11111
SL Education
New Syllabus Mathematics
8 th Edition
Secondary
Name:
Class
:
School:
1A
New Syllabus Mathematics
8 th Edition
Anglo Singapore International School 64 Campus
11111111111111 II
048476
Nautilus Shell
Secondary
1A
i)
SHINGLEE PUBLISHERS PTE LTD
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Kewalram Hillview Singapore 669594
Tel: 6760 1388 Fax: 6762 5684
e-mail: info@shinglee.com.sg
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©SHINGLEE PUBLISHERS PTE LTD
All rights reserved. No part of this book may be reproduced, stored in a retrieval
system or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording, orotherwise, without the prior permission of the Publishers.
First Published 2020
Second Reprint 2020
ISBN 978 981 32 4537 2
Printed in Singapore
think! Mathematics workbooks are specially designed to complement the textbooks as you acquire
mathematical concepts and skills.
Key Features
Questions that connect ideas within mathematics and between mathematics and the sciences through
applications of mathematics are included to allow you to better appreciate mathematics.
4.
An engineer calculated that the maximum current that can flow in a particular circuit before it is
considered unsafe is 10.52 amperes.
He is asked to indicate the maximum current correct to the nearest ampere.
What should the engineer indicate the maximum current as? Why?
One equation should be in the form hx + k 1, where hand k are proper fractions.
5
One equation should be in the form - - = 4, where p and q are rational numbers.
px+q
A Mid-year Checkpoint or an End-of-year Checkpoint consolidates the concepts covered in the textbook, and
is useful for assessing your learning and identifying areas that require further practice.
. .
h Sx+y
l
7. It1sg1ventat
x- y= .
7 3
2
Find the value of l...
X
Answer _ _ _ _ _ __
We hope that these workbooks provide sufficient practice as you build your confidence and gain success
in mathematics.
Chapter 1
Chapter 2
Chapter 3
Primes, Highest Common Factor and Lowest Common Multiple
Worksheet lA: Prime numbers
Worksheet lB: Square roots and cube roots
Worksheet 1C: Highest common factor and lowest common multiple
"'"''" "''" Exercise
Integers, Rational Numbers and Real Numbers
Worksheet 2A: Negative numbers
Worksheet 2B: Addition and subtraction involving negative integers
Worksheet 2C: Multiplication, division and combined operations involving
negative integers
Worksheet 2D: Fractions and mixed numbers
Worksheet 2E: Decimals
Worksheet 2F: Rational, irrational and real numbers
Review Exercise 2
1
1
3
9
15
19
19
23
27
31
35
39
41
Approximation and Estimation
Worksheet 3A: Rounding and significant figures
Worksheet 3B: Approximation and approximation errors in real-world contexts
Worksheet 3C: Estimation and estimation errors in real-world contexts
Review Exercise 3
45
Chapter 4
Basic Algebra and Algebraic Manipulation
Worksheet 4A: Basic algebraic concepts and notations
Worksheet 4B: Addition and subtraction of linear terms
Worksheet 4C: Expansion and factorisation of linear expressions
Worksheet 4D: Linear expressions with fractional coefficients
59
59
63
65
69
75
Chapter 5
Linear Equations
79
Worksheet SA: Linear equations with integer coefficients
79
Worksheet SB: Linear equations with fractional coefficients and fractional equations 81
Worksheet SC: Applications oflinear equations in real-world contexts
85
Worksheet 5D: Mathematical formulae
89
93
Chapter 6
Linear Functions and Graphs
Worksheet 6A: Cartesian coordinates
Worksheet 6B: Functions and linear functions
Worksheet 6C: Applications oflinear graphs in real-world contexts
Review Exercise 6
111
Number Patterns
Worksheet 7A: Number sequences
Worksheet 7B: Number sequences and patterns
Exercise 7
115
115
119
123
Chapter 7
45
49
51
55
97
97
101
107
Secondary 1 Express Mid-year Checkpoint A
Secondary 1 Express Mid-year Checkpoint B
127
135
Answer Keys
143
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Prime numbers
I.
1
2
4
12
7
21
29
Circle all the prime numbers in the above list.
2.
3.
Determine whether each of the following is a prime or a composite number.
(a) 26
(b)
53
(c) 108
(d)
247
(b)
3x5x5
(d)
5 x 7 x 7 x 7 x 11 x 11
Write each of the following in index notation.
(a) 2 X 2
X
2
X
2
(c) 3x3x3x7x19
4.
Find the prime factorisation of each of the following numbers, expressing your answer in index notation.
(a) 24
(b)
(c) 225
(d) 442
(e) 539
(f)
95
1200
5. Written as a product of its prime factors, 1584 = 2x x 3Y x 11.
Find the values of x and y.
6.
Written as a product of its prime factors, 7875 = x 2 x 53 x y.
Find the values of the integers x and y.
7. The whole number pis such that p x (p + 36) is a prime number. Find the prime number.
8.
Chen has 160 one-centimetre cubes.
He arranges all of the cubes into a cuboid.
Each side of the cuboid has a length greater than 1 cm.
Find the dimensions of two cuboids that he can make.
9.
Given that p and q are whole numbers such that p x q = 250, which of the following statements are true?
Explain your answer.
(a) It is not possible for either p or q to be a prime number.
(b) The largest possible value of p +
q is a prime number.
1.
The prime factorisation of each of the numbers is given. Find its positive square root.
(a) 400 = 24 x 52
(c)
2.
5625 = 32 X 54
(b)
3969 = 34 x 72
(d) 48 400 = 24
Written as a product of its prime factors, 91 125 = 36 x 53 •
Find the cube root of 91 125.
3.
4.
Use prime factorisation to find each of the following.
(a)
✓
784
(b)
.J1600
(c)
efsu
(d)
R.f 3375
The radius of a ball is R./ 1728 cm.
Without using a calculator, find the value of the radius.
X
52
X
11 2
5.
Use a calculator to evaluate each of the following, leaving your answer correct to 4 decimal places where
necessary.
(a)
16 +5 - ✓ 484
(b)
(c)
7.7x(3o
+/6
(d)
2
3
(e)
6.
The radius of a circle is
(f)
J½f
cm.
Find the radius of the circle, leaving your answer correct to 2 decimal places.
7.
The area of a square field is 650 m 2 •
Find the perimeter of the field, leaving your answer correct to 1 decimal place.
8.
The area of a square frame is 5.8 m 2 •
Sam wants to insert a square painting with sides oflength 2.4 m.
Determine if the frame is large enough to fit the painting.
9.
The square root of n is 32 x 5.
Find n as the product of its prime factors.
10. The cube root of n is 22 x 7.
Find n as the product of its prime factors.
11. (i)
Use prime factorisation to show that 1225 is a perfect square.
(ii) Hence, write down the value of
.J 1225 .
Express 140 as the product of its prime factors.
(ii) Write down the smallest positive integer, k, such that 140k is a perfect square.
13. Written as a product of its prime factors, 600
Find the smallest positive integer k such that
= 23 x 3 x 52•
6 0
~ is a square number.
14. Written as a product of its prime factors, 540 = 22 x 33 x 5.
Find the smallest positive integer k such that 540k is a perfect cube.
15. Written as a product of its prime factors, 130 977
(i)
= 35 x 72 x 11.
Find the smallest positive integer a such that 130 977 a is a perfect square.
(ii) Find the smallest positive integer b such that 130 977b is a perfect cube.
(iii)
. . mteger
.
130-977
. a pnme
.
Sugges t a poss1'ble positive
va1ue o f c sue h t h at - 1s
numb er.
C
16. When written as the product of their prime factors,
A is 23 x 33,
B is 22
x 32 x 52,
C is 2 x 3 x 72 •
Find
(i)
the value of the cube root of A,
(ii) the value of the square root of B,
(iii) the smallest positive integer k such that kC is a perfect square.
17. (i)
Express 216 000 as the product of its prime factors.
(ii) Hence, find the cube root of 216 000.
18. Written as a product of its prime factors, 15 092
(i)
(ii)
=2
2
x 7x x 1lY.
Find the values of the integers x and y.
Write down two possible positive integer values of k such that 15 092k is a multiple of 21.
19. Plastic sticks are attached to form the framework of a cube.
The volume of the cube is 21 952 mm 3 •
Find
(i)
the length of each plastic stick,
(ii) the total length of plastic sticks used.
20. When written as the product of their prime factors,
pis 24 X 33 X 7,
q is 22 X 3 X 53,
r is 23 x 32 x 72 •
(i)
Given that p x a is a perfect square, find the smallest positive integer a.
(ii) Given that
f; is a perfect cube, find the largest positive integer b.
(iii) Is p x q x r a perfect cube? Explain your answer.
21. (i)
Express 2704 as a product of its prime factors.
(ii) Using your answer to part (i), explain why 2704 is a perfect square.
(iii) m and n are both prime numbers.
Find the values of m and n so that 2704x m is a perfect cube.
n
Highest common factor and lowestcomrnon multiple
1.
2.
Find the highest common factor of each of the following sets of numbers.
(a)
18 and 45
(b)
55 and 132
(c)
95 and 361
(d)
378 and 1050
Find the highest common factor of each of the following sets of numbers.
(a) 15, 90 and 225
(c)
3.
147,189 and 231
(b)
98, 126 and 238
(d)
165, 198 and 429
Find the highest common factor of each of the following, giving your answer as the product of its prime
factors.
(a) 22 x 33 x 5 and 23 x 3 x 52
(b)
32 x 7 x 11 3 and 34 x 5 x 11 2
4.
Find the lowest common multiple of each of the following sets of numbers.
(a) 20 and 25
(c)
5.
104 and 130
54 and 72
(d)
168 and 224
Find the lowest common multiple of each of the following sets of numbers.
(a) 32, 88 and 242
(c)
6.
(b)
110, 132 and 176
(b)
63, 105 and 315
(d)
136,204 and 272
Find the lowest common multiple of each of the following, giving your answer as the product of its prime
factors.
(a) 2 x 33 x 52 and 24 x 32 x 5
(b)
2 x 32 x 53 x 11 and 33 x 5 x 7
7.
A number is less than 200. It is also a multiple of 15. What is the largest possible number that this can be?
8.
A number is more than 300. It is also a multiple of 28. What is the smallest possible number that this
can be?
9.
fa Write down a possible multiple of 12 that lies between 130 and 190.
10. A number has 11 factors. Do you agree with each of the following statements? Explain why.
(a) The number is a composite number.
(b) The number is a perfect square.
(c)
The number is a multiple of 2.
© Shing Lee Publishers Pte Ltd
Primes, Highest Common Factor and lowest Common Multiple
11
11. (a) Write 390 as the product of its prime factors.
(b) Find the highest common factor of 390 and 234.
12. The numbers 1200 and 1960, written as the products of their prime factors, are
1200 = 24
1960
X
3 X 52,
= 23 X 5 X 72.
Find
(i)
the highest common factor of 1200 and 1960,
(ii) the lowest common multiple of 1200 and 1960.
13. 504 expressed as a product of its prime factors is 23 x 32 x 7.
(a) Express 540 as a product of its prime factors.
(b) Find
(i)
the highest common factor of 504 and 540,
(ii) the lowest common multiple of 504 and 540.
12
Primes, Highest Common Factor and Lowest Common Multiple
© Shing Lee Publishers Pte Ltd
14. Written as the product of its prime factors, 4356
= 22 x 32 x 11 2 •
(a) Express 440 as the product of its prime factors.
(b) Hence write down
(i)
the LCM of 4356 and 440, giving your answer as the product of its prime factors,
(ii) the greatest integer that will divide both 4356 and 440 exactly.
15. When written as the product of their prime factors,
A is 22 x 34,
B is 23
x 32 x 5,
C is 2 x 32 x 52 x 7.
Find
(i)
the value of the square root of A,
(ii) the LCM of A, B and C, giving your answer as the product of its prime factors,
(iii) the greatest number that will divide A, B and C exactly.
16. When written as the product of their prime factors,
pis 26 X 33,
q is 22 X 3 X 5,
r is 2 x 32 x 7.
Find
(i)
the value of the cube root of p,
(ii) the LCM of p, q and r, giving your answer as the product of its prime factors,
(iii) the greatest number that will divide p, q and r exactly.
17. (a) Express 180 as the product of its prime factors.
(b) Find the HCF of 180 and 48.
(c) Toothpicks are sold in packs of 180.
Satay sticks are sold in packs of 48.
Alice buys the same number of toothpicks as satay sticks.
Find the least number of packs of toothpicks that she could have bought.
18. 214 boys and 230 girls signed up to participate in The Amusing Race. Two days before the event, 4 boys and
6 girls withdrew from the event. The remaining boys and girls had to be equally divided into as many
groups as possible such that every group had the same number of boys and girls. Find
(i)
the greatest number of groups that can be formed,
(ii) the number of boys and the number of girls in each group.
19. A number has exactly nine factors.
Two of the factors are 6 and 18.
List all the factors of the number.
Date: _ _ _ _ _ __
1.
Find the largest prime factor of 2820.
2.
The whole number pis such that p x (p + 42) is a prime number. Find the prime number.
3.
Written as a product of its prime factors, 550 = 2 x 52 x 11.
Find the smallest positive integer k such that
4.
(i)
550k is a square number,
• .)
( 11
550.1s a square num b er.
T
(a) Express 882 as the product of its prime factors.
(b) Find two numbers, both smaller than 150, that have a lowest common multiple of 882 and a highest
common factor of 21.
5.
(a) Express 360 as the product of its prime factors.
(b) Find the lowest common multiple of 360 and 270.
(c) Explain why a prime number cannot be a perfect square.
The lowest common multiple of p, 63 and 81 is 1134. Find a possible value of p.
6.
7.
The numbers 700 and 1050, written as the products of their prime factors, are
700 = 22
1050 = 2
X
X
52
X
7,
3 X 52 X 7.
Find
the smallest positive integer h for which 700h is a multiple of 1050,
10 0
(ii) the smallest positive integer k for which
is a factor of 700.
(i)
J
8.
Written as the product of its prime factors, 2660
= 22 x 5 x 7 x
19.
(a) Express 570 as the product of its prime factors.
(b) Hence write down
(i)
the LCM of 2660 and 570, giving your answer as the product of its prime factors,
(ii) the greatest integer that will divide both 2660 and 570 exactly.
When written as the product of their prime factors,
A is 23 x 53,
Bis 2 x 3 x 5 x 7,
C is 22 x 52 x 72 •
Find
(i)
the value of the cube root of A,
(ii) the value of the square root of C,
(iii) the smallest positive integer k such that k x B x C is a perfect square.
10. When written as the product of their prime factors,
pis 26
q is 2
X
X
3 3,
32 X 53 X 11,
r is 2 2 X 3 X 52 X 7.
Find
(i)
the value of the cube root of p,
(ii) the LCM of p, q and r, giving your answer as the product of its prime factors,
(iii) the greatest number that will divide p, q and r exactly.
11. In a hotel, the management conducts a hygiene check in the kitchen every 18 days and one in the laundry
room every 30 days.
Both areas undergo a check on pt July.
When is the next date that both areas undergo a check together?
12. Joe has 784 one-centimetre cubes.
He arranges all of the cubes into a cuboid.
The perimeter of the top of the cuboid is 30 cm.
Each side of the cuboid has a length greater than 2 cm.
Find the height of the cuboid.
13. Two security cameras monitor the trophy cabinet of a sports club.
The first camera scans it every 28 seconds.
The second camera scans it every 70 seconds.
In 10 minutes, how many times do both cameras scan the trophy cabinet together?
Date: _ _ _ _ _ __
Negative numbers
-0.4
2
7
65
1
-3-
0
9.8
2
In the above list, write down the numbers that are
(a) positive integers,
(c)
2.
positive numbers,
negative integers,
(d)
negative numbers.
Fill in each box with'>' or'<'.
(a) 24
49
(b)
5.1
1.5
(c)
7
(d)
10
-10.1
-3
1
6
(e) 0
3.
(b)
(f)
Represent the numbers on a number line.
1
(a) 4 , -4, 0, 0.28
5
(b) -6.8, 0.86,
7
1
, -5, 10
3
(c) integers between -2 and 5
(d) positive integers less than 8
1
8
1
9
-16
1
and Oona number line.
10
(ii) Hence, arrange the given numbers in ascending order.
4.
(i)
Represent the numbers 1.1, -1, 10
5.
(i)
Represent the numbers -3, 3.03, ½ and -3½ on a number line.
(ii) Hence, arrange the given numbers in ascending order.
6.e(i) Write down two numbers that are greater than -15.
(ii) Express the relationships using the inequality sign">".
Write down two numbers that are less than -24.
(ii) Express the relationships using the inequality sign"<".
8.
Alex says, "Since 10 is less than 12, then -10 is also less than -12."
Do you agree with Alex? Explain your answer.
9.
The lowest point of the Dead Sea Depression is about 413 metres below sea level.
Represent this altitude using a negative number.
10. It is believed that the lowest natural temperature recorded at ground level on Earth is 89.2 °C below zero.
Represent this temperature using a negative number.
11. When an investment portfolio increases in value, it is considered to be a positive number.
Represent each of the transactions using a positive or a negative number.
(a) The value increases by $5000.
(b) The value decreases by $3600.
12. If 5 m represents 5 m above ground level, what does - 7 m represent?
13. If20 km represents 20 km due East, what does -16 km represent?
14. If 39° represents a clockwise rotation of 39°, what does -51° represent?
15. If -4.8 °C represents a 4.8 °C drop in temperature, what does 13.5 °C represent?
16. What does the following statement mean?
"Mrs Lee's monthly salary is adjusted by $80."
17. The altitudes of some places are as shown:
Cascade Mountain, Canada: 2998 m
Death Valley, USA: -86 m
Bukit Timah Hill, Singapore: 164 m
Karagiye Depression, Kazakhstan: -132 m
Arrange these places in ascending order of their altitudes.
Class: _ _ _ _ __
Date: _ _ _ _ __
Addition and subtraction involving negative integers
1.
2.
3.
Evaluate each of the following.
(a) 2 + 9
(b)
-2+9
(c) 2+(-9)
(d)
(-2)+(-9)
(a) 4 - 7
(b)
-4 - 7
(c) 4 - (-7)
(d)
(-4)-(-7)
Evaluate each of the following.
Find the value of each of the following.
(a)
12+(-15)
(b)
-80 + 70
(c)
26 - 91
(d)
-28 - 28
(e)
17 + (-53)
(t)
-49 + (-34)
(g) 84-(-20)
(h)
-36 - (-36)
4.
Find the value of each of the following.
(a) 6 + (-3) + (-2)
(c)
(b)
40 - 90 - 30
(d)
12-(-31)-(-45)
(f)
-33 - 33 + (-87)
(h)
-27 - (-19) - (-24)
(b)
19 +
=0
(d)
-17 +
= -6
(f)
42-
=-10
(h)
21-
=-1
18-(-25)+(-70)
(e) -24 + 16 + (-10)
(g)
5.
-49-(-21)+(-15)
Fill in each box with the correct number.
(a) 8 +
(c) -24 +
(e)
30 -
(g) -15 -
=2
= -27
= -3
= -15
6.
7.
Fill in each box with the correct number.
(a)
+ 16 = 8
(b)
+ (-20) =4
(c)
+ 70 =-20
(d)
+ (-39)
(e)
- 54 = -2
(t)
- (-16) = 0
(g)
- (-31)
(h)
- (-18)
= 14
= -93
= -12
The temperature of Sapporo on a particular night is -11 °C.
The next morning, the temperature rises by 4 °C.
What is the temperature in the morning?
8.
The temperature of Helsinki on a particular morning is -20 °C.
The temperature of Helsinki later in the day is -15 °C.
Does the temperature increase or decrease in the day?
By how many °C does it increase or decrease?
9.
A spinner rotates 48° clockwise, then 25° anticlockwise, followed by another 33° clockwise.
Describe its final position from the starting position.
10. The highest point of Qixia Mountain in China is 286 m above sea level.
Amsterdam's Schipol Airport in Holland, is 4 m below sea level.
Baku, the capital of Azerbaijan, is believed to be the lowest lying national capital in the world.
(i)
Find the difference in altitude between the highest point of Qixia Mountain and Schipol Airport.
(ii) Baku is believed to have an altitude of 24 m below Schipol Airport.
Represent the altitude of Baku using a negative number.
11.
A bank account with an overdraft facility allows the account holder to withdraw more money than the
account holds.
Mr Wang's bank account has an overdraft facility.
In January, his bank balance was $3000.
In February, he withdrew $4750.
In March, he withdrew $2200.
In April, he deposited some money so that his balance exceeded $500.
How much money could Mr Wang have deposited in April?
A number x lies between -50 and -30.
A number y lies between -80 and -20.
Give an example of a pair of numbers, x and y, such that
(a)
the sum of x and y is -63,
(b)
the difference between x and y is -36.
Name:---,-,-------------
Class: , - - - - - - - -
Date: _ _ _ _ __
Multiplication, division and combined operations
involving negative integers
1.
Evaluate each of the following.
(a) 7
X
8
(c) 7x(-8)
2.
(b)
(-7) x 8
(d)
(-7) x (-8)
(b)
(-24) + 4
(d)
(-24) + (-4)
(b)
(-12) x (-10)
(d)
(-99)x0
Evaluate each of the following.
(a) 24 + 4
(c)
24+(-4)
Find the value of each of the following.
(a) 15
X
(c)
(-1)
(e)
-34
2
0
(g)
(-6)
X
(-84)
(f)
(h)
75
-5
-96
-4
Find the positive and negative factors of each of the following numbers.
(a) 4
(c)
-15
(b)
20
(d)
-1
Write down a positive multiple and a negative multiple of each of the following numbers.
5.
(a) 3
(c)
-5
(b)
12
(d) -11
6.
7.
Find the value of each of the following.
(a)
(-1) 2
(b)
(-1) 3
(c)
(-8) 2
(d)
(-8) 3
(e)
/64
(f)
ef64
(g)
-/is
(h)
~-125
(-1)
(b)
(-10) x (-2) x (-4)
(c) 5x(-6)+(-2)
(d)
28+(-7)x9
(e)
(f)
(-9)2x(-l)
(h)
(-4) + (-2) 2 + (-1) 3
= -35
(b)
(-23) x
9 = -54
(d)
Find the value of each of the following.
(a) 4 X (-3)
8.
X
(-60)+(-5)+(-3)
Fill in each box with the correct number.
(a) 7 X
(c)
(e) 80 +
(g)
X
= -16
+ (-5)
= -20
(f)
(h)
X
(-4)
(-76)+
+ (-6)
=-92
= 28
= -19
= 15
9.
Evaluate each of the following.
(a) 6x(-3)2+(-10)
(c)
-19+(-2)x(-5)-;-(-1)
(e) (-4)x25-Yx(-4)
(b)
(-7-5)-;-(-7+4)
(d)
[(-8) + (-3)]
X
(-s)2-(-s)
(f)
29 (-s)+(-7)
(-2) 3
10. Michael has $80 in his wallet.
He spends $7 every day.
When he runs out of money, he borrows from his sister.
(i)
How much does he spend altogether in 20 days?
(ii) How much money has he borrowed from his sister after 20 days?
(iii) On which day does he start borrowing money from his sister?
11. Sarah rolls a six-sided die.
If it shows a prime number, the number shown on the die will be added to her score.
If it shows any other number, the number shown on the die will be deducted from her score.
The table shows the number of times each number is obtained when Sarah rolls the die 15 times.
1
2
3
4
5
6
3
0
7
0
4
1
What is Sarah's score after rolling the die 15 times?
12. Min sells clothes on her blog shop.
She plans to conduct a sale before the holidays.
The table shows the amount of profit or loss she makes on each item sold.
Blouse
$6, profit
Scarf
$3, loss
Skirt
$4, loss
(a) On a particular day, Min sells 4 blouses, 3 scarves and 5 skirts.
Does she make a profit or a loss? How much is the profit or the loss?
(b)
Give an example of the number of blouses, scarves and skirts Min could have sold if she made an
overall loss of between $10 and $20.
Class: _ _ _ _ __
Fractions and mixed numbers
1.
Find the value of each of the following.
1 1
(a)
4+3
(e)
i+(-2..)
(g)
(k)
7
(b)
5
14
1
2
(h)
_2.__(_.!.)
12
6
14
1
1
15
1 4
-6-+-
2 5
Date: _ _ _ _ __
2.
Evaluate each of the following.
3 20
(a) 10 X 27
12 ( 34)
15
(e)
-x - -
(g)
(-32-)xs~
11
9
3. Find the value of each of the following.
(a)
(½
½+¼)x(-j)
(d)
[+HlH-½r
(f)
4.
A company incurs monthly expenses.
Rental takes up
3
8
of the monthly expenses and wages take up
I
3 of the monthly expenses.
The remaining amount is used to purchase goods.
Find the fraction of the monthly expenses used to purchase goods.
5.
Andy, Bob and Charlie take
(i)
¾hour, } hour and ¾hour to code a simple program.
Arrange the fractions in ascending order.
(ii) The average time taken is calculated by dividing the total time taken by three.
Find the average time taken by the three boys.
6.
At a country club, ;
0
Of the remaining area,
of the land is occupied by a golf course.
¼is occupied by a club house.
Find the fraction of the land not occupied by the club house.
7.
The diagram shows two identical containers, P and Q, containing different amounts of water.
p
Q
Find the fraction of a container of water that should be transferred from P to Q so that both containers
have the same amount of water.
Class: _ _ _ _ __
Decimals
1.
2.
Find the value of each of the following.
(a) 23.8 + 5.49
(b)
10.1 - 7.64
(c) 3.07 + (-4.18)
(d)
-6.2 + 3.49
(e) -0.92 + (-1.5)
(f)
-8.8 - (-0.12)
(b)
0.61 x 0.25
(d)
400.8 x 0.36
(f)
0.43
Evaluate each of the following.
(a) 6.1
(c)
X
7.004
(e) 0.4 2
2.5
X
2.9
Date: _ _ _ _ __
3.
Find the value of each of the following.
(a) 21.6 + 0.3
(c)
4.
17.01 + 4.5
(b)
0.216 + 0.3
(d)
0.2496 + 0.48
(b)
-0.9 - (-0.9) 2 X (-1) 3
Evaluate each of the following.
(a) 5.95 + 0.33 - (-4.27)
X
(c) o.1 2 x [(-7.6)+(-0.4)]
(-2)
5. Sandy and Tessa each described how they would find the value of 43.2 x 0.65.
(a) Sandy said, "First, I align the decimals. Then, I multiply 432 by 5. Next, I multiply 432 by 6.
Finally, I add the numbers and place the decimal point:'
Show Sandy's method.
(b) Tessa said, "I convert 43.2 into an improper fraction with 10 as the denominator. I also convert 0.65
into an improper fraction with 100 as the denominator. Then I multiply the numerators and the
denominators separately. Finally, I place the decimal point:'
Show Tessa's method.
6.
Paul described how he would find the value of
~~i~ .
(a) He said, "I find it easier to convert the denominator to an integer. So I multiply both the numerator
and the denominator by 100. Then it becomes easier to divide:'
Show Paul's working.
(b) Paul then said, "I can use a similar method to find the value of
Show Paul's working.
~:~! :'
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Rational, irrational and real numbers
-5.1
22
7
✓
85
0
2
-3-
~
9
TI
In the above list, write down the numbers that are
(a) rational numbers,
2.
irrational numbers.
22
5
174.06
-0.2.
6
9
In the above list, write down the numbers that are
(a) terminating decimals,
3.
(b)
(b)
3
8
7
0.3125
recurring decimals.
Use a calculator to evaluate each of the following, leaving your answer correct to 3 decimal places.
(a)
¾
r (-¾)
( )\(-5¾
(b)
-11.7 + (-0.16)2 x TI
3
()
C
(e)
Fs
(
10+(25 )
(d)
(f)
22
/44
?
xl.5- -2xTixl.5x~
2
4.5 + (-5.5
✓ 92-82
r
9
4.
5.
Express each of the following in the form
yj ,where a and b are integers and b -:t 0.
(a)
0.8
(b)
0.49
(c)
0.515
(d)
0.049
Express each of the following in the form
(a)
9.87
yj ,where a and b are integers and b -:t 0.
(b)
3.456
Class: _ _ _ _ _ __
1.
Fill in each box with'>' or'<'.
(a) -9 + (-12)
(c)
2.
Date: _ _ _ _ _ __
5 8
2x2~
4
Express ;
2-+6 9
6
5
4-(-20)
(b)
5 x (-3)
(d)
(-2)3-(-2)x(-2) 2
(-17)
X
3
4.8 + (-1.2)
as a decimal.
The numbers p, q, rand s are represented on the number line.
0
p q
2
l
r
s
The values of p, q, rand s are listed below.
0.75
15
7
3n
5
4
7
Find p, q, rand s.
4.
(i)
Represent the numbers 6, -0.6, 3¾, 0.6 2 and -
~i
on a number line.
(ii) Hence, arrange the given numbers in descending order.
3
s.
(a)
12.7x4
Calculate the exact value of _ _ _ .
81 49
(b) Calculate
.J 3.5
2
+4.5.
Give your answer correct to 1 decimal place.
6.
Consider the rational numbers
(a)
¾and ¾ .
Which number is greater?
(b) Consider a third rational number
i~~.
Without using a calculator, how can you tell whether
(c)
Write down two rational numbers between
(d) How many rational numbers lie between
7.
¾, ¾ or i~~ is the greatest?
¾and ¾.
¾and ¾?
¾of the books are fiction books.
Of the remaining books, ¾are assessment books and ¼are magazines.
At a bookstore,
The rest are non-fiction books.
(i)
What fraction of the books are non-fiction?
(ii) Arrange the books in ascending order of their quantities.
8.
The diagram shows a vertical post used to measure the level of water in a catchment area.
lm
0
-1 m
(a) Estimate the level of water shown.
(b) When the water level reaches 1.5 m, an alert is raised.
How much higher can the water level increase before the alert is raised?
9.
The table shows the highest and lowest temperatures one day in Beijing, Singapore and Seoul.
(i)
Beijing
Singapore
Seoul
Highest
2°c
32 °C
-3°C
Lowest
-10°C
26 °C
-9°C
Find the difference between the highest temperature in Singapore and the lowest temperature in
the three cities.
(ii)
The lowest temperature in Osaka is greater than the highest temperature in Seoul and lower than
the highest temperature in Beijing.
Suggest a value for the lowest temperature in Osaka.
10. A check was conducted on five batteries to determine the typical lifespan of each battery when used in a
neck massager.
Each battery is expected to last for 4 hours.
The results are recorded in the table.
(i)
Battery
A
B
C
D
E
Number of hours greater or less than the expected lifespan
-0.2
+1.4
+0.8
-1.3
-0.5
Find the actual lifespan of battery D.
(ii) Find the total lifespan of all the batteries.
11. Countries in the world follow different time zones.
There are 24 main time zones in the world.
In January, the local time in Copenhagen is - 7 hours relative to the local time in Singapore.
The local time in Melbourne is +3 hours relative to the local time in Singapore.
(a) When it is 6 a.m. in Singapore on 5th January, find the local time and day in
(i)
Melbourne,
(ii) Copenhagen.
(b) When it is 2.25 p.m. in Singapore, it is 9.25 a.m. in Kenya.
What can you say about the local time in Kenya relative to the local time in Singapore?
Class: _ _ _ _ _ __
Rounding and significant figures
I.
Round off each of the following to the nearest integer.
(a) 50.7
2.
398.4
Round off each of the following to the nearest 100.
(a) 4279
3.
(b)
(b)
1605.5
Round off each of the following to 1 decimal place.
(a) 63.184
(b)
956.02
Round off each of the following to 2 decimal places.
(a) 22.0536
5.
0.197 414
Round off each of the following to the nearest 0.001.
(a) 15.2784
6.
(b)
(b)
3.001 009
Round off each of the following to the nearest ten thousandth.
(a) 97.132 565
(b)
0.406081
Round off each of the following to 1 significant figure.
8.
(a) 3916
(b)
504
(c) 2.807
(d)
0.3099
Round off each of the following to 2 significant figures.
(a) 71 448
(b)
239
19.506
(d)
0.080 75
(c)
Date: _ _ _ _ _ __
9.
Round off each of the following to 3 significant figures.
(a) 254.810
(c)
3.9986
(b)
72 167
(d)
0.580 339
10. State the number of significant figures in each of the following.
(a) 48017
(c)
3000.15
11 . Ca1cul ate
(b)
0.026 590
(d)
703.0
204.16 , g1vmg
. . your answer correct to 1 s1gm
· 'fi cant fi gure.
13.8-1.07 2
12. (a) Convert
5
into a decimal. Write down all the figures shown on your calculator.
14
(b) Express the answer in part (a) correct to 2 significant figures.
13. (a) Evaluate
~::i:.s
_
as a decimal. Write down all the figures shown on your calculator.
8
70
(b) Give your answer in part (a) correct to
(i)
3 significant figures,
(ii) 3 decimal places.
14. The number 524 000 is correct to k significant figures.
(i)
Explain why k cannot be 2.
(ii) Write down the possible values of k.
A number is rounded to 0.0408, correct to 3 significant figures.
15.
Give three examples of this number.
16. The distance between Singapore and Vancouver is 12 813 km.
Write 12 813 to the nearest thousand.
17. The mass of a baby is 3.4 kg, correct to 1 decimal place.
What is the minimum mass of the baby?
18. The population of Finland was 5 503 000 in 2017.
(a) This value has been rounded to the nearest 1000.
(i)
What is the largest possible value of the population of Finland in 2017?
(ii) What is the smallest possible value of the population of Finland in 2017?
(b)
This value has been rounded to 5 significant figures.
Suggest two possible values of the population of Finland in 2017.
19. In 2017, the land area of Singapore was 722.5 km 2 •
(i)
How many significant figures are there in 722.5?
(ii) Write 722.5 correct to 2 significant figures.
20. The dimensions of a tennis court for singles matches are 23.77 m by 8.23 m.
Find the area of the court, giving your answer correct to 2 significant figures.
21. A metal block has a mass of 50 grams, correct to the nearest gram.
(i)
Find the least possible mass of the metal block.
(ii) The volume of the metal block is 13 cm3, correct to the nearest cubic centimetre.
Find the greatest possible mass of 1 cubic centimetre of the metal.
22. The total bill at a restaurant is $70.3846.
If the customer pays using a credit card, the amount is rounded to 2 decimal places.
If the customer pays in cash, the amount is rounded to the nearest 10 cents.
Would you advise the customer to use a credit card or cash to pay?
Show working to explain your answer.
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Approximation and approximation errors in real-world contexts
1.
The area of a square is 352 cm2 • Find
(i)
the length of the square,
(ii) the perimeter of the square.
2.
The circumference of a circle is 248 cm. Find
(i)
the radius of the circle,
(ii) the area of the circle.
3.
A bag of corn chips costs $3.95.
Susie has $30.
What is the greatest number of bags of corn chips she can buy?
4.
An engineer calculated that the maximum current that can flow in a particular circuit before it is
considered unsafe is 10.52 amperes.
He is asked to indicate the maximum current correct to the nearest ampere.
What should the engineer indicate the maximum current as? Why?
5.
Members of the public voted for one of three finalists in a dance competition.
The table shows the votes each finalist received.
Amazing ballerina
3551
Dance addict
2036
Young flamingo
3748
Total
9335
(a) (i)
38.0%
38.04%
Complete the third column of the table.
(ii) What do you notice about the total percentage of votes?
(b) (i)
Complete the fourth column of the table.
(ii) Why is the total percentage of votes in the fourth column different from that in the third column?
6.
Employees in an organisation voted for the destination for a staff retreat.
The table shows the votes each destination received.
Bintan
24
30.0%
Phuket
35
43.8%
Sentosa
21
26.3%
Total
80
100.1%
Why is the total percentage of votes 100.1 % instead of 100%?
Explain why there is an additional 0.1 %.
Class:-'--------
Date: _ _ _ _ _ __
Estimation and estimation errors in real-world contexts
1.
Estimate the value of each of the following.
(a)
2.
(b)
~1002
Estimate the value of each of the following, giving your answer correct to 1 significant figure.
(a)
3.
-J63.8
4.03x4.97
10.2
(b)
fiCu+fiX99
,j48.88
(a) Express each of the following correct to 2 significant figures.
(i)
718.469
(ii) 23.858
.
. . your answer correct to 1 s1gm
. •fi cant fi gure.
(b) H ence, estimate
t h e va1ue of 718.469
_
, g1vmg
23 858
4.
A student estimated the value of 42.8 + 3.16 - 5.907 to be 4.
(i)
Use estimation to check whether her answer is reasonable.
(ii) Use a calculator to evaluate 42.8 + 3.16 - 5.907.
Is your estimated value in part (i) close to the actual value?
5.
Which one of the following is likely to be the height of a double-decker bus?
40 cm
6.
4000 cm
180 g
1800 g
A publishing company sold 279 566 copies of a magazine from
(i)
pt
Write down a calculation you could do mentally to estimate the average number of copies sold each day.
Is your estimated value dose to the actual value?
A watch costs 9000 Thai baht.
The conversion rate is 1 baht = S$0.043 251.
Without using a calculator, estimate the price of the watch in S$.
9.
18 000 g
January to 7th January.
(ii) Use a calculator to find the actual value.
8.
40 000 cm
Which one of the following is likely to be the mass of a tray of 30 eggs?
18 g
7.
400 cm
A student has $20.
He buys 4 pens at $2.90 each and 5 highlighters at $1.95 each.
Use estimation to determine if he has enough money to pay for the stationery.
10. Fang attends dance classes.
She wrote the following note.
What misconception has Fang made?
Write the correct working to estimate the area of the dance studio.
11. Grandma Lucy orders her groceries online.
Baby carrots (bag)
2
$2.10
Chicken stock (bottle)
3
$3.90
Frozen chicken thigh (bag)
4
$7.25
Portobello mushrooms (packet)
4
$4.15
Potatoes (bag)
2
$2.85
Show how you can estimate the total amount of money Grandma Lucy has to pay.
12. George is visiting Singapore.
He sees the following advertisements at the airport.
Plan A
PlanB
20 GB of local data for 7 days
35 GB oflocal data for 7 days
$29.96
$55.64
Without using a calculator, help George decide which plan is better value for money.
13. The following shows two options for popcorn at a cinema.
Buddy Combo
Family Combo
150 g of popcorn
$7.90
150 g + 50 g of popcorn
$9.90
(a) Without using a calculator, decide which option is better value for money.
(b) Jasmine says that she uses the concept oflowest common multiple (LCM) to decide which option is
better value for money.
Show Jasmine's method.
Date: -----,-----.---
1.
10.59 2
_ _
16 33 224
Write down the first five digits on your calculator display.
(a) Calculate
(b) Write your answer to part (a) correct to 2 significant figures.
2.
By writing each value correct to 1 significant figure, estimate the value of
4.26x 8.53
97.1
Show your working.
3.
(a) Convert (
r
~
into a decimal. Write down all the figures shown on your calculator.
(b) Express the answer in part (a) correct to
(i)
1 significant figures,
(ii) 3 significant figures.
4.
The mass of jelly beans in a bottle is listed as 0.6 kg, correct to 1 decimal place.
What is the minimum mass of the jelly beans?
5.
The number 37 600 is correct to k significant figures.
(i)
Write down the possible values of k.
(ii)
For one of the values of k, find the difference between the greatest and smallest integer values of
the number.
6.
Which one of the following is likely to be the mass of a watermelon?
0.8 kg
7.
80 kg
800kg
Which one of the following is likely to be the length of a badminton court?
0.13m
8.
8kg
1.3m
13m
130m
A pair of in-ear headphones costs 150 Malaysian ringgit (RM).
The conversion rate is S$1
= RM3.0141.
Without using a calculator, estimate the price of the pair of in-ear headphones in S$.
9.
In Indonesia, a particular leather bag costs 900 000 Indonesian rupiah (Rp).
The conversion rate is RplO00 = S$0.094 766.
(i)
Without using a calculator, estimate the price of the leather bag in S$.
In South Korea, the same bag costs 80 000 South Korean won (W).
The conversion rate is S$1
= W821.64.
(ii) Without using a calculator, find out whether the bag is more expensive in Indonesia or in South Korea.
10. In 2017, there were approximately 17 423 000 visitor arrivals into Singapore, excluding Malaysian arrivals
by land.
This value has been rounded to 5 significant figures.
(i)
Write down the greatest possible number of visitor arrivals.
(ii) Write down the least possible number of visitor arrivals.
11. 1he amount of money tourists spent on accommodation in 2012, 2014 and 2016 is shown in the table.
Amount of money spent, in millions of dollars
(i)
2012
2014
2016
5038
5309
5916
Round off the amount of money spent on accommodation in 2012 to the nearest ten million dollars.
(ii) Find the average amount of money spent in a month in 2014, giving your answer correct to
2 significant figures.
(iii) Find the maximum amount of money spent on accommodation in 2017, such that the increase in the
amount of money spent on accommodation from 2016 to 2017 is not more than 100 million dollars.
12.
Amanda and Beth eat at the same restaurant.
The bill for each of them shows the same amount.
Amanda pays using a credit card, so the amount is rounded to 2 decimal places.
Beth pays in cash, so the amount is rounded to the nearest 10 cents.
Beth pays $29.00.
Beth pays more than Amanda.
Suggest
(i)
the amount calculated in the bill before it is rounded to 2 decimal places,
(ii) how much Amanda could have paid.
13. Two supermarkets are conducting a promotion for a new brand of washing detergent.
Twin Pack
Plus Pack
2 x 850 ml of detergent
$16.80
850 ml+ 150 ml of detergent
$10.80
Without using a calculator, decide which option is better value for money.
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Basic algebraic concepts and notations
1.
Write down the algebraic expression for each of the following.
(a) sum of 4x and Sy
(b)
subtract 3x from lOy
(c) product of 2xy and Sz
(d)
divide 7x by 9yz
(e) square root of xy
(f)
cube root of 3z
Write down the algebraic expression for each of the following.
(a) sum of Bx, 7y and 6z
(b)
subtract Sx from the sum of 9y and 2z
(c) subtract 4x from the product of 5 and 3y (d) product of2y, 9 and Sz
(e) divide the sum of x and 4y by z
3.
4.
(f)
divide 3y by the sum of 2x and 6z
Write down in words the meaning of each algebraic expression.
(a) 6x+ lly
(b)
(c) xy2
(d)
9x - 4y
Sx
Mr Chan is x years old.
(a) His daughter is 28 years younger than him.
Find an expression, in terms of x, for his daughter's age.
(b) Mr Chan is 3 years older than his wife.
Find an expression, in terms of x, for his wife's age.
5.
The diagram shows a rectangle.
The shorter side of the rectangle is x cm long.
The longer side of the rectangle is three times as long as the shorter side.
Fill in the blanks.
2
.____A_r_ea_=_ _ _c_m__
__.I Width= _ _ cm
Length= _ _ cm
6.
7.
Given that x = 5 and y
= -2, find the value of each of the following expressions.
(a) 4x + 9y
(b)
(c) 3xy
(d)
Given that x
X
y
xy
3
= -4 and y = 7, find the value of each of the following expressions.
(a) -llx - 2y
(c)
4x- 9y
-+Sy 5x
(b)
5x- 3(7x + y)
(d) x 2 + y2
8.
Given that x = -5 and y =
¼, find the value of each of the following expressions.
(a) 3y-2x
(c)
x-y
x+y
(b)
.!._.!.
(d)
✓ -Sxy
y
X
Given that x =
½and y = -¼ ,find the value of each of the following expressions.
(a) 7 - l2xy
(b)
2+.±-6
y
X
(c) S(x + 2y) - 9x
10. Given that x =
-½, y = 0 and
(a) 99xyz
(c)
z
= 4, find the value of each of the following expressions.
yz) 3
(b)
(x2
(d)
(x + z)(z2
-
2
x z _ 3z-y
5
2x+z
-
xz + z) 2
11. Given that x = 3, y = -5 and z =
½, find the value of each of the following expressions.
(a) xy-yz+xz
(b)
(d)
(x+ y-z) 2
x+z
-xSxz
2
y
-
12. Thomas says, "2x + 3y is the same as 3y + 2x, so 2x - 3y is the same as 3y - 2x."
Do you agree with Thomas? Explain why.
There are more than four ways to interpret the algebraic expression 12pq.
One of them is the product of 12p and q.
Write down as many as you can.
Addition and subtr~ction6f linearterms
1.
Simplify each of the following.
(a) 7x + 4x
2.
Sx + (-3x)
(d)
-6x+x
(e)
-2x- 9x
(f)
-4x - (-5x)
(a) 3x + 10 - 6x + 5
(b)
9x-4+(-x)-(-1)
(c) 5x +Sy+ 7x - y
(d)
4y-10x-3y+6x
(e) -9x- 2y + (-x) - (-2y)
(f)
-Sy+ (-3x) - 5y + (-4x)
(h)
3 - 6y + (-2x) - S - (-7y) + x
(b)
-7x - 15 and 2x - 3y + 9
(b)
4x- Syz from 7xy- Syz + 3x.
Simplify each of the following.
+ 12 + Sy- 9 - 9x
Find the sum of each of the following.
(a) 24x + 5y and y- 10x
4.
5x - 2x
(c)
(g) 4x- y
3.
(b)
Subtract
(a) 6xy- 11 from 7yx + 5 - xy,
5.
Consider three consecutive even integers.
(a) Given that the smallest of the three integers is x, find the sum of the three integers.
(b) Given that the largest of the three integers is x, find the sum of the three integers.
6.
Hannah has $50.
She buys 2 cups of bubble tea at $x each.
She also buys 5 pancakes at $y each.
(i)
How much does Hannah spend altogether?
(ii) How much money does she have left?
7.
A tube of toothpaste costs $x.
A roll of dental floss costs twice as much as a tube of toothpaste.
An electric toothbrush costs four times as much as a roll of dental floss.
Is it correct to conclude that the electric toothbrush costs six times as much as a tube of toothpaste? Show
your working to explain your answer.
8.
A salesman receives a fixed monthly salary of $x.
For every van he sells, he collects a commission of $y.
(a) In a particular month, he sells 6 vans.
How much does he receive in total in that month?
(b) The next month, he sells 5 vans.
As a result, he receives $240 less in that month.
How much commission would he have received ifhe sold 7 vans instead?
Class:-'----~--
Date: _ _ _ _ __
Expansion and factorisation of linear expressions
1.
Expand each of the following expressions.
(a) S(x + 7)
(b)
6(x-10)
(c) 4(3x + 8)
(d)
9(2x - 1)
(e) -(x + 12)
(f)
-2(x- 7)
(h)
-4(5x - 3)
(j)
5(12 - 4x)
(k) -2(8 - 3x)
(1)
-9(9 - lOx)
(m) 7(5x + 4y)
(n) 8(3x - 8y)
(o) -3(6y + x)
(p)
(g) -6(2x
(i)
+ 9)
3(1 + 6x)
-5(2y- 9x)
2.
Simplify each of the following.
(b)
llxxx2xy
(d)
4x x lOy
(f)
(-12x)
(h)
(-3x) x 8y x (-4z)
(a) 2a(9x + 4y)
(b)
7a(8x- 3y)
(c) -a(Sx + y)
(d)
-6a(x - IOy)
(e) 4bc(6y- llx)
(f)
-5bc(3y + 16x)
(a) 8
(c)
XX X y
3xxx(-9)xy
(e) Sx x (-6y)
(g) 6x
3.
x y x 7z
X
(-12y)
Expand each of the following expressions.
4.
Expand and simplify each of the following.
(b)
(c) 8x+ 3y- (3x+ 8y)
(d) -(4x - 9y) - 9y
(e)
6(2x - y) + 4(x - Sy)
(g) 6x + 2(x - 3y + 2z) + Sy
5.
-(Sx + 7y) + Ilx
(a) 12(3x + y) - lOy
(f)
3(3x + 8y) - 2(4x - 9y)
(h) -4x - 3(2x + 12y - 3z) - 9z
Expand and simplify each of the following.
(a) -2{Sx - 6a[y- (14y + x)]}
(b)
10{7y + (-3a)[8x- 3y- 2(x - 4y)]}
6.
Add 7x - 3y + S to the product of 2y - 9x + 11 and 4.
7.
Subtract the product of-6 and [x - Sy+ 2(3y- 4x)] from the sum of7x + 2y and-(4y - lOx).
8.
Factorise each of the following completely.
(a)
16x + 12
(b)
9x - 45
(c)
10 - lSx
(d)
-33x - 44
(e)
I4ax + 6ay
(f)
-2Iax + 56ay
(h)
-Sax + IObx + I2cx
(b)
Sax - 20a(y - z)
(g) 24x- 27y + 3z
9.
Factorise each of the following completely.
(a) 9x + ISx(a + b)
(c)
-3x(4y + 7z) - I2x
(d)
-17x2
(e)
7a(I - 4x) + 3a(Sx - 6)
(f)
-2x(9a + b) - 8x(3a - Sb)
10. Factorise -34a4 bx2 - 85a 3 b2x 2 - 68a3 bc2x 2 completely.
-
34xy
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Linear expressions with fractional coefficients
2.
1
2
1
6
2
5
-x--y -x+-y
(d)
-¾[(12x+28y )-s(7x-y )]
Expand and simplify each of the following.
(a)
3.
2
3
(b)
¾[ 2(11x+7)-4+23x]
Express each of the following as a single fraction in its simplest form.
(a)
1+
Sx
14
(b)
6x
11 -3
4.
Express each of the following as a single fraction in its simplest form.
(a)
3x+7x
4 12
(b)
2x_2x
3
9
(C)
Sx-3
x+-6-
(d)
1-Sx -6x
(e)
1+ 4x-l
.)
(1
9x+l lOx-3 1
-6---7-+3
8
(h)
7x-4 8x-3
-9-+-5-
(j)
2x+5 x 7-6x
-----4
6
8
5.
Express each of the following as a single fraction in its simplest form.
Sx+ y 4x+9y
(b) lOy 3x 8y
(a) -6-+
3
7+ 2
(c)
3y-10x
4
(e)
-6-+4+
(g)
x+
(i)
2x-y+ 6x-9y
9x+ y
x+2y
5
3x
(d)
2y-5x
12
(t)
7x-4y IOy+3x
3
+
5
2
5x+2y
8
7x _ 4y-5x
8
3
5y-6x 9y 4x+ y
---'---+---3
2
9
h)
(
4 y-
(j)
3y x-
3x+2y
7
+
2x-3y
4x-5y
10
4
7y+2x
4
6.
Express each of the following as a single fraction in its simplest form.
(a)
9(x+y) 2x-y
12 +-4-
(b)
7(4x-y) 2y
20
+5
8x+8y - 2(6x-y)
(d)
(f)
(g)
l
_y-_x -[----'-4( 3_x_+2~y) _ ---'-3
( x_9--'-y)
4
3
6
9
45
u( 2y-x)
3(7x-4y)
6
8
s(5 y-4x)
(h)
15
[
l
3(x+7y) +3x-4y
2
It is given that ¼x+½ y+%x+ ~y
1
7.
.
ax+ 7y
It is given that
+
3
9.
(i)
. .)
(11
=
= {x+ 1~ y. Suggest a set of values for h, k, p and q.
ex 29y
+
. Suggest a set of values for a, band c.
12 12
.
..
.
19x+l0y 4x-15y
Without wntmg any workmg, how can we tell that
+
can be expressed in the form
12
18
ax+by
ax+by
3 6 but not - -5- , where a and bare constants?
Sh ow th at 19x+l0y + 4x 15y can b e expresse d.mth e 1orm
c
ax+by h
db
12
3 6 , w ere a an are constants to
18
be determined.
10. A student wrote the following working to express
its simplest form.
s(4x-y) 11(x-2y) 6(3y-2x)
6
-
21
+
7
as a single fraction in
Do you agree with the student? If you do not, identify the error(s) and write the correct working to express
s( 4x-y) 11(x 2y) 6(3y-2x)
6
-
21
+
7
as a single fraction in its simplest form.
Class: _ _ _ _ _ __
1.
Given that x = 2 and y =
-½ ,find the value of each of the following expressions.
(a) 4x + 9y
(c)
2.
_.'.2'.__
x+y
(b)
1
X
_..!:_
y
(d)
(3x - y) 2
(b)
4[7y + 2(-Sx)] - 3(8x - 9y)
(b)
-40ay - 56y - 24bcy
Expand and simplify each of the following.
(a) -5(6x - lly) + 2(x- 3y)
Factorise each of the following completely.
(a) 3abxy - 12acxz + 6ax
4.
Find the product of-8 and the sum of-2(10x + 3y) and -5(6y- 7x).
Date: _ _ _ _ _ __
5.
Express each of the following as a single fraction in its simplest form.
(a)
3(7x-y) s(x+9y)
5
8
(b)
4Y-
u=2x-1
6.
(6x+Sy 8x-2y)
7 +
3
v=x+3
Write an expression in its simplest form, in terms of x, for
(i)
. .)
( 11
7.
3(v-2)-3u,
U
V
5+3.
(a) By substituting suitable values of x and y, show that lOx - 7y is not always the same as 7y- I Ox.
(b) Are there any non-zero values of x and y for which lOx- 7y has the same value as 7y- lOx?
8.
Consider three consecutive odd integers.
(a) Given that the smallest of the three integers is x, find the sum of the three integers.
(b) Given that the largest of the three integers is x, find the sum of the three integers.
(c) What can you say about the difference between the largest and the smallest of any three consecutive
odd integers?
9.
The perimeter of a rectangle is (6x + 12y- 10) cm.
The length of the rectangle is (2x + 4y - 3) cm.
(i)
Find an expression, in terms of x and y, for the breadth of the rectangle.
(ii) It is given that x
= 5 and y = 1.
Find the area of the rectangle.
10. There are six players in an "Under-14" volleyball team.
The average height of the two 12-year-old players is h m.
The average height of the four 13-year-old players is km.
(i)
Find the sum of the heights of all six players.
(ii) Find the average height of the six players.
11. Grandpa Charlie is x years old.
His son is 30 years younger than him.
His granddaughter is half the age of his son.
Find an expression, in terms of x, for
(a) his son's age,
(b) his granddaughter's age,
(c)
the sum of the ages of Grandpa Charlie, his son and his granddaughter.
12. Donny buys p comic books.
Of these, q books are in English and the rest are in Japanese.
The English books cost $8 each.
The Japanese books cost $9 each.
Find an expression, in terms of p and q, for
(a) the number ofJapanese books Donny buys,
(b) the total amount of money Donny spends on the comic books.
13. A fruit seller weighs a bag of apples and a bag oflemons.
Each bag of 20 apples has an average mass of x kg.
Each bag of 10 lemons has an average mass of y kg.
He places 120 apples and 50 lemons in a basket.
The empty basket has a mass of 2 kg.
(i)
What is the total mass of the basket of fruits?
(ii) State any assumption you have made.
14. Julia has $100.
She buys 3 identical water bottles and 2 identical lunch boxes.
A water bottle costs $x.
A lunch box costs $6 more than a water bottle.
(i)
How much does Julia spend altogether?
After buying the water bottles and lunch boxes, Julia realises that she does not have enough money to buy
another water bottle.
(ii)
Suggest how much a water bottle costs.
Date:--'-----'-----
Linear equations with integer coefficients
1.
Solve each of the following equations.
(a) x+4= 15
(b)
x
+ 7 = -7
(c)
x- 6 = 10
(d) x-12=-l
(e)
3x=27
(f)
5x = -55
(h)
-6x = -84
(j)
4x- 3 = 25
(k) 7x + 10 = -4
(1)
2x- 9 = -9
(m) 2- 5x = 67
(n)
11 - 3x = -16
(p)
8x- 0.7 = 4.9
(g) -2x = 42
(i)
9x + 1 = 82
(o) 4x + 1 =
-¾
2.
Solve each of the following equations.
(a) 3x = 2x + 25
(c)
9x-14=2x
(e)
10x- 3
= 8x -
(g) 2(4x + 1)
(i)
30
(m) 3(x+ 1)
17
= 16
= 5(3x -
(k) 4(9x + 4)
(b) 7x=x+ 12
2)
= 19x -
=
(o) 7(2x - 3.9)
1
(d)
5x + 1 = 4x + 6
(f)
11 - 2x = 5 - 3x
(h)
-6(x - 1)
(j)
14 = -7(2x + 7)
(l)
2Ix = 3(8x + 5)
(p)
5(2x + 0.4) + 6(x - 1.5) = 0
= 18
s( x-¼)
= 4(x + 1.3)
Class: _ _ _ _ __
Date: _ _ _ _ __
Linear equations with fractional coefficients and
fractional equations
1.
Solve each of the following equations.
1
(a)
x=3
6
(c)
¼x+S = 9
(d)
¾x-1=2
(e)
-¾x+7=1
(f)
10--x = -5
(g)
X
(h)
yx=3x+I9
=
¾x-6
3
4
2.
Solve each of the following equations.
(a)
x+i
= 27
(b)
Sx-1
(e)
4x-l
9
=3
(t)
7x+2_ 4 =0
(h)
3x+7 _2x+l =0
8
5
(j)
-4-+
(1)
2 x+-7-
(i)
(k)
l-~
4
= 5x+8
7(x-l)
2
8
4(8x+l)
9
1
= 24
= 42
6
9-2x
3(x+3)
2-Sx
10
=
=x
20x-9
6
+ 11
3.
Solve each of the following equations.
(a)
.J.Q_ = 5
(b)
(c)
_.3.l_ =-7
(d)
(e)
~-1
5x+3
(g)
x+7
x+9
(i)
_8_
x-3
x+l
4x+6
=8
32
x-
(f)
1
-6 = --+1
= 19
(h)
3x-2
4x+2
=
(j)
4
5x+I2
(1)
12
13
---=--ISx-14
lSx-14
=I
15
3
x+
4
7
x+6
=6
2
= 2x-ll
4.
Determine if x = ~: is the solution of the equation 2( x+½) = ¾x-¼,
5.
3x + 8 = 20 and 10 - x = 6 are equivalent equations because they have the same solution.
(i)
(ii)
6.
What is the solution?
State another two equivalent equations that have the same solution as the one in part (i).
It is given that Sx + 2y = 3x + Sy.
(a)
Can we conclude that the solution of the equation is x = 6 and y = 2?
(b) Find the value of ; ; .
7.
Show thatthe equation ¼(7 +3x)
= 0.75x + 2.25 has no solution.
State three equivalent equations that have the same solution x
= -2.
One equation should be in the form ax + b = c, where a, b and c are integers.
One equation should be in the form hx + k = I, where h and k are proper fractions.
5
One equation should be in the form - = 4, where p and q are rational numbers.
px+q
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Applications of linear equations in real-world contexts
1.
The sum of three consecutive integers is 138.
Find the integers.
2.
The sum of three consecutive odd integers is 165.
Find the integers.
3.
The sum of three consecutive even integers is always a multiple of 3.
Why is this so?
Six months ago, Paul was 8 kg heavier than he is now.
His mass now is
¾of his mass six months ago.
What is his present mass?
5.
Wayne has three fewer $SO-notes than $10-notes.
The total value of the notes is $330.
How many of each note does Wayne have?
6.
The price of a dining table is $320 more than three times the price of a chair.
Let the price of the chair be $x.
(i)
Find an expression, in terms of x, for the price of the dining table.
Mr Chua pays $860 for a dining table and six chairs.
(ii) Write down an equation in x to represent this information.
(iii) Solve the equation in part (ii) to find the price of the dining table.
7.
Kate is one-third as old as Mrs Wong.
After 6 years, the sum of their ages will be 60 years.
(i)
How old is Kate now?
(ii) How old was Mrs Wong when Kate was born?
8.
A conference hall has four entrances.
The number of people who used each entrance to attend a conference is shown in the table.
A
B
C
D
X
2x
2x + 70
3x + 20
(a) Based on the table, is it accurate to say that the most number of people used entrance D? Why?
(b) A total of 850 people attended a conference.
How many people used entrance C?
9.
Mrs Low has some money to buy plates and bowls for her new house.
If she buys p plates at $12 each, she will have $3 left.
If she buys (p + 4) bowls at $7 each, she will have $5 left.
(i)
Find the value of p.
(ii) How much money does Mrs Low have to buy plates and bowls?
(iii) Mrs Low decides to buy 5 plates.
She uses the remaining money to buy bowls.
How many bowls can she buy?
10.
Q
(3x- l) cm
P""'---------------~R
(8x- 3) cm
PQR is a triangle.
PQ = (Sx + 2) cm, QR= (3x - 1) cm and PR= (8x - 3) cm.
(i)
Find an expression, in terms of x, for the perimeter of triangle PQR.
The perimeter of triangle PQR is 30 cm.
(ii) Form an equation in x and solve it.
(iii) Find the length of the longest side of the triangle.
11. James is choosing between two options to travel from Singapore to Beijing.
Option A (non-stop): 3x hours
Option B (transit in Hong Kong): ( Sx
-¾)
hours
A non-stop flight takes 3 hours 20 minutes less than flying via Hong Kong.
(i)
Form an equation in x and solve it.
(ii) For his outbound flight, James chooses option B.
What is his total flight time in hours and minutes?
12.
Write a real-world application problem such that the equation to be formed to solve the problem is
= 10.
7x - S(x + 8)
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Mathematical formulae
1.
It is given that y
= ~x -
3.
Find the value of y when x
= 56.
2
a+b )
it is given that w =
10
Find the value of w when a= -2 and b = -3.
(
3.
The average, A, of three numbers p, q and r, is given by the formula A =
p+q+r
.
3
Two of the numbers are 10 and 12.
Suggest a third number and find the corresponding average of the three numbers.
4.
(i)
Find a formula for the sum S of any four consecutive even numbers.
You must define the variable used.
(ii) Hence, find the sum of the four consecutive even numbers when the smallest number is -22.
5.
In some countries, the degree Fahrenheit (°F) is the unit used to measure temperature.
To convert from f°F to c degree Celsius (°C), we apply the formula c =
To convert from c °C to f°F, we apply the formula f
= 1.8c + 32.
s(!-32)
9
.
(a) Convert 35 °C into °F.
(b) Convert 104 °Finto 0 C.
6.
The formula for finding the circumference C of a circle is given by C = 2nr, where r is the radius of the circle.
(a) The radius of a circle is 6 cm.
Find the circumference of the circle.
(b) The circumference of a circle is 24 cm.
Find the radius of the circle, giving your answer correct to 3 decimal places.
7.
An object that is moving has kinetic energy.
Its kinetic energy, E joules, is given by E =
½mv
2
,
where the mass of the object is
object is v mis.
(a) An object of mass 0.2 kg is moving at a speed of 0.3 mis.
Find its kinetic energy.
(b) An object moving at a speed of 0.4 mis has 0.16 joules of kinetic energy.
Find the mass of the object.
mkg and the speed of the
8. The time taken to print an A3 poster is 7 seconds.
The time taken to print an A4 poster is 5 seconds.
(i)
Find a formula for the total time taken to print m A3 posters and n A4 posters.
(ii) Hence, find the total time taken to print 24 A3 posters and 10 A4 posters.
Give your answer in minutes and seconds.
Jean buys m identical plant pots for her garden.
Each plant pot costs $x.
(i)
Find a formula for the total cost $C of m plant pots.
(ii)
Hence, suggest how many plant pots she could buy with $48 if each plant pot costs more than $4
and she does not receive any change.
10. In 2018, the number of students studying Mathematics in a college was n.
This number rose by x in 2019 and by another 3x in 2020.
(i)
Write down an expression, in terms of n and x, for the number of students studying Mathematics in
the college in 2020.
(ii) Given that x = 28 and that there were 622 students studying Mathematics in the college in 2020, find
the value of n.
11. The table shows the travel insurance premiums payable for individual and family coverage.
(i)
Individual
Family
Premium for first three days
$32
$80
Premium for each subsequent day
$5
$11
Find the total premium Alan has to pay if he buys a 7-day policy for himself.
(ii) Find a formula for the total premium Mr Chong has to pay for a family policy for n days, where n > 3.
(iii) Is it possible for a family of three to pay less premiums by selecting the individual policy? Explain your
answer with clear working.
12. Vivien wants to cook fish and chips for a party.
A packet of fish costs $a.
A sack of potatoes costs $b.
Vivien estimates that one packet of fish is enough for three persons and one sack of potatoes is enough for
eight persons.
Let x be the number of people attending the party.
(a) Help Vivien derive a formula to calculate how much she will spend on fish and potatoes.
(b) Does the formula in part (a) apply for all real values of x? Explain your answer.
1.
Solve each of the following equations.
(a) 4x + 21
=5
3.
1 - 3x =
1
4
(c)
-8(x - 2) + 9 = 0
(d)
2(7x - 3)
= 6x + 15
(e)
5 _ 2x = 4x+l
(t)
2x+l
3x-l
4
3
2.
(b)
Determine if x
6
= I is the solution of the equation
=7
½( 4x- l) = ½x + / 0 .
State two equivalent equations that have the same solution x =
-¾ .
4.
When a number is tripled, it gives the same result as when 32 is added to it.
What is the number?
5.
The sum of two numbers, one of which is three-quarters of the other, is 49.
Find the product of the numbers.
6.
I am thinking of a number.
When I subtract 7 from the number and then multiply the result by 4, the answer is the same as subtracting
the number from 40 and then multiplying the result by 8.
Let the number be x.
Write down an equation in x and solve it.
7.
The numerator of a fraction is 3 less than the denominator.
When 7 is added to the numerator and to the denominator, the fraction becomes
Find the original fraction.
~.
8.
It is given that 4x - 7y = x + 3y.
Find the value of each of the following.
If it is not possible to find it, explain why.
(a)
2'.._..!.
X
(b)
2
y-1
x-2
9.
p~_2(_2_x_-_l_)_o_n_ _!11---------.Q
-
(Sx+ 1) cm
s
(7x + 3) cm
PQRS is a rectangle.
QR= (5x + 1) cm and RS= (7x + 3) cm.
Tis a point on PQ such that PT= 2(2x - 1) cm.
(i)
Find an expression, in terms of x, for QT.
The length of PQ is
~
times the length of QR.
(ii) Form an equation in x and solve it.
(iii) Find the area of rectangle PQRS.
R
IO. The diagram shows a cylinder.
The total surface area, A cm2 , of the cylinder is given by A= 2nr(r + h), where r cm and h cm are the radius
and height of the cylinder respectively.
Find the total surface area of a cylinder with radius 6 cm and height 11 cm.
rem
hem
11. Nora works in a shop.
She is paid $12 for each hour she works.
She is also paid a commission of $20 for each oven she sells.
One week she works for m hours and sells n ovens.
(i)
Find a formula for the amount of money she earns in that week.
(ii) In a particular week, Nora works 22 hours and sells 3 ovens.
How much does Nora earn in that week?
(iii)
In another week, Nora earns a total of $460.
Find two possible sets of values for m and n.
Class: _ _ _ _ _ __
Cartesian coordinates
Write down the coordinates of each point shown.
A
B
C
D
E
F
Date: _ _ _ _ _ __
2.
Plot and label each of the following points.
A(-3, 6)
B( 4, 5)
C(O, 2)
D(5, -1)
E(-4, -5)
F(-2, O)
3. Plot and label each set of points.
Join the points in order with straight lines and identify each geometrical shape obtained.
(a) A(6, 1), B(6, 4), C(2, 4), D(2, 1)
(b) E(-5, 0), F(O, 1), G(-1, 6), H(-6, 5)
(c) P(-2, O), Q(-2, -6), R(-5, -3)
(d) W(l, -1), X(O, -5), Y(4, -5), Z(5, -1)
4.
The vertices of a right-angled triangle are P(-3, 1), Q(4, 1) and R(4, 3).
(i)
Plot the points P, Q and R.
(ii) Find the area of triangle PQR.
The vertices of a triangle ABC are A(-8, -1) and B(O, -1).
The area of triangle ABC is 16 units 2 •
By plotting the points A and B, or otherwise, find three possible pairs of coordinates of C.
Class: _ _ _ _ _ __
Functions and linear functions
1.
The equation of a function is y = 4x + l.
Find
(i)
the value of y when x = 2,
(ii) the value of x when y = -3.
2.
The equation of a function is y
=7 -
2x.
Find
(1.)
1
t h e va1ue of y wh en x = - ,
2
(ii) the value of x when y = 5.
3. The equation of a function is y
= ¼x -
3.
Find
the value of y when x = -6,
(i)
(1·1·) t he va1ue of x wh en y
4
= 15 .
The equation of a function is y = 10 - 0.9x.
Find
(i)
the value of y when x
(ii) the value of x when y
5.
= 8.5,
= 0.
The equation of a function is y =ax+ b, where a and bare non-zero constants.
When x = 4,y = -5.
Give an example of the values of a and b that satisfy the above condition.
Date: _ _ _ _ _ __
6.
(i)
(ii)
Show that the point (1, 7) does not lie on the line y = Sx - 2.
The point Plies on the line y = Sx - 2.
Write down a possible pair of coordinates for P.
7.
Complete the table.
(a) y
= 3x- 1
(b) y
= -x+ 8
(c) y=7+x
(d) y=l0-4x
(e) y
1
= -x-5
(f) y
= -l.2x + 9
4
....
(g)
2
(h)
-6
(i)
(j)
1
3
0
.)
4
0
6
-2.7
8.
(i)
Using a scale of 1 cm to represent 1 unit on each axis, draw a horizontal x-axis for values of x from
-6 to 6 and a vertical y-axis for values of y from -8 to 8.
Draw the graph of each of the following functions for values of x from -6 to 6.
(a) y=
(c)
1
2 x+l
(b) y=
1
2 x+5
1
y= -x-5
2
(ii) What do you notice about the lines you have drawn in part (i)?
9.
(i)
Using a scale of 1 cm to represent 1 unit on each axis, draw a horizontal x-axis for values of x from
-6 to 6 and a vertical y-axis for values of y from -8 to 6.
Draw the graph of each of the following functions for values of x from -6 to 6.
=x -
(a)
y
(c)
1
y=-x-1
4
1
(b) y
= -x -
(d) y
= -0.5x -
1
I
(ii) What do you notice about the lines you have drawn in part (i)?
10. (i)
Using a scale of 1 cm to represent 1 unit, draw a horizontal x-axis for values of x from -5 to 5.
Using a scale of 1 cm to represent 2 units, draw a vertical y-axis for values of y from -8 to 14.
Draw the graph of each of the following functions for values of x from -5 to 5.
(a)
y=2x+3
(b) y=6-x
(ii) The point (a, O) lies on y = 2x + 3.
Write down the value of a.
(iii) The point (4, b) lies on y = 6 - x.
Write down the value of b.
(iv) The point (h, k) lies on bothy= 2x + 3 and y = 6 - x.
Write down the values of hand k.
11. Find the gradient and they-intercept of each of the following lines.
(b)
(a)
j
,
l
(c)
I
i
2
(d)
Name: _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Applications of linear graphs in real-world contexts
I.
The graph shows the cost, $C, of sending a parcel of mass m kg overseas.
f-30•+-+---+--+----+----~--------+--+---+-i
i
I
~0-1---+--i---+----l---t---+--i---+-~~----t------i
'
I'--2-0•+·- + - - - + - - - - - - + - - - - - + - - - - + - - - + - - - + - ~ ~ - - - + - - - - + - - - - + - - - - - - 1
IP
l
i
$0
fO
I
Use the graph to find
(i)
the cost of sending a parcel overseas if its mass is
(a)
40 kg,
(b)
76 kg,
(ii) the mass of the parcel if the cost of sending it overseas is
(a)
$50,
(b)
$260.
2.
The graph shows the exchange rate between pounds(£) and Singapore dollars($).
,__.--+----+-----+--+---+----+-----,.----+----+----+--+-----+---+----<
Use the graph to estimate how much Ann will receive
(i)
in pounds if she changes
(a)
$170,
(b)
$50,
(b)
£24.
(ii) in dollars if she changes
(a)
£60,
Singapore
dollars($)
3. The length, y cm, of a spring when a mass, x g, is attached to it is given by y
(i)
= 0.2x + 4.
Complete the table.
20
30
(ii) Using a scale of 4 cm to represent 10 g, draw a horizontal x-axis for values of x from Oto 30.
Using a scale of 1 cm to represent 1 cm, draw a vertical y-axis for values of y from Oto 10.
Draw the graph of y = 0.2x + 4 for values of x from Oto 30.
(iii) Find the length of the spring when each of the following masses is attached to it.
(a)
25g
(b)
18g
(iv) Explain the significance of they-intercept of the graph.
4.
The graph shows Jimmy's journey from home to school.
He left home at 08 30, walked to the bus stop and travelled the rest of the way on the bus.
Distance from home (km)
Ho
08 45
(i)
08!50
I
How long did Jimmy wait at the bus stop?
(ii) How far was he from school at 08 45?
(iii) Find the gradient of the graph between 08 40 and 08 50.
What is the significance of this value?
(iv) Jimmy's neighbour, Ken, left home at 08 35.
He cycled to school at a constant speed and reached school I minute before Jimmy.
Show Ken's journey on the above graph.
Date: _ _ _ _ _ _ __
Class: ________
y
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Write down the coordinates of each point shown.
A
B
C
D
E
F
(ii) State the coordinates of the point on the circle that
(iii)
(a)
has the same x-coordinate as D,
(b)
has the same y-coordinate as E.
Write down the coordinates of two possible points on the circle that are each 5 units away from the
point (O, -1).
2.
The equation of a function is y
=5
-¾ x .
Find
(i)
the value of y when x
= -8,
(1·1·) th e va1ue of x when y =
1
2.
= 7 x+ 2} pass through the point
(-1,,½)? Explain your answer.
3.
Does the line y
4.
Find the gradient and they- intercept of each of the following lines.
(a)
(b)
5.
Draw each of the following lines on the graph paper provided.
Label each line clearly.
(a) Line 1: a line passing through (0, 2) and (4, 1)
(b) Line 2: a line with gradient
½and y-intercept -3
(c) Line 3: a line with gradient O and passing through (5, -4)
(d) Line4:y= 4-½x
6.
Doreen plans to cater food for a family gathering.
The graph shows the cost, $C, of catering food for n people.
$C
!---o-+---+---t----+----t--+----+--~!----+---+--t-----11»1 n
~
1
(i)
r.,:
1
.
15
2;0
2Jf
i
3,?
I
3i5
4;()1'
i
I
Complete the table.
20
40
(ii) Doreen has a budget of $600.
Does she have enough money to pay for the catering of food for 45 people?
(iii) Doreen is puzzled by the observation from the graph that there is a cost even though the value
of n is 0. Explain to Doreen the significance of this cost.
(iv) Doreen notices that the cost of catering food for each person decreases when there are more people.
Explain to Doreen why this is so.
Name: _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Class: _ _ _ _ _ __
Date: _ _ _ _ _ __
Number sequences
1.
2.
For each of the following sequences, state a rule on how to obtain the sequence and write down the next
two terms.
(a)
15, 22, 29, 36, 43, ...
(b)
50, 46, 42, 38, 34, ...
(c)
8, 16, 32, 64, 128, ...
(d)
-2400, 1200, -600, 300, -150, ...
Write down the next two terms of each of the following sequences.
(a)
3, 7, 11, 15, 19, ...
(c) 49, 43, 37, 31, 25, ...
(b)
2, 6, 18, 54, 162, ...
(d)
288, 144, 72, 36, 18, ...
(e) 64, -32, 16, -8, 4, ...
Write down the next two terms of each of the following sequences.
(a)
10, 9, 12, 11, 14, ...
(b)
4, 5, 9, 14, 23, ...
(c)
2, 5, 10, 50, 500, ...
(d)
3, 4, 8, 9, 18, ...
(e)
6,5,3,-1,-9, ...
(f)
-8.1, -7.1, -3.1, 5.9, 21.9, ...
4.
Given the n th term, write down the first three terms of each of the following number sequences.
(b)
(a) 3n+8
8n-9
(d)
(e)
5.
10
(f)
n3
4n
n+5
The first term in a sequence is 48.
Each following term is found by subtracting 9 from the previous term.
(i)
Write down the second and third terms.
(ii) Write down an expression, in terms of n, for the nth term.
4, 11, 18, 25, 32, ...
6.
(i)
Find an algebraic expression for the nth term in the sequence.
(ii) Show that 333 is a term in the sequence.
7.
The first four terms of a sequence are 5, -1, - 7 and -13.
(i)
Write down the next two terms in the sequence.
(ii) Find an expression for the nth term of the sequence.
(iii) The kth term of the sequence is-163.
Find the value of k.
8.
The first four terms of a sequence are 3, 7, 11 and 15.
(i)
Write down the 8th term of the sequence.
(ii) Find an expression, in terms of n, for the n th term of the sequence.
(iii) One term in the sequence is 231.
Find the value of n for this term.
9.
Each term in this sequence is found by adding the same number to the previous term.
a, 15, b, c, 36, ...
(i)
Find the values of a, b and c.
(ii) Write down an expression, in terms of n, for the nth term.
(iii) Is 112 is a term of this sequence? How can you tell?
IO. Given that n is an integer,
(i)
write down expressions for the next two odd numbers after 2n - 3,
(ii) (a)
(b)
find, in its simplest form, the expression for the sum of these three odd numbers,
explain why the sum is a multiple of 3.
I 1.
The first three terms in a sequence are -5, k and -19, where k is an integer.
(i) Suggest a value for k.
(ii) Using your answer in part (i), find an expression, in terms of n, for the n th term of the sequence.
12. (a) The n th term of a sequence is given by 3n2 - 1.
Write down the first four terms of the sequence.
(b) The first four terms of another sequence are 4, 13, 28 and 49.
(i)
By comparing this sequence with the sequence in part (a), write down an expression, in terms
of n, for the n th term of this sequence.
(ii) Hence, find the 25 th term.
13. The first five terms in a sequence are 2, 6, 46, 446 and 4446.
Find a relationship, in terms of n, between the nth term and the (n + l) th term of the sequence.
Class: _ _ _ _ _ __
Date: _ _ _ _ _ _ __
Number sequences ahd patterns
1.
The diagram shows some patterns made from identical white and grey unit squares.
Figure 1
(i)
Figure 2
Figure 3
Figure 4
Complete the table.
1
2
1
3
4
2
3
4
5
9
3
4
n
(ii) A large square has a length of 65 units.
How many white unit squares does it have?
(iii) Will there be a figure in this sequence that has 920 grey unit squares? Explain your answer.
2.
The first four figures in a sequence are as shown .
•
• •
• •
• •
•
• •
• • • • • • • • • • •
•
Figure 1
(i)
Figure 2
Figure 4
Figure 3
Draw the next two figures of the sequence.
(ii) Complete the table.
1
3
2
6
3
4
(iii) As the figure number increases, how does the number of dots change?
(iv) Belle thinks that the number of dots in Figure n is given by the formula n2 + 3n + 2.
Do you agree with her? If you do not agree, find a formula for the number of dots in Figure n.
3. The first three terms in a sequence of numbers, T1, T2 , T3 ,
2
T I =1 +4=5
T2 = 22 + 8 = 12
T3 = Y + 12 = 21
(i)
Find T 4 •
(ii) Find an expression, in terms of n, for Tn.
(iii) Evaluate T 50 •
4.
Consider the pattern:
1
1
3+2
5
1
1
4+3
7
-+=
4 5
1
1
5+4
= -20
1
1
6+5
11
2+3 = 2X3 = 6
3+4 = 3X4 = 12
9
5+6 = 5x6 = 30
.!.+l.. = 2l+k = p__
k
(i)
21
kx21
Write down the 5th line in the pattern.
(ii) Find the values of k, p and q.
(iii) Find the value of
:o + : 1 .
q
•••
are given below.
5.
Consider the pattern:
1=1 2 =1 3
9
= 32 = 13 + 23
36 = 62 = 13 + 23 + 33
3025 = a2 = 13 + 23 + J3 + ... + b3
(i)
Write down the 4 th line in the pattern.
(ii) Find the value of a and of b.
6.
Consider the pattern:
5 = 1 X 5 = 1 X (1 + 4)
12 = 2 X 6 = 2 X (2 + 4)
21 = 3
X
7 = 3 X (3 + 4)
32 = 4
X
8=4
X
(4 + 4)
396 = a x b = a x (a+ 4)
(i)
th
Write down the 5 line in the pattern.
(ii) Find the values of a and b.
(iii) Write down the 20 th line in the pattern.
(iv) Will there be a line with 2204 on the left-hand side? Explain your answer.
Name: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
1.
Class: _ _ _ _ _ __
The nth term of a sequence is ¼n(n + 1)( n + 2).
Write down the first 3 terms in the sequence.
2.
TI1e n th term of a sequence is given by Tn
= 100 -
3n 2•
Complete the table.
1
97
3.
4
16
-143
The first term of a number sequence is 1.
This sequence follows the rule:
Add 3, then multiply by 2.
(i)
Write down the second and third terms.
(ii) A second sequence follows the same rule.
Its third term is 54.
Find the first term.
Date: _ _ _ _ _ __
4. The first four terms of a sequence are 2
(i)
1
1
1
2 , 16 , - 6
and -1
1
2.
Write down the 7th term of the sequence.
(ii) Find an expression, in terms of n, for the nth term of the sequence.
(iii) One term in the sequence is -18¾.
Find the value of n for this term.
5.
Given that n is an integer,
(i)
write down expressions for the next two even numbers after 2n,
(ii) (a)
(b)
6.
find, in its simplest form, the expression for the sum of these three even numbers,
determine whether the sum is always a multiple of 4.
The first three terms in a sequence of numbers, T1, T 2, T3,
3
T=l
-2=-1
1
T2 = 23 - 3 = 5
T3 = 33 - 4 = 23
(i)
Find T4 •
(ii) Find an expression, in terms of n, for T.
"
(iii) Evaluate T18 •
•••
are given below.
7.
(a) The n th term of a sequence is given by 2n 2 + 5.
Write down the first four terms of the sequence.
(b) The first four terms of another sequence are 8, 15, 26 and 41.
(i)
By comparing this sequence with the sequence in part (a), write down an expression, in terms
of n, for the n th term of this sequence.
(ii) Hence, find the 36th term.
8.
Consider the pattern:
22 + 2
X
1 + 12 = 7 = 23 - 13
32 + 3 X 2 + 22 = 19 = 33 - 23
42 + 4
X
3 + 32 = 37 = 4 3 - 33
52 + 5 X 4 + 42 = 61 = 53 - 4 3
(i)
Write down the next line in the pattern.
(ii) Write down the n th line in the pattern.
(iii) Hence, find the value of 242 + 24 x 23 + 23 2 •
9.
A hexagon is a six-sided figure.
The diagram shows some patterns made from identical white and grey hexagons.
Figure 1
(i)
Figure 2
Figure 3
Figure 4
Complete the table.
1
1
0
1
2
1
1
2
3
4
12
(ii) Find an expression, in terms of n, for the number of grey hexagons in Figure n, where n is an odd number.
(iii) Which figure has the same number of white hexagons as Figure 156?
Date: _ _ _ _ _ __
1 hour 15 minutes
READ THESE INSTRUCTIONS FIRST
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an approved scientific calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Section A
1
2
16
33
49
511
1000
List
(a) the composite numbers,
Answer - - - - - - [l]
(b) the perfect cubes.
Answer - - - - - - [1]
2.
(-14.8)2
(a) Calculate 23.56 x 0.979 ·
Write down the first five digits on your calculator display.
A nswer - - - - - - - [l]
(b) Write your answer to part (a) correct to 3 significant figures.
A nswer - - - - - - - [l]
3.
Without using a calculator, find the value of 5
¾x f + (-½)
2
Answer _ _ _ _ _ _ _ _ _ [2]
4. Expand and simplify 5[5x - 2y(4x - 3) + 6y] - I Ox.
Answer - - - - - - - [2]
5. A straight line is drawn passing through the points (-3, O) and (5, -4).
Find the gradient of the line.
Answer - - - - - - [2]
6.
(a) Factorise 12abc - 28abcx + 36acx completely.
Answer _ _ _ _ _ _ _ _ _ [l]
(b) Without using a calculator, find the value of 169 x 1003 - 3 x 169.
Answer - - - - - - - [l]
7.
It is given that
5x+y
1
= 2·
7 x- 3y
Find the value of l
X
.
Answer _ _ _ _ _ _ _ _ _ [2]
8.
Two different sizes of tins of tomatoes are shown.
The mass of the tomatoes and the price are given on the tins.
LARGE
SMALL
425 g
$1.45
925 g
$2.95
Which size of tin gives the better value?
Answer The _ _ _ _ _ _ _ _ _ tin gives better value. [2]
9.
A student wrote the following to solve the equation
~:~~ = ¾.
What misconception has the student made? Show working to explain your answer.
Answer
[2]
10. Solve
3
x+
7
1
= 2; .
Answer x
11. (a)
= _________ [3]
Give two examples of irrational numbers.
Answer - - - - - - [l]
..
a
(b) Express 0.39 in the form b , where a and b are integers and b ;t:. 0.
Answer - - - - - - [2]
12. A plot ofland is used to grow flowers.
½of the land is allocated for orchids.
After the orchids have been planted, } of the remaining land is allocated for roses.
After orchids and roses have been planted, 0.75 of the remaining land is allocated for tulips.
What fraction of the plot ofland is not occupied by the flowers?
Answer - - - - - - [3]
Section B
13. (a) Use prime factors to explain why 68 x 153 is a perfect square.
[3]
(b) The number 68k is a perfect cube.
Find the smallest positive integer value of k.
[2]
(c) Find the lowest common multiple of 68 and 153, expressing your answer in index notation.
[1]
(d) One-third of the product of 68 and 108 is the same as 2x x 3Y x 17z_
Find the values of x, y and z.
[2]
14. In a sequence, the same number is subtracted each time to obtain the next term.
The first five terms of the sequence are
p
41
q
r
13
(a) Find the values of p, q and r.
[2]
(b) Write down an expression for the nth term of this sequence.
[2]
(c) Explain why-201 is not a term of this sequence.
[1]
(d)
Suggest a term of this sequence, other than those listed above, that is greater than -250.
[l]
The first five terms of a second sequence are
1
41
1
p
(e) Find the sum of the 8th term and the 9th term.
1
q
1
r
1
13
[2]
15. A manager plans to distribute journals to his staff at the end of the year.
He sends an email to a printing firm and obtains the following quotation:
Quotation for journals
• $240 for 10 journals
• $330 for 15 journals
• $420 for 20 journals
The more you order, the less you pay!
(i)
By letting x represent the number of journals and $y represent the cost, write down three pairs of
coordinates from the information provided.
[l]
(ii) Using a scale of 1 cm to represent 2 journals, draw a horizontal x-axis for values of x from 0 to 20.
Using a scale of 2 cm to represent $100, draw a vertical y-axis for values of y from $0 to $500.
Plot the points in part (i) and join them with a straight line.
[3]
(iii) Extend the line in part (ii) such that it intersects the y-axis.
Explain the significance of they-intercept of the graph.
[l]
(iv) The manager is not sure whether the caption "The more you order, the less you pay!" is a true statement.
Explain clearly whether you agree with the caption. Show calculations to support your answer.
[2]
1 hour 15 minutes
READ THESE INSTRUCTIONS FIRST
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an approved scientific calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Section A
1.
Write the following numbers in order of size, starting with the smallest.
0.62,
3
5'
(¾J
3
4
Answer _ _ , _ _ , _ _ , _ _ [l]
2.
Are the numerical values of TT,
22
and 3.142 identical? Explain your answer.
7
Answer _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [l]
3.
It is given that p = _3a - b3 •
Find the value of p when a= 4 and b = -2.
Answer - - - - - - [2]
4.
Expand and simplify S(x + 6y) - 2[4(3y- x) + 7x].
Answer - - - - - - - [2]
2
5.
7a -y can be written in the form -hay(
The expression -8axy +- 7 a+ kx ) .
5
5
Find the values of h and k.
Answer _ _ _ _ _ _ _ _ _ [2]
Can :
!
be expressed as a terminating decimal or a recurring decimal?
Show how to, or explain why not.
Answer
[2]
7.
A student wrote, "If 2 and 8 are factors of a number, then 16 is also a factor of that number:'
Do you agree with the student?
Explain your answer.
Answer _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
- - - - - - - - - - - - - - - - - - - - - - - - - - [2]
8.
Shan has 315 one-centimetre cubes.
She arranges all of the cubes into a cuboid.
The perimeter of the top of the cuboid is 24 cm.
Each side of the cuboid has a length greater than 3 cm.
Find the height of the cuboid.
Answer _________ cm [2]
Answer - - - - - - [3]
10. Simplify
¼ax(-6b)+3ba-(-½b
)+(-/a).
Answer - - - - - - [3]
11.
A
p
B
(2.8x + 0.1) m
R
Q
(3x- 2.5) m
C
ABCD is a kite and PQR is a triangle.
The lengths of the sides of ABCD and PQR are shown in the diagram.
The perimeter of PQR is 0.5 m greater than the perimeter of ABCD.
Form an equation in x and solve it.
Answer x
= _________ [3]
12. (a) Express 198 as the product of its prime factors.
Answer - - - - - - [1]
(b) The number 198k is a perfect square.
Find the smallest positive integer value of k.
Answer k
= - - - - - - [1]
Answer x
=- - - - - - [2]
(c) xis a number between 200 and 300.
The highest common factor of x and 198 is 33.
Find the smallest possible value of x.
Section B
13. Amanda produces the newsletter for an editorial club.
Each newsletter contains pages printed on both sides in colour and in black.
The graphic designer informs Amanda that the upcoming issue of the newsletter will have 4 pages printed
in colour and 20 pages printed in black.
Every 4 pages printed in colour will cost x cents.
Every 4 pages printed in black will cost y cents.
(i)
(ii)
Find an expression for the cost of printing one copy of the newsletter.
[1]
Each newsletter costs 14 cents to print.
Give an example of the cost of printing 4 pages in colour and the cost of printing 4 pages in black. [ l]
An educational institution in Shanghai wants to subscribe to this newsletter.
In Singapore, each newsletter costs S$2.20.
The conversion rate is ¥1=S$0.19845.
(iii) Without using a calculator, estimate the price of one newsletter in¥.
[2]
Each sheet of paper has a mass of 4.5 g.
(iv) How heavy is one copy of the newsletter?
[I]
14. (a) These are the first four terms in a sequence.
7
(i)
13
19
25
Find an expression, in terms of n, for the nth term of the sequence.
(ii) Explain why it is not possible for a term in the sequence to be a multiple of 3.
(b) The n th term of a different sequence is given by T,,
(i)
[2]
[l]
= 1:;~~n.
Use the formula to find L
"
Give your answer as a fraction.
[l]
(ii) The value of Tk can be simplified to { .
3
Find the value of k.
[3]
(iii) Find the least value of n for which Tn is greater than 1.
[3]
15. The graph shows Aaron's journey from the office to his home.
He left the office at 17 30, walked to the train station and travelled the rest of the way on the train.
(i)
How long did Aaron wait at the train station?
(ii) How far was he from home at 18 00?
[l]
[l]
(iii) Find the gradient of the graph between 17 50 and 18 10.
What is the significance of this value?
[2]
(iv) Aaron's brother, Darren, works 13 km away from home.
Darren left his office at 17 30.
He travelled home at a constant speed and reached home at the same time as Aaron.
(a)
Show Darren's journey on the above graph.
(b)
The two graphs intersect at one point.
Make two comments about this point of intersection in the context of the question.
[2]
[2]
20. (i)
(ii) 1500
21
8. (a) 2 X 3 X 5 X 19
(b) (i) 22 X 3 X 5 X 7 X 19
(iii) Yes
21. (i)
24 X 13 2
(ii) 190
(iii) m = 13, n = 2
9. (i)
1. 2, 7,29
(iii) 210
2. (a) Composite number
10. (i)
(b) Prime number
(d) Prime number
3. (a) 24
(c)
7 X 19
4. (a) 23 X 3
(c) 32 X 52
(e)
72 X 11
X 4,y
(b) 3
X 52
(d) 5
X 73 X
11 2
(b) 5 X 19
(ii) 26 X 33 X 53 X 7 X 11
(c) 19
(d) 42
(iii) 6
2. (a) 15
(b) 14
11. 29 th September
(c) 21
(d) 33
12. 14 cm
24 x3x5 2
(b)
4. (a) 100
(b) 216
(c) 520
(d) 672
6.
7. 37
8.
(d) 816
(c) 2640
(a)
24 X
J3
X 52
(b) 2 X J3 X 53 X 7 X 11
8 cm x 4 cm x 5 cm or
20 cm x 4 cm x 2 cm
9. (a) False
7. 195
2.
(b) True
144
9.
11. (a) 2x3x5xl3
(b) 63
(b) 78
(d) 220
12. (i) 40
2. 45
13. (a) 22 X 33 X 5
3. (a) 28
(b) 40
(b) (i) 36
(d) 15
(c) 8
(b) (i) 23 X 32 X 5 X 11 2
(a) 359
(b) 0.3567
(c) 9.7676
(d) 8.5338
5.4663
(f)
(ii) 44
15. (i)
10.2294
6. 3.01 cm
16. (i)
(ii) 26 X 33 X 5 X 7
12
(iii) 6
8. Yes
17. (a) 22 X 32 X 5
9.
52
10.
26 X
73
(b) 12
(c) 4
18. (i)
11. (ii) 35
12. (i)
(ii) 23 X 34 X 52 X 7
18
(iii) 18
7. 102.0m
34 X
(ii) 7560
14. (a) 23 X 5 X 11
4. 12cm
(e)
(ii) 58 800
(ii) 15 boys, 16 girls
14
19. 1, 2, 3, 4, 6, 9, 12, 18, 36
22 X 5 X 7
(ii) 35
33
(ii) 2541
13. 6
(b) -16
1
(d) -0.4, -32, -16
(a) <
(b) >
(c) <
(d) >
(e)
(b) Yes
(c) Insufficient information to conclude
(c) 75
1. (a) 65
2
(c) 7 '65, 9.8
8. 308
10. (a) Yes
1. (a) 20
13. 5
(b) 315
5. (a) 3872
2
x= 3,y= 7
32 X 11 2
3. (a) 22 X 3 X 5
(d) 2xl3xl7
(f)
12
(b) 11
1. (a) 9
(c) Composite number
33 X
(ii) 70
10
(f)
>
<
1
4. (ii) -1,0,1.1, 1010
1
1
5. (ii) -33, -3, 3' 3.03
6. (i)
-12,4
-12 > -15, 4 > -15
(ii)
7. (i)
-25,-30
-25 < -24, -30 < -24
(ii)
8. No
9. -413m
10. -89.2 °C
11. (a) +$5000
(b) -$3600
12. 7 m below ground level
13. 16 km due West
14. An anticlockwise rotation of 51 °
15. A 13.5 °C increase in temperature
16. Mrs Lee's monthly salary is either
increased or decreased by $80.
17. Karagiye Depression, Kazakhstan;
Death Valley, USA; Bukit Timah Hill,
Singapore; Cascade Mountain, Canada
14. 50
15. (i)
(iii)
16. (i)
11907
6
(ii) 30
(iii) 6
17. (i)
26 X 33 X 53
3. (i)
22
4. (a)
2x3 2
(ii) 22
x72
5. (a) 23 X 32 X 5
(ii) 60
18. (i) X= 3,y= l
(ii)
19. (i)
(ii) 336mm
28mm
1. 47
2. 43
3 and 9
(i)
3
1. (a) 11
(b) 1080
2.
2
6.
7.
(b) 126 and 147
(ii) 3
(b) 7
(c) -7
(d) -11
(a) -3
(b) -11
(c)
(d) 3
11
(b) -10
(e) -5
(f)
(c) -65
(d) -56
(g) 100
(h) -90
3. (a) -3
4.
6.
(f)
(h) 0
(c) -29
(d) 88
(a) 1
(b) -80
(e) -64
(f)
-83
9. (a) 44
(d) 88
(g) 10
(h) 0
(i)
150
(j)
(g) -43
(h) 16
$140
(ii) $60
(a) -6
(b) -22
(c) 24
(d) 2
11. 32
(e) 36
(f)
12. (a) Loss, $5
(g) 12
(h) 22
-153
10. (i)
(a) -8
(b) 24
(c) -90
(d) -54
(e) 52
(f)
(g) -17
(h) -30
(i)
(k)
7
- 5 10
(c)
(g)
(ii) -28 m
11.
x=-40,y=-23
X=-40,y=-76
2.
(a)
(c)
(e)
(b) -56
(c) -56
(d) 56
(g) -28
(a) 6
(b) -6
(i)
(c) -6
(d) 6
(a) -90
(b) 120
(c) 84
(d) 0
(e) -17
(f)
(g) 0
(h) 24
2
(a)
1
- 42
(c)
1
(a) ±1, ±2, ±4
(e)
614
15
(b) ±l,±2,±4,±5,±10,±20
7
24
3.
-15
4.
±1, ±3, ±5, ±15
5. (i)
5. (a)
(c)
12,-9
(b)
60,-36
25, -10
(d)
44,-33
(a) 1
(b) -1
(c) 64
(d) -512
(e) 8
(f)
(g) -5
(h) -5
(b) -80
(c)
15
(d) -36
(e)
-4
(f)
-5
(h)
7.
2 2 2
9'5'3
(a) -2.26
(b) -0.09
(c) 0.19
(d) 1
()
a -5.1, 22 , 0, -3 2 , ~
'-64
7
9
--71
5
(f)
(h)
(d)
132_
3
(f)
-4
(h) 21
3
-48
1
(I) 46
1
(b) 14
1
(d) 162
(ii)
7
. 22
(c) 0.027
(d) -13.768
(e) 2.420
(f)
4. (a)
8
9
515
999
889
90
12.248
49
(b)
99
49
(d)
990
1711
(b)
495
s.
(a)
I.
(a) <
(b) >
(c) <
(d) >
-3
(l) 117
18
1
(b) 22
(f)
, 0.3125
(b) -11.620
(c)
19
20
1
4
3
8
(a) -196.808
3
(d)
(j)
(a) 4.06,
4
58
135 h
37
40
15 o f a contamer
.
176
2. 0.28
3.
p=
5
15
7 , q = 0.75, r = 7 , s =
-81
I.
(a) 29.29
-rr
8. (a) -1.7 m
(b) 3.2m
9. (i)
42 °C
(ii)
10. (i)
2.7h
(ii) 20.2 h
-1 °C
11. (a) (i) 9 a.m. on 5th January
(ii) 11 p.m. on 4th January
(b) The local time in Kenya is
-5 hours relative to the local time
in Singapore.
(b) 2.46
(c) -1.11
(d) -2.71
(e) -2.42
(f)
2. (a) 15.25
3rr
8
2
12
(ii) 6, 3 , o.6 , -o.6,
9
5. (a) 15.875
(b) 4.1
5
100
6. (a)
(b)
6
W1
97 33
(c)
120' 40
(d) Infinite number of rational
numbers
3
7. (i)
32
(ii) Non-fiction books, magazines,
assessment books, fiction books
4.
4
7. (a) 12
(g) -216
6.
(b)
(j)
12.
3
(k) -25
(d) ±1
8.
2
- 11s
(a) 56
(c)
6.
2
9
2
3
(d) 0.52
(b) -6, 19' -0.2, 7
5 blouses, 7 scarves, 7 skirts
7
12
1
2
9
14
2
5
1
110
(a)
10. (i)
4.
2.
-16
I.
(c) 3.78
(b) fe,rr
3.
(e)
290m
I.
(iii) 12 day
(b)
(b) 0.72
10
th
52
(a) 72
1
(f)
-27
9. 56° clockwise from the starting
position
3.
4.
(e) -18
8. Increase, 5 °C
2.
(b) 4
(e) -36
7. -7°C
1.
3.
(g) 104
(c)
5.
4
-8.68
(b) 0.1525
(c) 20.3116
(d) 144.288
I.
(a) 51
(e) 0.16
(f)
2.
(a) 4300
0.064
(b) 398
(b) 1600
3.
(a) 63.2
4. (a) 22.05
(b) 956.0
8. S$360
(b) 0.20
9. No
8.
5.
(a) 15.278
(b) 3.001
10. 190 m 2
6.
(a) 97.1326
(b) 0.4061
11. $66
7. (a) 4000
(b) 500
12. Plan A
(c) 3
(d) 0.3
13. (a) Family Combo
(a) 71 000
(b) 240
(c) 20
(d) 0.081
(a) 255
(b) 72 200
I.
(c) 4.00
(d) 0.580
2.
10. (a) 5 s.f.
(b) 5 s.f.
3. (a) 0.629 737 6093
(c) 6 s.f.
(d) 4 s.f.
8.
9.
(c)
(a) 7.9594
(b) 8.0
(ii) 0.042
(b) (i) 0.6
(i)
3, 4 or 5
(c)
(ii) 0.630
(ii)
(d) 0
(b)
4
-35
-641
(d) 1694
(b) 62.
4
1
(d) 1
20
-22.
6
13.
product of p and l2q, product of
6p and 2q, product of 3p and 4q
I.
(a) llx
7. 13m
8. S$50
9. (i)
S$90
(ii) South Korea
0.040 75, 0.040 80, 0.040 818
10. (i)
17 423 499
(ii) 17 422 500
11. (i)
5040 million dollars
17. 3.35 kg
(ii) 440 million dollars
18. (a) (i) 5 503 499
(iii) 6016 million dollars
(ii) 5 502 500
5 503 012, 5 502 999
12. (i)
(c)
$28.9542 (ii)
(b) 3x
(d) -Sx
Sx
(f)
(e) -llx
(b) 8x- 3
l2x+ 7y
(d) y-4x
(f)
(e) -l0x
$28.95
13. Twin Pack
(ii) 720
(ii) 4.04 g
22. Credit card
X
2. (a) -3x+ 15
(c)
20. 200 m 2
49.5 g
(b) -13
5
-36
12. No
99
3, 4, 5 or 6
16. 13 000
21. (i)
1
(d) 22
6. 8 kg
13. (a) 0.041 639 387 77
4 s.f.
2
19
11. (a) -11
3
5
5.
19. (i)
1
(a) 8
(c)
12. (a) 0.357 142 8571
(b)
(b)
10. (a) 0
4. 0.55 kg
(b) (i) 0.0416
(c)
9.
11. 20
(b) 0.36
3
(a) 104
-13y- 7x
(g) -Sx + 4y + 3 (h) -x+ y-5
3.
(a) 14x+ 6y
(b) -5x-3y-6
4.
(a) 16
(b) 7xy-x
5.
(a) 3x+6
(b) 3x- 6
6.
(i)
(ii) $(50 - 2x - Sy)
$(2x+ Sy)
7. No
I.
(i)
18.8 cm
(ii) 75.0 cm
(i)
39.5 cm
(ii) 4890 cm 2
3. 7
2.
(a) 4x+ Sy
(c)
16xyz
(d)
7x
9yz
(e)
.[xi
(f)
Fz
(a) 8x+7y+6z (b) 9y+ 2z- Sx
(d) 90yz
15y 4x
x+4y
3y
(e)
(f)
z
2x+6z
3. (a) sum of 6x and llY
(c)
4.
10 amperes
5.
(a) (i) 21.8%, 40.1 %, 99.9%
(b) (i) 21.81 %, 40.15%, 100%
(b) subtract 4y from 9x
(c) product of x and the square of Y
(d) divide Sx by 8
4.
I.
(a) 8
(b) 10
2. (a) 2
(b) 2
3. (a) (i) 720
(ii) 24
(b) 30
4. (i)
No
(ii) Yes
5. 400cm
6.
1800 g
7. (i)
280 000 + 7 = 40 000
(ii) 39 938, yes
(a) (x - 28) years
5. Length = 3x cm, Width
Area 3x2 cm 2
xcm,
(a) 2
(b) 38
(c) -30
(d) -33
7. (a) 30
13
(c) -28
(a) $(x+ 6y)
(b) $1680
I.
(a) Sx+ 35
(b) 6x- 60
(c)
12x + 32
1
(b) 43
(d) 65
(d) 18x- 9
(e) -x-12
(f)
(g) -12x- 54
(h) -20x + 12
(i)
(j) 60- 20x
3 + 18x
-2x + 14
(k) -16 + 6x
(1)
(m) 35x + 28y
(n) 24x- 64y
(o) -l8y-3x
(p) -lOy + 45x
2. (a) 8xy
(b) (x - 3) years
6.
8.
(b) l0y- 3x
-81 + 90x
(b) 22xy
(c) -27xy
(d) 40xy
(e) -30xy
(f)
(g) 42xyz
(h) 96xyz
3. (a) l8ax+ Say
(c)
-Sax - ay
(e) 24bcy
(f)
(b) 56ax- 2lay
(d) -6ax + 60ay
44bcx
-15bcy- 80bcx
4. (a) 36x+ 2y
(c)
l44xy
Sx- Sy
(b) 6x-7y
(d) -4x
(e)
16x - 26y
(f)
(g) 8x- y + 4z
5.
x + 42y
(e)
(h) -lOx- 36y
(f)
(a) -lOx - 156ay - 12ax
(b) 70y - 180ax - 150ay
(g)
6. Sy - 29x + 49
(a) 4(4x + 3)
(b) 9(x - 5)
(c) 5(2-3x)
(d) -11(3x+ 4)
(e) 2a(7x + 3y)
(f)
7a(8y - 3x)
(g) 3(8x - 9y + z)
(h) 2x(-4a + Sb+ 6c)
9.
7.
8.
a=5,b=3,c=l7
1. (a) 5
(d) -17x(x+2y)
-2x(2la - 19b)
(h) -7
(i)
(j) -45
9
(b) 55y- 64x
(a) 3ax(by - 4cz + 2)
2.
19
3
5x-2y
7. (b) Yes
(d)
19
11
3
12 x-4 y+2z
9. (i)
3.
(a)
(c)
4. (a)
(c)
(e)
(g)
(i)
5. (a)
lOx- 7y
14+5x
7-8x
-84x
8.
(d) 33x- 27y
(b)
(d)
17x-4
4x-35
10
x+ 13
14
13x+l9y
6
(d)
6x-33
11
2+36x
9
(c)
4x
9
l-53x
8
(6x +Sy+ 2) kg
14. (i)
$(5x + 12)
8.
(h)
26x+9
24
2lx-36y
14
6lx-32 y
24
(a) 11
(c)
(h)
(g)
2x+83y
28
(j)
35y-38x
20
28x+ y
(o)
(c)
3x-15y
14
(d)
(f)
2
7
3.!_
7
(h) 27
(i)
0
(k)
-111
1
2
(a) 1
(j)
2~
(1)
_4.!_
8
2
(b) 7
(c)
-2.!..
(d) -2
(e)
225
(f)
-13
(g)
1
2
(h)
2
3
(i)
-8.!..
5
(j) -34
(k)
-2
4
3
11
(I)
No solution
(i)
4
Sx + 1 = 21, 7 - x = 3
(a) No
(b)
2½
2
1
Sx+l7=7,- x- =1,
3
3
(i)
2.
(b) -14
(d) 11
(f)
(g) -21
20
28x+42y
45
16
(e) 9
59x+l0y
15
(b)
(d) 72
9
5
=4
0.125x+l.5
109y-44x
18
4
$16
20
107x-47
45
(d)
7'!_
(ii)
(ii)
6.
(f)
(a)
5.
7 x-5
(b)
(b) 9
819
4. No
(b) $(9p - q)
13. (i)
Ilx+2y
--6
35x 46y
8
5x+2y
(%x-4s)years
12. (a) p - q
(e)
(i)
6.
7 y-54x
20
+ 2y - 2) cm
(x
h+2k
10. (i) (2h + 4k) m (ii) - - m
3
11. (a) (x - 30) years
1
(b)
(x-30) years
2
1.
(c)
(b) 3x- 6
3.
(f)
(j)
(a) 24
(g)
llx+l2
15
7
(I)
(e) 7
(ii) 55 cm2
(b) llx- 6y
(b)
llx-3
6
(a) 3x + 6
18
20
(k) 5
(c)
143x~ 249y
0
(b) 83y-74x
21
(ii)
6. (i) 6 - 3x
(c)
(c)
(d) 8
(g) -30
40.!_
3.
(b) 10
16
(f)
(d)
(a) -28x + 49y
18
(c)
(e) 9
5 _ (a)
2. (a) 36x+ 8
16
1
3
4. 288y - 120x
1
1
(b) 2x-10y
(p)
(a)
(b) -8y(Sa + 7 + 3bc)
12
3
35x+8y
4
7
2
(b)
2.
10. -17a3 bx2 (2a +Sb+ 4c2)
1. (a)
2~
9. (ii) a= 65, b = o
46x+ll7y
lO.
42
(b) Sa(x - 4y + 4z)
(f)
(i)
(h) -2
(o) 3.25
1.
(e) -a(ll + 13x)
124
(m) 1
(a) 9x(l + 2a + 2b)
(c) -3x(4y+7z+4)
(g)
3
(k) -1
515y-109x
30
h=2,k=7,p=5,q=9
(h)
7. -25x + 4y
8.
82x-13y
15
124y-107x
24
45x+83y
12
9
-11
(h) 14
(j)
7
(k) -2
(I)
O
(m) -13
(n) 9
2
5
1. 45, 46 and 47
2. 53, 55 and 57
4. 64 kg
5. 8 $10-notes and 5 $SO-notes
6. (i)
$(3x + 320)
(ii) 3x + 320 + 6x = 860
(p) 0.7
(iii) $500
(a) 25
2
(c) 2
5
7. (i)
8.
12 years old (ii) 24 years old
(a) No
(b) 260
9. (i)
6
s.
(ii) $75
8
2.
(-8, 3), (-4, 3), (0, 3)
(iii) 2
10. (i)
Worksheet 6B: Functions and linear
functions
(16x - 2) cm (ii) 2
(iii) 13 cm
11. (i)
2
(ii) 9 h 20 min
9
(ii) -1
2. (i)
8
(ii)
3.
-4
(ii)
2.35
1
(ii) 119
5.
3. (a 11,11
6. (ii)
4.
(i)
4.
4x+ 12
(ii) - 76
(b) 40 °C
6. (a) 37.7 cm
(b) 3.820 cm
7. (a) 0.009 joules (b) 2 kg
(i)
C =mx
(ii)
n +4x
(ii) 510
$52
(ii) $(lln + 47)
es
9.
(iii) No
$(a;+
b;)
(b)
1
(c)
(e)
2. Yes
e
.!_
•4 •
• • 5·
38
- \d~ -:.: s
3~
8
(f)," ;"-
•
l
·~ra.ia.·
f
3.
lOx + 9 = \
4. 16
. _,
9.
-
~--
29
11
14
,.,,
"
(c) 3, 6, 11
(d) 2- 5- 8-
1 10
(e) 10, 14, 27
(f)
5.
(i)
39,30
(ii) 57-9n
6.
(i)
7n- 3
7.
(i)
-19,-25
1, 7
(d) -4, 10
(e)
.!.4' -5
(f)
-1.2, 9
..
2.
3.
4.
8.
(ii) They have the same y-intercept.
(iii) 2
a = 8, b = 22, c = 29
(i)
(iii) No
(ii) 1 + 7n
-½,-1
2n - 1, 2n + 1
10. (i)
(ii) (a) 6n - 3
11. (i)
8
(ii) 8
-12
2-?n
12. (a) 2, 11,26,47
(a) $140
(b) $230
(ii) (a) 4 kg
(b) 88 kg
(i)
(a) £100
(b) £30
(ii) (a) $100
(b) $40
(i)
(i)
13. T rl + I = T II + 4(10" - 1)
Worksheet 7B: Number sequences
and patterns
6, 8, 10
1. (i)
(iii) (a) 9 cm
(b) 7.6cm
(i)
(ii) 2.6km
?min
Fig. 3: 4, 9, 7, 16
Fig. 4: 5, 16, 9, 25
Fig. n: n + 1, n' , 1 + 2n,
n' + 2n + 1 or (n + 1) 2
(ii) 129
Review Exercise 6
1. (i)
(3x + S) cm (ii) 3
$(12m+20n)
2.
(ii) $324
(iii) e
(iii) No
A( 4, 2), B(S, -1), C(3, -5),
D(-4, -4), E(- 3, 3), F(0, 4)
(ii) (a) (-4, 2)
10. 641 cm'
m= 10, n = 17; m = 30, n = 5
(iii)
(a (-5, -1) and (0, -6)
(i)
11
4. (a)
6.
(i)
~4, - 30
2.
3.
(ii) 6
(b)
1
- 12, 12
(c) Triangle
(ii) 7 units'
© Shing Lee Publishers Pte Ltd
=
7 +6
6x7
=
13
42
( "') 161
lll 1296
(ii) No
5.
(i)
(i)
= 10' = l3 + 23 + y + 43
a = 55, b = 10
45 = 5 X 9 = 5 X (5 + 4)
100
(ii)
6.
(ii) a= 18, b = 22
Worksheet 7A: Number sequences
(b) Square
(d) Parallelogram
(ii) n' + 4n
32
(ii) k = 20,p = 41, q = 420
200,320,560
Chapter 7 Number Patterns
(a) Rectangle
(i)
(iii) 2700
1 1
4 · (i) 6+7
Worksheet 6A: Cartesian coordinates
1. A(S, 2), B( 4, -2), C(- 3, -5), D(-5, 4),
E(O, 6), F(l, 0)
(ii) 10, 15, 66
(iv) No, ½(n' +3n+2)
(b) (3, 3)
3. Yes
Chapter6 Linear Functions and Graphs
4.
(ii) 4n - 1
(iii) 58
9.
(b) -1,4
(d)
1
1
2' 2
2 1 1
1 1
3' 7' 2
(ii) 11 - 6n
31
(i)
(iii) 31.2 km/h
(iii) 384 cm'
3.
1
2'
(iii) 29
~
5
(b) Not possible to find
11. (i)
(b) -1, 8
(b) (i) 3n' + 1 (ii) 1876
1.
(a)
(i)
(b) -1, 7, 15
Worksheet 6C: Applications of linear
graphs in real-world contexts
'. ..
5. 588
8.
~
46.9, 82.9
(a) 11,14,17
(c)
(c) 0, -3
Review Exercise 5
(a) -4
(f)
4.
P(l, 3)
11. (a) 2, 1
1.
(e) -25,-57
(iv) h = 1, k = 5
(b) No
1
1
45' 135
(f)
(d) 19,38
2, b = -13
10. (ii) - 1.5
1
(c) 25 000, 12 500 000
4
3
= 2x+ 4 (h) y= -6x
1
(i) y= -- x +6 (j) y= -2.7
3
8. (ii) They are parallel.
(7m + Sn) s (ii) 3 min 38 s
(d) 9, 42
19, 13
(e) -2, 1
285
(g) y
(i)
12. (a)
ea=
e
7. (a) 3, -1
5. (a) 95 °F
8.
(i)
(c)
(b) 37,60
Worksheet SD: Mathematical formulae
1. 29
1
2. 22
(b) 486, 1458
3. (a) 13, 16
1. (i)
(i)
(a) 23, 27
1.
(a) 50, 57
(b) 30, 26
(c) 256, 512
(d) 75, -37
(iii) 480
= 20 X 24 = 20 X (20 + 4)
(iv) No
1
2
Answer Keys
147
Review Exercise 7
1.
Secondary 1 Express Mid-year
Checkpoint B
1
12' 6, 15
2
2. 9;52,-668
3.
(i)
4.
(i)
5.
(i)
1.
8,22
(ii) 9
23 4
(ii) 6-3n
-5~
2
(iii) 17
11. 3
62 + 6
12. (a) 2 x 32 x 11
X
5 + 52 = 91 = 6 3 - 53
(iii) 1657
(iv) 54 g
1 + 6n
21
(b) (i)
170
(iii) 22
14. (a) (i)
(iii) Figure 155
Secondary l Express Mid-year
Checkpoint A
(a) 16,33,49,511, 1000
(b) 1, 1000
(a) 9.4965
3.
2~
3
15x - 40xy + 60y
5.
6.
(b) 9.50
1
2
(a) 4ac(3b - 7bx + 9x)
(b) 169 000
7.
3
5
8.
large
10. -3
11. (a)
12.
8 fi,,rr
(b)
~~
~
15
13. (b) 578
(c) 2 2 x Y x 17
(d) x=4,y=2,z=l
14. (a)
p = 34, q = 27, r = 20
(b) 48 - 7n
(d)
8 -92
(e)
23
120
15. (i)
+ Sy) cents
(iii) ¥11
n-1
2
2.
(x
fa 6.5 cents for colour and
1.5 cents for black
Fig. 12: 6, 6, 12
4.
13. (i)
Fig. 3: 2, 1, 3
(ii)
(b) 22
(c) 231
(ii)
Fig. 4: 2, 2, 4
1.
1~
20
(ii) 2633
(ii) (n + 1) 2 + (n + 1) x n + n 2
= (n + 1) 3 - n3
(i)
7. No
10. -2ab
(b) (i) 2n 2 + n + 5
9.
3. -73
9.
(a) 7, 13,23,37
8. (i)
2. No
8. 9cm
(iii) 5813
7.
43
6. Yes, a recurring decimal
(ii) n 3 - n - 1
59
0.62,
5. h = 1, k = -40
(b) No
(i)
3
' 5'
(5
4. 6y-x
2n + 2, 2n + 4
(ii) (a) 6n + 6
6.
3)
(10, 240), (15, 330), (20,420)
15. (i)
13 minutes
(iii) 27.3 km/h
(ii) 6
(ii) 4.6 km
New Syllabus Mathematics
8 th Edition
think! Mathematics workbooks are specially
designed to complement the textbooks as you
acquire mathematical concepts and skills.
Key features
• Questions that connect ideas within mathematics
and between mathematics and the sciences
through applications of mathematics are included
to allow you to better appreciate mathematics.
~ Challenging questions require you to apply
thinking, reasoning and metacognitive skills.
~ Open-ended questions reinforce
comprehension and encourage reflection
through the con5truction and analysis of
possible answers.
• A Mid-year Checkpoint or an End-of-year
Checkpoint consolid 9 tes the concepts covered
in the textbook, and is useful for assessing your
learning and identifying areas that require
further practice.
ISBN 978 981 32 4537 2
SL Education
9
789813 245372