
× p(100 – 49)
2
4
25
1
= 50p +
× p(51)
p+
2
4
25
51
= 50p +
p+
p
2
4
301
=
p
4
= 236 cm2 (to 3 s .f .)
200
1
15. (i) Perimeter of shaded region =
× 2p
+ 2(10)
2
2
=
1
1
× 20 × 21 –
× 10 × 10 .5
2
2
= 210 – 52 .5
= 157 .5 m2
=
1
1
× AC × BD =
× CD × AE
2
2
1
1
× 20 × BD =
× 22 × 16
2
2
10 × BD = 176
BD = 17 .6
Length of BD = 17 .6 cm
11. Area of nACD =
12
12. (i) Area of surface of circular pond = p
2
200
p + 20
2
= 42 .2 m (to 3 s .f .)
=
2
(ii) Area of shaded region
= area of semicircle BCD – area of nBCD
= p(6)
= 36p
= 113 m2 (to 3 s .f .)
2
2
1
1
× p 200 –
× 10 × 10
2
2
2
= 25p – 50
= 28 .5 m2 (to 3 s .f .)
=
(ii) Area of path = p(6 + 2)2 – 36p
= p(8)2 – 36p
= 64p – 36p
= 28p m2
211
1
1
× (35 .5 + 20) × 15
2
1
=
× 55 .5 × 15
2
= 416 .25 cm2
(ii) Perimeter of trapezium = 35 .5 + 18 + 20 + 16
= 89 .5 cm
5. (i) Area of trapezium =
0.785
cm
p
0.785
1
Area of shaded region =
×2
×
p
2
16. Radius of each circle =
0.785
p
0.785
p
= 0 .250 cm2 (to 3 s .f .)
17. Area of grass within the goat’s reach = p(1 .5)2
= 2 .25p m2
Time the goat needs = 2 .25p × 14
= 99 .0 minutes (to 3 s .f .)
=
6. (i) Area of trapezium =
Exercise 13B
1. (a) Area of parallelogram = 12 × 7
= 84 cm2
= 150 m2
= 150
= 150
= 30
= 18
Length of RS = 18 m
(ii) Perimeter of trapezium = PQ + QR + RS + PS = 54 .7 m
12 + QR + 18 + 13 = 54 .7
43 + QR = 54 .7
QR = 11 .7
Length of QR = 11 .7 m
7. Area of shaded regions = area of trapezium ABCD – area of nBCE
42
6
=7m
(b) Base of parallelogram =
42.9
7.8
= 5 .5 mm
(c) Height of parallelogram =
1
1
× (10 + 14) × 12 –
× 14 × 12
2
2
1
=
× 24 × 12 – 84
2
= 144 – 84
= 60 cm2
=
1
2. (a) Area of trapezium =
× (7 + 11) × 6
2
1
=
× 18 × 6
2
= 54 cm2
126
(b) Height of trapezium =
1
× (8 + 10)
2
126
=
1
× 18
2
126
=
9
= 14 m
702
2
= 351 m2
8. Area of parallelogram ABFG =
351
27
= 13 m
Height of parallelogram ABFG with reference to base FG =
1
× (2 × 27) × 13
2
1
=
× 54 × 13
2
= 351 m2
9. (a) Total area of shaded regions
= area of rectangle – area of parallelogram
– area of circle – area of triangle
= (12 + 14) × (15 + 10)
1
– (12 + 14 – 5 – 2) × 10 – p(4)2 –
× 12 × 15
2
= 26 × 25 – 19 × 10 – 16p – 90
= 650 – 190 – 16p – 90
= 370 – 16p
= 320 cm2 (to 3 s .f .)
(b) Area of shaded region = area of trapezium – area of circle
Area of shaded region =
72
–5
1
×8
2
72
=
–5
4
= 18 – 5
= 13 mm
(c) Length of parallel side 2 of trapezium =
3. (i) Area of parallelogram = 6 × 9
= 54 cm2
(ii) Perimeter of parallelogram = 2(10 + 6)
= 2(16)
= 32 cm
4. Area of parallelogram = PQ × ST = QR × SU
PQ × 8 = 10 × 11 .2
PQ × 8 = 112
PQ = 14
Length of PQ = 14 m
1
1
× (PQ + RS) × PT
2
1
× (12 + RS) × 10
2
5 × (12 + RS)
12 + RS
RS
1
× (35 + 18) × 18 – p(6)2
2
1
=
× 53 × 18 – 36p
2
= 477 – 36p
= 364 cm2 (to 3 s .f .)
=
212
10. Area of figure = area of trapezium ABCE
– area of parallelogram GHDE – area of semicircle
2
15
1
1
= × (12 + 13 + 15) × 24 – 13 × 16 – × p
2
2
2
(b) Total area of shaded regions
= area of circle – area of triangle – area of rectangle
1
× (2 × 13 .6) × 13 .6 – 16 × 11
2
1
= 184 .96p –
× 27 .2 × 13 .6 – 176
2
= 184 .96p – 184 .96 – 176
= 184 .96p – 360 .96
= 220 cm2 (to 3 s .f .)
(c) Total area of shaded regions
= p(13 .6)2 –
1
1
× 40 × 24 – 208 –
× p(7 .5)2
2
2
= 480 – 208 – 28 .125p
= 272 – 28 .125p
= 184 cm2 (to 3 s .f .)
=
11. Area of nAED =
1
× AE × ED = 25 cm2
2
AE × ED = 50
1
1
× (48 + 16) × 20 +
× (30 + 20) × 16
2
2
1
1
=
× 64 × 20 +
× 50 × 16
2
2
= 640 + 400
= 1040 cm2
(d) Area of shaded region = area of trapezium – area of triangle
=
1
× (EB + DC) × ED
2
1
=
× (3AE + 4AE) × ED
2
1
=
× 7AE × ED
2
7
=
× AE × ED
2
7
=
× 50
2
= 175 cm2
12. (i) Let the height of the parallelogram ABCD with reference to the
base BC be h cm .
Area of parallelogram ABCD = BC × h = 80 cm2
Area of trapezium BCDE =
1
1
× (17 + 9) × (2 × 6) –
× 17 × 6
2
2
1
=
× 26 × 12 – 51
2
= 156 – 51
= 105 cm2
28
28
1
× 2p + 2p
2. (i) Perimeter of shaded region =
4
2
2
=
1
× 2p(14) + 2p(7)
2
= 14p + 14p
= 28p
= 88 .0 cm (to 3 s .f .)
=
1
× BE × h
Area of nABE =
2
1
=
× 2BC × h
2
= BC × h
= 80 cm2
(ii) Let the height of the parallelogram ABCD with reference to the
base DC be h′ cm .
Area of parallelogram ABCD = DC × h′ = 80 cm2
(ii) Area of shaded region
= area of big semicircle – area of two small semicircles
1
× p(14)2 – p(7)2
2
= 98p – 49p
= 49p
= 154 cm2 (to 3 s .f .)
3. Area of shaded region = area of one square of sides (2 × 12) cm
= (2 × 12)2
= 242
= 576 cm2
4. (i) Area of parallelogram = 9 × 25
= 225 m2
(ii) Perimeter of parallelogram = 2(9 + 30 .8)
= 2(39 .8)
= 79 .6 m
5. Let AB = BC = CD = DE = EF = AF = x cm .
(x + x) × x = 24
2x × x = 24
2x2 = 24
x2 = 12
Since x > 0, x = 12
Area of parallelogram BCEF = 12 × 12
= 12 cm2
=
1
× DF × h′
2
1
1
=
× DC × h′
2
2
1
=
× DC × h′
4
1
=
× 80
4
= 20 cm2
Area of nADF =
Review Exercise 13
1. (a) Area of shaded region
= 11 × 13 + 7 × (14 + 13) + 8 × (35 – 20) + 9 × 35 – 12 × 9
= 143 + 7 × 27 + 8 × 15 + 315 – 108
= 143 + 189 + 120 + 315 – 108
= 659 cm2
213
1
1
× (8 + 8 ÷ 2) × (6 ÷ 2)
2
1
=
× (8 + 4) × 3
2
1
=
× 12 × 3
2
= 18 cm2
6. Area of trapezium ABPQ =
Challenge Yourself
1. Let the length of AB be x cm .
Then the length of BC = the length of AC = 2x cm .
Area of nABC
= area of nABD + area of nBCD + area of nACD
1
1
1
×x×9+
× 2x × 7 +
× 2x × 7
2
2
2
= 4 .5x + 7x + 7x
= 18 .5x cm2
Case 1: The base of nABC is taken to be AB .
7. Area of figure
= area of rectangle ABCF + area of trapezium FCDE
= 20 × 15 +
=
1
× (20 + 3 .5) × 7
2
1
× 23 .5 × 7
2
= 300 + 82 .25
= 382 .25 m2
= 382 .25 × 0 .0001 ha
= 0 .038 225 ha
36
8. (i) x + y =
1
×6
2
36
=
3
= 12
(ii) Since x = 2y,
2y + y = 12
3y = 12
y =4
\x=2×4
=8
9. Length of each side of square = 1
=1m
Perimeter of square = 4 × 1
=4m
1
Radius of circle =
m
p
1
Circumference of circle = 2p
m
π
1
× x × h1 = 18 .5x
2
\ h1 = 37 cm
Case 2: The base of nABC is taken to be BC or AC .
= 300 +
1
× 2x × h2 = 18 .5x
2
\ h2 = 18 .5 cm
2. (i) Perimeter of figure = pr1 + pr2 + pr3 + pr4 + pr5 + AB
= p(r1 + r2 + r3 + r4 + r5) + AB
AB
+ AB
2
70
=p×
+ 70
2
= 35p + 70
= 180 cm (to 3 s .f .)
(ii) Perimeter of figure = pr + AB
=p×
AB
+ AB
2
70
=p×
+ 70
2
= 35p + 70
= 180 cm (to 3 s .f .)
(iii) Given a line segment AB of fixed length, regardless of the number
of semicircles drawn on the line segment, the perimeter of the
figure will be the same.
=p×
1
Required difference = 4 – 2p
π
= 0 .455 m (to 3 s .f .)
21
10. Circumference of drum = 2p
2
= 21p cm
Number of complete turns of handle required
9.89 × 100
21π
989
=
21p
= 15 (rounded up to the nearest whole number)
=
1
214
3.
C
D
A
E
B
BD
5
ED
1
=
⇔
=
BE
4
BE
4
Area of n AED
ED
1
=
=
Area of n ABE
BE
4
Area of nAED
1
=
4
20
\ Area of nAED = 5 cm2
Since nACD shares the same base AD and the same height as nABD,
area of nACD = area of nABD .
Since nAED is a common part of nACD and nABD,
area of nDCE = area of nABE = 20 cm2 .
Area of n BCE
BE
4
=
=
Area of n DCE
ED
1
Area of nBCE
4
=
20
1
\ Area of nBCE = 80 cm2
Area of trapezium
= area of nABE + area nAED + area of nDCE + area of nBCE
= 20 + 5 + 20 + 80
= 125 cm2
215
1
Chapter 14 Volume and Surface Area of Prisms and Cylinders
TEACHING NOTES
Suggested Approach
Students have learnt the conversion of unit area and perimeter and area of plane figures in the last chapter. This chapter will be
dealing with the conversion of unit volumes and the volume and surface area of solids, which is a natural transition from the last
chapter, from two-dimensional to three-dimensional. To assist in the students’ understanding, teachers should continually remind
students to be aware of the linkages between both topics, as well as introducing real-life applications that can reinforce learning.
Section 14.1: Conversion of Units
Teachers should recap the unit conversion of lengths and areas, proceed to introduce of volume by stating actual
applications (see Class Discussion: Measurement in Daily Lives), and then stating the different units associated
with volume (e.g. m, cm3 and m3).
Students should recognise how the number of dimensions and the unit representation for lengths, areas and
volumes are related (e.g. cm, cm2 and cm3). Students should recall calculations such as 1 cm3 = 1 cm × 1 cm ×
1 cm and solve problems involving conversion of unit volumes.
Section 14.2: Nets
Teachers should first define and explain that nets are basically flattened figures that can be folded to its threedimensional solids.
Teachers should show the nets of the various solids. Students are encouraged to make their own nets and form
the different three-dimensional solids. They should also be able to visualise the solids from different viewpoints.
Section 14.3: Volume and Surface Area of Cubes and Cuboids
Teachers can state that the volume of an object refers to the space it occupies, so the greater the volume, the
more space the object occupies.
Students should be informed and know that the volume of cubes and cuboids is the product of its three sides
(base × height = (length × breadth) × height).
The formulas for the total surface area of cubes and cuboids can be explored and discovered by students (see
Class Discussion: Surface Area of Cubes and Cuboids). It is important for the students to observe that the total
surface area is the total area of all its faces.
Section 14.4: Volume and Surface Area of Prisms
Teachers can introduce prisms to the students by stacking a few cubes to form a prism and show them how a
prism looks like. Students should know terms like lateral faces and cross-sections, and learn that prisms are
solids with uniform polygonal cross-sections. Teachers can ask the students to name some real-life examples of
prisms and use this opportunity to get them to explain why certain objects are not prisms so that they can get a
better understanding about prisms.
Observant students should realise that cuboids are prisms. Teachers can highlight to the students that prisms do
not necessarily have square bases and challenge students to think of bases of other possible shapes (see Fig. 14.2
on page 348).
Teachers should illustrate and derive the formulas for the volume and total surface area. Students need to
understand the definitions of volume and total surface area rather than memorise the formulas.
1
216
Section 14.5: Volume and Surface Area of Cylinders
Similar to the last section, teachers can introduce cylinders by stacking coins or showing students real-life
examples of cylindrical objects. Only right circular cylinders are covered in this syllabus.
Some students may think that cylinders are also prisms since both have uniform cross-sections. Teachers need
to impress upon students that this is not the case even though cylinders and prisms share similarities (see
Investigation: Comparison between a Cylinder and a Prism).
Teachers should also cover the formulas for the volume and total surface area of cylinders. Again, students need
to understand the definitions of volume and total surface area rather than memorise formulas.
Section 14.6: Volume and Surface Area of Composite Solids
Teachers should go through Worked Example 10 closely with students. Other than assessing their understanding,
teachers can inform students to be aware of any sides that should be omitted in finding total surface areas.
Challenge Yourself
Teachers should challenge students to think how the cross-section of the cuboid looks like in finding the volume
and surface area.
217
1
Part III:
WORKED SOLUTIONS
Name
Class Discussion (Measurements in Daily Lives)
Figure
Net
Other
4%
1. (i)
Wash
Basin
10%
Triangular
Prism
Showers
29%
Laundry
19%
Kitchen sink
22%
Flushing
16%
Cylinder
Source:
http://www.pub.gov.sg/conserve/Households/Pages/Watersavinghabits.aspx
Class Discussion (Surface Area of Cubes and Cuboids)
The activity which requires the greatest amount of water is shower .
(ii) –
Some measures:
• Take shorter showers.
• Turn off the shower tap while soaping.
• Use a tumbler when brushing your teeth.
• Do not thaw food under running water. Let it defrost
overnight inside the refrigerator instead .
• Wash vegetables and dishes in a sink or container filled
with water .
• Install thimbles or water saving devices at taps with high
flow rate.
• Turn off taps tightly to ensure they do not drip.
• Do not leave the tap running when not in use.
2. (i) The volume of one teaspoon of liquid is 5 ml .
(ii) This corresponds to 2 litres of water.
1. A cube has 6 surfaces. Each surface is in the shape of a square .
The area of each face is equal .
\ The total surface area of a cube is 6l 2 .
A cuboid has 6 surfaces. Each surface is in the shape of a rectangle .
\ The total surface area of a cube is 2(b × l + b × h + l × h) .
2. The total surface of the object is equal to the total area of all the
faces of the net .
Thinking Time (Page 349)
1. (i) The shape of all the lateral faces of a right prism is a rectangle.
(ii) The shape of all the lateral faces of an oblique prism is a
parallelogram.
2.
Investigation (Cubes, Cuboids Prisms and Cylinders)
Part II:
Name
Figure
Net
Cube
3. Examples of building structures and items are European-style houses
and chocolates. They are shaped as prisms as they have a uniform
cross-section.
Cuboid
Thinking Time (Page 354)
Examples are can drinks, toilet rolls and iron rods. They are shaped as
cylinders as they have a uniform circular cross-section.
1
218
(ii)
Investigation (Comparison between a Cylinder and a
Prism)
1 cm3 = 1 ml
165 000 cm3 = 165 000 ml
165 000
l
1000
=
165 l
1. The polygon will become a circle.
2. The prism will become like a cylinder.
=
Thinking Time (Page 358)
Practise Now 2
1. (i) Volume of the cuboid = l × 18 × 38 = 35 568
35 568
l=
18 × 38
= 52
(ii) Volume of each small cube = 2 × 2 × 2 = 8 cm3
Number of cubes to be obtained
r
Circumference = 2pr
h
r
35 568
8
= 4446
2. Volume of the open rectangular tank
=
55 × 35 × 36
= 69 300 cm3
Volume of water in the open rectangular tank initally
=
r
Total outer surface of a closed cylinder
= pr2 + 2prh + pr2
= 2pr2 + 2prh
1
× 69 300
2
=
34 650 cm3
Total volume of water in the open rectangular tank after 7700 cm3
of water are added to it
=
34 650 + 7700
=
42 350 cm3
Let the depth of water in the tank be d cm .
55 × 35 × d = 42 350
1925d = 42 350
d = 22
Depth of water = 22 cm
=
Class Discussion (Total Surface Area of Other Types of
Cylinders)
(a) an open cylinder
Total outer surface of an open cylinder = 2prh + pr2
(b) a pipe of negligible thinkness
Practise Now 3
External volume = (180 + 30 + 30) × (80 + 30 + 30) × (120 + 30)
= 240 × 140 × 150
= 5 040 000 cm3
Internal volume = 180 × 80 × 120
= 1 728 000 cm3
Volume of concrete used = 5 040 000 – 1 728 000
= 3 312 00 cm3
Total outer surface of a pipe of negligible thickness = 2prh
Practise Now 1
(a) (i) 1 m3 = 1 000 000 cm3
10 m3 = 10 × 1 000 000 cm3
= 10 000 000 cm3
(ii)
1 cm3 = 1 ml
10 000 000 cm3 = 10 000 000 ml
10 m3 = 10 000 000 ml
(b) (i) 1 000 000 cm3 = 1 m3
165 000
165 000 cm3 =
m3
1 000 000
Practise Now 4
1. (i) Volume of cuboid = 8 × 5 × 10
= 400 cm3
(ii) Surface area of the cuboid = 2(8 × 5 + 8 × 10 + 5 × 10)
= 340 cm2
= 0.165 m3
219
1
2. (i) Volume of water in the tank
= 16 × 9 × 8
=
1152 cm3
=
1152 ml
Practise Now 7
1. Base radius = 18 ÷ 2 = 9 cm
Height of the cylinder = 2.5 × 9 = 22.5 cm
Volume of the cylinder = pr2h
= p(9)2(22.5)
= 5730 cm3 (to 3 s.f.)
2. Base radius = 12 ÷ 2 = 6 cm
Volume of the cylinder = p(6)2h = 1000
1000
h=
p (6)2
h = 8.84 cm (to 3 s.f.)
1152
l
1000
= 1.152 l
(ii) Surface area of the tank that is in contact with the water
=
(16 × 9) + 2(16 × 8 + 9 × 8)
=
544 cm2
3. Let the length of the cube be l cm .
l × l × l = 27 cm3
l 3 = 27
=
Practise Now 8
l = 3 27
l =3
Total area of the faces that will be coated with paint
=
6(3 × 3)
=
54 cm2
1. Since petrol is discharged through the pipe at a rate of 2.45 m/s, i.e.
245 cm/s, in 1 second, the volume of petrol discharged is the volume
of petrol that fills the pipe to a length of 245 cm.
In 1 second, volume of petrol discharged
=
volume of pipe of length 245 cm
= pr2h
= p(0.6)2(245)
= 88 .2p cm3
In 3 minutes, volume of petrol discharged
= 88 .2p × 3 × 60
= 49 900 cm3
= 49 .9 l (to 3 s.f.)
2. Base radius = 0 .036 ÷ 2 = 0 .018 m
Since water is discharged through the pipe at a rate of 1.6 m/s, i.e.
in 1 second, the volume of water discharged is the volume of water
that fills the pipe to a length of 1.6 m.
In 1 second, volume of water discharged
=
volume of pipe of length 1.6 m
= pr2h
= p(0.018)2(1.6)
=
0.000 518 4p cm3
Volume of the cylindrical tank
= pr2h
= p(3.4)2(1.4)
= 16 .184p cm3
Time required to fill the tank
16.184 p
=
0.000 518 4 p
Practise Now 5
1. Base area = area of square
=4×4
= 16 m2
Volume of the prism = base area × height
= 16 × 10
= 160 m3
2. Base area = area of triangle
1
× 5.6 × x
2
= 2 .8x cm2
Volume of the prism = base area × height = 2 .8x × 12 = 151.2
33 .6x = 151.2
x = 4.5
=
Practise Now 6
(i) Volume of the prism = base area × height
1
= × 3 × 4 + (6 × 5) × 4.5
2
= 36 × 4.5
= 162 cm3
(ii) Total surface area of the prism
=
perimeter of the base × height + 2 × base area
=
(3 + 4 + 6 + 5 + 6) × 4.5 + 2 × 36
= 180 cm2
11
s
81
=
520 min (to the nearest minute)
= 3 1219
Practise Now 9
1. (i) Total surface area of the can
= 2pr2 + 2prh
= 2p(3.5)2 + 2p(3.5)(10)
=
24.5p + 70p
=
94.5p
=
297 cm2 (to 3 s.f.)
1
220
(ii) Area of the can that is painted
= pr2 + 2prh
(An open cylinder has only one
= p(3.5)2 + 2p(3.5)(10)
base and a curved surface)
=
12.25p + 70p
=
82.25p
Ratio of the area of the can that is painted, to the total surface
area found in (i) .
=
94.5p : 82.25p
=
94.5 : 82.25
=54 : 47
2. (i) Area of the cross section of the pipe
= p(2.5)2 – p(2.1)2
=
6.25p – 4.41p
= 1 .84p cm2
(ii) Internal curved surface area of the pipe
= 2p(2.1)(12)
=
50.4p
=
158 cm2 (to 3 s.f.)
(iii) Total surface area of the pipe
=
2(1.84p) + 50.4p + 2p(2.5)(12)
= 3 .68p + 50.4p + 60p
= 114 .08p
=
358 cm2 (to 3 s.f.)
Exercise 14A
1. (a) (i) 1 m3 = 1 000 000 cm3
4 m3 = 4 × 1 000 000 cm3
= 4 000 000 cm3
(ii) 1 m3 = 1 000 000 cm3
0.5 m3 = 0.5 × 1 000 000 cm3
=
500 000 cm3
(b) (i) 1 000 000 cm3 = 1 m3
250 000
m3
250 000 cm3 =
1 000 000
= 0.25 m3
3
(ii) 1 000 000 cm = 1 m3
67 800
67 800 cm3 =
m3
1 000 000
= 0.0678 m3
2. (a) (i)
1 m = 1 000 000 cm3
0 .84 m3 = 0 .84 × 1 000 000 cm3
= 840 000 cm3
(ii)
1 cm3 = 1 ml
840 000 cm3 = 840 000 ml
(b) (i) 1 000 000 cm3 = 1 m3
2560
2560 cm3 =
m3
1 000 000
3
= 0.002 56 m3
(ii)
1 cm = 1 ml
2560 cm3 = 2560 ml
Practise Now 10
3
1. (i) Volume of the container
= 20 × 9 × 14 +
1
× p(14)2(20)
4
=
2520 + 980p
=
5600 cm3 (to 3 s.f.)
(ii) Total surface area of the container
1
2
= 2 4 × π (14) + 2(9 × 14) + 2(20 × 9) + 2(14 × 20)
1
+
× 2p(14)(20)
4
= 98p + 252 + 360 + 560 + 140p
= 238p + 1172
= 1920 cm2 (to 3 s.f.)
2. (i) Volume of the solid
= 6 × 12 × 8 –
2560
l
1000
=
2.56 l
Volume of the cuboid = 6 × 8 × 10
= 480 cm3
Surface area of the cuboid = 2(6 × 8 + 8 × 10 + 6 × 10)
= 376 cm2
Volume of the cuboid = 7 × 12 × 5
= 420 cm3
Surface area of the cuboid = 2(7 × 12 + 5 × 7 + 5 × 12)
= 358 cm2
Volume of the cuboid = 120 × 10 × 96
= 115 200 mm3
Surface area of the cuboid
=
2(120 × 10 + 96 × 10 + 120 × 96)
=
27 360 mm2
=
3. (a) (i)
(ii)
(b) (i)
(ii)
(c) (i)
1
× p(3)2(12)
2
(ii)
=
576 – 54p
= 406 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
(d) (i) Volume of the cuboid = 1
1
1
×
× 10
2
2
1
= 7 cm3
2
(ii) Surface area of the cuboid
1
1 1 1
= 2 1 × + × 10 + 1 × 10
2
2 2 2
1
1
= 2 8 × 6 – × π (3)2 + 2(8 × 12) + × 2p(3)(12) + 6 × 12
2
2
=
96 – 9p + 192 + 36p + 72
=
360 + 27p
=
445 cm2 (to 3 s.f.)
= 41
221
1
cm2
2
1
6. Volume of the rectangular block of metal
= 0 .24 × 0 .19 × 0.15
= 0 .00684 m3
Let the length of the cube be l cm .
Volume of each small cube = l × l × l = 0 .00684 m3
l3 = 0 .00684
(e) (i) Volume of the cuboid = 1
2
3
5
×
×
5
8
8
21
3
=
cm
64
(ii) Surface area of the cuboid
2 5
2 3 3 5
= 2 1 × + × + 1 ×
5 8
5 8 8 8
l = 3 0.00684
= 0.190 (to 3 s.f.)
\ Length of each side = 0 .190 m
7. Volume of the open rectangular tank
= 4 × 2 × 4 .8
= 38 .4 m3
Volume of water in the open rectangular tank initally
43
cm2
160
(f) (i) Volume of the cuboid = 3.9 × 0.7 × 1.5
= 4.095 cm3
(ii) Surface area of the cuboid
=
2(3.9 × 0.7 + 0.7 × 1.5 + 3.9 × 1.5)
= 19 .26 cm2
=3
Length
Breadth
Height
Volume
Total surface
area
(a)
24 mm
18 mm
2160 mm3
1284 mm2
(b)
5 cm
5 mm
4.
3
3 cm
8 cm
120 cm
52.5 cm3
(c)
2.5 cm
6 cm
3.5 cm
(d)
12 m
8m
6m
576 m3
3
× 38 .4
4
= 28 .8 m3
4000 l = 4000 × 1000 ml
= 4 000 000 ml
= 4 000 000 cm3
4 000 000 3
=
m
1 000 000
=
158 cm2
89.5 cm2
432 m2
= 4 m3
Total volume of water in the open rectangular tank after 4000 litres
of water are added to it
= 28 .8 + 4
= 32 .8 m3
Let the depth of water in the tank be d m.
4 × 2 × d = 32 .8
8d = 32 .8
d = 4 .1
\ Depth = 4.1 m
8. External volume = (3.2 + 0.2 + 0.2) × (2.2 + 0.2 + 0.2) × (1.5 + 0.2)
= 3 .6 × 2 .6 × 1.7
= 15.912 m3
Internal volume = 3.2 × 2 .2 × 1.5
= 10.56 m3
Volume of wood used = 15.912 – 10.56
= 5.352 m3
9. External volume = 15 × 10 × 45
= 6750 cm3
Internal volume = 3 × 2 × 45
= 270 cm3
Volume of the hollow glass structure = 6750 – 270
= 6480 cm3
10. (i) Volume of water in the tank
= 0 .2 × 0.15 × 0 .16
= 0 .0048 m3
= 0 .0048 × 1 000 000 cm3
= 4800 cm3
= 4800 ml
(a) Volume = 24 × 18 × 5
= 2160 mm3
Surface area = 2(24 × 18 + 24 × 5 + 18 × 5)
= 1284 mm2
(b) Let the height of the cuboid be h cm .
Volume = 5 × 3 × h = 120 cm3
120
= 8 cm
5×3
Surface area = 2(5 × 3 × 5 × 8 + 3 × 8)
= 158 cm2
(c) Let the length of the cuboid be l cm .
Volume = l × 6 × 3.5 = 52.5 cm3
\h =
52.5
= 2.5 cm
6 × 3.5
Surface area = 2(2.5 × 6 + 6 × 3.5 + 2.5 × 3.5)
= 89.5 cm2
(d) Let the breadth of the cuboid be b m .
Volume = 12 × b × 6 = 576 m3
\l =
576
=8m
12 × 6
Surface area = 2(12 × 8 + 6 × 8 + 12 × 6)
= 432 m2
(i) Volume of the cuboid = 28 × b × 15 = 6720 cm3
\b =
5.
6720
28 × 15
= 16
\ Breadth = 16 cm
(ii) Volume of each small cube = 4 × 4 × 4 = 64 cm3
Number of cubes to be obtained
\b =
6720
64
=
105
4800
l
1000
= 4 .8 l
=
1
=
222
16. In one minute, the water will flow through 22 × 60 = 1320 cm
along the drain .
Amount of water that will flow through in one minute
= 30 × 3.5 × 1320
= 138 600 cm3
= 138 600 ml
(ii) Surface area of the tank that is in contact with the water
=
(0.2 × 0.15) + 2(0.2 × 0.16 + 0.15 × 0 .16)
= 0 .142 m2
11. (i) Volume of water in the tank
= 80 × 40 × 35
= 112 000 cm3
= 112 000 ml
138 600
l
1000
= 138 .6 l
17. (i) Let the height of the cuboid be h cm .
Surface area of the cuboid = 2(12 × 9 + 12 × h + 9 × h)
= 426 cm2
2(108 + 12h + 9h) = 426
2(108 + 21h) = 426
108 + 21h = 213
21h = 213 – 108
21h = 105
h =5
\ Height of cuboid = 5 cm
(ii) Volume of the cuboid
= 12 × 9 × 5
=
540 cm3
18. (i) Floor area of Room A = 26 × 1
= 26 m2
Volume of Room A = 26 × 1 × 3
= 78 m3
Floor area of Room B = 5 × 5
= 25 m2
Volume of Room B = 5 × 5 × 3
= 75 m3
Floor area of Room C = 6 × 6
= 36 m2
Volume of Room C = 6 × 6 × 1 .8
= 64 .8 m3
(ii) No. If both rooms, A and B, have the same height, then we will
use the floor area as the gauge. If the rooms do not have the
same height, then we will use the volume to decide.
=
112 000
=
l
1000
= 112 l
(ii) Surface area of the tank that is in contact with the water
=
(80 × 40) + 2(80 × 35 + 40 × 35)
= 11 600 cm2
11 600 2
m
=
10 000
= 1 .16 m2
12. Let the length of the cube be l cm .
l × l × l = 64 cm3
l 3 = 64
l = 3 64
=4
Total area of the faces that will be coated with paint
=
6(4 × 4)
= 96 cm2
13. Let the length of the cube be l cm .
6(l × l) = 433.5
6l 2 =433.5
l 2 =72.25
l = 72.25
= 8.5
Volume of the cube
=
8.5 × 8.5 × 8.5
=
614.125 cm3
14. (i) Number of trips required to fill the entire quarry
2.85 × 1 000 000
6.25
=
456 000
(ii) Cost to fill the quarry
=
456 000 × $55
=
$25 080 000
(iii) 3 hectares = 30 000 m2
Cost to fill 1 m2 of the land
25 080 000
=$
30 000
=
Exercise 14B
1. (a) Volume of the prism = base area × height
1
= × (75 + 59) × 46 × 120
2
= 3028 × 120
= 369 840 cm3
(b) Volume of the prism
=
base area × height
1
= × (16 + 28) × (18 – 7) + 7 × 28 × 38
2
= 438 × 38
= 16 644 cm3
= $836
15. Volume of wood used to make this trough
=
(185 × 45 × 28) – [(185 – 2.5 – 2.5) × (45 – 2.5 – 2.5) × (28 – 2.5)]
=
(185 × 45 × 28) – (180 × 40 × 25.5)
=
233 100 – 183 600
=
49 500 cm3
49 500
=
m3
1 000 000
=
0.0495 m3
223
1
(c) Volume of the prism = base area × height
= [9 × 5 + 9 × 3 + (16 – 8) × (9 – 6)] × 10
= 96 × 10
= 960 cm3
(d) Volume of the prism = base area × height
1
= × (14 + 18) × 6 × 12
2
= 96 × 12
= 1152 cm3
(e) Volume of the prism = base area × height
1
= × 6 × 8 + 13 × 10 × 5
2
= 154 × 5
= 770 cm3
(f) Volume of the prism = base area × height
(d) Area of nABC
1
× 7.8 × 24 .6
2
=
95.94 cm2
Volume of prism
=
95.94 × CD = 38 376
CD = 400 cm
3. Air space in the hall = Volume of the prism
= base area × height
=
= 1 × 42 × (38 – 23) + 42 × 23 × 80
2
= 1281 × 80
= 102 480 m3
4. (a) (i) Volume of the prism = base area × height
1
= × 6 × 4 × 15
2
= 12 × 15
= 180 cm3
(ii) Total surface area of the prism
=
perimeter of the base × height + 2 × base area
=
(5 + 5 + 6) × 15 + 2 × 12
= 264 cm2
(b) (i) Volume of the prism = base area × height
= [2 × 7 + (5 – 2) × (7 – 6)] × 9
= 17 × 9
= 153 cm3
(ii) Total surface area of the prism
=
perimeter of the base × height + 2 × base area
=
(7 + 2 + 6 + 3 + 1 + 5) × 9 + 2 × 17
=
250 cm2
5. (i) Volume of water in the pool when it is full
=
Volume of the prism
=
base area × height
1
= × 18 × (12 – 3) + 3 × 18 × 35
2
= 135 × 35
= 4725 cm3
AB
BC
BC
Area of
nABC
Volume of
prism
(a)
3 cm
4 cm
7 cm
6 cm2
42 cm3
(b)
9 cm
14 cm
11 cm
63 cm2
32 cm
15 cm
2.
(c)
(d) 24 .6 cm
7.8 cm
2
300 cm
240 cm
400 cm
95.94 cm2
693 cm3
72 000 cm3
38 376 cm3
(a) Area of nABC
1
×4×3
2
= 6 cm2
Volume of prism
=6×7
= 42 cm3
(b) Area of nABC
=
1
= × (1.2 + 2) × 50 × 25
2
= 80 × 25
= 2000 m3
(ii) Area of the pool which is in contact with the water
=
[(1.2 + 50 + 2 + 50.01) × 25 + 2 × 80] – (25 × 50)
=
1490.25 cm2
1
× BC × 9 = 63
2
4.5BC = 63
BC = 14 cm
Volume of prism
= 63 × 11
= 693 cm3
(c) Volume of prism
= Area of nABC × 300 = 72 000
Area of nABC = 240 cm2
Area of nABC
=
=
Exercise 14C
1. (a) (i) Volume of the closed cylinder
= pr2h
= p(7)2(12)
=
1850 cm3 (to 3 s.f.)
(ii) Total surface area of the closed cylinder
= 2pr2 + 2prh
= 2p(7)2 + 2p(7)(12)
= 98p + 168p
= 266p
= 836 cm2 (to 3 s.f.)
1
× 15 × AB = 240
2
7.5AB = 240
AB = 32 cm
1
224
(c) r = 4 ÷ 2 = 2 cm
Volume = 528 cm3
p(2)2h = 528
(b) Base radius = 1 .2 ÷ 2 = 0 .6 m
(i) Volume of the closed cylinder
= pr2h
= p(0.6)2(4)
=
4.52 m3 (to 3 s.f.)
(ii) Total surface area of the closed cylinder
= 2pr2 + 2prh
= 2p(0.6)2 + 2p(0.6)(4)
=
0.72p + 4 .8p
=
5.52p
=
17.3 m2 (to 3 s.f.)
(c) (i) Volume of the closed cylinder
= pr2h
= p(15)2(63)
=
44 500 mm3 (to 3 s.f.)
(ii) Total surface area of the closed cylinder
= 2pr2 + 2prh
= 2p(15)2 + 2p(15)(63)
=
450p + 1890p
= 2340p
=
7350 mm2 (to 3 s.f.)
528
4p
=
42.0 cm (to 3 s.f.)
Total surface area
= 2pr2 + 2prh
= 2p(2)2 + 2p(2)(42.02)
=
553 cm2 (to 3 s.f.)
(d) d = 4 × 2 = 8 m
Volume = 1056 m3
p(4)2h = 1056
h=
1056
16 p
=
21.0 m (to 3 s.f.)
Total surface area
= 2pr2 + 2prh
= 2p(4)2 + 2p(4)(21.01)
= 629 m2 (to 3 s.f.)
3. Base radius = 0 .4 ÷ 2 = 0 .2 m
h=
Diameter
Radius
Height
Volume
Total
surface area
(a)
8 .00 cm
4 .00 cm
14 cm
704 cm3
453 cm2
(b)
28 .0 cm
14 .0 cm
20 cm
12 320 cm3
2990 cm2
(c)
4 cm
2 cm
42 .0 cm
528 cm3
553 cm2
(d)
8m
4m
21 .0 m
2.
1056 m3
3
× 0.2 = 0.15 m
4
2
Volume of the cylinder = pr h
= p(0.2)2(0.15)
= 0 .006p m3
= 6000p cm3
Height of the cylinder =
6000 p
l
1000
= 18 .8 l
4. Let the depth of water in the drum be d cm .
Base radius = 48 ÷ 2 = 24 cm
150 l = 150 000 ml = 150 000 cm3
Volume of water in the drum = pr2d = 150 000 cm3
p(24)2d = 150 000
150 000
d=
p (24)2
= 82 .9
\ Depth of water = 82.9 cm
5. Base radius = 15 ÷ 2 = 7.5 cm
Capacity of the drink trough
=
629 m2
(a) Volume = 704 cm3
pr2(14) = 704
r2 =
r=
704
14 p
704
14 p
=
4.00 cm (to 3 s.f.)
\ d = 2 × 4.00 = 8.00 cm (to 3 s.f.)
Total surface area
= 2pr2 + 2prh
= 2p(4.001)2 + 2p(4.001)(14)
=
453 cm2
(b) Volume = 12 320 cm3
pr2(20) = 12 320
r2 =
1
× p × (7.5)2 × 84
2
=
7420 cm3 (to 3 s.f.)
=
7.42 l
6. 35 mm = 35 ÷ 10 = 3.5 cm
Base radius = 3.5 ÷ 2 = 1.75 cm
Total surface area that need to be painted for 1 wooden closed
cylinder
= 2pr2 + 2prh
= 2p(1.75)2 + 2p(1.75)(7)
=
6.125p + 24.5p
=
30.625p
=
12 320
20 p
12 320
20 p
=
14.0 cm (to 3 s.f.)
\ d = 2 × 14.0 = 28.0 cm (to 3 s.f.)
Total surface area
= 2pr2 + 2prh
= 2p(14.00)2 + 2p(14.00)(20)
= 2990 cm2 (to 3 s.f.)
r =
225
1
Total surface area that need to be painted for 200 wooden closed
cylinders
= 200 × 30.625p
= 19 200 cm2 (to 3 s.f.)
7. Base radius = 2 .4 ÷ 2 = 1 .2 m
Volume of the tank = pr2h
= p(1.2)2(6.4)
= 9 .216p m3
= 9 216 000p cm3
Volume of the cylinder container = pr2h
= p(8.2)2(28)
= 1882.72p cm3
Number of completed cylindrical container which can be filled by
the oil in the tank
11. Base radius of the pipe = 64 ÷ 2 = 32 mm
Since water is discharged through the pipe at a rate of 2.05 mm/s,
i.e. in 1 second, the volume of water discharged is the volume of
water that fills the pipe to a length of 2.05 mm.
In 1 second, volume of water discharged
=
volume of pipe of length 2.05 mm
= pr2h
= p(32)2(2.05)
= 2099 .2p mm3
Base radius of the cylindrical tank = 7.6 ÷ 2 = 3.8 cm = 38 mm
2 .3 m = 230 cm = 2300 mm
Volume of the cylindrical tank
= pr2h
= p(38)2(2300)
= 3 321 200p cm3
Time required to fill the tank
9 216 000 p
1882.72 p
=
4895 (to the nearest whole number)
8. External base radius = 28 ÷ 2 = 14 mm = 1.4 cm
Internal base radius = 20 ÷ 2 = 10 mm = 1 cm
Volume of the metal used in making the pipe
= p(1.4)2(35) – p(1)2(35)
= 68 .6p – 35p
= 33 .6p
= 106 cm3 (to 3 s.f.)
9. Base radius of the copper cylindrical rod = 14 ÷ 2 = 7 cm
Volume of the copper cylindrical rod
= pr2h
= p(7)2(47)
= 2303p
Let the length of the wire be l .
Base radius of the wire = 8 ÷ 2 = 4 mm = 0 .4 cm
Volume of the wire = p(0.4)2 l = 2303p
(0.4)2 l = 2303
2303
l=
0.16
= 14 400 cm (to 3 s.f.)
= 144 m
10. Base radius = 2 .4 ÷ 2 = 1 .2 cm
Since water is discharged through the pipe at a rate of 2.8 m/s, i.e.
280 cm/s, in 1 second, the volume of water discharged is the volume
of water that fills the pipe to a length of 280 cm.
In 1 second, volume of water discharged
=
volume of pipe of length 280 cm
= pr2h
= p(1.2)2(280)
= 403 .2p cm3
Half an hour = 30 minutes
In 30 minutes, volume of water discharged
= 403 .2p × 30 × 60
=
725 760p cm3
= 2 280 000 cm3 (to 3 s.f.)
= 2280 l
=
1
3 321 200 p
2099.2 p
83
=
1582
s
656
=
26 min (to the nearest minute)
12. (i) Volume of water in the tank = 18 × 16 × 13
= 3744 cm3
(ii) Let the height of water in the cylindrical container be h .
Base radius of the cylindrical container = 17 ÷ 2 = 8.5 cm
Volume of water in the cylindrical container = p(8.5)2h
= 3744
3744
h=
p (8.5)2
= 16.5
\ Height of water = 16.5 cm
(iii) Surface area of the cylindrical container that is in contact with
the water
= pr2 + 2prh
= p(8.5)2 + 2p(8.5)(16.49)
=
72.25p + 280 .33p
=
352.58p
= 1110 cm2 (to 3 s.f.)
13. (i) Base radius = 186 ÷ 2 = 93 mm = 9 .3 cm
=
1
× 93 = 31 mm = 3 .1 cm
3
Total surface area of the container
= 2pr2 + 2prh
= 2p(9.3)2 + 2p(9.3)(3.1)
=
172.98p + 57.66p
= 230 .64p
=
725 cm2 (to 3 s.f.)
Height =
226
(ii) Area of the container that is painted
= pr2 + 2prh
(An open cylinder has only one
= p(9.3)2 + 2pr(9.3)(3.1) base and a curved surface)
= 86 .49p + 57.66p
=
144.15p
Fraction
(ii) Total surface area of the solid
=
2(5 × 12) + 2(5 × 8) + 2(12 × 8) + 2(2 × 7) + 2(3 × 2)
= 120 + 80 + 192 + 28 + 12
= 432 cm2
2. (i) Volume of the solid
= p(2.5)2(8) + 7 × 11 × 3
=
50p + 231
= 388 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
=
7 × 11 + 2p(2.5)(8) + 2(3 × 7) + 2(3 × 11) + (7 × 11)
=
77 + 40p + 42 + 66 + 77
= 262 + 40p
= 388 cm2 (to 3 s.f.)
3. (i) Volume of the solid
= p(5)2(3) + p(12.5)2(2)
=
75p + 312.5p
=
387.5p
= 1220 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
= p(12.5)2 + 2p(5)(3) + 2p(12.5)(2) + p(12.5)2
=
156.25p + 30p + 50p + 156.25p
=
392.5p
= 1230 cm2 (to 3 s.f.)
4. (i) Volume of the glass block
144.15 p
230.64 p
5
=
8
14. (i) Base radius = 23 ÷ 2 = 11.5 mm = 1.15 cm
Height = 4 mm = 0 .4 cm
Volume of water and metal discs in the tank
=
(32 × 28 × 19) + 2580[p × (1.15)2 × 0.4]
=
(17 024 + 1364.82p) cm3
Let the new height in the tank be h .
Volume in the tank = 32 × 28 × h
= 17 024 + 1364.82p
896h = 17 024 + 1364.82p
=
17 024 + 1364.82 π
896
= 23.8 (to 3 s.f.)
\ New height of water in the tank = 23 .8 cm
(ii) Surface area of the tank that is in contact with the water after
the discs have been added
=
2(32 × 23.79 + 28 × 23.79) + 32 × 28
=
3750 cm2 (to 3 s.f.)
15. Total surface area of the pipe
=
2[p(3.8 + 0.8)2 – p(3.8)2] + 2p(3.8 + 0.8)(15) + 2p(3.8)(15)
= 13 .44p + 138p + 114p
=
265.44p
= 834 cm2 (to 3 s.f.)
16. 124 mm = 12 .4 cm = 0 .124 m
28 km2 = 28 000 000 m2
Volume of the rain
= 28 000 000 × 0 .124
=
3 472 000 m3
Volume of each channel
= 18 × 26 .4
=
475.2 m3
Time required for the channels to drain off the rain
h=
1
× p(24)2(56) + 24 × 56 × 40
4
= 8064p + 53 760
=
79 100 cm3 (to 3 s.f.)
(ii) Total surface area of the glass block
1
2
= 2 × π (24) + 2(40 × 24) + 2(40 × 56) + 2(24 × 56)
4
1
+ × 2p(24)(56)
4
= 288p + 1920 + 4480 + 2688 + 672p
= 960p + 9088
= 12 100 cm2 (to 3 s.f.)
5. (i) Volume of the solid
=
= 10 × 12 × 7 –
1
× p(2)2(12)
2
=
840 – 24p
=
765 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
1
= 2 7 × 10 – × π (2)2 + 2(7 × 12) + 2(3 × 12)
2
1
+ × 2p(2)(12) + 10 × 12
2
=
140 – 4p + 168 + 72 + 24p + 120
=
500 + 20p
=
563 cm2 (to 3 s.f.)
6. (i) Volume of the remaining solid
= p(12)2(32) – p(5)2(14)
= 4608p – 350p
=
4258p
= 13 400 cm3 (to 3 s.f.)
3 472 000
475.2 × 2
59
=
3653
s
297
=
61 minutes (to the nearest minute)
=
Exercise 14D
1. (i) Volume of the solid
=
7 × 3 × 2 + 12 × 8 × 5
= 42 + 480
=
522 cm3
227
1
(ii) Area that will covered in paint
= 2p(12)(32) + 2p(5)(14) + 2[p(12)2]
=
768p + 140p + 288p
= 1196p
=
3760 cm2 (to 3 s.f.)
7. (i) Volume of the solid
1
= × (40 + 88) × 70 × 25 – p(15)2(25)
2
(c) (i) Volume of the solid
= 4 × 5 × 1 – 2(1 × 3)
=
20 – 6
= 14 cm3
(ii) Total surface area of the solid
=
2(1 × 4) + 8(1 × 1) + 2(1 × 3) + 2[4 × 5 – 2(1 × 3)]
=
8 + 8 + 6 + 40 – 12
=
50 cm2
(d) (i) Volume of the solid
=1×1×5+2×4×1+1×1×3
=
5+8+3
= 16 cm3
(ii) Total surface area of the solid
=
2(1 × 5) + 2(1 × 1) + 2(1 × 3) + 2(1 × 4)
+ 2[1 × 5 + 2 × 3 + 1 × 5]
= 10 + 2 + 6 + 8 + 32
=
58 cm2
2. 4.5 m = 450 cm, 3.6 m = 360 cm
Number of bricks required
450
18
360
=
×
×
18
9
6
= 3000
3. Volume of the rectangular block of metal
=
256 × 152 × 81
=
3 151 872 mm3
Let the length of the cube be l mm .
l 3 = 3 151 872
=
112 000 – 5625p
= 94 300 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
1
2
=
(74 + 40 + 74 + 88) × 25 + 2 × (40 + 88) × 70 – π (15)
2
+ 2p(15)(25)
=
6900 + 8960 – 450p + 750p
=
15 860 + 300p
= 16 800 cm2 (to 3 s.f.)
8. (i) Volume of the solid
1
× [p(6 + 1.5)2 – p(6)2] × 8
2
=
4(56.25p – 36p)
=
4(20.25p)
= 81p
=
254 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
=
=2×
1
1
× [p(7.5)2 – p(6)2] +
× 2p(7.5) × 8
2
2
1
× 2p(6) × 8 + 2(1.5 × 8)
2
=
20.25p + 60p + 48p + 24
=
24 + 128.25 p
=
427 cm2 (to 3 s.f.)
+
l = 3 3151 872
= 147 (to 3 s.f.)
\ Length of each side = 147 mm
4. Let the length of the cube be l cm .
l 3 = 343
l = 3 343
=7
Total surface area of a cube
= 6l 2
=
6(7)2
= 294 cm2
5. (i) Its volume
Review Exercise 14
1. (a) (i) Volume of the solid
= 6 × 3 × 2 + 12 × 2 × 3
=
36 + 72
= 108 cm3
(ii) Total surface area of the solid
=
2(2 × 12) + 2(3 × 2) + 2(3 × 3) + 2(2 × 6) + 2(3 × 2)
+ 3 × 6 + 3 × 12
= 48 + 12 + 18 + 24 + 12 + 18 + 36
= 168 cm2
(b) (i) Volume of the solid
=6×8×2–2×2×2
=
96 – 8
= 88 cm3
(ii) Total surface area of the solid
=
2(2 × 6) + 2 × 8 + 2(3 × 2) + 3(2 × 2) + 2(6 × 8 – 2 × 2)
= 24 + 16 + 12 + 12 + 88
=
152 cm2
1
1
× (20 + 22.5) × 17 × 45.5
2
=
16 436.875 cm3
(ii) Volume of a gold bar with a mass of 200 g
16 436.875
=
× 200
250 000
=
=
13.1495 cm3
=
13.1495 × 1000 mm3
=
13 149.5 mm3
228
(iii) Volume of the gold bar weighing 200 g = 13 149.5 mm3
9. (i) Volume of the solid
= p(6)2(14) + 22 × 18 × 8
=
504p + 3168
=
4750 cm3 (to 3 s.f.)
(ii) Total surface area of the solid
=
2(22 × 18) + 2p(6)(14) + 2(8 × 22) + 2(8 × 18)
=
792 + 168p + 352 + 288
= 1432 + 168p
= 1960 cm2 (to 3 s.f.)
10. (i) Volume of the remaining solid
=
15 × 24 × 16 – p(4)2(7)
=
5760 – 112p
=
5410 cm3 (to 3 s.f.)
(ii) Area that will be covered in paint
= 2p(4)(7) + 2(15 × 24) + 2(16 × 24) + 2(16 × 15)
=
56p + 720 + 768 + 480
=
56p + 1968
= 2140 cm2 (to 3 s.f.)
1
× (20 + x) × 15 × 50 = 13 149.5
2
375(20 + x) = 13 149.5
13149.5
375
13149.5
x=
– 20
375
49
\ x = 15
750
6. Base radius of the cylindrical barrel = 70 ÷ 2 = 35 cm
Volume of water in the cylindrical barrel that is drained away
= p(35)2(6)
=
7350p cm3
0 .2 l = 200 ml = 200 cm3
Time taken for the water level in the barrel to drop by 6 cm
7350 p
=
200
20 + x =
=
115 minutes (to 3 s.f.)
7. (i) Volume of water in the pail
= p(32)2(25)
=
25 600p
= 80 400 cm3 (to 3 s.f.)
(ii) Volume of water in the pail after 2000 metal cubes
are added to it
=
25 600p + 2000(2 × 2 × 2)
=
25 600p + 16 000
Let the new height of water in the pail be h cm .
p(32)2h = 25 600p + 16 000
25600 π + 16000
h=
π (32)2
=
30.0 (to 3 s.f.)
\ New height = 30 .0 cm
8. (i) Internal radius = 4.2 ÷ 2 = 2.1 cm
External radius = 5 ÷ 2 = 2.5 cm
Volume of metal used in making the pipe
=
[p(2.5)2 – p(2.1)2] × 8 .9
=
16.376p
=
51.4 cm3 (to 3 s.f.)
(ii) 51.45 cm3 = 0.00 005 145 m3
Cost of the pipe = 0.00 005 145 × 2700 × $8
= $1 .11
Challenge Yourself
(i) Volume of the solid
=
50 × 70 × 30 – 10 × 10 × 70 – 2(10 × 10 × 10) – 2(10 × 10 × 20)
=
105 000 – 7000 – 4000 – 2000
= 92 000 cm3
(ii) Total surface area of the solid
=
2(30 × 70 – 10 × 10) + 2(50 × 30 – 10 × 10) + 2(50 × 70 – 10 × 10)
+ 4(10 × 60) + 8(10 × 10) + 8(10 × 20)
= 4000 + 2800 + 6800 + 2400 + 800 + 1600
= 18 400 cm2
229
1
Chapter 15 Statistical Data Handling
TEACHING NOTES
Suggested Approach
In primary school, students have learnt statistical diagrams such as pictograms, bar graphs, pie charts and line graphs. Here,
students revisit what they have learnt and they are expected to know and appreciate the advantages and disadvantages of each
diagram. With such knowledge, students can choose the most appropriate diagram given a certain situation. Teachers may want
to give more examples when introducing the various stages of a statistical study and engage with students in evaluating and
discussing the issues involved in each stage. Knowledge from past chapters may be required (i.e. percentage).
Section 15.1: Introduction to Statistics
Teachers should define statistics as the collection, organisation, display and interpretation of data. Teachers may
want to briefly cover each stage of a statistical study and give real-life examples for discussion with students,
in the later sections. Students are expected to solve problems involving various statistical diagrams.
Section 15.2: Pictograms and Bar Graphs
Using the example in the textbook, teachers can show how each stage is involved in a statistical study, where
the data is displayed in the form of a pictogram and bar graph. Students should appreciate what happens in each
stage, cumulating in the conclusion through the interpretation of the data. Through the example, students should
also learn to read, interpret and solve problems using information presented in these statistical diagrams.
Students should know the characteristics of pictograms and bar graphs and take note of the merits and limitations
of pictograms and bar graphs (see Attention on page 370 and Thinking Time on page 371).
Section 15.3: Pie Charts
Some students may still be unfamiliar with calculating the size of the angle of each sector in a pie chart. As such,
teachers may wish to illustrate how this is done. Students need to recall the characteristics of a pie chart (see
Attention on page 376).
Other than the examples given in the textbooks, teachers may give more examples where a data set is represented
by a pie chart, such as students’ views on recent current affairs.
Section 15.4: Line Graphs
Teachers may want to recap how line graphs are drawn. Students need to know the advantage, disadvantage and
the cases line graphs are best used in. (see Attention on page 378).
Teachers can discuss some situations where pictograms, bar graphs, pie charts or line graphs are most suitable and
assess students’ understanding of statistical diagrams (see Class Discussion: Comparison of Various Statistical
Diagrams).
Section 15.5: Statistics in Real-World Contexts
Teachers can use the examples given in the textbooks and further illustrate in detail how each stage in a statistical
study is carried out using real-life examples.
Teachers can get the students to discuss and think of more ways to collect data besides conducting questionnaires.
Other ways can include telephone interviews, emails, online surveys etc.
Teachers may want to assign small-scale projects for students where they conduct their own statistical studies.
Such projects allow students to apply what they have learnt about statistical data handling in real-world contexts.
Section 15.6: Evaluation of Statistics
Teachers should go through the various examples in the textbook and discuss with students the potential issues
that can arise at each stage of a statistical study. The importance of not engaging in any unethical behaviors,
ensuring objectivity and providing the complete picture without omitting any forms of misrepresentation need
to be inculcated into students.
1
230
(b) A line graph should be used to display the data as we need to
display the trend of the change in the population of Singapore
from the year 2004 to the year 2013 .
(c) A pie chart cannot be used to display the data as we will
not be able to directly determine the exact number of
Secondary 1 students who travel to school by each of the 4 modes
of transport . A line graph is inappropriate as it is used to display
trends over time . Hence, a pictogram or a bar graph should be
used to display the data . Since there are only 4 categories, we
may wish to use a pictogram instead of a bar graph as it is more
visually appealing and is easier to read .
(d) A pie chart should be used to display the data as it is easier to
compare the relative proportions of Secondary 1 students who
prefer the different drinks .
WORKED SOLUTIONS
Thinking Time (Page 371)
1. Michael is correct . In a pictogram, each icon represents the same
number . Hence, since there are 3 buses and 4 cars, more students
travel to school by car than by bus .
2. To avoid a misinterpretation of the data, we can replace each bus
and each car in the pictogram with a standard icon . Alternatively,
we can draw the buses and the cars to be of the same size .
Class Discussion (Comparison of Various Statistical Diagrams)
1.
Statistical
Diagram
Advantages
• It is more colourful and
appealing .
• It is easy to read.
• It is difficult to use icons to
represent exact values .
• If the sizes of the icons are
inconsistent, the data may
easily be misinterpreted .
• If the data has many
categories, it is not
desirable to use a
pictogram to display it as it
is quite tedious to draw so
many icons .
• The data sets with the
lowest and the highest
frequencies can be easily
identified.
• It can be used to compare
data across many
categories .
• Two or more sets of data
with many categories can
be easily compared .
• If the frequency axis
does not start from 0, the
displayed data may be
misleading .
• The categories can be
rearranged to highlight
certain results .
• The relative size of each
data set in proportion to
the entire set of data can be
easily observed .
• It can be used to display
data with many categories .
• It is visually appealing.
• The exact numerical value
of each data set cannot be
determined directly .
• The sum of the angles of
all the sectors may not
be 360° due to rounding
errors in the calculation of
the individual angles .
• It is not easy to compare
across the categories of
two or more sets of data .
• Intermediate values can be
easily obtained .
• It can better display trends
over time as compared to
most of the other graphs .
• The trends of two or more
sets of data can be easily
compared .
• Intermediate values may
not be meaningful .
• If the frequency axis
does not start from 0, the
displayed data may be
misleading .
• It is less visually appealing
as compared to most of the
other graphs .
Pictogram
Bar graph
Pie chart
Line
graph
Disadvantages
Performance Task (Page 381)
1. Collection of Data
Guiding Questions:
• What are the types of food that are sold in your current school
canteen?
• What other types of food would students like to be sold in the
school canteen? How many choices would you like to include
in the questionnaire?
• What should be the sample size? How do you ensure that the
sample chosen is representative of the entire school?
• How many choices would you like each student surveyed to
select?
2. Organisation of Data
Guiding Questions:
• How can you consolidate the data collected and present it in a
table?
• How should you organise the data such that it is easy to
understand?
3. Display of Data
Guiding Question:
• Which statistical diagram, i.e. pictogram, bar graph, pie chart
or line graph, is the most suitable to display the data obtained?
4. Interpretation of Data
Guiding Questions:
• How many more food stalls can your school canteen
accommodate?
• What is the conclusion of your survey, i.e. based on the statistical
diagram drawn, which types of food stalls should your school
engage for the school canteen?
Teachers may wish to refer students to pages 380 and 381 of the textbook
for an example on how they can present their report.
2. (a) A bar graph should be used to display the data as we need to
compare data across 12 categories . The categories with the
lowest and the highest frequencies can also be easily identified.
231
1
Part IV: Interpretation of Data
1. The conclusion was obtained based on a simple majority, i .e . since
more than 50% of the employees were satisfied with working in the
company, the survey concluded that the employees were satisfied
with the company and that the company was a good place to work
in .
Class Discussion (Evaluation of Statistics)
Part I: Collection of Data
1. Teachers to conduct poll to find out the number of students who
know Zidane, Beckenbauer and Cruyff. It is most likely that some
students will know who Zidane is, but most (if not all) students will
not know who Beckenbauer and Cruyff are.
2. It is stated in the article that the poll was conducted on the UEFA
website . As such, the voters who took part in the poll were
most likely to belong to the younger generation who are more
computer-savvy and hence, the voters were unlikely to be
representative of all football fans .
3. As shown in the article, the number of votes for the three footballers
were close, with 123 582 votes for Zidane, 122 569 votes for
Beckenbauer and 119 332 votes for Cruyff . This is despite the fact
that most of the younger generation, who were most likely to have
voted in the poll, may not know who Beckenbauer and Cruyff are
as they were at the peak of their careers in the 1970s . Hence, if older
football fans were to participate in the poll, Zidane would probably
not have come in first place.
4. The choice of a sample is important as if the sample chosen for
collection of data is not representative of the whole population, the
figures that are obtained may be misleading. Hence, a representative
sample should be chosen whenever possible .
2. 40% × 300 = 40 × 300
100
= 120 employees
It is stated in the article that 40% of the employees, i .e . 120 employees
were not satisfied with working in the company. As such, even
though a simple majority of the employees was satisfied with working
in the company, it cannot be concluded that most of the employees
were satisfied. This shows that we should not use simple majorities
to arrive at conclusions or make decisions .
3. The amendment of the constitution of a country is a very serious
matter where the agreement of a simple majority is insufficient,
therefore there is a need for a greater percentage of elected Members
of Parliament (MPs) to agree before the constitution can be amended .
As a result, the Singapore government requires the agreement of at
least a two-third majority before the constitution can be amended .
Teachers may wish to take this opportunity to get students to search
on the Internet for some laws that have been passed in the Singapore
Parliament that resulted in a constitutional amendment.
4. It is important to have a basis or contention in order to decide on an
issue, and that in some occasions, it is insufficient to make decisions
based on a simple majority .
Part II: Organisation of Data
1. Banks and insurance firms, timeshare companies and motor vehicle
companies received the most number of complaints .
2. The article states that banks and insurance firms, which were
grouped together, received the most number of complaints . If banks
and insurance firms were not grouped together, it is possible that
timeshare companies received the most number of companies . For
example, if the 1416 complaints were split equally between banks
and insurance firms, they would have received 708 complaints each,
then the number of complaints received by timeshare companies,
i .e . 1238 complaints, would have been the greatest .
3. This shows that when organising data, it is important to consider
whether to group separate entities as doing so might mislead
consumers and result in inaccurate conclusions .
Teachers may wish to ask students whether a simple majority,
i.e. more than 50% of the votes, is necessary to decide on an issue.
For example, in the 2011 Singapore Presidential Elections, Dr Tony
Tan was elected President of the Republic of Singapore with 35.2%
of the total valid votes cast.
Part V: Ethical Issues
It is unethical to use statistics to mislead others as it is essentially a form
of misrepresentation and people may arrive at the wrong conclusions or
make the wrong decisions .
The rationale for teaching students to be aware of how statistics can be
used to mislead others is so that the students will be more discerning
when they encounter statistics and will not be misled by others. Teachers
should also impress upon students that they should not use statistics to
mislead others because it is unethical to do so.
Part III: Display of Data
1. Although the height of the bar for Company E appears to be
twice that of the bar for Company C, Company E’s claim is not
valid as the bars do not start from 0 . By reading off the bar graph,
Company E sold 160 light bulbs in a week, which is not twice as
many as the 130 light bulbs sold by Company C in a week .
2. For bar graphs, if the vertical axis does not start from 0, the height
of each bar will not be proportional to its corresponding frequency,
i .e . number of light bulbs sold by each company in a week . Such
display of statistical data may mislead consumers .
1
Practise Now (Page 371)
1. (a) (i) Profit earned by the company in 2010 = 5.5 × $1 000 000
= $5 500 000
(ii) Profit earned by the company in 2012 = 7 × $1 000 000
= $7 000 000
(b) The company earned the least profit in 2009. The profit decreased
by 1 .5 × $1 000 000 = $1 500 000 in 2009 as compared to 2008 .
232
2. (a)
Practise Now (Page 376)
Number of television sets
Sales of Television Sets in 5 Shops
120
100
Farhan’s total expenditure on the holiday
= $1000 + $1200 + $400 + $1200 + $200
= $4000
November
December
80
60
Item
Angle of sector
40
20
0
Shop 1
Shop 2
Shop 3
Shop 4
Shop 5 Shop 6
Shop 7
Food
$1000
× 360° = 90°
$4000
Shopping
$1200 × 360° = 108°
$4000
Hotel
$400
× 360° = 36°
$4000
Shop
(b) (i) Total number of television sets sold in the seven shops in
November
= 60 + 30 + 50 + 70 + 40 + 64 + 70
= 384
(ii) Total number of television sets sold in the seven shops in
December
= 90 + 48 + 80 + 112 + 80 + 88 + 96
= 594
384
(c) Required percentage =
× 100%
384 + 594
384
=
× 100%
978
43
= 39
%
163
70 + 96
(d) (i) Required percentage =
× 100%
978
166
=
× 100%
978
476
= 16
%
489
(ii) No, I do not agree with the manager . Since Shop 2 sold the
least number of televsion sets in November and December,
it should be closed down .
(e) The company performed better in terms of sales in December .
This could be due to the fact that Christmas is in December
when people buy television sets as gifts for others .
Air
Ticket
$1200
× 360° = 108°
$4000
$200
× 360° = 18°
$4000
Others
Others
18°
Food
Air Ticket
108°
108°
36°
Shopping
Hotel
Practise Now 1
1. (i) 4x° + 2x° + 237 .6° = 360° (/s at a point)
4x° + 2x° = 360° – 237 .6°
6x° = 122 .4°
x° = 20 .4°
\ x = 20 .4
4(20.4° )
(ii) Required percentage =
× 100%
360°
81.6°
× 100%
360°
2
= 22 %
3
360°
(iii) Amount of fruit punch in the jar =
× 759 ml
237.6°
= 1150 ml
=
233
1
2. (i) The least popular colour is black .
(ii) Total number of cars sold
= 2000 + 3500 + 5000 + 6000 + 1500
= 18 000
Angle of sector that represents number of blue cars sold
2000
=
× 360°
18 000
= 40°
Angle of sector that represents number of grey cars sold
3500
=
× 360°
18 000
= 70°
Angle of sector that represents number of white cars sold
5000
=
× 360°
18 000
Exercise 15A
1. (i) The greatest number of buses registered was in 2012 .
Number of buses registered in 2012 ≈ 6 .5 × 40 000
= 260 000
(ii) Total number of buses registered from 2008 to 2012
≈ 24 × 40 000
= 960 000
(iii) Total amount the Registry of Vehicles collected in 2010
≈ 4 .5 × 40 000 × $1000
= $180 000 000
(iv) Percentage increase in number of buses registered
from 2011 to 2012
1
× 100%
5.5
2
= 18 %
11
=
= 100°
Angle of sector that represents number of red cars sold
6000
=
× 360°
18 000
= 120°
Angle of sector that represents number of black cars sold
1500
=
× 360°
18 000
= 30°
(iii) No, I do not agree with her . This is because the number of cars
indicated on the y-axis is in thousands, thus 3500 grey cars and
1500 black cars are sold .
2. (i)
Students who Play Volleyball, Basketball or Tennis
Volleyball
Basketball
Tennis
Each circle represents 10 students .
(ii) Required ratio = 4 : 5
5
× 100%
6
1
= 83 %
3
(iii) Required percentage =
Practise Now 2
3.
(i) The number of fatal road casualties was the highest in 2008 .
Year
2005
2006
2007
2008
2009
Number of fatal
road casualties
173
190
214
221
183
300
Number of copies (in thousands)
(ii)
(iii) Percentage decrease in number of fatal road casualties
from 2008 to 2009
221 – 183
=
× 100%
221
=
Newspaper Distribution to Households
350
38
× 100%
221
43
%
221
(iv) There are traffic cameras installed along more roads.
= 17
250
200
150
100
50
0
1
234
2008
2009
2010
Year
2011
2012
(iv) The percentage of successful candidates increases over the six
years as they practise past-year papers and learn from their
mistakes .
6. (i) Total number of workers employed in the housing estate
=4×1+6×2+5×3+3×4+2×5
= 4 + 12 + 15 + 12 + 10
= 53
(ii) Total number of shops in the housing estate
=4+6+5+3+2
= 20
Number of shops hiring 3 or more workers = 5 + 3 + 2
= 10
10
\ Required percentage =
× 100%
20
= 50%
(iii) Some shops have more customers as they are located at places
with higher human traffic, thus they need to employ more
workers .
4. (a)
Class
Class
1A
Class
1B
Class
1C
Class
1D
Class
1E
Number of students
who score a distinction
in Mathematics
9
11
16
12
20
Number of students
who score a distinction
in Science
8
13
12
16
15
Number of students
Students who Score a Distinction in
Mathematics or Science
25
Mathematics
Science
20
15
10
5
0
1A
1B
1C
Class
1D
1E
Exercise 15B
(b) (i) Total number of students in the 5 classes who score a
distinction in Mathematics
= 9 + 11 + 16 + 12 + 20
= 68
(ii) Total number of students in the 5 classes who score a
distinction in Science
= 8 + 13 + 12 + 16 + 15
= 64
1. Total number of students surveyed = 768 + 256 + 64 + 192
= 1280
12
× 100%
68
11
= 17
%
17
(d) Percentage of students in Class 1D who score a distinction in
Science
(c) Required percentage =
16
× 100%
40
= 40%
(e) No, Jun Wei is not correct to say that there are 35 students in
Class 1E . There may be students in the class who do not score
distinctions in both Mathematics and Science . There may also be
students in the class who score distinctions in both Mathematics
and Science .
5. (i) Number of candidates who sat for the examination in 2009
= 950
(ii) Number of candidates who failed the examination in 2012
= 500
(iii) Total number of candidates who failed the examination in the
six years
= 400 + 350 + 350 + 400 + 450 + 500
= 2450
Mode of transport
Angle of sector
Bus
768
× 360° = 216°
1280
Car
256
× 360° = 72°
1280
Bicycle
64
× 360° = 18°
1280
Foot
192
× 360° = 54°
1280
=
Foot
Bicycle
18°
54°
72°
Car
216°
Bus
2. (i) Angle of sector that represents number of students who prefer
yam = 90°
(ii) Angle of sector that represents number of students who prefer
vanilla
= 360° – 120° – 90° – 50° (/s at a point)
= 100°
500
× 100%
2450
20
= 20
%
49
\ Required percentage =
235
1
(ii) Percentage increase in mass of the baby from the 4th to 6th month
5 – 4.2
=
× 100%
4.2
0.8
=
× 100%
4.2
1
= 19
%
21
6. (a) Total angle of sectors that represent number of female students
and teachers in the school
= 360° – 240° (/s at a point)
= 120°
Angle of sector that represents number of teachers in the school
1
=
× 120°
6
= 20°
(b) (i) Number of female students in the school = 5 × 45
= 225
100°
× 100%
360°
7
= 27 %
9
(iii) Required percentage =
360°
×5
50°
= 36
(iv) Total number of students in the class =
180°
× 100%
360°
= 50%
72°
(ii) Required percentage =
× 100%
360°
= 20%
1
17
2 × 360°
(iii) x° =
100
= 63°
\ x = 63
4. (i) Total number of cars in the survey = 20 + 25 + 20 + 30 + 25
= 120
(ii) Total number of people in all the cars
= 20 × 1 + 25 × 2 + 20 × 3 + 30 × 4 + 25 × 5
= 20 + 50 + 60 + 120 + 125
= 375
(iii) Number of cars with 4 or more people = 30 + 25
= 55
3. (i) Required percentage =
240°
× 45
20°
= 540
(ii) Number of male students in the school =
(c) Total school population = 45 + 225 + 540
= 810
2
× 45
3
= 30
Number of females in the school = 225 + 30
= 255
255
\ Required percentage =
× 100%
810
13
= 31
%
27
5
7.
× 360° = 120°
1+ x + 5
5
120°
=
6+x
360°
5
1
=
6+x
3
15 = 6 + x
\x =9
8. (i) The town had the greatest increase in the number of people from
2011 to 2012 .
(ii)
Number of female teachers in the school =
55
× 100%
120
5
= 45 %
6
(iv) Angle of sector that represents number of cars with 1 people
20
=
× 360°
120
= 60°
Angle of sector that represents number of cars with 2 people
25
=
× 360°
120
= 75°
Angle of sector that represents number of cars with 3 people
20
=
× 360°
120
= 60°
Angle of sector that represents number of cars with 4 people
30
=
× 360°
120
= 90°
Angle of sector that represents number of cars with 5 people
25
=
× 360°
120
= 75°
\ Required percentage =
5. (i)
Month
Mass (kg)
0
1
2
3
4
5
6
3 .2
3 .4
3 .8
4
4 .2
4 .4
5
Year
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Number of
people (in
thousands)
6
9
9 .5
12
14
15
16
18
19
25
(iii) Percentage increase in number of people in the town from 2009
to 2012
25 000 – 16 000
=
× 100%
16 000
9000
=
× 100%
16 000
1
%
4
(iv) There are more new immigrants in the town .
= 56
1
8
236
(ii) Number of students who read more than 4 books = 5 + 1
=6
Total number of students in the class = 2 + 5 + 9 + 8 + 6 + 5 + 1
= 36
6
\ Required percentage =
× 100%
36
2
= 16 %
3
(iii) Number of students who read fewer than 3 books = 2 + 5 + 9
= 16
Angle of sector that represents number of students who read
fewer than 3 books
16
=
× 360°
36
= 160°
3. Percentage of students who are enrolled in the Arts course
= 100% – 25% – 30% – 15%
= 30%
9. (i)
Temperature (°C)
Temperature of Patient
40
39
38
37
36
35
1500
1800
2100
0000
0300
0600
0900
Time (hours)
10.
11.
12.
13.
(ii) Temperature of the patient at 1700 hours ≈ 39 °C
Temperature of the patient at 0100 hours ≈ 38 °C
The majority of the respondents in Kate’s survey are most likely
females while those in Khairul’s survey are most likely males . Kate
and Khairul may have conducted each of their surveys at a different
location, e .g . Kate may have conducted her survey at Orchard Road
while Khairul may have conducted his survey at a housing estate .
No, I do not agree with Nora . The temperatures in both countries
range from 24 °C to 35 °C . The temperatures in Country X seem to
change more drastically than those in Country Y because the vertical
axis of the line graph which shows the temperatures of Country X
starts from 23 °C instead of 0 °C .
(i) Based on the 3-dimensional pie chart, Raj spends the most on
luxury goods .
(ii) Based on the 2-dimensional pie chart, Raj spends the most on
rent and luxury goods .
(iii) In a 3-dimensional pie chart, the sizes of the sectors will look
distorted . The sectors towards the back of the pie chart will
appear smaller than those towards the front .
No, I do not agree with Amirah . As there are more cars than
motorcycles in Singapore, it is not surprising that there are more
accidents involving cars than motorcycles . Moreover, there may be
a higher chance of accidents involving motorcycles occurring due
to the nature of the vehicle .
Type of course
Angle of sector
Science
25
× 360° = 90°
100
Engineering
30
× 360° = 108°
100
Business
15
× 360° = 54°
100
Arts
30
× 360° = 108°
100
Science
Arts
108°
54°
Review Exercise 15
Business
108°
Engineering
1. (i) Required ratio = 6 : 3
=2:1
7
× 100%
4
= 175%
2. (i) Total number of books read by the students in the class in a
month
=2×0+5×1+9×2+8×3+6×4+5×5+1×6
= 0 + 5 + 18 + 24 + 24 + 25 + 6
= 102
(ii) Required percentage =
4. (i) Total angle of sectors that represent amount Devi spends on
clothes and food
= 360° – 36° – 90° – 90° (/s at a point)
= 144°
Angle of sector that represents amount Devi spends on food
1
× 144°
4
= 36°
=
36°
× 100%
90°
= 40%
\ Required percentage =
237
1
(ii) Devi’s monthly income =
360°
× $400
36°
= $4000
Devi’s annual income = 12 × $4000
= $48 000
5. (i)
Year
Number of laptops
2008
2009
2010
2011
2012
70
30
44
90
26
(ii) Percentage decrease in number of laptops purchased by the
company from 2008 to 2009
70 – 30
× 100%
70
40
=
× 100%
70
1
= 57 %
7
(iii) The company might have had a tighter budget in 2009 .
=
Challenge Yourself
The better way to display the data using a bar graph is as follows:
Breathing rate
150
50
40
100
30
20
50
1
Swimming
Jogging
Running
Walking
10
Breathing rate (per minute)
Pulse rate (per minute)
Pulse rate
238
(ii) Volume of wood used = 72 × 54 × 48 – 158 355
= 186 624 – 158 355
= 28 269 cm3
(iii) Mass of box = 0 .9 × 28 269
= 25 442 .1 g
Revision Exercise D1
1. Let the radius of the quadrant be x cm .
1
× 2px + 2x = 71 .4
4
1
px + 2x = 71 .4
2
150°
× 100%
360°
2
= 41 %
3
72°
(ii) Required percentage =
× 100%
360°
= 20%
15
(b) x° =
× 360°
100
= 54°
\ x = 54
6. (a) (i) Required percentage =
1
x π + 2 = 71 .4
2
71.4
\x=
1
π+2
2
= 20 .00 (to 4 s .f .)
1
× p(20 .00)2
4
= 314 cm2 (to 3 s .f .)
1
12
Perimeter of shaded region =
× 2p(12) + 2p
2
2
= 12p + 2p(6)
= 12p + 12p
= 24p
= 75 .4 cm (to 3 s .f .)
Area of shaded region = area of big semicircle
1
=
× p(12)2
2
= 72p
= 226 cm2 (to 3 s .f .)
Area of trapezium ABEF
= area of rectangle ACEF – area of nBCE
1
= area of rectangle ACEF –
× area of nBDE
2
1
= 12 × 8 –
× 24
2
= 96 – 12
= 84 cm2
Volume of solid = base area × height
= [12 × 12 + (18 – 12) × 3] × 6
= (144 + 6 × 3) × 6
= (144 + 18) × 6
= 162 × 6
= 972 cm3
Total surface area of solid
= perimeter of base × height + 2 × base area
= [12 + 12 + 18 + 3 + 6 + (12 – 3)] × 6 + 2 × 162
= (12 + 12 + 18 + 3 + 6 + 9) × 6 + 324
= 60 × 6 + 324
= 360 + 324
= 684 cm2
(i) Capacity of box
= (72 – 1 .5 – 1 .5) × (54 – 1 .5 – 1 .5) × (48 – 1 .5 – 1 .5)
= 69 × 51 × 45
= 158 355 cm3
= 158 .355 l
Area of quadrant =
2.
3.
4.
5.
239
1
Revision Exercise D2
1. Area of photograph = 40 × 25
= 1000 cm2
Area of margin = (40 + 4 + 4) × (25 + 4 + 4) – 1000
= 48 × 33 – 1000
= 1584 – 1000
= 584 cm2
2. (i) Perimeter of figure
1
= 24 + 15 + (24 – 10) +
× 2p(10) + (15 – 10)
4
= 24 + 15 + 14 + 5p + 5
= 58 + 5p
= 73 .7 cm (to 3 s .f .)
(ii) Area of figure = area of rectangle – area of quadrant
1
= 24 × 15 –
× p(10)2
4
= 360 – 25p
= 281 cm2 (to 3 s .f .)
3. Area of parallelogram = PQ × ST = QR × SU
10 × ST = 7 × 9
10 × ST = 63
ST = 6 .3
Length of ST = 6 .3 cm
4. Volume of prism = base area × height
1
= × (8 + 3 + 8 + 3) × 4 × 20
2
1
= × 22 × 4 × 20
2
= 44 × 20
= 880 cm3
Total surface area of prism
= perimeter of base × height + 2 × base area
= (8 + 5 + 3 + 8 + 3 + 5) × 20 + 2 × 44
= 32 × 20 + 88
= 640 + 88
= 728 cm2
5. Volume of cylinder = p(62 – 52)(2 .4 × 100)
= p(36 – 25)(240)
= p(11)(240)
= 2640p cm3
Mass of cylinder = 7 .6 × 2640p
= 20 064p
= 63 000 g (to 3 s .f .)
6. (i) The attendance was the greatest in the 4th week .
(ii) The Drama Club stopped its weekly meeting in the 9th week .
45 – 15
(iii) Required percentage =
× 100%
45
30
=
× 100%
45
2
= 66 %
3
(iv) Most of the Drama Club members were busy preparing for the
school examination .
1
240
is a soft metal, diving cylinders made of aluminium are more prone
to physical damage. On the other hand, as steel is a tough metal,
diving cylinders made of steel are more durable and are less prone
to physical damage. However, steel comprises of iron, which is
more susceptible to corrosion, thus steel diving cylinders are more
difficult to maintain.
3. (a) Percentage increase in the annual mean surface temperature in
Singapore from 1948 to 2011
Problems in Real-World Contexts
1. (i) A suitable unit for the measurements in the floor plan is the
millimetre (mm) .
(ii) Length of AB = (800 + 4800 + 3200 + 1600 + 1400 + 2500)
– (4400 + 1775 + 400)
= 14 300 – 6575
= 7725 mm
$500 000
(iii) Price per square metre =
110
= $4545 (to the nearest dollar)
(iv) 1 foot ≈ 0 .3048 m
1 square foot = 0.092 90 m2 (to 4 s .f .)
\ $1000 psf = $1000 ÷ 0.092 90
= $10 760 per square metre (to 4 s.f.)
The condominium unit is $10 760 ÷ $4545 = 2.37 (to 3 s.f.) times
as expensive as the flat that Mr Lee is interested to purchase.
2. (i) Volume of water the diving cylinder can contain
= p(6 .75)2(85)
= 3872 .8125p cm3
= 3 .872 812 5p l
= 12 .2 l (to 3 s .f .)
(ii) Volume of gas that the diving cylinder can hold
volume of cylinder × pressure in cylinder
=
atmospheric pressure
3.872 812 5 π × 200
=
1.01
=
2409 l (to 4 s .f .)
= 2410 l (to 3 s .f .)
(iii) Duration the diver can stay underwater
volume of gas consumed
=
breathing rate × ambient pressure
27.6 – 26.8
× 100%
26.8
0.8
=
× 100%
26.8
=
2.99% (to 2 d.p.)
(b) (i) Majority of Singapore’s emissions in 2020 is expected to
come from the industry sector.
Amount of emissions contributed by the industry sector
= 60 .3% × 77.2 MT
=
46.5516 MT
(ii) Reasons for the likely increase in emissions from 2005 to
2020:
• Due to rapid urbanisation, there will be an increase
in the demand for expansion of petrochemicals and
manufactured products from Asian countries such as
Singapore.
• The growth of the population and the economy results
in an increased use of transportation. In addition, the
expansion of port activities causes an increase in the
emissions from domestic maritime transport.
• There will be an increase in the demand for commercial
spaces as well as a greater intensity in the usage of
space.
• Due to increases in population and household income,
there will be an increase in the demand for electrical
appliances.
=
2409
15
20 × 1.01 +
10
2409
=
20 × (1.01 + 1.5)
=
Teachers may wish to note that the list is not exhaustive.
(c) Measures that have been put in place by the Singapore
government to reduce emissions and to mitigate the effects of
climate change:
• The Energy Conservation Act, which was implemented in
2013, mandates companies in the industry and transport
sectors to adopt energy efficient technologies and processes.
• The implementation of a Carbon Emissions-based Vehicle
(CEV) Scheme in 2013 seeks to encourage consumers to
buy low carbon emissions cars. Moreover, in the next few
years, the rail network will be increased to about 280 km.
This encourages people to use public transport, which is
more carbon efficient.
• All new buildings and existing ones undergoing extensive
renovation are required to adhere to the Green Mark
standards, i.e. the buildings should be sustainable and
environmentally-friendly.
2409
20 × 2.51
= 48 .0 minutes (to 3 s .f .)
=
Teachers may wish to ask students to state an assumption that
they have made in their calculations, e.g. the volume of gas
consumed by the diver as he descends to a depth of 15 m is
negligible.
(iv) Since the amount of time the diver can stay underwater if he
uses the diving cylinder is 48.0 minutes, which is less than one
hour, the diving cylinder is not suitable for the diver.
Diving cylinders are usually made of aluminium or steel. Teachers
may wish to ask students to find out an advantage and a disadvantage
of a diving cylinder made up of aluminium and of steel.
For example, aluminium cylinders are easier to maintain as
aluminium is more resistant to corrosion. However, as aluminium
241
1
•
•
The Minimum Energy Performance Standards Scheme
which was implemented in 2011 restricts the sale of energy
inefficient appliances.
The Singapore government is moving away from the
disposal of waste in landfills to the incineration of waste.
B. Solving
• How do you solve the problem mathematically?
• What are the different ways that you can use to present the
results from your calculations? Which way should you use
to present the results to your classmates so that they will
be able to make an informed decision?
C. Interpreting
Students should be able to interpret the mathematical solution in
the context of the real world, i.e. they should be able to advise
their classmates to choose one of the three price plans based on
the mathematical solution that they have obtained.
D. Reflecting
• After finding the most value-for-money plan for their
classmates, students should check whether there are any
other issues that their classmates may have to take into
consideration before subscribing to a plan.
• Students should review the chosen plan to determine
whether it is the ideal plan for their classmates.
• Students should also review the method that they have used
and consider whether there are other methods that can be
used to solve the problem.
Teachers may wish to note that the list is not exhaustive.
4. The Mathematical Modelling Process consists of the following
steps:
A. Formulating
• Students are required to understand the information given
in the question and a discussion may be carried out to help
them comprehend the problem. Some guiding questions are
as follows:
(i) Based on the information in the table given, what does
your classmates’ choice of plan depend on? Should
your classmates choose a plan based on the price alone,
i.e. would you advise your classmates to choose
Plan A because the monthly subscription fee is the
lowest?
(ii) Other than the monthly subscription fee, what are some
other factors that may affect the amount a user has to
pay each month? How can you simplify the problem
so that it is easier to carry out a comparison?
• Some examples of assumptions that can be made to simplify
the problem are as follows:
(i) The subscription contract spans a period of two years
and within the two years, the monthly subscription fees,
as well as the terms and conditions of the price plans,
do not change .
(ii) The charges for overseas incoming calls and overseas
outgoing calls are the same for all three price plans.
(iii) Other mobile services, e.g. caller ID, multimedia
messaging service (MMS) etc., provided are set at the
same price.
(iv) The amount of data used each month is the same .
(v) The average download speeds are the same.
•
For higher-ability students, teachers may wish to remove one
assumption from the assumptions made, and get them to come up
with another model for the problem.
5. For this problem, teachers may wish to first set the budget and the
fundraising target for the students.
The Mathematical Modelling Process consists of the following
steps:
A. Formulating
• Students are required to understand the information given
in the question and a discussion may be carried out to help
them comprehend the problem. Some guiding questions are
as follows:
(i) How many cookies do you estimate your class will
be able to sell? How many cookies should your class
make? Does the number of cookies your class makes
have to be a multiple of 48? How would this affect the
ingredients required?
(ii) Based on the number of cookies your class decides to
make, what is the total amount of money required to
purchase the ingredients?
Teachers may wish to ask students to state other assumptions
that may be made.
In this problem, we shall consider data usage to be the only
variable.
Teachers may wish to ask students what is meant by ‘data usage
is the only variable’ and get them to give some other examples
of variables. They may also wish to guide students to classify
the information into three categories, i.e. when data usage does
not exceed 12 GB, when data usage exceeds 12 GB but does
not reach the cap of $30 and when data usage reaches the cap
of $30.
1
Teachers may wish to ask students whether they are
able to buy the exact quantities of ingredients required
to bake the cookies, e.g. 20 teaspoons of baking soda,
and how this may affect the total cost.
(iii) What is the budget for the fundraising event? Are there
other costs that need to be taken into consideration?
Does the total cost lie within the budget? If not, what
can be done to ensure that the total cost lies within the
budget?
242
•
(iv) What is the fundraising target that needs to be met?
Besides the fundraising target, what are some other
factors which need to be considered before pricing the
cookies? How should you price your cookies?
Some examples of assumptions that can be made to simplify
the problem are as follows:
(i) Miscellaneous costs, e.g. transport costs for travelling
to buy the ingredients and cost of electricity used to
bake the cookies, are not taken into consideration when
calculating the total cost .
(ii) There is no wastage of ingredients .
Teachers may wish to ask students to state other assumptions
that may be made.
B. Solving
• How do you solve the problem mathematically?
• How should you present the results from your calculations?
C. Interpreting
Students should be able to interpret the mathematical solution in
the context of the real world, i.e. they should be able to advise
the class on the number of cookies that need to be made and the
price at which they should sell the cookies in order to maximise
their profit based on the mathematical solution that they have
obtained .
D. Reflecting
• After finding the number of cookies that need to be made
and the price at which they should sell the cookies in order
to maximise their profit, students should check whether
there are any other factors that may affect the profit made,
e.g. wastage at the end of the day.
• Students should also review the method that they have used
and consider whether there are other methods that can be
used to solve the problem.
For higher-ability students, teachers may wish to remove one
assumption from the assumptions made, and get them to come up
with another model for the problem.
243
1
NOTES
CONTENTS
1
244
mughal 55