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