54 © Noordhoff Uitgevers bv
2
Numbers and
Formulas
A popular attraction at fairs and amusem*nt parks is the
Octopus, also known as Spider. While sitting in a gondola,
you spin at high speed, with the gondolas moving up and
down as well. There are different types of Octopuses. If an
Octopus has six arms, with five gondolas for two people on
each arm, you can use a product of three factors to calculate
the number of people each ride can accommodate.
© Noordhoff Uitgevers bv 55
2 Arithmetic Operations
Learning objectives
- You can use the operations addition, subtraction, multiplication, and division
####### when making calculations.
- You can apply the order of operations.
O 1 Mover Tom is standing by the lift and wants to
take some boxes to the top floor. See the picture.
a How many kg can Tom weigh at the most?
b Circle what you did in your calculation.
- multiply • divide
- add • subtract
Theory A Sum, difference, product and quotient
You do not only do arithmetic at school. Consider the following
questions.
- What is cheaper: a subscription with 5 GB for € 20 a month or a
subscription with 8 GB for € 25 a month?
- Will eight buses with 45 seats each suffice to transport 352 pupils and
ten supervisors?
Some calculations contain a multiplication. A different
word for multiplication is product.
The product of 3 and 8 is 3 × 8 = 24.
3 and 8 are called the factors of the product.
3 × 8 means 8 + 8 + 8.
Division has to do with multiplication.
24 ÷ 3 = 8, since 8 × 3 = 24.
The quotient of 24 and 3 is 24 ÷ 3 = 8.
The sum of 8 and 11 is 8 + 11 = 19.
8 and 11 are called the terms of the sum.
The difference between 12 and 7 is 12 − 7 = 5.
Addition, subtraction, multiplication, and division are
examples of operations.
####### Learning objective You can use the operations addition,
####### subtraction, multiplication, and division when making calculations.
####### O 1
3 × 8 = 24
product
factors
12 − 7 = 5
difference
24 ÷ 3 = 8
quotient
8 + 11 = 19
sum
terms
© Noordhoff Uitgevers bv 2 Arithmetic Operations 57
2 a Rewrite 7 + 7 + 7 + 7 as a product and provide the answer.
b Which multiplication has to do with 165 ÷ 15?
c Calculate the quotient of 56 and 14.
d Calculate the product of 11 and 8.
e Calculate the difference of 11 and 8.
3 a Calculate the sum of the product of 4 and 5
the quotient of 8 and 2.
b Calculate the quotient of the product of 3 and 8 and
the difference of 104 and 100.
c Calculate the product of the sum of 3 and 5 and
the difference of 17 and 9.
4 For the following questions, first write down the product or quotient.
Then give the answer.
a At a fair there is an Octopus with six arms. There are five gondolas on
each arm. Each gondola can accommodate two people. What is the
maximum number of people the Octopus can accommodate per ride?
b Ted takes part in a 500m swimming competition.
A lane is 25 meters long. How many laps does
he need to swim?
c The tutor of class B1g buys each student a soda
that costs 90 cents and a bag of crisps that costs
60 cents. In total, the tutor spends 42 euros.
How many students are there in B1g?
E 5 Matthew: ‘How old are your three children?’
Natasha: ‘The product of their ages is 36.’
Matthew: ‘That’s not enough information.’
Natasha: ‘The sum of their ages is your house number.’
Matthew: ‘Okay, but then I’m still not sure.’
Natasha: ‘The twins are not going to school yet.’
Matthew: ‘Ah! Now I know.’
How old are Natasha’s children? Tip: make a list
with all possible ages.
####### c Check I can use the addition, subtraction, multiplication, and division
####### operations when making calculations.
####### Not quite mastered this learning objective yet? Then study theory A and
####### exercise 4, and do exercise L1.
L 1 a There are 5 different flavours in a box of tea. There are 12 teabags of each flavour.
A teabag has 2 grams of tea. How many grams of tea does the box contain?
b Chantal works for four weeks during the summer holidays. She works
thirty hours a week. She earns a total of 720 euros.
How much does Chantal earn per hour?
####### 2
Always write down the
calculation and the answer.
####### 3
####### 4
####### E 5
58 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
Example
Calculate (20 − (20 − 8)) · 3 − 18 ÷ 9.
Solution
(20 − (20 − 8)) · 3 − 18 ÷ 9 =
(20 − 12) · 3 − 18 ÷ 9 =
8 · 3 − 18 ÷ 9 =
24 − 2 = 22
R 7 On the right, in the calculation of 14 + 2· (7 − 2) − 9,
on each line, what is being calculated is marked.
Mark what is calculated on each line in the elaboration
of the example.
8 Calculate. Do not forget to write down the intermediate steps.
a 9 + 6 · 5 e (8 + 3) · (8 ÷ 2)
b (9 + 3) · 7 − 80 f 20 + 64 ÷ (8 ÷ 4)
c 20 − 2 · 8 − 4 g 20 − 64 ÷ 8 ÷ 4
d 8 + 3 · (7 + 2) h 6 − 3 · (16 ÷ (2 + 6))
A 9 Calculate.
a 128 ÷ 4 − (25 − 17)· 4 + 48 ÷ 12 − 4 d 1800 ÷ (600 − (2 · 250 − 200))
b (9 · 6 − 18 − 8 · 3) ÷ 6 + 5 · 3 e 45 − 3 · (8 − 4 · (5 − 2 · (3 − 1)))
c 800 − (300 − (200 − 150) · 2) − 450 f 8000 + 20 · 30 · (50 − 6 · (45 − 40))
A 10 Fill in the missing number so the calculations are correct.
a (8 + ) · 5 − 20 = 60 c 27 ÷ 3 + · 4 − 7 = 38
b − 3 · 4 + 12 ÷ 3 = 21 d 5 · (18 − ) + 7 · 4 − 3 = 45
E 11 Add two brackets so the calculations are correct.
a 400 ÷ 50 − 10 + 2 · 3 = 16 b 12 · 8 − 4 + 2 · 5 = 66
Add two brackets to make the result as big as possible.
c 2 · 2 + 4 · 2 + 7 d 3 · 5 + 4 · 5 + 2
####### c Check I can apply the order of operations.
####### Not quite mastered this learning objective yet? Then study theory B and do
####### exercise L2.
L 2 Calculate.
a 8 + 7 · 6 − 3
b 27 − (48 − 15) ÷ 3
c 700 − (240 − 80 ÷ 2) + 120
Here you see how you write down
the solution in your notebook.
####### R 7
14 + 2 · (7 − 2) − 9 =
14 + 2 · 5 − 9 =
14 + 10 − 9 =
24 − 9 = 15
####### 8
####### A 9
####### A 10
####### E 11
60 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
2 The hcf and the lcm
Learning objective
- You can calculate the hcf and the lcm.
O 12 There are 24 pupils in B1a. For a project, the teacher divides the students
into equally sized groups.
a In how many ways can this be done? Write down all the possibilities.
b There are 29 students in B1h. The teacher also wants to divide this
class into equally large groups. What problem does the teacher
encounter?
Theory A Factors and prime numbers
The numbers 1, 2, 3, 4, ... are called natural numbers.
Integers are all the natural numbers including zero. One is the smallest
natural number, but there is no largest natural number.
The number 3 is a factor of 24, since the solution of the
division 24 ÷ 3 is a natural number.
1 is also a factor of 24, because the result of 24 ÷ 1 is a
natural number.
The highest factor of 24 is 24 itself, since 24 ÷ 24 = 1.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
The number 13 has only two factors: 1 and 13.
The number 13 is an example of a prime number.
Other examples of prime numbers are 2, 5, 23 and 67.
A natural number with exactly two factors is a prime number.
Every natural number greater than 1 that is not a
prime number can be written as a product of
prime numbers.
To the right you can see that 60 = 2 · 2 · 3 · 5.
We say that 60 is written as the product of
prime factors.
13 List all the factors of the following numbers.
a 36 b 60 c 53
Write these numbers as the product of prime factors.
d 42 e 96 f 126
####### O 12
A factor is always a
natural number.
So 60 = 2 · 2 · 3 · 5.
60
30
2
2
15
3
5
####### 13
© Noordhoff Uitgevers bv 2 The hcf and the lcm 61
R 16 See Theory B and the example.
a Why is it useful when calculating the hcf to start with the greatest
factor of the smallest number?
b Why is it useful when calculating the lcm to start with the smallest
multiple of the greatest number?
17 Calculate.
a hcf(36, 48)c hcf(28, 88)e hcf(7, 13)g hcf(10, 15, 45)
b lcm(9, 15)d lcm(18, 24)f lcm(54, 72)h lcm(5, 7, 10)
18 You can also calculate the hcf and lcm of two numbers with prime
factors. Below you can see how that is done for 24 and 60.
Calculating the hcf and lcm of 24 and 60 with prime factors
Write both numbers as the product of prime factors.
24 = 2 · 2 · 2 · 3
60 = 2 · 2 · 3 · 5
hcf(24, 60) = 2 · 2 · 3 = 12
lcm(24, 60) = 2 · 2 · 2 · 3 · 5 = 120
Use the same method to calculate the
hcf and the lcm of
a 21 and 28 c 390 and 650
b 56 and 72 d 20, 30 and 40
A 19 A sailor watches for the flashes of lighthouses A and B.
Every 30 seconds he sees a flash from lighthouse A;
the light from B shines every 40 seconds.
At a certain moment, he sees the lights from A and B
at the exact same time.
a After how many seconds will this happen again?
b How often would this occur if A’s light flashed
every 50 seconds and B’s every 60 seconds?
A 20 A rectangular terrace is 120 by 192 cm.
Mr. Tree wants to tile the terrace with identical
square tiles, without breaking or cutting any
of the tiles.
a Can the tiles be 15 by 15 cm? Or 12 by 12 cm?
b What are the dimensions of the biggest possible
tile?
####### c Check I can calculate the hcf and the lcm.
####### Not quite mastered this learning objective yet? Then study theory A and B and do exercise L3.
L 3 a Calculate hcf(36, 60).
b Calculate lcm(12, 15).
####### R 16
####### 17
####### 18
Start with the prime factors of 24 and
multiply by the prime factors of 60
that you haven’t used yet.
Take the common prime factors.
####### A 19
Check whether you can
use the hcf or the lcm.
####### A 20
####### figure 2.
© Noordhoff Uitgevers bv 2 The hcf and the lcm 63
2 Fractions
Learning objectives
- You can simplify fractions.
- You can turn mixed numbers into improper fractions and vice versa.
- You can add and subtract fractions.
- You can multiply fractions.
O 21 The chocolate cake on the right is divided into ten equally
sized slices. During a coffee break, six people each take a
slice.
Tessa says: ‘ 106 of the cake has been eaten.’
Sheila says: ‘ 35 of the cake has been eaten.’
Why are they both right?
Theory A Fractions
The pizza shown here is divided into 8 equally big slices.
Each slice is one eighth. You write this down as 18.
There are mushrooms on 3 of the 8 slices.
For 3 out of 8 you write 38.
The number 38 is an example of a fraction.
In 38 , 3 is called the numerator: you count the
number of slices you have.
In 38 , 8 is called the denominator: every slice is
called one eighth.
The denominator tells you in how many equal pieces something is
divided, and the numerator tells you how many of those pieces you have.
Here you see that 128 = 46 = 23.
The fraction 128 is simplified to 23.
812462= = 3
####### figure 2.
####### O 21
####### figure 2.
38 denominator numeratorfraction64 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
Theory B Adding and subtracting fractions
Fractions with the same denominator have
common denominators.
When adding and subtracting fractions with common
denominators, the denominator does not change.
Therefore, 27 + 37 = 57 and 89 − 59 = 39 = 13.
The fractions 14 and 15 do not have a common denominator.
In order to add them up, you have to create common
denominators for them.
####### 1
4 +
####### 1
5 =
####### 5
20 +
####### 4
20 =
####### 9
####### 20
The new denominator 20 is the product of denominators 4 and 5.
For 103 + 158 you do not have to take 10 · 15 as the new
denominator. It also works with the denominator 30,
since lcm(10, 15) = 30. So 103 + 158 = 309 + 1630 = 2530 = 56.
Agreement in doing arithmetic with fractions
Simplify the result as much as possible.
####### Learning objective You can add and subtract fractions.
Example
Calculate.
a 145 + 13 c 2 13 − 56
b 103 + 127 d 8 − 1
Solution
a 145 + 13 = 95 + 13 = 2715 + 155 = 3215 = 2 152 c 2 13 − 56 = 73 − 56 = 164 − 56 = 96 = 32 = 1 12
b 103 + 127 = 1860 + 3560 = 5360 d 8 − 134 = 6 14
Fractions with a common
denominator have the
same denominator.
Simplify the fraction.
× 5× 5 × 4 × 4
When adding fractions
that do not have a
common denominator,
you take the lcm of the
old denominators as the
new denominator.
Take lcm(10, 12) = 60 as
the new denominator.
You can do this mentally.
In this chapter we write
down fractions as mixed
numbers, but that is not
mandatory.
66 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
24 Calculate.
a 12 + 13 c 112 − 14 e 4 − 123
b 34 − 13 d 2 13 + 1 14 f 5 34 − 1 121
A 25 Calculate.
a 37 + 58 c 78 − 684 e 2 15 − 34 + 107
b 1
####### 5
8 −
####### 5
12 d
####### 3
5 +
####### 2
3 +
####### 1
6 f 3
####### 2
5 − (
####### 3
10 + 1
####### 1
4 )
A 26 Calculate.
a 5 38 − ( 114 + 2 12 ) b 12 34 − ( 6 78 − 2 163 ) c 28 − (3 + 4 23 )
A 27 The Husseini family use 13 of their monthly salary to pay the rent,
####### 1
15 for their telephone bills, and
####### 11
60 for groceries.
Calculate what fraction of their monthly salary is left.
A 28 Anna Ant and Lil Ladybug walk along a
long branch. Anna moves from left to right
and has walked 23 of the length of the
branch. Lil moves from right to left and
has walked 34 of the length of the branch.
What part of the length of the branch are the
two bugs apart?
####### c Check I can add and subtract fractions.
####### Not quite mastered this learning objective yet? Then study theory B and do exercise L5.
L 5 Calculate.
a
####### 5
6 −
####### 3
4 b 2
####### 3
4 +
####### 11
12 c 2
####### 1
6 −
####### 1
2 +
####### 2
####### 3
O 29 In figure 2, 34 of the rectangle is red.
In figure 2, 12 of the red part in figure 2 is blue.
a What fraction of the rectangle in figure 2 is blue?
b What is 12 · 34?
####### 24
####### A 25
####### A 26
####### A 27
####### A 28
####### O 29
####### figure 2.
ab© Noordhoff Uitgevers bv 2 Fractions 67
33 Calculate.
a half of 34 b 15 of 60 c a quarter of 109 d 18 of 1000
34 a Calculate 15 of 60.
b Calculate 35 of 60.
c 15 of a number is 60. Calculate the number.
d 35 of a number is 60. Calculate the number.
e Calculate 34 of 84.
f 34 of a number is 84. Calculate the number.
g 23 of a number is 200. Calculate the number.
A 35 There are 120 guests in a hotel. One morning, 13 of them went to a
museum, 14 went for a walk, and 16 went to the pool. The rest of the
guests stayed in the hotel.
a How many guests stayed in the hotel?
b At 12 noon, 45 of the swimmers come back to the hotel. Half of the
walkers also come back, but the guests who went to the museum are
still out. What fraction of the guests are in the hotel at noon?
A 36 There are 540 lower secondary students at the South Sea College.
That is 209 of the total number of students. 59 of the lower secondary
students is in HAVO. Of the senior students, 115 is in HAVO.
Calculate which part of the total number of students is in HAVO.
A 37 Ella has a box of 60 chocolates. She gives 101 to Alex, then 19 of what is
left to Bilal, then 18 of what is left to Carly, then 17 of what
is left to Demi, and so on, until she has given her last friend half of what
is left. How many chocolates does Elly have left for herself?
E 38 In a class, each student is either taking swimming lessons or dance
classes, or both. Three fifths of the students swim and three fifths dance.
Five students participate in both swimming and dancing.
How many students are there in this class?
####### c Check I can multiply fractions.
####### Not quite mastered this learning objective yet? Then study theory C and do exercise L6.
L 6 Calculate.
a 37 · 16 b 113 · 2 34 c 16 · 23 · 35 d 5 · 29
####### 33
####### 34
Notice the difference
####### 1
3 of 60 is
####### 1
3 · 60 =
####### 60
3 = 20.
####### 1
3 of a number is 60,
which means the number is
3 · 60 = 180.
####### A 35
####### A 36
####### A 37
####### E 38
© Noordhoff Uitgevers bv 2 Fractions 69
2 Negative Numbers
Learning objectives
- You can put positive and negative numbers in order.
- You can add and subtract with negative numbers.
O 39 In North America, considerable temperature
differences occur during winter. See the
weather map for a winter day in February.
a What is the lowest temperature on the
weather map?
b Which places on the map had a lower
temperature than Memphis?
c It was colder in Boston than it was in
Columbia.
What was the difference in temperature?
Theory A Positive and negative numbers
On a thermometer, there are numbers above and below 0.
The numbers above 0 are positive numbers.
The numbers below 0 are negative numbers.
Below you see a number line with positive and negative numbers.
- 5 – 4 – 3 –2 – 1 0 1 2 3 4 5 smallergreater
- 4 45 – 3 12 – 231 14 3 25 4
####### figure 2.
Positive numbers are to the right of zero on a number line.
Negative numbers are to the left of zero on a number line.
As you move to the right on the number line, the numbers become greater.
As you move to the left on the number line, the numbers become smaller.
####### O 39
####### figure 2.
201510 5 0− 5− 10
I recognise negative numbers from the
minus sign. This minus sign is a bit shorter
than the minus sign used in subtraction.
70 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
O 44 Weatherman Al checks the temperature several
times per day. In the figure to the right you can
see the temperature at three points in time during
a day in December.
a What is the temperature at 3 a.? And at 6 a.?
By how many °C has the temperature dropped
in this period?
b What is the temperature at 6 a.? And at 9 a.?
By how many °C has the temperature risen in
this period?
c At 12 p. (noon) it is 5 °C warmer than it was
at 9 a.
What is the temperature at noon?
d At some point it is −3 °C. Then the temperature
drops 2 °C.
What is the new temperature?
Theory B Addition and subtraction with arrows
When the temperature is 1 °C and drops by 7 °C, it becomes −6 °C.
The calculation is 1 − 7 = −6.
When the temperature is − 6 °C and it rises by 4 °C, it becomes −2 °C.
The calculation is − 6 + 4 = −2.
You can illustrate these kinds of calculations with arrows on a number
line.
The number line below illustrates the addition −3 + 5 = 2.
- 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
####### figure 2 Add 5: the arrow moves 5 to the right.
The number line below illustrates the subtraction − 1 − 7 with the use of
arrows. You start at −1 and move 7 to the left. You arrive at −8.
So − 1 − 7 = −8.
- 9 – 8 –7 – 6 – 5 – 4 – 3 –2 –1 0 1 7
####### figure 2 Subtract 7: the arrow moves 7 to the left.
When you calculate the sum of − 4 and 4, the result is 0.
The numbers − 4 and 4 are opposite numbers.
Two numbers with 0 as their sum are called opposite numbers.
####### O 44
− 10 − 5 0 510153 a. 6 a. 9 a−− 5051015− 10−051015
####### figure 2.
72 Chapter 2 Numbers and Formulas © Noordhoff Uitgevers bv
Opposite numbers are at the same distance from 0 on a number line.
− 4 − 3 − 2 − 1 0 1 2 3 opposite numbers opposite numbers
####### figure 2.
The opposite of 8 is −8. You can find the opposite of 8 by
putting a minus sign in front of it.
You can replace the words opposite of with the minus sign.
So the opposite of 10 is − −10 = 10.
And so − −5 = 5.
45 Which number is the additive inverse, or opposite number, of:
a 12 c − 38
b − 75 d 0
46 Which of the following statements are true? Explain your answe
a The opposite of a number is always negative.
b Opposite numbers are always different.
c The opposite of the opposite of a number is the same as the original number.
d The opposite of a number can be greater than the number.
47 Calculate. You can use arrows on a number line or do it by heart.
a − 4 + 6 d −3 + 3
b −9 + 5 e − 8 − 26
c − 6 − 4 f 6 − 8
48 Calculate.
a 27 − d 31 − 76 − 29 g −59 + 83
b 12 − 10 e 57 − 62 h − 213 − 0
c −6 + 33 f −13 + 23 i −131 + 67
49 Check the examples on the right.
Calculate in the same way.
a − 5 − (13 − 7) d 35 − (12 + 43)
b − 12 − 8 − 1 e 5 − 8 − (12 − 7)
c − 12 − (8 − 1) f 13 − 48 − (11 + 27)
The opposite of − 5 is 5,
so − − 5 = 5.
####### 45
####### 46
####### 47
####### 48
####### 49
− 17 − 19 + 40 =
− 36 + 40 = 4
− 6 − (5 − 3) =
− 6 − 2 = − 8
© Noordhoff Uitgevers bv 2 Negative Numbers 73
