Mathematics-MYP1-1

worked example 3 (extension)

Find the highest common factor
(called the HCF) of 102 and 153.

Solution

Number Factors

102 1, 102, 2, 51 , 3, 34, 6, 17

153 1, 153, 3, 51 , 9, 17

The common factors are in colour.
The highest common factor is 51.

worked example 4 (extension)

Find the lowest common multiple
(called the LCM) of 15 and 9.

Solution

Number Multiples

15 15, 30, 45 , 60, 75, 90, . . .

9 9, 18, 27, 36, 45 , 54, . . .

There would be other common multiples but the lowest common multiple is seen to be 45.

Exercise 3:05

1 Complete the table below.
Product 24 24 24 24 20 20 20 12 12 12 18 18 18 25 25
Factor 1 2 3 4 4 2 1 12 6 4 6 9 18 5 1
Factor 24 12

2 Use the table above to list all factors of: Write each
factor only
a 24 b 20 c 12 d 18 e 25
once.
3 List all the factors of: e7
j 40
a 10 b8 c 15 d9 o 152
h 45 i 36
f 100 g 27 m 132 n 121

k 144 l 160

4 Answer true or false.

a 3 is a factor of 313. b 99 is a multiple of 9.

c 3 is a factor of 300. d 2 is a factor of 11 012. e 210 is a factor of 210.

f 210 is a multiple of 210. g 35 is a multiple of 5. h 42 is a multiple of 6.

i 1 is a factor of all counting numbers. j 1 is a multiple of 9.

78 INTERNATIONAL MATHEMATICS 1

5 List the first nine multiples of:

a3 b5 c2 d 10 e4
h9 i7 j8
f 11 g6

6 Complete the tables and find the highest common factor (HCF) in each case.

a Number Factors b Number Factors

36 90 The HCF is the
24 135 largest number
The HCF is . . . . . The HCF is . . . . .
that is a
factor of both.

c Number Factors d Number Factors

116 105

348 165

The HCF is . . . . . The HCF is . . . . .

7 Find the highest common factor of:

a 16 and 22 b 45 and 60 c 80 and 100 d 28 and 42
g 240 and 144 h 64 and 27
e 102 and 132 f 700 and 750

8 Complete the tables and find the lowest common multiple (LCM) in each case.

a Number Multiples b Number Multiples

7 10 The LCM is the
smallest number

5 12 that is a
multiple of both.

The LCM is . . . . . The LCM is . . . . .

c Number Multiples d Number Multiples

6 30

15 25

The LCM is . . . . . The LCM is . . . . .

9 Find the lowest common multiple of:

a 6 and 4 b 25 and 40 c 12 and 18 d 4 and 14
g 20 and 8 h 22 and 33
e 8 and 10 f 24 and 30

10 a Find the multiples of 5 between 60 and 90.
b List all common factors of 160 and 200.
c What are the multiples of 2 between 1 and 20?
d Which of the multiples of 5 that are less than 100 are also multiples of 6?
e Find the multiples of 6 between 20 and 50.

79CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

3:06 | Prime and Composite

Numbers

prep quiz 1 6×8 2 4 × 12 3 48 × 1 4 24 × 2

3:06 5 List all the factors of 48.

6 Complete: 6 × 1 = . . . , 6 × 2 = . . . , 6 × 3 = . . . , 6 × 4 = . . . , 6 × 5 = . . .

7 List the first five multiples of 6.

8 List the factors of 30.

9 List the factors of 48.

10 The larger of two numbers is 601 and their difference is 20. What is their sum?

A prime number is a counting number that has exactly two factors, itself and 1.
For example, 7 has only two factors, 7 and 1, so 7 is a prime number.

A composite number has more than two factors.
For example, 25 has three factors, 25, 1 and 5, so 25 is a composite number.

The number 1 has only one factor, 1, so 1 is neither prime nor composite.

worked example 1

Use the Sieve of Eratosthenes to list all primes less than 60.

Solution

Eratosthenes was a Greek mathematician who lived from 276 to 194 BC. He invented a way

of sorting the prime numbers from the composites.

Step 1 Cross out 1 as it is not a

prime.
1 2 3 4 5 6 Step 2 Circle the prime number

7 8 9 10 11 12 2 then cross out all

13 14 15 16 17 18 multiples of 2.

Step 3 Circle the next number,

19 20 21 22 23 24 3. This is a prime. Now

25 26 27 28 29 30 cross out any multiples of

3 not already crossed out.
31 32 33 34 35 36 Step 4 Circle the next number, 5.
Eratosthenes
37 38 39 40 41 42 This is a prime. Now cross (276?–194 BC)

43 44 45 46 47 48 out any multiples of 5 not Greek mathematician,
already crossed out. historian, astronomer, poet

49 50 51 52 53 54 Step 5 Continue using this and geographer. Born at
55 56 57 58 59 60 method until all composite Cyrene in northern Africa,
he went to Alexandria to
numbers have been
take charge of the great

crossed out. library there.

The primes less than 60 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.

80 INTERNATIONAL MATHEMATICS 1

worked example 2

Use factor trees to write as products of prime numbers:

a 24 b 252

Solutions b 252

a 24

46 4 63

2 22 3 2 23 21

∴ 24 = 2 × 2 × 2 × 3 2 23 3 7
= 23 × 3
252 = 2 × 2 × 3 × 3 × 7
I’m playing with = 22 × 32 × 7
factor trees.

I’m paying the
tractor fees!

Exercise 3:06

1 Copy and complete the table below, listing all factors of the numbers 1 to 20.

Number Factors Number Factors

11 11

2 1, 2 12 a List all numbers from
3 1, 3 13 1 to 20 that have only
4 1, 4, 2 14 two factors. These are
5 15 prime numbers.
6 16
7 17 b List all numbers from
8 18 1 to 20 that have more
than two factors. These
are composite numbers.

9 19

10 20 1, 20, 2, 10, 4, 5

81CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

2 Look back to example 1 and use the Sieve of Eratosthenes to list all prime numbers that are less
than 120. Start by listing all the numbers from 1 to 120 in rows of six numbers.

3 a Write down all of the even prime numbers.
b How many odd prime numbers are less than 20?
c List all the odd composite numbers that are less than 20.
d How many composite numbers are between 20 and 40?
e Write each of the even numbers from 6 to 24 as the sum of two prime numbers,
eg 12 = 5 + 7. (The mathematician Goldbach suggested that every even number
greater than 4 is the sum of two odd prime numbers.)

4 Complete these factor trees. b 20
a 12

34 4

12 = . . . . × . . . . × . . . . 20 = . . . . × . . . . × . . . .
c 18 d 16

4

18 = . . . . × . . . . × . . . . 16 = . . . . × . . . . × . . . . × . . . .

5 Use factor trees to write these numbers as products of their prime factors.

a 30 b 36 c 100 d 144 e 96 f 225

6 Another way to write a number as a product of its prime factors is to ■ Example: 360

continue to divide by prime factors (starting from the smaller ones) 2 ) 360
2 ) 180
until you get 1 as a result. Use this method, which is shown to the 2 ) 90
3 ) 45
right, to write these numbers as products of prime factors. 3 ) 15
5) 5
a 216 b 675 c 735 d 504
1
e 1125 f 1232 g 1323 h 3375

6 is called a perfect number because the sum of all of its
factors, other than 6 itself, is equal to 6, ie 1 + 2 + 3 = 6.

360 = 23 × 32 × 5

Appendix C 3:06 Finding factors game
C:01 HCF and LCM by prime factors

Challenge worksheet 3:06 HCF and LCM by prime factors

82 INTERNATIONAL MATHEMATICS 1

3:07 | Divisibility Tests

1 List all the factors of 40. prep qu iz
2 Write the next number after 60 that is divisible by 10.
3 Write the next three multiples of 6 after 12. 3:07
4 What is the largest even number that is less than 1000?
5 What is the next odd number after 999?

255 ÷ 5 = 51 6 In this number sentence, which number is the quotient?

7 Which is the divisor? 8 Which is the dividend?

9 318 813 ÷ 11 10 102 102 102 ÷ 9

If 4 divides a number and leaves no remainder, then the number is divisible by 4. Here are some
simple divisibility tests you should know.

Divisor Divisibility test Example

2 The number must be even, 4136 is divisible by 2

ie it must end in 0, 2, 4, 6 or 8. as it is even. Can you explain
why a number is
3 The sum of the digits is divisible 30 012 is divisible by 3 divisible by 2

by 3. as (3 + 0 + 0 + 1 + 2) is if it ends in
0, 2, 4, 6 or 8?
divisible by 3.

4 The number formed by the last 76 112 is divisible by 4
two digits must be divisible by 4. as 12 is divisible by 4.

5 The last digit must be 5 or 0. 11 225 is divisible by 5
as it ends with a 5.

6 The number must be divisible by 40 002 is even and its digit

both 2 and 3. sum is divisible by 3.

8 The number formed by the last 963 216 is divisible by 8 By using the tests
three digits is divisible by 8. as 216 is divisible by 8. in the table, we
can construct
9 The sum of the digits is divisible 142 128 is divisible by 9 other tests.

by 9. as (1 + 4 + 2 + 1 + 2 + 8) is

divisible by 9.

10 The last digit must be 0. 814 710 is divisible by 10
as it ends with 0.

11 The sum of the digits in odd- 7 081 426 is divisible by • Divisibility by
numbered places will be equal to 11 as (7 + 8 + 4 + 6) – 12 is the same
the sum of the digits in even- (0 + 1 + 2) is 22, which as divisibility
numbered places, or will differ is divisible by 11. by both 3 and 4.
by a multiple of 11.
• Divisibility by
25 The last two digits will be 00, 80 925 is divisible by 25 as 50 is the same
as divisibility
25, 50 or 75. it ends in 25. by both 2 and
25.
100 The last two digits will be 00. 81 700 is divisible by 100
as it ends in 00.

83CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

Exercise 3:07 Foundation Worksheet 3:07

1 Which of these numbers are divisible by 2? Divisibility tests
571, 3842, 5816, 2221, 887, 9000, 374 555, 1 Write the first 10 even
8774, 8166, 7008
numbers.
2 Which of these numbers are divisible by 3? 2 Write the first 10
7114, 830, 3006, 7110, 21 441, 30 031, 8145,
60 001, 211 002, 78 multiples of:
a2 b3
3 Write the numbers below 30
that are divisible by:
a2 b3

3 Which of these numbers are divisible by 4?

1004, 67 814, 7118, 2222, 6124, 8156, 98, 61 852, 934

4 Which of these numbers are divisible by 9?
9994, 31 024, 30 024, 81 810, 41 238, 3333, 727, 411, 765, 936

5 Which of these numbers are divisible by 11?
4115, 8003, 6633, 7228, 860, 74 186, 10 406, 92 180, 999, 61 809

6 Find which of the numbers 3, 4, 5, 6, 8, 9 and 11 will exactly divide:

a 440 b 3883 c 3916 d 1485 e 3993

7 Which of the numbers 1078, 7600, 13 476 and 33 885 are divisible by:

a 2? b 3? c 4? d 6? e 9?
j 50?
f 10? g 11? h 25? i 100?

8 What divisibility test could we use for 24?

9 The test for divisibility by 6 is that the number must be divisible by both 2 and 3.

a What divisibility test could we use for 15?
b What divisibility test could we use for 18?
c What divisibility test could we use for 12?

10 Use the divisibility tests where possible to list all

of the factors of:

a 720 b 346 c 270 d 1260

e 567 f 1575 g 729 h 5280

11 Find the smallest number that is greater than 1000 and:

a is divisible by 3 b is divisible by 8 A number
is divisible
c is divisible by 9 d is divisible by 11 by all of
its factors.
e is a multiple of 25 f is a multiple of 100

g is a multiple of 15 h is a multiple of 12

12 a Find the lowest common multiple of 2, 5, 9, 10 and 11.
b Find the highest common factor of 810, 324 and 1944.

c Find the lowest common multiple of 2, 3, 4, 5, 9 and 100.
d List all factors of 3 × 52 × 7. (There are twelve.)
e List all factors of 33 × 11.

3:07 Multiples patterns

84 INTERNATIONAL MATHEMATICS 1

3:08 | Square and Cube Roots

Give the basic numeral for: prep quiz

1 52 2 42 3 142 4 23 5 33 6 43 3:08

7 Find the number that has a square of 36.

8 Find the number that has a square of 100. Foundation Worksheet 3:08

9 If 1764 = (2 × 3 × 7) × (2 × 3 × 7), what is Square and cube roots
the number that has a square of 1764? 1 a If 4 × 4 = 16, then 16 = .
2 a If 72 = 49, then 49 = .
10 What number has a square of 10 000? 3 a If 5 × 5 × 5 = 125 then 3 125 = .
4 a If 63 = 216 then 3 216 = .

If the square of 15 is 225, then the square root of 225 is 15.

If 152 = 225, then 225 = 15. √ means
‘the square root of’
If the cube of 8 is 512, then the cube root of
512 is 8. 3√ means
‘the cube root of’.

If 83 = 512, then 3 512 = 8.

The number that gives 225 when squared is called the square root of 225.
The number that gives 512 when cubed is called the cube root of 512.

Exercise 3:08

1 Complete: worked examples

a If 7 × 7 = 49, b If 11 × 11 = 121,

then 49 = then 121 = 1 i If 3 × 3 = 9,
then 9 = 3
c If 13 × 13 = 169, d If 5 × 5 × 5 = 125,

then 169 = then 3 125 = ii If 4 × 4 × 4 = 64,
then 3 64 = 4
e If 9 × 9 × 9 = 729, f If 12 × 12 × 12 = 1728,

then 3 729 = then 3 1728 = iii If 52 = 25,
then 25 = 5
g If 162 = 256, h If 212 = 441,

then 256 = then 441 = iv If 63 = 216,
then 3 216 = 6
i If 142 = 196, j If 73 = 343,

then 196 = then 3 343 =

k If 43 = 64, l If 93 = 729, v If (2 × 7) × (2 × 7) = 196

then 3 64 = then 3 729 = then 196 = 2 × 7
= 14
m If (2 × 9) × (2 × 9) n If (3 × 7) × (3 × 7)

= 324, = 441, continued ➜➜➜

then 324 = . . . × . . . then 441 = . . . × . . .

==

o If (9 × 11) × (9 × 11) = 9801,

then 9801 = . . . × . . . =

85CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

p If (2 × 3 × 5) × (2 × 3 × 5) = 900, vi If (2 × 3 × 7) × (2 × 3 × 7)
then 900 = . . . × . . . × . . . = 1764,
=
then 1764 = 2 × 3 × 7
q If (7 × 11 × 13) × (7 × 11 × 13) = 1 002 001, = 42
then 1 002 001 = . . . × . . . × . . .
= vii If (2 × 2 × 7)3 = 21 952,
then 3 21 952 = 2 × 2 × 7
r If (5 × 5 × 11)3 = 20 796 875, = 28
then 3 20 796 875 = . . . × . . . × . . .
= 2 2 × 2 × 2 × 2 × 13 × 13
= (2 × 2 × 13) × (2 × 2 × 13)
2 Find the square root of each product. = 2 × 2 × 13
= 52
a 2×2×3×3×3×3
b 3×3×5×5 3 3 2×2×2×7×7×7
c 3×3×3×3×5×5 = 3 (2 × 7) × (2 × 7) × (2 × 7)
d 5 × 5 × 7 × 7 × 11 × 11 =2×7
e 2×2×2×2×5×5×7×7 = 14
f 2×2×3×3×5×5
4 i 25 = 5 ,
3 Find the cube root of each product. Since 5 × 5 = 25

a 3×3×3×5×5×5 ii Find 576
b 2 × 2 × 2 × 11 × 11 × 11 576 = (2 × 2 × 2 × 3)2
c 2×2×2×3×3×3 576 = 24
d 2×2×2×2×2×2
e 3×3×3×3×3×3 5 i 3 125 = 5 ,
f 2×2×2×5×5×5 since 5 × 5 × 5 = 125

4 Find the basic numeral for: ii Find 3 8000 .
By product of primes,
a9 b1 c4 8000 = (2 × 2 × 5)3
3 8000 = 20
d 16 e 64 f 36
6 135 is between 11 and 12,
g0 h 100 i 81 as 135 is between 112 and 122.

j 144 k 324 l 400

m 1296 n 1764 o 225

5 Find the basic numeral for:

a 38 b 3 27 c 31
f 30
d 3 64 e 3 1000 i 3 1331
l 3 5832
g 3 216 h 3 512

j 3 3375 k 3 4096

6 Use the table of squares below to give the two counting
numbers between which each square root lies,

Number 123 4 5 6 7 8 9 10 11 12 13 14
Square 149 16 25 36 49 64 81 100 121 144 169 196

a 13 b 72 c 20 d 40 e 90 f3
g 180 h 30 i 132 j 17 k 147 l 116

86 INTERNATIONAL MATHEMATICS 1

7 First write an estimate for the square root and 7 576
then use a calculator to find the answer. 576 = 24

a 256 b 324 c 676 8 3 2744
3 2744 = 14
d 1521 e 1089 f 484

8 First write an estimate for the cube root and
then use a calculator to find the answer.

a 3 125 b 3 729 c 3 1331

d 3 8000 e 3 4913 f 3 1728

3:09 | The Binary System:

Mathematics for Machines
(Extension)

Write in expanded form: 1011two is prep qu iz
1 31 627 one-zero-one-one base two,
2 15 038 3:09
ie (1 × 23) + (0 × 22) + (1 × 21) + (1 × 1)
Write as a basic numeral:
= 8 + 0 + 2 + 1 = 11

3 (6 × 104) + (3 × 103) + (7 × 102) + (8 × 101) + (9 × 1)

4 (1 × 104) + (0 × 103) + (5 × 102) + (2 × 101) + (0 × 1)

Complete:
5 21 = . . ., 22 = . . ., 23 = . . ., 24 = . . ., 25 = . . ., 26 = . . .

Simplify: That is not
6 (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) one thousand
7 (1 × 23) + (1 × 22) + (0 × 21) + (1 × 1) and eleven!
8 (1 × 16) + (1 × 8) + (1 × 4) + (1 × 2) + (0 × 1)
9 (1 × 24) + (1 × 23) + (1 × 22) + (1 × 21) + (0 × 1)

10 (1 × 24) + (0 × 23) + (1 × 22) + (0 × 21) + (0 × 1)

To use a number system based on ten, we need ten digits: Looks pretty
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Machines that work by using base-ic to
electrical currents can only use two digits: ‘off’ and ‘on’. me!
These are their digits for 0 and 1. They use a
base two system that has place value.

worked examples

1 Change 111011two into a base ten numeral.
2 Write 54ten as a numeral in binary numerals (base two).

continued ➜➜➜

87CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

Solutions

1 25 24 23 22 21 1 The column values are powers of 2.

32s 16s 8s 4s 2s 1s (1 × 25) + (1 × 24) + (1 × 23) + (0 × 22)
+ (1 × 21) + (1 × 1)
111011 = 32 + 16 + 8 + 0 + 2 + 1
= 59
∴ 111011two = 59ten

2 Starting with the larger column values, 26 25 24 23 22 21 1
distribute 54 across the columns.

54 is made up of 32 and 16 and 4 and 2. 64s 32s 16s 8s 4s 2s 1s

∴ 54 = (1 × 32) + (1 × 16) + (1 × 4) + (1 × 2) 110110
Put zeros in the other columns.

Method

There are no 64s in 54.
There is one 32 in 54, ie 54 = (1 × 32) + 22. Place the one in the 32s column.
There is one 16 in the 22 that remains, ie 22 = (1 × 16) + 6. Place the one in the 16s column.
There is one 4 and one 2 in the 6 that remains, ie 6 = (1 × 4) + (1 × 2).
Place a one in both the 4s column and the 2s column.
∴ 54ten = 110110two

Exercise 3:09

1 Change these base two numerals to numerals in base ten.

a 10two b 11two c 100two d 101two e 110two
f 111two g 1000two h 1001two i 1010two j 1011two
k 101101two l 11000two m 100011two n 111111two o 1101two

2 Write these base ten numerals as binary numerals. e 40

a 17 b 20 c 64 d 35

3 Complete the table below.

a Numeral 26 25 24 23 22 21 1 Numeral
(base ten) 64s 32s 16s 8s 4s 2s 1s (base two)

1010001two
1101011two
1000111

80

100

97

125

88 INTERNATIONAL MATHEMATICS 1

Fun Spot 3:09 | Making magic squares fun spot

5 10 9 24 The rows, columns and diagonals of a magic square have 3:09
12 8 4 24 the same sum. In this magic square, each row, column
7 6 11 24 and diagonal has a sum of 24.

24 24 To make a ‘three-by-three’ magic square you need nine
24 24 24 numbers. These numbers, when placed in order, must
form a pattern with the same difference between each
pair of numbers. The numbers used here are:
4, 5, 6, 7, 8, 9, 10, 11 and 12.

Example
2, 4, 6, 8, 10, 12, 14, 16, 18. These have a difference of 2. We will now make a magic square
using these numbers.

Step 1 Step 2 Step 3
Write the numbers in Join the half-way points Place each number which
rows of three. as shown. is outside a square into
the opposite empty square.

246 2 46 2 46
8 10 12 8 10 12 8 10 12
14 16 18 14 16 18
14 16 18

Step 4 Activity
Rotate the square 1 Use the method shown to make a magic square using:
anticlockwise to give it
the appearance below a 3, 4, 5, 6, 7, 8, 9, 10 and 11.
b 5, 10, 15, 20, 25, 30, 35, 40 and 45.
4 14 12 c 36, 33, 30, 27, 24, 21, 18, 15 and 12.
2 What is the sum of a column in each magic square
18 10 2 of part 1?
3 Use the pattern 1, 2, 4, 8, 16, 32, 64, 128, 256 to make a
8 6 16 square using the method above. What is the product of
numbers in each row, column and diagonal?

89CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

hematical terms Mathematical terms 3 divisibility tests
mat • Rules used to find factors of a number.
3 average (arithmetic mean)
• The result of ‘evening out’ a set of
even numbers
numbers. • Whole numbers that are divisible by 2.
• To find the average, divide the sum of the
They end in 0, 2, 4, 6 or 8.
terms by the number of terms.
eg The average of 3, 5, 3, 7, 10 and 2 factor
• A factor of a counting number divides it
= (3 + 5 + 3 + 7 + 10 + 2) ÷ 6
= 30 ÷ 6 exactly.
=5 eg The factors of 6 are 1, 2, 3 and 6.
• common factor
basic numeral A number that is a factor of all numbers
• The simplest way of writing a number. being considered.
eg 7 is a common factor of 14, 21 and 70.
eg The basic numeral for (4 + 8) × 2 is 24. • highest common factor, HCF (extension)
The largest of the common factors.
cardinal numbers eg 18 and 24 have common factors, 2, 3 and
• Zero and the counting numbers.
6 but the highest common factor is 6.
ie {0, 1, 2, 3, . . .}
Fibonacci numbers
composite number • The set of numbers 1, 1, 2, 3, 5, 8, . . .
• A number that has more than two factors.
The next number in the pattern is the sum
eg 9 is composite as it has three factors, of the two before it.
1, 3 and 9.
grouping symbols
counting numbers
• The numbers {1, 2, 3, 4, . . .}. • They are used to tell us which operation is

cube (numbers) to be done first.
• The answer when a whole number is
• There are 3 types.
used 3 times as a product.
eg 53 = 5 × 5 × 5 = 125 parentheses brackets braces

cube root (3 ) () [] {}
• To find the cube root of a number, say 8,
• When one set is within another, do the
find the number that needs to be cubed
to give 8. operation in the innermost grouping
ie 23 = 8 so 3 8 = 2
symbols frist.
decrease
• To make smaller. increase
• To make larger.
difference
• The result of subtracting one number multiple
• A multiple of a counting number is found
from a larger number.
by multiplying it by another counting
distributive property number.
• This shows how multiplication eg The multiples of 5 are 5, 10, 15, 20, . . .
• common multiple
(or division) of a number can be done A number that is a multiple of all numbers
by breaking the number into an addition being considered.
(or subtraction) of separate parts. eg 50 is a common multiple of 2 and 5.
eg 6 × 108 = 6 × (100 + 8) • lowest common multiple, LCM (extension)
The smallest of the common multiples.
= (6 × 100) + (6 × 8) eg 10 is the LCM of 2 and 5.
or 8 × 99 = 8 × (100 − 1)
20 is the LCM of 2, 5 and 10.
= (8 × 100) − (8 × 1)

90 INTERNATIONAL MATHEMATICS 1

number properties quotient
• Those properties that belong to all • The result of dividing one number by

numbers. another.
eg Multiplying any number by 1 leaves
square number
the number unchanged. • The result of multiplying a counting

odd numbers number by itself.
• Whole numbers that are not divisible eg 1, 16, 25

by 2. They end in 1, 3, 5, 7 or 9. square root ( )
• The square root of a number, say 64, is
order of operations
• The set of rules that give the order in the number that must be squared to give
64. The square root of 64 is 8.
which the operations are performed. eg 64 = 8
ie 1 ( ) 2 × and ÷ 3 + and −
sum
palindromic numbers • The result of adding numbers together.
• A number that is the same backwards as
triangular numbers
it is forwards. • Numbers that are the sum of consecutive
eg 929, 2002, 18 700 781
counting numbers beginning from 1.
prime number eg 1, 3 (that is 1 + 2),
• A counting number that has exactly two
6 (that is 1 + 2 + 3),
factors, itself and 1. 10 (that is 1 + 2 + 3 + 4), 15 etc.
eg 17, 31, 2

product
• The result of multiplying numbers

together.

Diagnostic Test 3: | Number: Its Order and Structure diagnos tic test

• Each section of the test has similar items that test a certain type of example. 3
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.

1 Write in words.   c 294 167 000 Section
a 2 456 325 b 97 005 000 1:02
1:02
2 Write the value of the 7 in each. 1:06

a 214 703 b 871 000 c 73 415 216 1:06

3 Write each as a numeral in its simplest form.
a (5 × 1000) + (4 × 100) + (9 × 10) + (6 × 1)
b (8 × 10 000) + (6 × 100) + (5 × 10) + (9 × 1)
c (3 × 1 000 000) + (2 × 100 000) + (1 × 10 000)

4 Write in expanded notation.

a 56 015 b 714 850 c 3 420 000

91CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

  Section
1:05
5 Write the basic numeral for each. 1:05
1:05
a 52 b 33 c 105 3:01
3:01
6 Write as a power. b 10 × 10 × 10 × 10 c 9 × 9 3:02
a 8×8×8
3:04
7 Write the basic numeral for each.
3:05
a 7 × 102 b 4 × 104 c 5 × 103 3:05
3:05
8 Simplify: b 33 − (16 − [4 + 10]) c 1----0----+-----4---0-- 3:05
a (6 + 8) × (11 − 9) 5 3:06

9 Simplify: b 12 + 18 ÷ (10 − 4) c 120 − 80 ÷ 8 − 6 × 10 3:06
a 18 − 3 × 4 + 5

10 Copy and complete to make each number sentence true.
a 6 × (900 + 8) = × 900 + × 8 b 7 × (600 − 3) = × 600 − × 3
c 4 × 398 = 4 × 400 − 4 ×

11 a List the first three square numbers.
b List the first three triangular numbers.
c List the first three odd numbers.

12 List all the factors of:

a 24 b 63 c 100

13 List the first five multiples of: c 11
a4 b8

14 Find the highest common factor of:

a 36 and 48 b 60 and 75 c 70 and 98

15 Find the lowest common multiple of:

a 12 and 9 b 6 and 8 c 20 and 14

16 a Write down all prime numbers that are less than ten.
b Write down all composite numbers that are less than ten.
c Which counting number is neither prime nor composite?

17 Complete these factor trees. 36 c 42
a 24 b

4 67

18 Find the highest common factor and the lowest common multiple of: 3:07
a 2 × 2 × 2 × 3 and 2 × 2 × 5 b 3 × 3 × 3 × 5 × 7 and 3 × 5 × 5 × 7
c 2 × 3 × 5 × 7 and 3 × 3 × 5 3:09
3:09
Find the simplest answer for:

19 a 25 b (7 × 11) × (7 × 11) c 2×2×3×2×2×3

20 a 3 27 b 3 11 × 11 × 11 c 32×2×2×3×3×3

92 INTERNATIONAL MATHEMATICS 1

assignm

Chapter 3 | Revision Assignment ent

1 Write the basic numeral in each case. g 23 × 11 − 22 × 11 3A
a 12 ÷ (28 − 24) h 31 × 98 + 69 × 98
b 84 − 10 × 8 i 98 × 37 + 2 × 37
c 100 − [50 − (20 − 7)]
d 23 + 5 − 2 + 3 6 Write true or false for:
e 6×5+6×3
f 3 + 7(0·7 + 1·3) a 8 × 3 ≠ 3 × 8 b 563 < 560
c 213 × 0 у 0 d 16 = 8
2 We sell buckets of blocks. In each bucket e 16 × 5 Ͽ 16 × 4 f 5 × 816 > 0
there are 157 blocks. We sold 93 buckets g 816 ÷ 1 Ͽ 816 × 1
of blocks on Monday and 7 on Tuesday. h 999 < 100 − 0 i 3 1 = 1
How many blocks did we sell on those
two days altogether? 7 Which of the numbers 25, 27, 41 214 and

13 are:

a odd? b even?

3 Given that 12 × 1256 = 15 072, what is c square? d cube?

the value of: e Fibonacci? f palindromic?

a 1256 × 12? b 13 × 1256? 8 a List the factors of 100.
b List the factors of 125.
c 11 × 1256? c List all the common factors of 100
and 125.
4 True or false?
a 2 × 3186 × 5 = 31 860 9 a List the first 10 multiples of 8.
b 9 × 888 = 8880 − 888 b List the first 10 multiples of 12.
c 555 × 12 = (555 × 6) × 2 c What is the lowest common multiple
d 18 158 ÷ 20 = (18 158 ÷ 2) ÷ 10 of 8 and 12?
e 4 × 186 × 25 = (4 × 25) × 186
f 16 × 100 − 1 = 16 × 99

5 Write as a basic numeral. 10 From the set 2, 5, 7, 12, 15, 36, 41 write:
a the prime numbers
a 1846 × 1 b 0 × 26 040 b the composite numbers

c 8145 + 0 d 446 − 0

e 0 ÷ 186 f 4186 ÷ 1

1 Order of operations
2 Number puzzles

93CHAPTER 3 NUMBER: ITS ORDER AND STRUCTURE

assignment Chapter 3 | Working Mathematically

3B 1 Use ID Card 4 on page xvi to give the 7 A school offers the following sports choices:
mathematical term for: Summer: cricket, water polo, basketball,
id a 14 b 15 c 16 d 17 volleyball, tennis
e 18 f 19 g 20 h 21 Winter: football, hockey, soccer, squash
3 i 22 j 23 How many different combinations are
available if one summer sport and one
2 Use the Arithmetic Card on page xix to winter sport are to be selected?
square the numbers in column B.
8 Production of Prestige Cars 2000-2005
3 A bookshelf has six shelves with 45 books
on each shelf. How many books are in the 2000 represents
bookshelf? 2001 200 cars
2002
4 A bus can hold 2003
58 people seated 2004
and 35 standing. 2005
How many of these
buses would be a In what years were:
needed to take a i the greatest number of cars produced?
school of 750 children ii the least number of cars produced?
and 30 teachers on a
school outing? b In what year were 600 cars produced?
c How many cars were produced in 2002?
5 A grocer bought 50 kg of tomatoes for $30. d How many cars do you think were
He sold 20 kg at $1.20 per kg and the
remainder, which were overripe, he sold for made in 2001? Can you be sure?
80c per kg. How much did he get for selling e Are the same symbols used to represent
the tomatoes? How much did he make?
a fraction of 200 cars in 2001 and 2004?
6 In Rugby Union, a team scores 5 points for f Do you think these graphs are intended
a try, 7 points for a converted try and
3 points for a penalty goal. In how many to convey accurate figures? Give a
ways could a team score 22 points? reason for your answer.

94 INTERNATIONAL MATHEMATICS 1

4

Decimals

Whoops! I forgot
to put in the decimal point.

Chapter Contents 4:08 Changing fractions to decimals
4:09 Rounding off
4:01 Review of decimals 4:10 Applications of decimals
4:02 Addition and subtraction of decimals
4:03 Multiplying a decimal by a whole number Fun Spot: World championship diving
4:04 Dividing a decimal by a whole number
4:05 Using decimals Investigation: Applying decimals
4:06 Multiplying a decimal by a decimal
4:07 Dividing by a decimal Mathematical Terms, Diagnostic Test, Revision

Fun Spot: Why did the orange stop Assignment, Working Mathematically

in the middle of the road?

Learning Outcomes

Students will operate with fractions, decimals, percentages, ratios and rates.

Areas of Interaction

Approaches to Learning, Homo Faber, Environment, Community

95

4:01 | Review of Decimals

see With our world’s increasing dependence on the calculator and the computer, decimals have become
more important. These machines use decimals in their displays and output. Where parts of a whole
number are to be shown, a decimal point is used.

1:09C

worked example

Write 734·625 in expanded notation. Hundredths This decimal has
Hundreds Tens Units Tenths 2 three decimal
7 3 4• 6
Thousandths places.

5

Solution

In expanded form this is:

(7 × 100) + (3 × 10) + (4 × 1) + (6 × --1--- ) + (2 × ----1---- ) + (5 × -----1------ )

10 100 1000

Exercise 4:01 Foundation Worksheet 4:01

Review of decimals

1 Write as a decimal:

1 Answer true or false for each. a --3--- b ----7----

10 100

a 0·3 means 3 of 10 equal parts. b 0·4 is the same 0·40. 2 Write as a fraction:

----1---- ----2---- a 0·7 b 0·03
100 100
c 0·8 is equal to --8--- . d 2× = 2 Which is larger:

10 a 0·7 or 0·08?

e 0·14 has two decimal places.

f The value of the 7 in 6·174 is ----7---- . g 0·7 is the same as 0·007.
100
h 0·35 is the same as --3---5--- .
100 i 7 = 7·00

j Ten lots of 0·1 gives 1.

worked example

----7---- + -----5------ 1 --1--- ----1---- -----1------ Fill the empty
spaces with
100 1000 10 100 1000
zeros.
= 0·075 0•0 7 5

2 Write each as a decimal.

a --6--- + ----4---- b --1--- + ----7---- c --2--- + ----3---- + -----8------

10 100 10 100 10 100 1000

d ----5---- + -----1------ e --3--- + -----9------ f ----6---- + -----6------
100 1000 10 1000 100 1000

g 2 + --3--- + ----5---- h 9 + --4--- i 15 + --3--- + ----6----
10 100 10 100
10

j 11 + ----2---- k 6 + ----8---- + -----1------ l 2 + --5--- + -----3------
100 1000 10 1000
100

3 Write these decimals in expanded form.

a 7·342 b 3·483 c 4·215
f 40·07
d 45·03 e 30·75

96 INTERNATIONAL MATHEMATICS 1

g 256·04 h 24·125 i 69·345 ■ To compare decimals,
j 246·379 k 482·375 l 238·667 give them the same
number of decimal
4 Which is smaller? places, eg

a 0·7 or 1 b 0·05 or 2 c 3 or 1·2 0·3 0·300
d 0·4 or 0·05 e 0·7 or 0·69 f 0·03 or 0·22 0·33 0·330
g 0·2 or 0·1 h 2·8 or 4·1 i 0·99 or 0·612 0·303 0·303
j 0·5 or 0·77 k 0·05 or 0·049 l 0·333 or 0·29

5 Arrange in ascending order. ‘Ascending order’ means
‘smallest to largest’.
a 0·3, 0·8, 0·6
c 0·8, 1, 0·55 b 3·85, 1·2, 7·5 On the number line,
e 0·3, 0·33, 0·03 d 3, 0·3, 0·03 the smaller decimal is
g 0·85, 0·9, 0·792 f 0·39, 0·41, 0·4
h 0·1, 0·09, 0·8 to the left.

6 Each ‘hundred square’ represents one whole.
What decimal is represented in each example?
ab

0 0·2 0·4 0·6 0·8 1

7 If the ‘large cube’ represents one whole, the ‘flat’ represents 0·1, the ‘long’ represents 0·01 and
the ‘short’ represents 0·001, what number is represented by the following?

8 How many decimal places has each of these? My car’s
odometer uses
a 8·14 b 23·123 c 963·5
one decimal
d 0·6 e 0·06 f 0·066 place.

g 5·068 h 15·03 i 900·1

j 463·05 k 6·004 l 0·7

9 Write as decimals. b --1---2--- c -----7------ d --1---2---5---
a --2---3---
100 1000 1000
100
f --6---0---4--- g 2 --9---3--- h 15 --1---0---3---
e ----1---5----
1000 100 1000
1000
j 7 --9---9---9--- k 11 ----3---3---- l 8 -----5------
i 10 ----9----
1000 1000 1000
100

97CHAPTER 4 DECIMALS

10 Write as fractions.

a 0·7 b 0·9 c 0·3 Fractions and
f 0·91 decimals are
d 0·21 e 0·77 i 0·853
l 9·007 rational
g 0·447 h 0·601 numbers.

j 2·99 k 8·3

11 Write each as a decimal.

a -1---4- b 8----6- c 7----3-

10 10 10

d 7----2---4- e 8----0---6- f -9---0---3-

100 100 100

g 1----1---8---6- h 7----4---0---6- i -8---0---1---4-

1000 1000 1000

j -4---5---0---4- k 6----5---0---4- l --8---4---2---

100 10 1000

12 Write in simplest fraction form.

a 0·30 b 0·3 c 0·300
f 0·700
d 0·7 e 0·70 i 0·0100 ■ 0⋅3 = 0⋅30 = 0⋅300
l 0·0010 Zeros at the end of a decimal
g 0·01 h 0·010 do not change its value.

j 0·090 k 0·030

13 What is the value of the 5 in each of these?

a 24 560·7 b 28·53 c 0·125 d 6·25
h 0·054
e 0·005 f 7·152 g 4·6125

14 True or false?

a 14 + ----6---- = 14·6 b 0·97 > 1 c 5 + 0·007 < 5 + 0·07

100 e 0·125 = 0·1 + 0·02 + 0·005 f 0·083 = 0·08 + 0·003

d 0·606 < 0·66

g 10 lots of 0·01 equals 0·1. h 10 lots of 0·001 equals 0·01.

4:02 | Addition and Subtraction

of Decimals

• The column in which a figure is placed determines its size.
• The decimal point separates the whole numbers from the parts of a whole, so the decimal points

must be under one another in question and answer.

To add or subtract decimals use the PUP rule: place Points Under Points.
An empty space may be filled by a zero.

worked examples PUP

1 14·6 + 3·15 + 9 Solutions 2 97·30 9 = 9·00
2 97·3 − 8·95 − 8·95 14·6 = 14·60
1 14·60
3·15 88·35
9·00

26·75

98 INTERNATIONAL MATHEMATICS 1

Exercise 4:02 Foundation Worksheet 4:02
+ and − of decimals
1 a 3·85 b6 c 9·175 1 a 6·4 see
1·6 5·8 4·7
0·41 4·25 9·36 + 0·2 Appendix
B:13 & B:14
d 26 459·6 e 54 000 f 3·1486 2 a 4·7
+ 9 006·8 + 12 673·5 + 9·852 − 1·1

g 9·08 + 12·16 h 4·018 + 0·582 3 a 6·2 + 0·1
i 6·9 + 3·1 j 12 + 0·12
k 6·3 + 8·05 l 150·86 + 9·1 We place tenths
m 0·9 + 15 + 3·125 n 8·6 + 9·3 + 0·75 under tenths,
o 8·25 + 6·85 + 9·125 p 0·185 + 0·25 + 0·03 hundredths under
hundredths, and
thousandths under
thousandths.

2 a 12·30 b 15·4 c 8·35 Empty spaces
− 9·85 − 6·75 − 1·148 may be filled
with zeros.
d 18·000 e 25 f9
− 6·375 − 19·625 − 0·025

g 9·68 h 40·125 i 37·889 Psst!
− 2·90 − 9·04 − 7·8

j 9·3 − 1·6 k 6·75 − 0·45 l 8·875 − 6·125 0·15 is
m 9 − 0·366 n 8·4 − 0·11 o 23·8 − 0·08 sometimes
p 8·799 − 3·89 q 1·086 − 0·7 r 3·467 − 0·35
s 689·5 − 18·25 t 634·8 − 115·85 u 4725 − 6·845 written
as ·15.

4:03 | Multiplying a Decimal by

a Whole Number

0·2 0·2 0·2 0·2 prep qu

4:03 iz

0 0·2 0·4 0·6 0·8 1 1·2 1·4 1·6 1·8 2 2·2 2·4 2·6 2·8 3 3·2 3·4

Use the number line above to find:

1 4 lots of 0·2 2 5 lots of 0·4 3 6 lots of 0·4
6 5 lots of 0·5
4 3 lots of 0·6 5 4 lots of 0·8 9 3 × 0·7
7 4 × 0·3 8 5 × 0·6

10 (7 lots of 0·2) + (3 lots of 0·6)

When multiplying decimals by a whole number, the number of figures after the
decimal points in the question and answer is the same.

99CHAPTER 4 DECIMALS

worked examples

1 2·8 × 5 2 $0·85 × 7 14·0 1·50
= 14 = 1·5
Solutions
One figure after the point. 2 $0.85
1 2·8 Two figures after the point. ×7
×5
$5.85
14·0

Multiplying by 10, 100 and 1000

3 0·81 × 10 4 0·031 × 100 5 1·015 × 1000 6 0·6 × 1000
= 1015·000 = 600·0
= 8·10 = 3·100 = 1015 = 600

= 8·1 = 3·1

From the examples above we can see an easier way to get the answer.

When we multiply by 10, 100 or 1000, we move the decimal point 1, 2, or 3 places
to the right.

That is, 4 0·0 3 1 × 100 5 1·0 1 5 × 1000 6 0·6 0 0 × 1000
= 3·1 = 1015 = 600
3 0·8 1 × 10
= 8·1

Exercise 4:03 Foundation Worksheet 4:03
Multiplying a decimal
1 a 3 × 0·2 b 2 × 1·2 c 4 × 1·1 1 a 0·3
d 5 × 0·08 e 7 × 0·03 f 8 × 0·06
g 0·11 × 6 h 0·7 × 3 i 0·04 × 9 ×2
2 a 3 × 0·4
2 a 2·4 b 1·9 c 0·25 3 a 0·6 × 10
×6 ×4 ×4
worked example
d 3·8 e 4·62 f 2·4
×5 ×5 ×5 0·125
×8
g 0·63 h 2·318 i 17·2 1·000
×3 ×4 ×8 ∴ 0·125 × 8 = 1

3 a 3·8 × 2 b 4·11 × 3 c 3·08 × 3 When using decimals, estimate
d 0·375 × 8 e 5·11 × 9 f 6·07 × 5 the size of your answer first.
g 16·43 × 7 h 26·08 × 6 i 78·46 × 4
eg 3·95 ϫ 7
4 a 1·25 × 10 b 9·125 × 10 c 84·35 × 10 ϭ: 4 ϫ 7 or 28
d 6·5 × 10 e 13·8 × 10 f 3·7 × 10
g 2·315 × 100 h 6·85 × 100 i 0·12 × 100
j 0·3 × 100 k 8·1 × 100 l 0·125 × 100
m 0·1111 × 1000 n 0·015 × 1000 o 8·1 × 1000
p 0·75 × 1000 q 30·05 × 1000 r 6·75 × 1000

100 INTERNATIONAL MATHEMATICS 1

4:04 | Dividing a Decimal by

a Whole Number

4 ) 3·2 4 lots of 0·8
make 3·2.
3·2 Ϭ 4

0 0·2 0·4 0·6 0·8 1 1·2 1·4 1·6 1·8 2 2·2 2·4 2·6 2·8 3 3·2 3·4 3·6 3·8 4

When dividing a decimal by a whole number, place the point in the answer above the
point in the question.

worked examples

1 1·624 ÷ 8 2 62·1 ÷ 6 3 0·0136 ÷ 4 Remember PUP:
Point
Solutions Under
Point
1 0·203 2 10·35 3 0·0034

8)1 ·16 224 6)6 2 ·2130 4)0 · 0 11316

Dividing by 10, 100 and 1000

4 8·13 5 0· 7 5 8 4

10)8 1 ·1330 100)7 5 ·758584840400

From examples 4 and 5 above, we can see an easier way to get the answer.

When we divide by 10, 100 or 1000, we move the decimal 1, 2 or 3 places to the

That is, 5 7 5·84 ÷ 100 6 0 0 0·6 ÷ 1000
= 0·7584 = 0·0006
4 8 1·3 ÷ 10
= 8·13

Exercise 4:04 Foundation Worksheet 4:04
Dividing a decimal
1 a 4)0·84 b 3)6·3 c 2)0·008 1 a 2)0·8
d 5)30·15 e 4)12·12 f 3)9·27 2 a 3)0·69
g 9)82·8 h 7)0·014 i 6)7·308 3 a 5)2·5
4 a 7)7·14

101CHAPTER 4 DECIMALS

2 a 0·35 ÷ 5 b 9·8 ÷ 2 c 5·4 ÷ 6 • When dividing
d 0·11 ÷ 11 e 0·125 ÷ 5 f 0·027 ÷ 9 a decimal, you
g 8·82 ÷ 7 h 243·2 ÷ 8 i 1·008 ÷ 8 may add extra
zeros to the
3 a 2)0·5 b 2)0·7 c 4)0·6 decimal.

d 5)1·3 e 8)2·8 f 4)1·34 • You can place
extra zeros
g 6)2·19 h 8)1·1 i 6)14·1 before the
whole number
j 4)3· k 6)9· l 8)5· part too.

4 a 34·1 ÷ 10 b 75·5 ÷ 10 c 64·8 ÷ 10 worked example
d 0·6 ÷ 10 e 0·15 ÷ 10 f 0·05 ÷ 10
g 751·6 ÷ 100 h 0·38 ÷ 100 i 6784·5 ÷ 100 0·31 ÷ 100
j 0·04 ÷ 100 k 0·35 ÷ 100 l 35 ÷ 100
m 876·4 ÷ 1000 n 3156·4 ÷ 1000 o 3437 ÷ 1000 = 0 0·31 ÷ 100
p 34·6 ÷ 1000 q 8·75 ÷ 1000 r 0·8 ÷ 1000 = 0·0031

5 a 1846 ÷ 1000 b 6·75 ÷ 10 c 87·3 ÷ 100
d 976·5 ÷ 10 e 346·8 ÷ 100 f 0·8 ÷ 10
g 34·6 ÷ 1000 h 4 ÷ 100 i 75·8 ÷ 10

4:05 | Using Decimals

Decimals are used in almost every walk of life. Measurement, money and almost every area where
numbers are used will demand an understanding of decimals.

Exercise 4:05 Foundation Worksheet 4:05

1 a $3.45 + $15.52 b $18 + $9.65 c $0.95 + $3.15 Using decimals
1 a 2 × 0·3 kg
d $8.95 − $3.58 e $20 − $6.72 f $50.00 − $0.35 2 a 0·8 m + 0·3 m
g $7.45 × 6 h $0.94 × 10 i $1.30 × 9 3 a 3)1·8 L
j $8.45 ÷ 5 k $7.60 ÷ 10 l $15.40 ÷ 7

2 Find short cuts in answering these by using properties of numbers. ■ Order of
Operations
a 3·125 × 1 b 17·94 × 0 c 3·008 + 0 1( )
2 × and ÷
d 0·9 + 1·37 + 0·1 e 9·83 × 5 × 0·2 f (0·9 × 6) + (0·9 × 4)

g 1 × 36·08 h 8·6 × 39 × 0 i 9·8 − 7·9 + 7·9

3 True or false? 3 + and −

a 0·91 + 0·89 = 0·89 + 0·91 b 0·8 × 6 = 6 × 0·8

c 15 − 7·218 = 7·218 − 15 d 8·4 ÷ 2 = 2 ÷ 8·4

e 3·45 × 30 = (3·45 × 10) × 3 f 0·86 × 200 = (0·86 × 100) × 2

g 0·9 × 8 + 0·1 × 8 = (0·9 + 0·1) × 8 h 10 × (1·3 + 0·7) = 10 × 1·3 + 10 × 0·7

4 Simplify. b 6·1 − 0·9 + 0·6 c 0·3 × 800 − 15 ■ To multiply by
e 1·6 − 8·8 ÷ 8 f 160 ÷ (1·8 + 8·2) 800, put down
a 4 + 5 × 0·8 h 10 − 0·6 × 5 + 7 i 8·1 + 3·95 ÷ 5 + 0·9 two zeros and
d 15 − 3·5 × 3 multiply by 8.
g 5 × 0·3 + 0·7 × 4

102 INTERNATIONAL MATHEMATICS 1

5 Dermot kept a record of the distance he travelled to football grounds during one weekend.
The trips were 68·4 km, 8·9 km, 42·4 km and 9 km.

a What was the total distance Dermot travelled?
b What was the difference between the longest and shortest trips?
c Two of the trips were to the same ground. Which two do you think they were, and why

would the numbers be different?

6 A duplicating machine takes 0·8 seconds to produce each copy. How long would it take the
machine to produce 100 copies?

7 It took 16·8 litres of paint to paint 4 identical rooms.
a How much paint would be needed to paint one room?
b How much paint would be needed to paint ten rooms?

8 BC = 0·9 m, DC = 3·55 m, DE = 1·1 m, FE = 2·225 m, AF = 2 m.

A B a What two lengths have a sum equal

Building 0·9 m to AB?

2m D 3·55 m C b What is the length of AB?
c What is the perimeter of the building?
1·1 m
d What is the difference in length between

F 2·225 m E FE and DC?

4:06 | Multiplying a Decimal

by a Decimal

Complete: 2 0·5 × 6 3 2·35 × 3 4 2 × 0·076 prep qu iz
1 8 × 0·3 6 10 × 7·1 7 0·01 × 8 8 15 × 0·1
5 0·35 × 10 10 0·3 × 70 4:06
9 3·105 × 4

We need to find a rule for multiplying a decimal by a decimal.

Let the large square here be 1 square unit of area. 1 unit
0·4
∴ 1 small square = ----1---- of the large square

100

= 0·01 of the large square

= 0·01 square units 0·3

Shade a rectangle 0·4 units long and 0·3 units wide. 1 unit
Area of shaded rectangle = length × breadth

= 0·4 × 0·3 square units

But the shaded rectangle has 12 small squares.

Area of shaded rectangle = --1---2--- of large square

100

= 0·12 square units

∴ 0·4 × 0·3 = 0·12

It looks as though we must have the same number of figures after the decimal point in the answer
as we had altogether after decimal points in the question.

103CHAPTER 4 DECIMALS

Let’s use a calculator to test this idea.

Copy the table into your workbook and use a calculator to complete it.

Question Number of Calculator Number of We experiment to
decimal answer decimal solve problems.

places in question places in answer

0·7 × 6 1

0·3 × 0·2 2

0·15 × 0·3 3

0·92 × 0·11 4

0·123 × 0·1 4

Your solution should show that what we suspected was true.

When multiplying decimals, the number of figures after the decimal point in
the answer must be the same as the total number of figures that come after the
decimal points in the question.

worked examples

1 0·008 × 0·5 ■ You can drop ‘trailing’ 2 5·5 × 1·7 9·35 has two
= 0·0040 zeros like this, but only = 9·35 decimal places.
= 0·004 after you have made sure
your decimal multiplication
is correct. 3 (0·03)2
= 0·03 × 0·03
= 0·0009

Exercise 4:06 Foundation Worksheet 4:06
Multiplying decimals
1 How many decimal places has each numeral? 1 a 0·41

a 9·35 g 0·7 c 0·014 ×2

d 0·0004 e 0·15 f 721·5 2 a 0·4 × 0·7
3 a 7·2
g 156·84 h 927·1 i 3·861
× 0·3
2 a 0·4 × 0·6 b 0·11 × 0·8 c 0·7 × 0·1
d 3·1 × 0·2
e 6 × 0·7 f 0·3 × 0·3 g 0·04 × 0·7 h 0·8 × 0·03
i 0·07 × 0·07 j 0·8 × 0·5 k 0·05 × 0·6 l 0·15 × 4
m 0·09 × 6 n 0·08 × 0·7 o 0·4 × 0·4
Multiply as though
3 a 0·125 × 0·3 b 2·14 × 0·6 c 6·7 × 0·4 there were no
decimal points,
d 3·825 × 700 e 0·426 × 80 f 7·6 × 900 then use the rule to
g 8·125 × 0·8 h 0·048 × 0·5 i 620 × 0·07 place the point in
j 6800 × 0·002 k 312 000 × 0·09 l 540 × 0·5 your answer.
m 0·01 × 346·7 n 0·6 × 87·2 o 0·11 × 73·1

104 INTERNATIONAL MATHEMATICS 1

■ 15·6 ← 1 decimal place Estimate the
× 1·7 ← 1 decimal place answer wherever

109 2 you can.
156 0
15·6 ϫ 1·7
26·52 ← 2 decimal places

ϭ• 16 ϫ 2 or 32
•

4 a 9·6 b 0·412 I would be happy with
× 3·1 × 2·4
an answer of 26·52.
c 1·33 d 36·51
× 8·3 × 0·14 g 654 × 0·18 h 0·16 × 0·16
k 365·1 × 1·4 l 365·1 × 0·14
e 15·6 × 1·7 f 23·8 × 0·12
i 3·5 × 3·5 j 0·25 × 0·25 c (0·6)2 ■ 0·032 = 0·03 × 0·03
f (0·05)2
5 a (0·2)2 b (0·03)2 i (0·17)2
d (0·8)2 e (1·1)2
g (0·021)2 h (1·5)2

4:07 | Dividing by a Decimal

1 0·3 + 0·14 2 6 − 0·05 3 6 + 0·04 + 0·3 4 2·5 × 6 5 (0·2)2 prep qu

6 0·3 + 0·6 × 4 7 2)2·08 8 2)2·09 9 7)38·5 10 4)0·0136 iz

worked example 1 0·0136 Ϭ 4 4:07

0·0136 ÷ 4 When dividing by a whole can be
number, place the point in the
0·00 3 4 answer above the point in the written as
0·0136 .
= 4)0·011316
4

To divide by a decimal, we change the question so that we are dividing by a
whole number.

worked example 2

0·0136 ÷ 0·04 ■ To multiply by 100, From this example we can see
= 0----·--0---1---3---6-- that as long as the point is moved
move the point two the same number of places in
0·04 each number, we will get the
right answer.
= -0---·--0-----1---3----6----×-----1---0---0-- places right. ∴ 0·0 1 36 ÷ 0·0 4
0·0 4 × 100 = 1·36 ÷ 4
-1---·--3---6-- = 0· 3 4
1----·--3---6-- 4 105CHAPTER 4 DECIMALS
= 4 4)1·1316

= 0·34

further examples

1 7·8 16 ÷ 0·4 2 8·6 0 0 ÷ 0·0 0 2 3 --8---7---5--- = -8---7---5---·--0------0- = -8---7---5---0---0-
= 78·16 ÷ 4 = 8600 ÷ 2 0·05 0·0 5 5

Exercise 4:07

1 Change each of these divisions into divisions by whole numbers.

a 8·35 ÷ 0·5 b 0·344 ÷ 0·08 c 0·651 ÷ 0·003

d 0·049 ÷ 0·7 e 0·8 ÷ 0·2 f 7·125 ÷ 0·05 ■ When moving the
point to the right, you
g 38 ÷ 0·04 h 18 ÷ 0·9 i 8 ÷ 0·008 may need to add zeros.

j 2·7 ÷ 0·03 k 4·72 ÷ 0·8 l 2·2 ÷ 0·11 8·600
You can put extra
2 Find the answers to each of the parts in question 1.
zeros here!
3 a 0·9 ÷ 0·3 b 3·6 ÷ 0·9 c 0·24 ÷ 0·6
d 0·08 ÷ 0·008 e 0·8 ÷ 0·04 f 42 ÷ 0·007 0·4 2 5
g 365 ÷ 0·04 h 14 ÷ 0·07 i 15 ÷ 0·3
j 49 ÷ 0·1 k 16 ÷ 0·01 l 2 ÷ 0·001 )7 3·42040

4 a 5·75 ÷ 2·5 b 11·11 ÷ 1·1 c 4·5 ÷ 0·15
d 7 ÷ 3·5 e 48 ÷ 2·4 f 8·76 ÷ 0·12
g 0·322 ÷ 0·04 h 0·081 ÷ 0·05 i 34·96 ÷ 2·3
j 7·23 ÷ 0·6 k 5·58 ÷ 0·18 l 0·47 ÷ 0·04

m -2---5---·--5-- n 3----1---·--7---1-- o -2---·--7---1---8--
0·5 0·04 0·03

fun spot
0·3
1·044:07
11·6
3·88
2·42
3·9

1
0·03
0·308
2·03

4·2
0·005

0·09
2·6

1·01
1·3

0·075
0·378

80
0·375
Fun Spot 4:07 | Why did the orange stop in the middle of the road?

Work out the answer to each part and put the letter for that part in the box above the correct
answer.

A 1·58 + 2·3 W 3 + 8·6 T 3·91 − 0⋅01

Y 8·6 − 6 P 0·7 × 6 A (0·3)2

E 5·2 × 0·2 H 0·02 × 15 O 4·06 ÷ 2

N 12·1 ÷ 5 U 0·3 ÷ 4 D 0·24 ÷ 8

T 0·3 + 0·008 E 0·25 × 4 S 0·08 × 1000

L 0·5 ÷ 100 S 11·11 ÷ 11 Q 0·7 + 0·2 × 3

H Write --3--- + ----7---- + -----5------ as a decimal. A Which is the smallest: 0·38, 0·378, 0·4?

10 100 1000

106 INTERNATIONAL MATHEMATICS 1

4:08 | Changing Fractions

to Decimals

Write as decimals: ■ --3--- ← numerator prep qu
10 ← denominator
1 --3--- 2 ----7---- 3 -1---9---6- 4 -----5------ 5 ----3---7---- 4:08 iz

10 100 100 1000 1000

Simplify: 8 -1---2---0- 9 1----4---8- 10 -8---3---1---5-

6 -1---6- 7 6-- 10 4 5

2 3

The Prep Quiz introduces two methods that we can use to change fractions to decimals.

METHOD 1 If the fraction is written as 10th, 100ths, 1000ths, etc, put the last
digit of the numerator in the matching column.

worked examples

Change these fractions to decimals.

1 ----3---7---- 2 -4---5---1- 3 1----6- = 3----2-

1000 100 5 10

1 --1--- ----1---- -----1------- 1 --1--- ----1----- -----1------ 1 --1--- ----1---- -----1------
10 100 1000
10 100 1000 10 100 1000

0•0 3 7 4•5 1 3•2

∴ ----3---7---- = 0·037 ∴ 4----5---1- = 4·51 ∴ 1----6- = 3·2

1000 100 5

METHOD 2 To change a fraction to a decimal, divide the numerator by the
denominator (divide the top by the bottom),

eg 3-- = 3 ÷ 4 or 4)3·00.
4

worked examples

Change these fractions to decimals. 0·83•

1 -3- = 3 ÷ 4 2 1 -5- = 1 + (5 ÷ 8) 3 -5- = 5 ÷ 6

4 8 6

0· 7 5 0· 6 2 5 0· 8 3 3 3 3 . . .

4)3·3020 8)5·502040 6)5·50202020202 . . .

∴ 3-- = 0·75 ∴ 1 -5- = 1·625 ∴ -5- = 0·83˙

4 8 6

Fractions can be written as either terminating or repeating decimals.
1 A terminating decimal comes to a definite end. It has a definite

number of decimal places; eg 3·6, 0·000125, 0·75.
2 A repeating or recurring decimal does not come to an end, but forms a pattern that is repeated

indefinitely (ie forever). To show that a set of digits is repeated, we place a dot over the first and
last of the repeating digits:
eg 0·666 . . . = 0·6˙ , 1·272727 . . . = 1·2˙ 7˙ , 32·4608608 . . . = 32·46˙ 08˙

107CHAPTER 4 DECIMALS

Exercise 4:08

1 Is each decimal terminating or repeating?

a 0·65 b 0·333 . . . c 21·9 d 8·7˙
g 0·7˙ h 7·7
e 0·777 f 0·777 . . . k 9·105 105 . . . l 2·1˙ 65˙
i 14·3˙ 5˙ j 4·0408

2 Write these repeating decimals in the shorter way.

a 0·999 . . . b 3·1444 . . . c 0·181818 . . . d 0·1666 . . .
h 0·142857142857 . . .
e 0·909090 . . . f 6·745745 . . . g 66·666 . . .

3 Write each as a decimal.

a --3--- b ----6---- c -----9------ If you can change the denominator
to 10, 100 or 1000, you can still
10 100 1000 use method 1,

d ----1---9---- e --5---5--- f 1 --1---

1000 100 10

g 3 --7---7--- h --7---0---9--- i 6 ----2---- eg -1--1-- ×5 = --5---5-- = 0·55
20 ×5 100
100 1000 100

j 1-- k -4---4- l -1---7---5-

2 50 500

m --7--- n 1-- o ----3---- ■ If using a calculator,
you would use method 2.
20 4 250

4 Use method 2 to change these to decimals.

a 4-- b -1- c 1--

5 8 4

d 3-- e 5-- f 7--

8 8 8

g --7--- h -3---3- i 3-- You continue to divide until
the answer terminates
20 50 4 or repeats.

j -1- k -2- l 1--

3 3 9

m -2- n 4-- o -7-

9 9 9

p 5-- q 1-- r 5--

6 6 9

5 Change these to decimals.

a 3 8-- b 2 3-- c 6 -2-

9 5 3

d --3--- e --5--- f 1----0-

11 11 11

g 2 --5--- h -1---1- i 1 --3--- 0·1 4 2 8 5 7 1 4 2...

12 15 16 )7 1·030206040501030206....

j -4- k -3- l --1--- b Express --9--- as a decimal.

7 7 14 11

6 a Convert 6 7-- to a decimal. d How would you write --5--- as a decimal?

8 16

c Change 12 1-- to a decimal. f Convert 2 1----2---4- to a decimal.

4 200

e Write 8 5-- as a decimal.

6

108 INTERNATIONAL MATHEMATICS 1

4:09 | Rounding Off

1 Is 16 closer to 10 or 20? 2 Is 153 closer to 150 or 160? prep qu

3 Is 153 closer to 100 or 200? 4 Is 4·1 closer to 4 or 5? 4:09 iz

5 Is 4·7 closer to 4 or 5? 6 Is $7.66 closer to $7.60 or $7.70?

7 Is 7·66 closer to 7·6 or 7·7? 8 Is 3·111 closer to 3·11 or 3·12?

9 Which two numbers does 14·63 lie between: 14·5 and 14·6 or 14·6 and 14·7?

10 What is the decimal halfway between 6·7 and 6·8?

To round off a decimal to the nearest whole number, we write down the whole number closest to it.

7·3 7·5 7·9 8·4 8·8

7 89

7·3 is closer to 7 7·9 is closer to 8 8·4 is closer to 8 8·8 is closer to 9

7·5 is exactly halfway between 7 and 8. In cases like this it is common to round up.
We say 7·5 = 8, correct to the nearest whole number.

To round off 72 900 to the nearest To round off 0·8134 to the nearest
thousand, we write the thousand thousandth, we write the thousandth
closest to it. closest to it.

72 900 0·8134

72 000 72 500 73 000 0·813 0·8135 0·814

72 900 is closer to 73 000 than to 72 000. 0·8134 is closer to 0·813 than to 0·814.
∴ 72 900 = 73 000, correct to the nearest ∴ 0·8134 = 0·813, correct to the nearest
thousand. thousandth.

To round off (or approximate) a number correct to a given place, we round up if
the next figure is 5 or more, and round down if the next figure is less than 5.

worked examples

Round off: 1 56 700 000 to the nearest million 2 0·0851 to the nearest hundredth
3 86·149 to one decimal place 4 0·66666 to four decimal places

Solutions 2 0·0851 has an 8 in the hundredths
column. The number after the 8 is 5
1 56 700 000 has a 6 in the millions or more (ie 5).
column. The number after the 6 is 5 ∴ 0·0851 = 0·09,
or more (ie 7). correct to the nearest hundredth.
∴ 56 700 000 = 57 000 000,
correct to the nearest million.

109CHAPTER 4 DECIMALS

3 86·149 has a 1 in the first decimal 4 0·66666 has a 6 in the fourth decimal
place. The number after the 1 is place. The number after the 6 is 5 or
less than 5 (ie 4). more (ie 6).
∴ 86·149 = 86·1, ∴ 0·66666 = 0·6667,
correct to one decimal place. correct to four decimal places.

Exercise 4:09 Foundation Worksheet 4:09

1 Round off these numbers to the nearest hundred. Rounding off
1 Round off to the nearest cent.
a 7923 b 1099 c 67 314
a 84·6c
e 609·99 f 350 g 74 932 2 Round off to the nearest dollar.

d 853·461 a $7.63
h 7850

2 Round off these numbers to the nearest whole number. I’m supposed
to get $496.496
a 9·3 b 79·5 c 45·1 d 2·7
per week,
e 2·314 f 17·81 g 236·502 h 99·5 but they
only gave
3 Round off these numbers to the nearest hundredth. me $496.50.

a 243·128 b 79·664 c 91·351 d 9·807
h 1·991
e 0·3046 f 0·0852 g 0·097

4 Round off these numbers to one decimal place.

a 6·70 b 8·45 c 2·119 d 6·092
h 9·99
e 0·05 f 246·739 g 29·88

5 The cost of petrol is 89·3 cents per litre. Angelika bought 12 litres of petrol. How much did this
cost, correct to the nearest cent?

6 Leonie bought 12 doors at $59.30 each. What is the ‘dec. pl.’ is short for
cost of the doors, rounded off to the nearest dollar? ‘decimal place’.

7 To change -4- to a decimal, Mel divided 9 into 4·000 . . .
Her answer9was 0·4˙ . Give this answer correct to:

a 1 dec. pl. b 2 dec. pl.
c 3 dec. pl. d 4 dec. pl.

8 Peter needed to divide a 3 metre length into 11 equal
parts. After dividing he got the answer 0·2˙ 7˙ . Give this

answer correct to: Remember that the
answer is in cents!
a 1 dec. pl. b 2 dec. pl.

c 3 dec. pl. d 4 dec. pl.

9 These calculator displays represent an answer in cents. 56c
Round the answers off to the nearest cent. or $0.56

a 1362.836 b 1941.341853

c 6118.91 d 509.5008

e 549.9342 f 1768049.5

10 Round off the answers shown on the displays in question 9 correct to the nearest dollar.

110 INTERNATIONAL MATHEMATICS 1

4:10 | Applications of Decimals

1 10 lots of 10c 2 10 × $0.10 3 10 × 0·1 prep qu

4 I have eleven coins in my pocket. There are three dollar coins, seven ten-cent coins and 4:10 iz

one five-cent coin. How much do I have?

5 0·97 + 0·033 6 $20 − $1.67 7 $10 − $0.32

8 $3.45 × 6 9 0·3 × 0·2 10 0·84 ÷ 0·4

Exercise 4:10 Foundation Worksheet 4:10

1 Round off each amount to the nearest 5 cents. Application of decimals
1 Round off to the nearest dollar.
a $34.27 b $8.52 c $0.99
a $105.56
d $17.11 e $85.44 f $9.55 2 Round off to the nearest 5 cents.

g $103.33 h $11.68 i $27.16 a 73c
2 Round off to the nearest 5 cents.

a $3.98

j $277.37 k $108.08 l $654.72

1 cent and 2 cent pieces were withdrawn from circulation in 1990.
The final bill is rounded to the nearest five cents.

2 Petrol is sold for 87·9 cents per litre. Find the price to be paid to the nearest 5 cents for buying:

a 8L b 3L c 6L d 7L

e 10 L f 30 L g 20 L h 100 L

i 27 L j 34 L k 52 L l 41 L

m 25 L n 19 L o 8·5 L p 0·5 L

3 Coola Soda Drinks sends out their bottles in containers 0·1 km is 100 m.
that hold 1000 bottles, 100 bottles or 10 bottles.
8360 bottles are to be sent out.

a What is the least number of full containers that
could be used?

b What is the greatest number of full containers
that could be used?

4 Katherine Collison owns a dress shop. She can purchase
pink ribbon on rolls that contain 0·1 km, 0·01 km or
0·001 km of ribbon. She needs to purchase 0·836 km
of pink ribbon.

a If she wanted to purchase as few rolls as possible,
how many of each type of roll would she purchase?
How many rolls is this altogether?

b What is the greatest number of rolls that she could
purchase in order to buy this ribbon?

111CHAPTER 4 DECIMALS

5 These are the winning times for the women’s 100 m sprint in the Olympic Games from 1964 to
1980. 1964: 11·4 s, 1968: 11·0 s, 1972: 11·07 s, 1976: 11·08 s, 1980: 11·06 s. Arrange these
times in order from least time taken to most time taken.

6 In the World Championship pairs skating finals, Jane and Christopher were given the scores
9·9, 9·8, 10, 10, 10, 10, 10, 10, 9·9, 9·1, 10 and 9·4 by the twelve judges. Two of the scores are
not counted, the lowest and one of the equal highest scores.

a What is the total of the remaining ten scores?
b To get the average score we divide the total found in a by 10. Write their average score

as a decimal.

7 When building his house, Peter drove 5 nails into a wall. The nails were in line, 0·75 metres
apart. What was the distance from the first nail to the last? (Hint: Draw a diagram.)

8 A box is to be filled with 100 pencils, each having a mass of 9·6 grams. If the box has a mass of
54·5 grams, what will be the mass of the box full of pencils?

9 Helen bought 6·6 litres of petrol at 91·3 cents per litre. How much did she pay?

10 A carton contains 0·6 litres of milk. How many 0·25 litre glasses can be filled from this carton?

11 In the 400-metres event, the runner who came second ran the distance in 44·37 seconds. The
race was won in a time of 43·79 seconds.

a What was the time difference between 1st and 2nd?
b If the winner broke the previous record by 0·07 seconds, what was the previous record?

12 A container holds less than 0·8 litres of water. Which of the following answers could be the

amount in four of these containers.

A 5 litres B 4·2 litres C 3·2 litres D 2·4 litres

13 Each tin of paint covers an area of 100 square metres. Joan wishes to cover an area of
460 square metres.

a How many tins of paint will she need?
b What is the cost of the paint if each tin costs $27.55?

14 Joel has a salary of $32 640 per year. He divided this amount by 365·25 to find his salary
per day, then multiplied by 7 to find his weekly salary. He calculated his weekly salary to
be $625.544. He was paid only $625.54 each week.

a How much short of his correct weekly salary was he paid?
b If there are 52 pays this year, how much will he have lost in payments?
c If 1000 employees lost an average of $0.005 per week, how much would be lost by them

altogether in 52 weeks?
d Who could profit by the loss (which would probably go unnoticed by the employees)?

Appendix E E:01 Changing repeating decimals to fractions

112 INTERNATIONAL MATHEMATICS 1

Fun Spot 4:10 | World championship diving fun spo t

At the championships there were seven diving judges. 6 5·5 7 7 7·5 8 7·5 4:10
Degree of difficulty 2·2
Scoring:
• Eliminate the highest and lowest

points awarded and find the total
of the rest.
• Multiply this total by the degree
of difficulty.
• Divide the score by 5 and multiply
by 3. (This converts the dive back
to what it would have been if only
five judges had been used with
two rejected as above.)

The more difficult dives
have a higher degree of
difficulty, eg DD ϭ 2·8.

worked example (seven judges)

Points Total × DD ÷ 5 then × 3 Score
(DD = degree of difficulty)
8, 7·5, 7·5, 7, 7, 6, 5·5 77 ÷ 5 × 3 = 46·2 46
9, 9, 8·5, 8·5, 8, 8, 7·5 35 × 2·2 = 77 117·6 ÷ 5 × 3 = 70·56 71

42 × 2·8 = 117·6

Calculate the score for each set of points.

1 8, 7, 7, 7, 7, 7, 6·5 DD = 2·0 2 6, 6, 6, 5·5, 5, 5, 4 DD = 2·4
4 9, 9, 8, 8, 8, 7·5, 7 DD = 1·8
3 4, 3·5, 3, 3, 3, 2, 2·5 DD = 2·8 6 7, 6·5, 6·5, 6, 6, 5, 4·5 DD = 2·6
8 2, 2, 1·5, 1·5, 1, 1, 0 DD = 2·8
5 6, 6, 6, 6, 6, 6, 6 DD = 2·2 10 8, 7, 6·5, 6·5, 6·5, 6, 6 DD = 2·0

7 8, 8, 8, 7, 6·5, 6·5, 6 DD = 2·4

9 9·5, 9·5, 9, 9, 9, 9, 9 DD = 2·6

4:10A Decimal magic squares
4:10B Currency conversion

113CHAPTER 4 DECIMALS

hematic investigation Investigation 4:10 | Applying decimals
4:10
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.

Calculate the cost of running the family car for one year.
You will need to estimate or discover the costs involved.

• distance travelled • petrol consumption
• petrol costs • oil and service costs
• tyres • repairs
• insurance • registration

matal termsMathematical terms 4 recurring (or repeating) decimal
• A decimal whose digits form a pattern that
4 ascending order
• From smallest to largest. continues to repeat.
eg 3·7777 . . . = 3·7˙
descending order
• From largest to smallest. 0·5454 . . . = 0·5˙ 4˙

decimal (or decimal fraction) terminating decimal
• A way of writing a common fraction • A decimal that has a definite number of

using place value. decimal places.
eg 3-- = 0·375 eg 0·375

8 round off (or approximate)
• To write in a simpler (less accurate) way.
decimal point
• The point that is used in a decimal to eg 0·94999 Ӏ 0·95

separate the whole number part from the estimate
fraction part. • To make a close guess or approximation.

decimal places
• The number of digits used in a decimal

that are to the right of the decimal place.

114 INTERNATIONAL MATHEMATICS 1

Assessment Grid for Investigation 4:10 | Applying decimals

The following is a sample assessment grid for this investigation. You should carefully read
the criteria before beginning the investigation so that you know what is required.

Assessment Criteria (B, C, D) Achieved ✓

a No systematic or organised approach has been used to 1
solve the problem. 2
Criterion B 3
Application & Reasoning b An organised approach has been used and the student has 4
attempted to find out the necessary data. 5
6
c An organised approach has been used and there is some 7
explanation of how the results were obtained. 8
9
d There is a full explanation of the how the results were 10
obtained. 1
2
e Full explanations are given and all the answers are 3
reasonable. 4
5
Criterion C a There is no working out and no explanations have been 6
Communication given. 1
2
b Working out is shown and presentation is good and easy
to follow. 3

c Working out and explanations are well communicated 4
using symbols or words.
5
Criterion D a Some attempt has been made to explain the method used. 6
Reflection & Evaluation 7
The method and majority of processes are justified and 8
b the answers have been checked for reasonableness with

some success.

c Reasoned explanations are given for the processes used
and all answers have been checked for reasonableness.

d Very concise and exact justifications are given for the
processes used and answers have been evaluated.

115CHAPTER 4 DECIMALS

diagnostic test Diagnostic Test 4: | Decimals

4 • Each section of the test has similar items that test a certain type of question.
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.

  Section
4:01
1 Write each decimal in expanded notation.
4:01
a 36·152 b 0·3333 c 7·039
4:02
2 Arrange in order, smallest to largest. 4:02
4:03
a 0·3, 0·9, 0·6 b 0·4, 0·11, 0·6 c 0·7, 0·77, 0·077 4:04
4:03
3 a 0·45 + 12·3 + 4 b 0·008 + 0·15 + 0·9 c 0·95 + 0·005 + 0·5 4:04
4:06
4 a 23·587 − 4·35 b 880·6 − 77·03 c 50 − 6·9 4:07
4:08
5 a 0·015 × 8 b 3·9 × 6 c 4·72 × 4
4:08
6 a 5·35 ÷ 5 b 6·2 ÷ 5 c 11·031 ÷ 6
4:08
7 a 0·4 × 100 b 3·375 × 10 c 1·56 × 1000
4:09
8 a 134·65 ÷ 100 b 9·135 ÷ 1000 c 0·045 ÷ 10

9 a 0·008 × 0·5 b 11·2 × 0·11 c (0·03)2
10 a 0·0136 ÷ 0·04 b 8·6 ÷ 0·002 c --8---7---5---

11 Is each decimal terminating or repeating? 0·05

a 0·75 b 1·272727 . . . c 1·2˙ 7˙

12 Write each as a decimal.

a 3 --7---7--- b 3 + --7--- + ----7---- c 6 -----7------

100 10 100 1000

13 Write each as a decimal. c 1 1----0-
a 5-- b 1--
11
86

14 Round off these decimals correct to two decimal places.

a 78·484 b 0·6666 . . . c 9·295

116 INTERNATIONAL MATHEMATICS 1

assignm

Chapter 4 | Revision Assignment ent

1 a Write the numeral for one million, 4 Are the following true or false?
three hundred and twenty-six
thousand, seven hundred and a 6 × 96 > 6 × 100 b 0·3 < 0·0333 4A
sixty-three.
c --1--- < -1- d -1---2- = --2---
b Write in words: 16 020.
c What is the value of the 1 in 716 250? 10 2 2 12
d Write 42 375 in expanded notation.
5 Copy and complete these number

sentences to make each sentence true.

a 6 × (300 + 6) = × 300 + × 6

b 73 × 10 = 10 ×

2 Simplify: c 6 × 199 = 6 × 200 − 6 ×
a (4 + 3) × (8 − 2)
b 17 − (12 − [2 + 10]) d 98 + 47 + 2 = 98 + + 47

6 Change these Roman numerals into our

c 2----5----+-----3---5-- d ------1---2------- own numerals.
5 (1 + 3)
a LXXXVII b CCXCIII

3 Simplify: c MDCLXIV d MCMXC
a 16 + 2 × 4 − 5
b 18 + 24 ÷ (10 − 4) 7 Write each in expanded form and as a
c 200 − 120 ÷ 4 − 3 × 5
d 18 ÷ [54 − (24 × 2)] basic numeral.

a 43 b 33 c 106

d 2 × 104 e 32 × 24

117CHAPTER 4 DECIMALS

gnment Chapter 4 | Working Mathematically
assi
4B Percentage1 Use ID Card 5 on page xvii to identify: 3 A school hall has 24 rows of chairs with 20
a1 b2 c3 d4 chairs in each row. If the chairs are reset in
id inflatione5 f 6 g7 h8 rows of 16, how many rows will there be?
Percentage
4 2 In an election one candidate received 9873 4 A publishing company wants to send some
inflationvotes, another received 7828 votes, and theadvertising material to 650 schools. If an
third 2314. If 215 voting papers were envelope costs 6 cents, the advertising
informal (not filled out properly) and there pamphlet costs 38 cents to produce and the
were 20 609 voters in the electorate, how stamp for the letter costs 55 cents, find
many did not vote? how much it costs to send the advertising
material to the schools.

Vote for me 5 A tennis tournament has 64 players. If only
and let me the winner of each match continues in
do it right the competition, how many matches
for you! must be played to obtain the winner of
the tournament?

6 A group of units has four different
buildings. Each building has five floors
with six apartments on the first four floors,
and four apartments on the fifth floor.
How many apartments are there in the
group of units?

7 Inflation in Highland Inflation in Lowland
20
10 15
10
5 5

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003
Year Year

a What was Highland’s inflation rate in 1999?
b What was Lowland’s inflation rate in 2001?
c What was Highland’s increase in inflation rate from 1999 to 2002?
d Which place appears to have the higher inflation rate? What is the real situation?
e In which year did Lowland first have an inflation rate of 11%?

1 Writing fractions as decimals

118 INTERNATIONAL MATHEMATICS 1

5

Directed
Numbers and
the Number Plane

This is the last time
I fly El Cheapo Airlines!

Chapter Contents 5:08 The number plane extended
5:09 Multiplication of directed numbers
5:01 Graphing points on the number line
5:02 Reading a street directory Investigation: Multiplying directed numbers
5:03 The number plane
Challenge: Using pronumerals
Mastery test: The number plane 5:10 Division of directed numbers
5:04 Directed numbers 5:11 Using directed numbers
5:05 Adventure in the jungle 5:12 Directed numbers and the calculator

Investigation: Directed numbers Fun Spot: Try this quick quiz!
5:06 Addition and subtraction of directed
Mathematical Terms, Diagnostic Test, Revision
numbers
5:07 Subtracting a negative number Assignment, Working Mathematically

ID Card

Learning Outcomes

Students will:
• Compare, order and calculate with integers.
• Create, record, analyse and generalise number patterns using words and algebraic

symbols in a variety of ways.
• Graph and interpret linear relationships on the number plane.

Areas of Interaction

Approaches to Learning, Homo Faber

119

5:01 | Graphing Points on the

Number Line

prep quiz Write down the number sentence represented by each number line.
12

0 2 4 6 8 10 0 2 4 6 8 10

5:01 . . . + . . . = 10 ...+...=9

34

0 2 4 6 8 10 0 2 4 6 8 10

56

0 2 4 6 8 10 0 2 4 6 8 10

78

0 2 4 6 8 10 0 2 4 6 8 10

9 10

0 2 4 6 8 10 0 2 4 6 8 10

The number line can be very useful. It can help us to perform additions, subtractions,
multiplications and even divisions. The number line can also be used to graph sets of numbers.

To graph a number on the number line, place a large dot at the position of that
number on the number line.

worked examples

Graph the following sets of points: We use braces to list
the numbers in a set.
1 {2, 4, 5} 01 23 4 56

2 {0, 1, 2, 3, 4} 0 1 2 3 4 5 6

3 {3, 6, 9} 0 2 4 6 8 10 12

4 {1 -1- , 3, 4 1-- } 01 23 4 56

22

5 {0·2, 0·5, 1·1} 0 0·2 0·4 0·6 0·8 1 1·2

120 INTERNATIONAL MATHEMATICS 1

Exercise 5:01

1 Use braces to show the set of numbers graphed A set of numbers
on each number line. is a collection of numbers.
a0 1 2 3 4 5 6 7

b0 1 2 3 4 5 6 7

c 8 9 10 11 12 13 14 15 16

d 11 12 13 14 15 16 17 18

e0 1 1 112 2 2 1 3 3 1 4
2 2 2

f0 1 1 3 1 1 1 1 1 1 3
4 2 4 4 2 4

g 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8

h 0 0·2 0·4 0·6 0·8 1 1·2 1·4 16

2 Graph each set of numbers below on a separate number line. 15

a {1, 3, 5} b {0, 2, 4, 6} c {2, 3, 5, 7} 14
d {0} e {1, 2, 3, 4, 5} f {10, 12, 14}

g {0·3, 0·5, 0·6} h { 1-- , -3- , 1} i {0, 0·5, 1, 1·5} 13

44

1, 2, 3, 4 , 5,... 12
are the counting numbers. 11

10

9

3 Graph each set on a separate number line. 8
7
a the counting numbers between 1 and 6 6
b the numbers two away from 5 5
c the odd numbers less than 8 4
d the counting numbers less than 4
e the even numbers between 11 and 17
f the even numbers between 8 and 12

3

2

• Number lines can be vertical. 1
0

121CHAPTER 5 DIRECTED NUMBERS AND THE NUMBER PLANE

5:02 | Reading a Street Directory

On a street
directory, a letter
and a number are
given. Come down from
the letter and across
from the number. For
example, J5 gives
Blacktown High School.

Exercise 5:02

1 Name the places or roads closest to the centre of the following reference coordinates.

a H4 b K3 c B3 d E5 e F8 f H1

g A6 h D9 i C5 j C2 k B5 l F9

m E2 n J5 o D8 p A3 q E4 r D5

2 Which reference coordinates could be used to show where to find:

a Ashlar Golf Course? b Harvey Park (top left)?

c Francis Park? d Crudge Road (top left)?

e Blacktown Showground? f Meroo Street (centre)?

g Jill Street (near Richmond Road)? h Second Avenue (off Sunnyholt Road)?

122 INTERNATIONAL MATHEMATICS 1

3 It is not always easy to decide on the best reference coordinate for a street.

a What three reference coordinates might be given for Osborne Road, which is on the top left
of the map?

b Name the two roads that meet at D4.
c What reference coordinates would be used for Third Avenue?

4 On this map 1 cm represents 250 m. 4 cm represents
1000 m
a Find the approximate length of Crudge Rd (B2). or 1 km.
b Find the approximate length of Newton Rd (D10).
c Find the approximate length of Ashlar Golf Course.
d Find (approximately) the shortest distance by road

from the intersection of Osborne and Davis Roads
(D4) to Blacktown High School.

5:03 | The Number Plane

On the street directory, a letter and a number were used to name a position. On the number plane,
two numbers are used. These are called coordinates, and are written in parentheses and separated
by a comma. The two axes used are called the x-axis (horizontal) and the y-axis (vertical).
The x-coordinate is always written before the y-coordinate.

(2, 5) is called a number pair or the coordinates of a point.
• The first number is the reading on the x-axis (how far right).
• The second number is the reading on the y-axis (how far up).
• The point (0, 0) is called the origin. (‘Origin’ means ‘starting point’ or ‘beginning’.)

(1, 4) is 1 y
to the right 5
and 4 up.
(4, 3) is above
(0, 0) is 4 ‘4’ on the
called the
origin. It x-axis and
across from ‘3’
is 0 to 3 on the y-axis.
the right
and 0 up. 2

1

0 1 2 3 4 5x

(3, 0) is 3 to
the right and

0 up.

123CHAPTER 5 DIRECTED NUMBERS AND THE NUMBER PLANE

worked examples

1 Write down the letter naming the points: Solutions

a (3, 4) b (2, 3) c (1, 0) 1 To find each point, start at the origin.
a (3, 4) is 3 to the right and 4 up.
2 What are the coordinates of: ∴ (3, 4) is N.
b (2, 3) is 2 to the right and 3 up.
a G? b L? c P? ∴ (2, 3) is I.
c (1, 0) is 1 to the right and 0 up.
y P Q ∴ (1, 0) is A.

5 2 a G is above 5, and to the right of 2.
∴ G is (5, 2).
4L M N
3H I JK b L is above 0, and on 4.
FG ∴ L is (0, 4).
2
c P is above 2, and to the right of 5.
1D E ∴ P is (2, 5).

A BC
0 1 2 3 4 5x

Exercise 5:03

1 Write down the letter used to name each of these points.

y V T S a (1, 3) b (2, 2)
7 c (5, 6) d (4, 7)
f (7, 5)
6Z X UR e (0, 0) h (6, 6)
g (3, 5) j (0, 2)
l (5, 0)
5Y W Q i (0, 6) n (6, 4) The x-coordinate
k (3, 0) p (1, 5) always comes first.
r (4, 4)
4B A P m (4, 3) t (7, 7)
o (7, 2) v (6, 1)
x (6, 7)
3G H q (2, 1) z (2, 6)
s (4, 2)

2F E C N u (0, 4)
w (7, 0)

1 DI M y (1, 1)

O JKL
0 1 2 3 4 5 6 7x

2 Use the number plane in question 1 to write down the coordinates of these points.

aA bB cC dD eE f F gG

hH iI jJ kK lL mM nN

oO pP qQ rR sS tT uU

124 INTERNATIONAL MATHEMATICS 1

3 Make a grid that has 10 equal units on the x-axis and 7 equal units on the y-axis. You may
draw your own or use graph paper. Plot the points below on your number plane. What shape
is formed when each set of points is joined in the given order?

a (7, 1) (7, 3) (9, 3) (9, 1) (7, 1) b (4, 1) (4, 4) (5, 4) (5, 1) (4, 1)

c (6, 0) (6, 3) (7, 0) (6, 0) d (0, 4) (2, 6) (5, 6) (3, 4) (0, 4)

e (1, 0) (0, 1) (0, 2) (1, 3) (2, 3) (3, 2) (3, 1) (2, 0) (1, 0)

4 This time, draw a grid with 10 units on the x-axis and 16 units on the y-axis. All units must be
the same size. Draw each shape on this number plane as you did in question 3.

a (2, 5) (2, 3) (1, 1) (1, 0) (2, 0) (3, 1) (4, 4)
b (8, 14) (7, 14) (7, 12) (8, 11) (8, 9) (9, 9)
c (2, 11) (3, 11) (4, 12) (5, 12) (5, 10) (4, 10) (1, 8) (0, 9) (0, 10) (1, 12) (5, 16) (6, 16)

(9, 13) (10, 10) (9, 9) (9, 7) (7, 4) (7, 3) (8, 1) (8, 0) (7, 0) (6, 1) (5, 4) (4, 4) (2, 5)
(1, 7) (1, 8)
d Put a big nose on your picture by starting from (5, 14) and ending at (6, 14).
e At (5, 15) and (6, 15) draw the pupils of Fred’s eyes. Draw the eyes around these.
f Draw in a small mouth and the fingers. Colour your picture.

5 Use 5 millimetre grid paper with 30 units on the x-axis and 24 units on the y-axis. (All units
must be the same size.) Draw each shape indicated by the coordinates as you have done above.
When you join the points, join them with curved lines.

a (6, 0) (8, 4) (10, 6) (12, 8) (10, 10) (8, 11) (2, 10) (4, 12) (8, 13) (12, 14) (6, 14) (2, 12)

(2, 10) (1, 12) (2, 16) (4, 18) (6, 18) (8, 20) (12, 22) (16, 23) (20, 23) (24, 22) (28, 20)

(26, 20) (29, 16) (28, 16) (29, 12) (28, 12) (29, 8) (28, 8) (30, 4) (29, 4) (30, 0)

b (8, 13) (6, 14)
c (11, 17) (11, 19) (12, 19 -1- ) (13, 19) (13, 17) (12, 16 1-- ) (11, 17)
d Colour the inside of the m2 outh to the right of (8, 13)2and above it.

e Draw a large black dot in the eye. f Add feathers and colour the eagle.

Mastery test 5:03 | The number plane

y Do this test again and again
How long until you get all the
questions right.
10 is CD?

6C D

5 8 What letter
2 Name this axis. is at (4, 1)?

4A Give 4 Give

5 coords. coords. 9 What letter
3 7 Give is at (1, 4)?

coords.
2

1B 1 Name this axis.

0 12 3456 7x
Give name 6 Give coords.

3 and coords.

125CHAPTER 5 DIRECTED NUMBERS AND THE NUMBER PLANE

5:04 | Directed Numbers

Many of the numbers we use represent situations that have direction as well as size. We call these
numbers directed numbers.

Once a direction is chosen to be positive (+), the opposite direction is taken to be negative (–).

■ Choosing directions for numbers I made a loss I made a profit
• If north is positive, then south is negative. of $380. of $500.
• If east is positive, then west is negative.
• If above zero degrees is positive, That’s (–380). That’s (+500).

then below zero degrees is negative.
• If to the right is positive, then to the left

is negative.

examples

1 Mount Everest is 8000 metres above sea level, so its height could be written as +8000 m
or just 8000 m.

2 The Dead Sea is 100 metres below sea level, so its height could be written as −100 m.
3 In cold countries the temperature is often below the freezing point of water (0°C).

The temperature on a cold day could be −5°C.

–20 is read ‘negative 20’ or ‘minus 20’.
+20 is read ‘positive 20’ or just ‘20’.
Note: +20 can be written more simply as 20.

Exercise 5:04

1 a Use a directed number to represent each of the following.

(A) a profit of $50 (B) 50 km south (C) a fall of 50 points

(D) a gain of 50 kg (E) $50 deposit (F) 50 m above sea level

(G) 50° below zero (H) 3 floors above ground (I) an increase of 50

(J) 50 cm to the right (K) 50 s after takeoff (L) 50 km east

(M) 50 s before takeoff (N) 50° above zero (O) 50 cm to the left

(P) a rise of 50 points (Q) a loss of $50 (R) a loss of 50 kg

(S) 50 km west (T) 50 m below sea level (U) a decrease of 50

(V) $50 withdrawal (W) 3 floors below ground (X) 50 km north

b Match up each of A to L in part a with its opposite chosen from M to X.

126 INTERNATIONAL MATHEMATICS 1

Directed numbers have both size and direction.

2 Write a directed number to show the size and direction of the number in each sentence.

a Jan’s Mathematics mark was 8 marks higher this time.

b Today, the temperature is 4°C below yesterday’s temperature.

c John took fourteen steps backwards.

d After playing marbles, Melissa had lost 15 marbles.

e The price of Microsoft shares rose by twelve cents Any whole number without

last week. a decimal point on the end
f The level of water in the reservoir has dropped is called an integer.

two metres.

g The price of fish has not changed this week.

h My profits have increased by $1800 this year.

i In the last hour we have travelled 60 km south.

j I have just discovered that my bank balance is

overdrawn by $135.

3 Write two different sentences that might be represented
by the directed number −3.

4 Write two different sentences that might be represented
by the directed number +7.

5 Write a directed number to show the change described in each sentence.

a The temperature fell from 37°C to 26°C.
b I had ten golf balls when I arrived. Now I have only 3.
c I was 4 km east of home. Now I am 2 km west of home.
d I bought shares for $23 per share and sold them for $19 per share.
e The diver was 2·4 m below the surface. She is now 4·4 m below the surface.
f The frog was 2 cm behind the line. After the jump it was 5 cm in front of the line.
g We entered the lift on the tenth floor. We left the lift one floor below ground level.

6 What temperature is shown on each thermometer below? e
a bc d

°C °C °C °C °C

40 40 40 40 40
30 30 30 30 30
20 20 20 20 20
10 10 10 10 10
0 00 0 0
–10 –10 –10 –10 –10
–20 –20 –20 –20 –20
–30 –30 –30 –30 –30

127CHAPTER 5 DIRECTED NUMBERS AND THE NUMBER PLANE