Focus PT3 2020 - Mathematics ( BI Version )

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CC037230
ISBN: 978-967-2375-04-3 Programme

Ng Seng How (Textbook Writer) • Ooi Soo Huat (Textbook Writer)
PELANGI Yong Kuan Yeoh (Textbook Writer) • Moy Wah Goon (Textbook Writer)
Chiang Kok Wei • Samantha Neo

CONTENTS












Form 1 4.5 Relationship between Ratios, Rates
and Proportions with Percentages,
Fractions and Decimals 59
LEARNING AREA Number and Operations PT3 Practice 4 63

Chapter Chapter
1 Rational Numbers 1 5 Algebraic Expressions 65


1.1 Integers 2 5.1 Variables and Algebraic
1.2 Basic Arithmetic Operations Expressions 66
involving Integers 4 5.2 Algebraic Expressions involving
1.3 Positive and Negative Fractions 8 Basic Arithmetic Operations 71
1.4 Positive and Negative Decimals 12 PT3 Practice 5 76
1.5 Rational Numbers 14
PT3 Practice 1 16 Chapter
6 Linear Equations 78
Chapter
2 Factors and Multiples 18 6.1 Linear Equations in One Variable 79
6.2 Linear Equations in Two Variables 82
2.1 Factors, Prime Factors and Highest 6.3 Simultaneous Linear Equations
Common Factor (HCF) 19 in Two Variables 85
2.2 Multiples, Common Multiples and PT3 Practice 6 88
Lowest Common Multiple (LCM) 23
PT3 Practice 2 26 Chapter
7 Linear Inequalities 90
Chapter Squares, Square Roots, Cubes
3 and Cube Roots 28 7.1 Inequalities 91
7.2 Linear Inequalities in One
3.1 Squares and Square Roots 29 Variable 94
3.2 Cubes and Cube Roots 36 PT3 Practice 7 99
PT3 Practice 3 44
Measuremant and
LEARNING AREA Geometry
LEARNING AREA Relationship and Algebra
Chapter Lines and Angles 100
Chapter 8
4 Ratios, Rates and Proportions 46 8.1 Lines and Angles 101

8.2 Angles related to Intersecting
4.1 Ratios 47 Lines 116
4.2 Rates 51 8.3 Angles related to Parallel Lines
4.3 Proportions 53 and Transversals 118
4.4 Ratios, Rates and Proportions 54
PT3 Practice 8 124





v

Chapter Measurement and
9 Basic Polygons 127 LEARNING AREA Geometry


9.1 Polygons 128 Chapter
9.2 Properties of Triangles and 13 The Pythagoras Theorem 195
the Interior and Exterior Angles
of a Triangle 129 13.1 The Pythagoras Theorem 196
9.3 Properties of Quadrilaterals 13.2 The Converse of Pythagoras
and the Interior and Exterior Theorem 199
Angles of a Quadrilateral 133 PT3 Practice 13 202
PT3 Practice 9 137

Chapter Form 2
10 Perimeter and Area 140
LEARNING AREA Number and Operations
10.1 Perimeter 141 Chapter
10.2 Area of Triangles, Parallelograms, 1 Patterns and Sequences 204
Kites and Trapeziums 146
10.3 Relationship between Perimeter 1.1 Patterns 205
and Area 154 1.2 Sequences 206
PT3 Practice 10 157 1.3 Patterns and Sequences 209
PT3 Practice 1 212


LEARNING AREA Discrete Mathematics LEARNING AREA Relationship and Algebra
Chapter Factorisation and Algebraic
Chapter 2 Fractions 215
11 Introduction to Set 161
2.1 Expansion 216
11.1 Set 162 2.2 Factorisation 220
11.2 Venn Diagrams, Universal 2.3 Algebraic Expressions and Laws
Sets, Complement of a Set of Basic Arithmetic Operations 224
and Subsets 166 PT3 Practice 2 226
PT3 Practice 11 173 Chapter
3 Algebraic Formulae 228


3.1 Algebraic Formulae 229
LEARNING AREA Statistics and Probability
PT3 Practice 3 234
Chapter
12 Data Handling 175 Measurement and
LEARNING AREA Geometry
12.1 Data Collection, Organization
and Representation Process, Chapter
and Interpretation of Data 4 Polygons 236
Representation 176 4.1 Regular Polygons 237
PT3 Practice 12 192
4.2 Interior Angles and Exterior
Angles of Polygons 239
PT3 Practice 4 245



vi

Chapter Chapter
5 Circles 248 9 Speed and Acceleration 306


5.1 Properties of Circles 249 9.1 Speed 307
5.2 Symmetrical Properties 9.2 Acceleration 312
of Chords 251 PT3 Practice 9 314
5.3 Circumference and
Area of a Circle 253 Chapter
PT3 Practice 5 259 10 Gradient of a Straight Line 316
10.1 Gradient 317
Chapter Three-Dimensional Geometrical PT3 Practice 10 323
6 Shapes 262


6.1 Geometric Properties Measurement and
of Three-Dimensional Shapes 263 LEARNING AREA Geometry
6.2 Nets of Three-
Dimensional Shapes 265 Chapter
6.3 Surface Area of Three- 11 Isometric Transformations 326
Dimensional Shapes 267
6.4 Volume of Three-Dimensional 11.1 Transformations 327
Shapes 274 11.2 Translation 328
PT3 Practice 6 280 11.3 Reflection 335
11.4 Rotation 340
11.5 Translation, Reflection
and Rotation as an Isometry 346
11.6 Rotational Symmetry 348
LEARNING AREA Relationship and Algebra PT3 Practice 11 349

Chapter
7 Coordinates 283
LEARNING AREA Statistics and Probability
7.1 Distance in the Cartesian
Coordinate System 284 Chapter
7.2 Midpoint in the Cartesian 12 Measures of Central Tendencies 352
Coordinate System 287
7.3 The Cartesian 12.1 Measures of Central Tendencies 353
Coordinate System 289 PT3 Practice 12 363
PT3 Practice 7 291
Chapter
13 Simple Probability 366
Chapter
8 Graphs of Functions 293 13.1 Experimental Probability 367
13.2 Probability Theory involving
8.1 Functions 294 Equally Likely Outcomes 369
8.2 Graphs of Functions 297 13.3 Probability of the Complement
PT3 Practice 8 304 of an Event 372
13.4 Simple Probability 374
PT3 Practice 13 376






vii

Form 3 Chapter
6 Angles and Tangents of Circles 436
LEARNING AREA Number and Operations 6.1 Angle at the Circumference and
Central Angle Subtended by
Chapter an Arc 437
1 Indices 378 6.2 Cyclic Quadrilaterals 440
6.3 Tangents to Circles 443
1.1 Index Notation 379 6.4 Angles and Tangents of Circles 449
1.2 Laws of Indices 380 PT3 Practice 6 451
PT3 Practice 1 386

Chapter Chapter
2 Standard Form 387 7 Plans and Elevations 453


2.1 Significant Figures 388 7.1 Orthogonal Projections 454
2.2 Standard Form 391 7.2 Plans and Elevations 457
PT3 Practice 2 395 PT3 Practice 7 468
Consumer Mathematics: Chapter
Chapter Savings and Investments, 8 Loci in Two Dimensions 472
3 Credit and Debt 397
8.1 Loci 473
3.1 Savings and Investments 398 8.2 Loci in Two Dimensions 474
3.2 Management of Credit and PT3 Practice 8 480
Debt 405
PT3 Practice 3 412



Measurement and LEARNING AREA Relationship and Algebra
LEARNING AREA Geometry
Chapter
Chapter 9 Straight Lines 483
4 Scale Drawings 414
9.1 Straight Lines 484
4.1 Scale Drawings 415 PT3 Practice 9 494
PT3 Practice 4 423
Answers 497
Chapter
5 Trigonometric Ratios 425

5.1 Sine, Cosine and Tangent of Acute
Angles in Right-angled Triangle 426
PT3 Practice 5 434












viii

Learning Area : Number and Operations
Chapter Form 1
Mathematics PT3 Chapter 2 Factors and Multiples
2 Factor and Multiples





















































Selina and Daniel work in the same office. Selina will take a rest every 30 minutes while Daniel
will take a rest every 45 minutes. If they take a rest at the same time at 10:00 a.m., what time
that they will take a rest together again?





KEYWORDS

• Factor • Multiple
• Prime factor • Common multiple
Concept • Common factor • Lowest common
Map • Highest common multiple (LCM)
factor (HCF) • Prime factorisation





18

Mathematics PT3 Chapter 2 Factors and Multiples

2.1 Factors, Prime Factors and TIPS
Highest Common Factor (HCF) Form Form
• 1 is a factor of all numbers.
• Each number has at least two factors, that
2.1.1 Determine and list the factors is 1 and the number itself. 1
of whole numbers


1. The factors of a whole number are the 2.1.2 Determine and list the prime
numbers that can divide the whole factors of a whole number
number exactly.
For example, 3 can divide 12 exactly. 1. The prime factors of a whole number are
Therefore, 3 is a factor of 12.
the prime numbers which are also the
2. A whole number has two or more factors factors of the whole number.
that can be listed. For example, 2 and 3 are the factors of 12
For example, 1, 2, 3, 4, 6 and 12 can divide and 2 and 3 are prime numbers. Therefore,
12 exactly. Therefore, 1, 2, 3, 4, 6 and 12 are 2 and 3 are the prime factors of 12.
the factors of 12.
3

1 List all the prime factors of 36.
Solution:
Determine whether 36 = 1 × 36
(a) 6 is a factor of 42, 36 = 2 × 18
(b) 8 is a factor of 52.
36 = 3 × 12
Solution: 36 = 4 × 9
(a) 42 ÷ 6 = 7 36 = 6 × 6
Therefore, 6 is a factor of 42. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

(b) 52 ÷ 8 = 6 remainder 4 Therefore, the prime factors of 36 are 2 and 3.
8 cannot divide 52 exactly.
Therefore, 8 is not a factor of 52. MATHS INFO

INFO
Try question 1 in Formative Practice 2.1 A prime number is a number that can only be
divided by the number 1 and the number itself.

2 2. A whole number can be expressed as a

List all the factors of 56. product of the prime factors.
Solution: 3. The process of expressing the whole
56 = 1 × 56 number as the product of prime factors is
56 = 2 × 28 known as prime factorisation.
56 = 4 × 14
56 = 7 × 8 TIPS
Therefore, the factors of 56 are
1, 2, 4, 7, 8, 14, 28 and 56. Prime factorisation can be carried out by
• repeated division
• factor tree method
Try question 2 in Formative Practice 2.1





19

Mathematics PT3 Chapter 2 Factors and Multiples
4 5

Express 84 in the form of prime factorisation. Determine whether
Solution: (a) 8 is a common factor of 32 and 48,
Form
Form
1 Method 1: Repeated division (b) 9 is a common factor of 18, 36 and 56,
(c) 13 is a common factor of 26, 52, 91 and
143.
2 84 1 Repeated division is performed
starting with the smallest prime Solution:
2 42 number. (a) 32 ÷ 8 = 4
3 21 2 The process of division with the 48 ÷ 8 = 6
prime numbers is continued until
7 7 the quotient is 1. Therefore, 8 is a common factor of 32 and
1 48.

(b) 18 ÷ 9 = 2
Therefore, 84 = 2 × 2 × 3 × 7 36 ÷ 9 = 4
The product of all divisors. 56 ÷ 9 = 6 remainder 2
56 cannot be divided exactly by 9.
Method 2: Factor tree Therefore, 9 is not a common factor of 18,
36 and 56.
84
(c) 26 ÷ 13 = 2 52 ÷ 13 = 4
84 = 12 × 7 12 7 91 ÷ 13 = 7 143 ÷ 13 = 11
Therefore, 13 is a common factor of 26, 52,
12 = 2 × 6 2 6 7 91 and 143.

2 2 3 7 Try question 4 in Formative Practice 2.1

84 is expressed as a product of
two factors and then performed 6
continuously until all the factors at
the bottom level are prime factors.
List all the common factors of 24, 36, 42 and
78.
Therefore, 84 = 2 × 2 × 3 × 7 Solution:
Factors of 24 : 1 , 2 , 3 , 4, 6 , 8, 12, 24
Try question 3 in Formative Practice 2.1
Factors of 36 : 1 , 2 , 3 , 4, 6 , 9, 12, 18, 36
Factors of 42 : 1 , 2 , 3 , 6 , 7, 14, 21, 42
Factors of 78 : 1 , 2 , 3 , 6 , 13, 26, 39, 78
2.1.3 Explain and determine the
common factors of whole Therefore, the common factors of 24, 36, 42 and
numbers 78 are 1, 2, 3 and 6.

1. The common factor of two or more whole Try question 5 in Formative Practice 2.1
numbers is the number which is a factor
of each whole number.
For example, 3 is a factor of 6 and 12. 2.1.4 Determine the highest common
Therefore, 3 is a common factor of 6 and factor (HCF)
12.

2. These common factors can be determined 1. The highest common factor (HCF) of
by using the method of listing all the factors several whole numbers is the largest
of each whole number. common factor of these numbers.





20

Mathematics PT3 Chapter 2 Factors and Multiples
For example, in Example 6, the common Therefore, the HCF of 24, 48 and 60 is
factors of 24, 36, 42 and 78 are 1, 2, 3 and 2 × 2 × 3 = 12. The product of all the
6. Therefore, the HCF of 24, 36, 42 and 78 common prime factors. Form
is 6.
Form
Try question 6 in Formative Practice 2.1 1
TIPS
HCF can be determined by Common mistake 1
• listing all the common factors
• repeated division INFO
• prime factorisation


7 2.1.5 Solve problems involving HCF

Find the highest common factor (HCF) of 24, 1. There are four steps that can be followed
48 and 60. in solving the problems involving HCF,
namely
Solution: 1 Understanding the problem
Method 1: Listing the factors 2 Planning a strategy
Factors of 24 : 1 , 2 , 3 , 4 , 6 , 8, 12 , 24 3 Implementing the strategy
Factors of 48 : 1 , 2 , 3 , 4 , 6 , 8, 12 , 16, 4 Making a conclusion
24, 48
Factors of 60 : 1 , 2 , 3 , 4 , 5, 6 , 10, 12 , 8 PAK-21 Daily Application
-
21
AK
P
15, 20, 30, 60
A piece of cardboard has length of 260 cm
The common factors of 24, 48 and 60 and width of 80 cm. Dania wants to cut the
are 1, 2, 3, 4, 6 and 12. So, the highest cardboard into several pieces of squares. What
common factor is 12.
is the largest measurement of side of the square
such that no cardboard is left?
Therefore, the HCF of 24, 48 and 60 is 12.
Solution:
Method 2: Repeated division 1 Understanding the problem
2 24 48 60 A cardboard with length of 260 cm and
width of 80 cm is cut into several pieces of
2 12 24 30
squares. Find the largest side of square that
3 6 12 15 Divide the numbers can be cut.
repeatedly with the
2 4 5 common factors until 2 Planning a strategy
it cannot be divided Find the HCF of 260 and 80.
further.
3 Implementing the strategy
Therefore, the HCF of 24, 48 and 60 is
2 × 2 × 3 = 12. 2 260 80 The HCF of 260 and
80 is 2 × 2 × 5 = 20.
The product of all divisors. 2 130 40
The largest
Method 3: Prime factorisation 5 65 20 measurement of
13 4
each side of square
24 = 2 × 2 × 2 × 3
is 20 cm.
48 = 2 × 2 × 2 × 2 × 3 4 Making a conclusion
60 = 2 × 2 × 3 × 5 260 ÷ 20 = 13
80 ÷ 20 = 4

1 Perform prime factorisation for each number. Try question 7 in Formative Practice 2.1
2 Identify all the common prime factors of the numbers.

21

Mathematics PT3 Chapter 2 Factors and Multiples
9 P PAK-21 Daily Application Solution:
21
-
AK
Separate the cakes into 2 portions where the
Alia divided 60 apples, 78 oranges and 90 kiwi number of cakes in each portion is a factor of 10.
fruits to several neighbours.
Form
Form
1 (a) What is the maximum number of neighbours Therefore, the division for 2 possible portions
is 2 and 5 cakes respectively.
to be given so that all the fruits are equally
distributed to them with none of the fruits
remained? 2 and 5 are the factors of 10.
(b) How many apples, oranges and kiwi fruits
received by each neighbour?
Portion 1: Portion 2:
Solution:
(a)
2 60 78 90
3 30 39 45

10 13 15

The HCF of 60, 78 and 90 is 2 × 3 = 6.
Therefore, the maximum number of For portion 1:
neighbours who received the fruits equally Cut 2 cakes into 10 slices of cake with equal size.
from Alia is 6. (5 slices for a cake)
(b) The number of fruits received by each
neighbour is 10 apples, 13 oranges and 15
kiwi fruits.
For portion 2:
Try questions 8 and 9 in Formative Practice 2.1 Cut 5 cakes into 10 slices of cake with equal size.
(2 slices for a cake)
TIMSS Challenge

The number of audience watching a mini orchestra
is more than 91 people but less than 100 people.
The number of audience is an even number and
can be divided exactly by 8. Calculate the number
of audience in the mini orchestra.
Therefore, each plate has a slice of cake from
portion 1 and portion 2 respectively.

HOTS Challenge
Praktis Formatif 1
Formative Practice 2.1


1. Determine whether each of the following
numbers is a factor of the numbers in brackets.
(a) 3 (33) (b) 6 (65)

(c) 7 (91) (d) 11 (121)
The diagram above shows 7 cuboid-shaped cakes (e) 5 (40) (f) 13 (182)
of equal size. How do you cut those 7 cakes so that
you can distribute them evenly on 10 plates where 2. List all the factors of each of the following
each plate has two slices of cake? numbers. (b) 12
(a) 34
(Hint: The two slices of cake do not have the same (c) 78 (d) 105
size) (e) 98 (f) 111






22

Mathematics PT3 Chapter 2 Factors and Multiples

3. Express each of the following numbers in the 2.2 Multiples, Common Multiples
form of prime factorisation. and Lowest Common Multiple Form
(a) 95 (b) 48
(c) 74 (d) 32 (LCM)
Form
(e) 65 (f) 124 1
2.2.1 Explain and determine the
4. Determine whether each of the following common multiples of whole
numbers is a common factor for the set of
numbers in brackets. numbers
(a) 4 (24, 60)
(b) 12 (84, 132) 1. The common multiple of several whole
(c) 5 (15, 30, 50) numbers is the number which is the
(d) 9 (27, 68, 108) multiple of all the whole numbers.
(e) 13 (26, 52, 78, 104) For example, 24 is a multiple of 6 and 12
(f) 8 (40, 64, 130, 168) respectively, thus 24 is a common multiple
of 6 and 12.
5. List all the common factors of each of the
following set of numbers.
(a) 12, 40 MATHS INFO

INFO
(b) 52, 84
(c) 24, 36, 50
(d) 35, 60, 75 A multiple of a whole number is the number
(e) 84, 132, 300, 420 which is the product of itself with another whole
(f) 42, 102, 144, 240 number.
6. Find the highest common factor (HCF) of each
of the following set of numbers. 10
(a) 36, 48
(b) 18, 72 Determine whether
(c) 30, 96 (a) 20 is a common multiple of 2 and 5,
(d) 20, 50 (b) 32 is a common multiple of 4, 8 and 12,
(e) 24, 90, 108 (c) 36 is a common multiple of 3, 9, 12 and 18.
(f) 144, 258, 348
Solution:
7. Wahid wants to install square tiles on a floor (a) 20 ÷ 2 = 10
measuring 84 m × 32 m. What is the largest
measurement of tiles that can be used so that 20 ÷ 5 = 4
the tiles would cover the entire floor? 20 can be divided exactly by 2 and 5.
Analysing Therefore, 20 is a common multiple of 2
and 5.
8. Madam Hani wants to wrap 171 pencils, 63
rulers and 27 pens into several gift packs. The (b) 32 ÷ 4 = 8 32 ÷ 8 = 4
pencils, rulers and pens are evenly distributed 32 ÷ 12 = 2 remainder 8
in each pack where none of them are left. What 32 cannot be divided exactly by 12.
is the most number of gift packs that could be
wrapped? Analysing Therefore, 32 is not a common multiple of
4, 8 and 12.
9. Three ribbons with length 160 cm, 256 cm and (c) 36 ÷ 3 = 12 36 ÷ 9 = 4
288 cm respectively were cut into strips of equal 36 ÷ 12 = 3 36 ÷ 18 = 2
length.
(a) What is the longest length of each strip 36 can be divided exactly by 3, 9, 12 and
such that there is no remaining ribbon? 18.
Therefore, 36 is a common multiple of 3, 9,
Analysing
12 and 18.
(b) What is the total number of strips of ribbon
that could be cut? Evaluating
Try question 1 in Formative Practice 2.2




23

Mathematics PT3 Chapter 2 Factors and Multiples
TIPS For example, in Example 11, the common
multiples of 6, 9 and 12 are 36, 72, 108,
If a number is a multiple of a whole number, 144, … thus the lowest common multiple
then the number can be divided exactly by of 6, 9 and 12 is 36.
Form
Form
1 the whole number.
TIPS
2. The list of common multiples of several
whole numbers can be determined as LCM can be determined by
• listing the common multiples
follows.
• repeated division
1 List the • prime factorisation
multiples of 2 Determine
each of the the first
given whole common 12
number. multiple.
Find the lowest common multiple (LCM) of 6,
12 and 15.
Solution:
3 The subsequent common
multiples can be listed Method 1: Listing the multiples
by multiplying the first Multiples of 6 : 6, 12, 18, 24, 30, 36, 42, 48, 54,
common multiple with other 60 , …
consecutive whole numbers.
Multiples of 12 : 12, 24, 36, 48, 60 , …
Multiples of 15 : 15, 30, 45, 60 , …
11
List the first five common multiples of 6, 9 Therefore, the LCM of 6, 12 and 15 is 60.
and 12.
Method 2: Repeated division
Solution:
Multiples of 6 : 6, 12, 18, 24, 30, 36, … The numbers that
Multiples of 9 : 9, 18, 27, 36, … 2 6 12 15 cannot be divided
Multiples of 12 : 12, 24, 36, … 2 3 6 15 exactly by the divisors
are brought down for
3 3 3 15
The first common multiples of 6, 9, 12 is 36. subsequent divisions.
5 1 1 5
Therefore, the first five common multiples of 1 1 1
6, 9 and 12 are
= 36 × 1, 36 × 2, 36 × 3, 36 × 4, 36 × 5 Division is performed continuously with
= 36, 72, 108, 144, 180 the divisor that can divide exactly at
least one of the numbers until all the
Try question 2 in Formative Practice 2.2 quotients become 1.



2.2.2 Determine the lowest common Therefore, the LCM of 6, 12 and 15 is
multiple (LCM) 2 × 2 × 3 × 5 = 60.
The product of all divisors.
1. The lowest common multiple (LCM) of
several whole numbers is the smallest Common mistake 2
common multiple of these numbers.
INFO



24

Mathematics PT3 Chapter 2 Factors and Multiples
Method 3: Prime factorisation Solution:
1 Understanding the problem
6 = 2 × 3 Form Form
12 = 2 × 2 × 3 The canteen serves curry noodles for every
15 = 3 × 5 4 days and ‘mi jawa’ for every 6 days. Find
which subsequent day where both types of 1

2 × 2 × 3 × 5 = 60 noodles will be served again.

2 Planning a strategy
Therefore, the LCM of 6, 12 and 15 is 60.
Find the LCM of 4 and 6.
3 Implementing the strategy
Alternative Method

2 4 6
Between the numbers 6, 12, and 15
2 2 3
1 Identify the largest number. 3 1 3
1 1
15, 30, 45, 60, 75, …
2 List the multiples of the The LCM of 4 and 6 is 2 × 2 × 3 = 12.
60 can be largest number. Both noodles will be served again on the
divided exactly 3 Determine the lowest
by 6, 12 and 15. multiple that can be divided same day after 12 days.
exactly by 6, 12 and 15.
4 Making a conclusion
12 ÷ 3 = 4
Therefore, the LCM of 6, 12 and 15 is 60.
12 ÷ 2 = 6

Try question 3 in Formative Practice 2.2 Try questions 4 and 5 in Formative Practice 2.2


2.2.3 Solve problems involving LCM Formative Practice 2.2


1. There are four steps that can be followed 1. Determine whether each of the following
in solving the problems involving LCM, numbers is a common multiple of the set of
namely numbers in brackets.
1 Understanding the problem (a) 108 (9, 18)
2 Planning a strategy (b) 180 (4, 9)
(c) 240 (6, 8, 24)
3 Implementing the strategy (d) 2 150 (3, 5, 7)
4 Making a conclusion (e) 1 320 (5, 6, 8, 15)
(f) 6 552 (8, 12, 18, 28, 36)
2. List the first five common multiples for each of
13 PAK-21 Daily Application the following set of numbers.
21
-
AK
P
(a) 3, 5
A canteen serves curry noodles for every 4 days (b) 4, 9
and ‘mi jawa’ for every 6 days. If the canteen (c) 16, 28
serves both curry noodles and ‘mi jawa’ noodles (d) 4, 8, 18
on a particular day, after how many days later (e) 2, 3, 5, 6
would both types of noodles be served again (f) 3, 6, 12, 20
on the same day? (g) 3, 18, 21, 28
(h) 4, 10, 15, 18, 28





25

Mathematics PT3 Chapter 2 Factors and Multiples

3. By using a suitable method, find the lowest Gopal wants to buy the same number of
common multiple for each of the following. balloons for both colours of the balloon for a
(a) 7, 8 birthday party. What is the least number of
(b) 12, 15 packets that should be bought by Gopal for
Form
Form
1 (c) 5, 15 each colour of the balloon?
(d) 18, 90 Analysing Evaluating
(e) 6, 15, 18
(f) 6, 16, 36 5. The bell in a secondary school rings every 45
(g) 7, 14, 49 minutes while the bell in a primary school rings
(h) 2, 15, 30 every 30 minutes. If both bells ring simultaneously
at a particular time, after how many minutes will
4. Two packets of red and purple balloons are sold both bells ring again simultaneously?
in 6 units and 8 units per packet respectively. Applying Evaluating




PT3 Practice 2



Section A 8. Which of the following statements is not true?
A 2 and 3 are factors of 36.
1. Which of the following numbers is not a factor B 24 and 36 are multiples of 12.
of 14? C 1, 2, 5 are prime factors of 10.
A 1 C 3
B 2 D 7 D 9 has three factors.
Section B
2. Which of the following numbers is the sum of all
the factors of 32? 1. The diagram below shows a bubble map.
A 30 C 63 Complete the bubble map to show the factors of
B 62 D 31 28. i-THINK [4 marks]
3. The highest common factor of 12, 48, 60 is
A 3 C 6
B 4 D 12
4. 14
b = 2 × 3 × a
a = 3 × 4 × c 28

Based on the above information, which of the
following is not true?
A 3 is a common factor of a and b. 7
B a is a factor of b.
C c is a multiple of a.
D 3 is a prime factor of a. 2. Mark (✓) for the numbers which are factors of
the given numbers.
5. Which of the following is a common multiple of
4, 8 and 10? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A 30 C 50 12
B 40 D 60
16
6. The number 24 is the lowest common multiple of
A 2, 3, 6 C 3, 4, 6 3. (a) Fill in the blanks. i-THINK [2 marks]
B 3, 6, 12 D 3, 6, 8
in the form
7. Given that 64 is a common multiple of 4 and x. of prime
Which of the following is not the value of x? factorisation 60 420
A 16 C 32 Relating as
B 18 D 64 factor



26

Mathematics PT3 Chapter 2 Factors and Multiples
(b) Complete the following blanks to show the (b) Find the smallest number that can be divided
determination of the lowest common multiple by 8 and 20 exactly. [2 marks]
(LCM) of the given numbers by using (c) Ruby wants to prepare a scrapbook about Form
repeated division. [2 marks] environmental pollution by using
Form 36 photos
and 54 printed articles from the Internet. If the
2 2 4 9 number of photos and the number of articles 1
2 1 2 9 for each page are the same and none of the
3 1 1 9 photos or articles are left,
3 1 1 3 (i) what is the most number of pages that
1 1 1 can be prepared for the scrapbook?
Evaluating [3 marks]
Therefore, the LCM of 2, 4 and 9 is (ii) how many photos and articles are there
= 2 × × 3 × 3 in each page of the scrapbook?
= Analysing [3 marks]
4. (a) List the first four multiples of 23. [4 marks]
4. (a) List all the prime factors of 36. [2 marks] (b) Find the largest number that divides 42 and
90 exactly. [2 marks]
(b) Complete the following blanks to show
the determination of the highest common (c) Three ropes A, B and C have the same
factor (HCF) of the given numbers by using length. Rope A, B and C are cut into 12 cm,
repeated division. [2 marks] 28 cm and 45 cm of sections, respectively.
It is found that all the ropes are completely
2 36 44 56 cut and none are left. What is the shortest
2 18 22 28 length, in cm, of the rope A, B and C?
9 11 14 [4 marks]
Therefore, the HCF of 36, 44 and 56 is 5. (a) Chong wants to buy souvenirs for his friends.
= 2 × The souvenirs that he chose are sold in 10
pieces per packet but the suitable wrappers
=
for the souvenirs are sold in 6 pieces per
packet. What is the least number of packets
Section C of souvenirs that should be bought by Chong
so that the wrappers he bought are exactly
1. (a) List the first four common multiples of 3, 4 enough for all the souvenirs? [3 marks]
and 6. [4 marks] (b) Mr. Muthi is provided with 120 erasers, 180
(b) Find the highest common factor (HCF) of 20, pencils and 240 pens to be wrapped as
28 and 32. [2 marks] souvenirs which will be given in a workshop.
If all the souvenirs wrapped are the same and
(c) Write the number 42 as the sum of two prime all the stationery provided have been used up
numbers. Give all the possible answers. for wrapping,
[4 marks] (i) how many maximum packets of souvenirs
that are prepared? [2 marks]
2. (a) Find the difference between the largest and (ii) how many erasers, pencils and pens are
the smallest prime factor of the number there in each packet of souvenirs?
1 505. [4 marks]
Analysing [3 marks]
(b) Find the lowest common multiple (LCM) of 5,
9 and 12. [2 marks] (c) State one example of two numbers where
(c) Find the sum of all the prime factors of the LCM of the two numbers is the product
of those numbers. Explain your answer.
1 050. [4 marks] [2 marks]
3. (a) The lowest common multiple of n and 12 is
60. Find the smallest value of n. [2 marks]





27

Mathematics
Mathematics PMR Answers
Mathematics PT3 Answers





6. (a) 491 (b) 530 7 1 5 1 3
Form 1 (c) 4 900 (d) 27 (d) (i) –1 8 , –1 2 , – , 1 4 , 2
8
(e) 2 804 (f) 120 3 1 5 1 7
Chapter 7. 15 m below sea level (ii) 2 , 1 4 , – , –1 2 , –1 8
8
1 Rational Numbers 8. Water level decreases by 6 m. 3
9. (a) Leong will not be awarded a prize 4. (a) –1 8 (b) 3
1
Formative Practice 1.1 because he answered 13 questions (c) –1 18 (d) – 3 4
correctly, 2 questions incorrectly
6
1. Positive numbers: and 1 question not answered; The (e) – (f) 1 7
2 , 7.3, 5, +3 total marks that he obtained is 34. 7 9
5 (b) John will likely receive a prize. If (g) –1 1 (h) –4 1
Negative numbers: he answered 13 questions correctly 3 8
and 3 questions not answered, so
3
–10, –67, – , –8.9 the total marks obtained is 36 marks. (i) 4
4 15
If he answered 13 questions
(Zero is not a positive number and not correctly, 1 question incorrectly and 5. 2 250 m below sea level
a negative number) 2 questions not answered, maybe 7
2. (a) –7 levels (b) +500 m the total marks obtained is 35 6. Not enough. Still short of 10 kg of
(c) –RM450 (d) +RM0.50 marks. sugar.
(e) –150 m (f) –3 cm
3. (a) –100 Formative Practice 1.4
(b) +38 or 38
(c) –45 Formative Practice 1.3 1. (a)
4. Positive integer: 4; 1. (a)
Negative integer: –8, –3
5. (a) –0.4 –0.2 –0.1 0.2 0.3
– – 4 – – 2 – 1 – 3
–3 –2 1 2 4 7 7 7 7 (b)
(b)
(b) –5.2 –3.9 –1.3 2.6 3.9
–15 –9 –3 3 6 –1– 1 3 – – 2 3 0 – 1 3 2. (a) 2.9
6. (a) 8 (b) –15 (c) (b) –5.6
7. (a) Largest integer: 6; (c) –1.11
Smallest integer: –10 3. (a) (i) –6.7, –3.31, –1.4, 3.87, 4.5
(b) Largest integer: 14; – – 1 2 – – 1 3 – 1 6 – 1 3 (ii) 4.5, 3.87, –1.4, –3.31, –6.7
Smallest integer: –16 (b) (i) –5.2, –3.0, –0.4, 0.9, 1.4
8. (a) –6, –5, –4, –2, 1, 3 (d) (ii) 1.4, 0.9, –0.4, –3.0, –5.2
(b) –10, –8, –6, 3, 7, 9 (c) (i) –4.11, –3.22, –1.44, 1.55, 2.33
(ii) 2.33, 1.55, –1.44, –3.22, –4.11
(c) –16, –14, –3, 0, 11, 18 3 1 1 1 (d) (i) –5.44, –5.42, –2.9, 0.03, 0.3
(d) –19, –13, –4, –3, 4, 19 – – 8 – – 4 – 8 – 2 (ii) 0.3, 0.03, –2.9, –5.42, –5.44
9. (a) 5, 3, 0, –1, –4, –6 4. (a) –1.66 (b) –1.2
(b) 7, 5, 3, –1, –2, –6 2. (a) 4 (b) –1 1
(c) 4, 2, 1, –1, –3, –7 5 2 (c) –9.35 (d) 8.985
(d) –1, –3, –5, –8, –10, –12 (e) 2 (f) –6.665
(h) –5.647
(g) –19.35
(c) –2 3 (d) –2 1
5 9 (i) –3.072
Formative Practice 1.2 1 3 1 1 5 5. –4°C
3. (a) (i) – , – , – , , 6. Danny gained a profit of RM185.
1. (a) 5 (b) 3 2 8 8 2 8
3
1
(c) –1 (d) –14 (ii) 5 , 1 , – , – , – 1
(e) –3 (f) 11 8 2 8 8 2 TIMSS Challenge
(g) –10 (h) –4 7 2 1 1 3 ‒; +; ‒
2. (a) –16 (b) –20 (b) (i) – 10 , – , , , 5
5 10 2
(c) 42 (d) 36 3 1 1 2 7
,
3. (a) –15 (b) –7 (ii) 5 , 2 10 , – , – 10 Formative Practice 1.5
5
(c) 7 (d) 12 1. All the numbers given are rational
5
4. (a) 0 (b) 1 (c) (i) –1 1 , –1 1 , – , 4 , 7 numbers.
(c) 12 (d) 27 3 6 6 9 9 –8 63 1 6
5
5. (a) 10 (b) 0 (ii) 7 , 4 , – , –1 1 , –1 1 –8 = 1 ; 3.15 = 20 ; 1 5 = 5
(c) 40 (d) 17 9 9 6 6 3 2. (a) –7.2 (b) 2.58
(e) 15 (f) –28 (c) –0.3125 (d) –3.905
(g) –6 (h) –4
497

Mathematics PT3 Answers
3. (a) –9 3 (b) –1 1 Chapter PT3 Practice 2
4 2 2 Factors and Multiples
(c) –5 4 (d) – 7 Section A
5 20 1. C 2. C 3. D 4. C 5. B

4. 3 3 kg TIMSS Challenge 6. D 7. B 8. C
8
96 people Section B
5. 1.805 m 1.
6. Tharishini scored the highest marks. Formative Practice 2.1 1
Emilia: 5 questions answered correctly, 1. (a) Yes (b) No
4 questions answered incorrectly and (c) Yes (d) Yes 4
1 question not answered. The marks (e) Yes (f) Yes
obtained is 8.5. 28
Ker Er: 5 questions answered correctly, 2. (a) 1, 2, 17, 34
5 questions answered incorrectly and (b) 1, 2, 3, 4, 6, 12 2 28
(c) 1, 2, 3, 6, 13, 26, 39, 78
0 question not answered. The marks (d) 1, 3, 5, 7, 15, 21, 35, 105
obtained is 7.5. (e) 1, 2, 7, 14, 49, 98
Tharishini: 4 questions answered (f) 1, 3, 37, 111
correctly, 0 questions answered
incorrectly and 6 questions not 3. (a) 5 × 19 2.
(b) 2 × 2 × 2 × 2 × 3
answered. The marks obtained is 9. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(c) 2 × 37
(d) 2 × 2 × 2 × 2 × 2
(e) 5 × 13 12 3 3 3 3 3 3
PT3 Practice 1 (f) 2 × 2 × 31 16 3 3 3 3
4. (a) Yes (b) Yes
Section A (c) Yes (d) No
1. D 2. C 3. A 4. D 5. B (e) Yes (f) No 3. (a) 2 × 2 × 3 × 5 = 60
5. (a) 1, 2, 4 2 × 2 × 3 × 5 × 7 = 420
Section B (b) 1, 2, 4 (b) Thus, LCM for 2, 4 and 9
1. (a) 7, 3.8, 3 1 , –4.8, –6 4 , –10 (c) 1, 2 = 2 × 2 × 3 × 3
5 5 (d) 1, 5
(b) = 36
(e) 1, 2, 3, 4, 6, 12
(f) 1, 2, 3, 6 4. (a) 2, 3
(b) Thus, HCF for 36, 44 and 56
6. (a) 12 (b) 18
(c) 6 (d) 10 = 2 × 2
2. (a) 52 (e) 6 (f) 6 = 4
3
(b) (i) 4 (ii) – (iii) – 23 7. 4 m × 4 m
1 2 10 8. 9 gift packs Section C
3. (a) – 4, –2.3, –1 1 , 0, 8 , 3.5 9. (a) 32 cm 1. (a) 12, 24, 36, 48
2 4 (b) 4
(b) (b) 22 strips of ribbon
(i) √7 7 (c) 5 + 37, 11 + 31, 13 + 29, 19 + 23
(ii) –3.5 3 Formative Practice 2.2 2. (a) 38
(b) 180
(iii) π 7 1. (a) Yes (c) 17
(b) Yes 3. (a) 5
Section C (c) Yes (b) 40
1. (a) 403 (d) No (c) (i) 18 pages
(e) Yes (ii) 2 photo and 3 paragraph
(b) (f) Yes 4. (a) 23, 46, 69, 92
8.5 + 3.75 16 – 3.75 2. (a) 15, 30, 45, 60, 75 (b) 6
(b) 36, 72, 108, 144, 180 (c) 1 260 cm
12 1 (c) 112, 224, 336, 448, 560 5. (a) 3 packs of souvenirs
4 (d) 72, 144, 216, 288, 360 (b) (i) 60 pack
(e) 30, 60, 90, 120, 150 (ii) 2 erasers, 3 pencils and 4 pens
(f) 60, 120, 180, 240, 300 (c) 3 and 4
7 × 7 49 ÷ 4 (g) 252, 504, 756, 1 008, 1 260
2 2
(h) 1 260, 2 520, 3 780, 5 040, 6 300
(c) –7 3. (a) 56 Chapter 3 Squares, Square Roots,
(b) 60
2. (a) (i) Step 2; 2 2 9 (c) 15 Cubes and Cube Roots
(ii) 0.5125 (d) 90
(b) RM680 (e) 90 Formative Practice 3.1
(c) Mr Foo made a profit of RM50. (f) 144
2
Mr. Foo bought the shares when the (g) 98 1. (a) 4 (b) 7 2
price of shares decreased because (h) 30 2.
the average cost of 1 unit of shares 4. 4 packets of red balloon and 3 packets
is lower. of purple balloon
5. 90 minutes
498

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