Mathematics 1 Key concepts is a collective work, conceived, designed and created by the Secondary Education department at Santillana, under the supervision of Teresa Grence.

WRITERS José Antonio Almodóvar Ana María Gaztelu Augusto González Pedro Machín Silvia Marín Carlos Pérez Domingo Sánchez

EDITORS José Antonio Almodóvar Isabel Checa Silvia Marín

CLIL CONSULTANT Guillermo Dierssen

EXECUTIVE EDITORS Nuria Corredera Carlos Pérez

PROJECT DIRECTOR Domingo Sánchez

BILINGUAL PROJECT DIRECTOR Margarita España

Do not write in this book. Do all the activities in your notebook.

SEC

ON

DA

RY

1MathematicsKey concepts

ES0000000126360 146700_Pags_Iniciales_98638.indd 1 16/04/2020 9:37:04

ES0000000126360 146700_Pags_Iniciales_98638.indd 2 16/04/2020 9:37:04

CONTENTS

1. Natural numbers .......................................................................... 4

2. Divisibility ..................................................................................... 12

3. Integers ........................................................................................ 18

4. Fractions ...................................................................................... 26

5. Decimal numbers ......................................................................... 32

6. Algebra ......................................................................................... 38

7. Metric system .............................................................................. 46

8. Proportionality and percentages ................................................. 52

9. Straight lines and angles ............................................................. 60

10. Polygons. Triangles ...................................................................... 68

11. Quadrilaterals and circumferences ............................................. 76

12. Perimeters and areas .................................................................. 82

13. Functions and graphs .................................................................. 88

14. Statistics and probability ............................................................. 96

ES0000000126360 146700_Pags_Iniciales_98638.indd 3 20/04/2020 11:52:59

Numbering systems1

Approximation of natural numbers2

The base-ten system uses ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is a positional system, meaning the digits have different values depending on their position within the number.

Example: 13 460 090 = 1 T. of millions + 3 U. of millions + + 4 H. of thousands+ 6 T. of thousands + 9 Tens

The Roman numbering system uses seven different letters:

I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1 000

EXAMPLE

1. Write down the value of each number.

a) MDXXIII = 1 523 b) CDXLI = 441 c) CXCIX = 199

Truncating a number to a certain order consists of replacing the digits of the lower orders with zeros.

In order to round a number to a certain order, we look at the digit of the following order:

• If it is higher than or equal to 5, we add 1 to the digit we are rounding to.

• If it is lower than 5, we leave the digit as it is.

Finally, we truncate the number we have obtained.

EXAMPLE

2. Round the number 5 178 463 to the nearest thousand.

5 178 463 4 < 5" Rounding = 5 178 000

ACTIVITIES

1 Rewrite the following in the base-ten system.

a) XXII b) DCLXIII

2 Truncate and round these numbers to the nearest hundred.

a) 3 729 b) 653 497

1 Natural numbers

The digit 0 is used to indicate a number has no units of the order where the 0 is placed.

Multiplication rule for Roman numerals.

If a Roman numeral has a line over it, its value is multiplied by a thousand.

LI = 51 000

H. of millions

T. of millions

U. of millions

H. of thousands

T. of thousands

U. of thousands

Hundreds Tens Units

4

ES0000000126360 146700_Unidad01_100696.indd 4 16/04/2020 9:37:37

ACTIVITIES

3 Calculate the dividend of a division where the divisor is 14, the quotient is 23 and the remainder is 2.

4 Compute and indicate how the following are read.

a) 24 b) 33

The distributive property also applies to subtraction.

3 ? (12 - 4) = 3 ? 12 - 3 ? 4

Properties of operations with natural numbers

3

3.1. Properties of the sum and multiplication

• Commutative property. The order of the addends or the products do not vary the result.

• Associative property. The order in which the summations are performed does not affect the result. The same thing applies for multiplication.

• Distributive property of multiplication over addition. A number multiplied by a sum of numbers is equal to the sum of the number multiplied by each of the addends.

3.2. Properties of subtraction and division

In a subtraction, the subtrahend plus the difference is equal to the minuend.

In a division, the dividend D is equal to the divisor d multiplied by the quotient c plus the remainder r, and the remainder must be smaller than the divisor.

D = d ? c + r r < d

Powers of natural numbers4

A power is an abbreviated way of writing a multiplication of equal factors.

an = a ? a ? a ? a ? … ? a 1444442444443

n times

a " is called the base and is the repeated factor.n " is called the exponent and indicates the number of times that the base is repeated.

The powers with an exponent equal to 2 are referred to as 'squared', and those with an exponent equal to 3 are referred to as 'cubed'.

EXAMPLE

3. Write down each power.

a) Seven to the power of six. 76 b) Five to the power of seven. 57

5

Natural numbers 1

ES0000000126360 146700_Unidad01_100696.indd 5 20/04/2020 11:53:36

Powers of base 10. Polynomial decomposition of a number

5

A power of base 10 that has a natural number as its exponent is equal to the unit followed by n zeros, where n is its exponent. Example: 105 = 10 ? 10 ? 10 ? 10 ? 10 = 100 000 144444424444443 1442443 5 times 5 zeros

The polynomial decomposition of a number is equal to the sum of the products of its digits multiplied by a power of base 10 corresponding to its order. Example:

5 064 209 = 5 ? 1 000 000 + 6 ? 10 000 + 4 ? 1 000 + 2 ? 100 + 9 = = 5 ? 106 + 6 · 104 + 4 · 103 + 2 · 102 + 9

Operations using powers6

6.1. Product and quotient of powers with the same base

• To multiply two or more powers with the same base we keep the base equal and add their exponents.

am ? an = am + n

• To divide two powers with the same base we keep the same base and subtract their exponents.

am : an = am - n

EXAMPLE

4. Compute.

a) 74 ? 72 = 7 ? 7 · 7 ? 7 ? 7 ? 7 = 74 + 2 = 76 1442443 123 4 times 2 times

b) 95 : 93 = (9 ? 9 · 9 ? 9 ? 9) : (9 ? 9 ? 9) = 95 - 3 = 92

6.2. Powers with an exponent equal to 1 or 0

• A power with an exponent equal to 1 is equal to the base " " a1 = a.

• A power with an exponent equal to 0 is equal to 1 " a0 = 1.

ACTIVITIES

5 Obtain the polynomial decomposition.

a) 7 854 b) 11 111

6 Perform these operations and write the solution as a single power.

a) 27 ? 24 b) 35 : 32

Do not forget to check before performing the operations whether both powers have the same base.

6

ES0000000126360 146700_Unidad01_100696.indd 6 16/04/2020 9:37:40

ACTIVITY

7 Express the following, whenever possible, as a single power.

a) 85 : 45 c) 214 ? 24

b) 74 ? 73 d) (35 ) 2 : (32 )4

6.3. Powers of a power

In order to raise a power to another power we keep the same base and multiply the exponents together.

(am)n = am ? n

EXAMPLE

5. Calculate.

(65)3 = 65 ? 65 ? 65 = 65 + 5 + 5 = 65 ? 3 = 615

6.4. Powers of products and quotients

• Raising the multiplication of two numbers to a power is equal to the product of each number raised to the exponent.

(a ? b)n = an ? bn

• Raising the division of two numbers to a power is equal to the quotient of each number raised to the exponent.

(a : b)n = an : bn

EXAMPLE

6. Express the following as the product of two powers.

(7 ? 2)3 = (7 ? 2) ? (7 ? 2) ? (7 ? 2) = 7 ? 7 ? 7 ? 2 ? 2 ? 2 = 73 ? 23

Express products and quotients of powers as a single power

Express, whenever possible, the following as a single power.

a) 73 ? 76 b) 86 : 83 c) 58 ? 28 d) 94 : 34 e) 49 ? 36 f ) 56 : 43

Follow these steps

1. If the bases are the same, add or subtract exponents.

a) 73 ? 76 = 73 + 6 = 79

b) 86 : 83 = 86 - 3 = 83

2. If the exponents coincide, we multiply or divide the bases.

c) 58 ? 28 = (5 ? 2)8 = 108

d) 94 : 34 = (9 : 3)4 = 34

3. If neither the bases nor the exponents coincide, it cannot be expressed as a single power.

e) 49 ? 36 " Cannot be expressed as a single power.

f ) 56 : 43 " Cannot be expressed as a single power.

KNOW HOW TO

In a product or a division of powers, if the base or the exponent coincide, you must perform the operation with the term that is different.

7

Natural numbers 1

ES0000000126360 146700_Unidad01_100696.indd 7 16/04/2020 9:37:41

Square root7

7.1. Exact square root

The exact square root of a number a is a number b, so when b is squared we obtain a.

a = b, when b2 = a

The radicand is the number a,

is the symbol of the root and we say b is the square root of a.

The numbers with an exact square root are called perfect squares.

a b=Symbol of the root

Radicand

RootF F

F

EXAMPLE

7. Calculate the roots of these perfect squares.

a) 4 = 2, as 22 = 4. b) 36 = 6, as 62 = 36.

7.2. Integer square root

If the radicand is not a perfect square, the square root is an integer square root.

The integer square root of a number a is the largest number b whose square is less than a. The remainder of the integer square root is the difference between the radicand a and the integer square root b squared.

Remainder = a - b2

ACTIVITIES

8 Compute these exact square roots.

a) 121 b) 144

9 Find the value of a in these non-exact square roots.

a) a . 5 and the remainder is 7.

b) a . 7 and the remainder is 3.

Calculate the square root of a number

Calculate the square root of 39.

Follow these steps

1. Find the largest number that when squared is less than, or equal to, the radicand.

52 = 25 " 25 < 39

62 = 36 " 36 < 39

72 = 49 " 49 > 39

2. If the square of the number is less than the radicand, that number is the integer square root. The difference between the radicand and the square of that number is the remainder.

b) 62 = 36 " 36 < 39

6 is the largest number whose square is less than 39.

The integer square root is 6 and the remainder is: 39 - 62 = 39 - 36 = 3

KNOW HOW TO

Taking the square root of a number and squaring it are inverse operations to each other.

If 49 = 7, then 72 = 49.

If 72 = 49, then 49 = 7.

8

ES0000000126360 146700_Unidad01_100696.indd 8 16/04/2020 9:37:47

Combined operations8

When multiple operations appear in the same expression, the order in which these operations must be performed are the following:

1. Operations appearing between brackets and square brackets.

2. Multiplications and divisions from left to right.

3. Addition and subtraction from left to right.

EXAMPLE

8. Perform.

a) 8 + 16 : 2 - 9 = 8 + 8 - 9 = 16 - 9 = 7

b) 3 ? (8 + 2) : 2 - 9 = 3 ? 10 : 2 - 9 = 30 : 2 - 9 = 15 - 9 = 6

c) 36 : 4 : 3 - (8 + 2) : 5 = 36 : 4 : 3 - 10 : 5 = 9 : 3 - 2 = 3 - 2 = 1

d) 10 - (6 + 3) + (11 - 5) : 3 ? 15 : 5 = 10 - 9 + 6 : 3 ? 15 : 5 = = 10 - 9 + 2 ? 15 : 5 = = 10 - 9 + 30 : 5 = = 10 - 9 + 6 = 1 + 6 = 7

ACTIVITIES

10 Compute.

a) 17 + (4 ? 2 - 7) ? 3

b) (11 - 7) ? 4 + 2 ? (8 + 2)

11 Compute.

a) :?( )15 3 2 9 32 3- +

b) ?25 36 3 4 82+ - +_ i

Perform combined operations with powers and roots

Calculate the result of the following expression.

10 - (4 + 2)2 : 16 + 5 ? (7 - 4) + 23

Follow these steps

1. Perform the operations that are inside the brackets.

2. Compute the powers and the roots.

10 - (4 + 2)2 : 16 + 5 ? (7 - 4) + 23 =

= 10 - 62 : 16 + 5 ? 3 + 23 =

= 10 - 36 : 4 + 5 ? 3 + 8 =

= 10 - 9 + 15 + 8 =

= 1 + 15 + 8 =

= 16 + 8 =

= 24

3. Perform the multiplications and the divisions.

4. Compute the additions and subtractions.

KNOW HOW TO

In order to be able to operate with powers and roots we must first compute their value.

F

F

F

F

F

F

F

FF F

F

F F F

F

9

Natural numbers 1

ES0000000126360 146700_Unidad01_100696.indd 9 16/04/2020 9:37:51

FINAL ACTIVITIES

12 Indicate the positional value of the digit 3.

a) 5 396 b) 12 463 c) 303 030 d) 3 532 001

13 Write down the following in the base-ten system.

a) XVIII b) LXXI c) XCVII d) MDCXXVIII

14 Approximate these numbers by truncating and rounding them to the nearest hundred.

a) 24 536 b) 200 664 c) 456 283

15 Apply the distributive property and compute.

a) 2 ? (5 - 3) b) (12 - 7 + 3) ? 8

16 Write down how the following powers are read.

a) 32 b) 75 c) 43 d) 1417

17 Write down the numbers whose polynomial decomposition is given by:

a) 6 · 104 + 7 · 103 + 9 ? 10 + 7

b) 3 · 105 + 4 · 102 + 1

c) 8 · 103 + 102

d) 2 · 106

18 Write down the following as a single power.

a) 24 ? 26 : 27 c) 102 ? 106 : 103

b) 35 : 33 ? 32 d) 76 : 73 ? 74

19 Write down the following as a single power.

a) 52 ? 32 c) 86 : 26 e) 210 ? 1010

b) 47 ? 27 d) 207 : 107 f ) 124 : 44

20 Compute.

a) (24)3 b) (52)5 c) (34)6 d) (75)3

21 Compute the integer square root and remainder for the numbers written down by Anne.

22 Perform the following operations.

a) 28 - 3 ? 2 ? 4 d) 14 : 2 + 3 ? 9 - 5

b) 5 ? 9 : 3 + 7 e) (42 - 6) : 6 + 5 ? 3

c) 25 + 4 ? 2 - 7 ? 3 f ) 15 ? (7 - 3) : (3 - 1)

23 Compute.

a) ?3 9 3 33 2 3- -

b) :?12 3 25 3 492+ +_ i

c) :7 64 5 52 3+ -

a) 79

b) 32

c) 140

d) 853

24 A boat was carrying 502 people. It made three stops. On the first stop 256 people left the boat. On the second stop 162 people got on the boat and on the third stop 84 people got off the boat. How many people are left on the boat after the three stops?

25 In one piggy bank there are 246 € and in a different one there are 114 €.

a) If all the money is in 2 € coins, what is the total number of coins in the piggy banks?

b) If all the money was in 5 € notes, how many would there be?

26 How much money is there in a wallet containing two 20 € notes, three 10 € notes, six 5 € notes and four 2 € coins?

27 Six people have 1 000 € to spend on a trip. They must travel by both plane and train. The train ticket costs 38 € and the plane ticket costs 125 €. Do they have enough money to go on the trip?

28 In an ethnic music festival there are artists from three different continents. 350 come from Asia. There are 157 more artists from Africa than there are from Asia and 98 fewer artists from Europe than there are from Asia. What is the total number of artists?

29 An orange tree has produced 40 kg of oranges this year but only 27 kg last year. If last year the price per kilogram was 3 € and this year it is 2 €, have profits increased or decreased compared to last year?

30 Raquel had 12 €. She spent half at the cinema and the other half on a winning lottery ticket that won her 15 € for each euro she paid for it. How much money did she win?

31 Two flowers cost 3 € and to make a bouquet you need 12 flowers.

a) How many bouquets can you make for 90 €?

b) If you want to make a profit of 40 €, how much should you charge for each bouquet?

10

ES0000000126360 146700_Unidad01_100696.indd 10 16/04/2020 9:37:53

Base-ten system

H. of millions

T. of millions

U. of millions

H. of thousands

T. of thousands

U. of thousands

Hundreds Tens Units

Roman numbering system

I = 1 V = 5 X = 10 L = 50

C = 100 D = 500 M = 1 000

Approximation of natural numbers

458 173 695

Truncating to the nearest thousand: 458 173 000

Rounding to the nearest thousand: 458 174 000

Powers

a = a ? a · a ? … ? a 1442443 n times

Base F an F Exponent

52 is read as '5 to the power of 2' or '5 squared'.

Square root

a = b, where b2 = a

a b=Root symbol

Radicand

RootF F

F

9 = 3 is read as 'the square root of 9 is 3'.

Numbering systems

1 Decompose the following into their unit orders.

a) 124 325 c) 2 743 005

b) 65 906 185 d) 601 020 304

2 Write down a number consisting of 6 tens of thousands, 2 units of thousands, 3 hundreds, and 3 times more units than hundreds.

3 Write down the following in the base-ten system.

a) XXIV b) CDXIV c) MCMI

4 Write down the following in the Roman numbering system.

a) 54 c) 643 e) 7 499

b) 124 d) 7 981 f ) 15 613

5 Truncate the following to the nearest hundred.

a) 1 462 b) 67 529 c) 19 999

6 Round these numbers to the nearest ten thousand.

a) 10 805 b) 1 372 154 c) 509 843

Operations with natural numbers

7 Copy and complete, indicating the property used in each case.

a) 12 + 39 = 4 + 12

b) 7 ? (4 - 3) = 7 ? 4 - 7 ? 4c) 5 ? 2 ? 4 = 5 ? (2 ? 3)

8 How many units must be added to the dividend of the division 186 : 24 in order for the remainder to be 0?

Powers and roots

9 Express the following in the form of a power whenever possible.

a) 2 ? 2 ? 2 ? 2 ? 2 b) 4 ? 5 ? 4 ? 5 ? 4 ? 5

10 Copy and complete in your notebook.

a) 47 = 4 : 57 c) (34)4 = 312

b) 35 = 97 : 4 d) 43 ? 53 = 353

11 Compute.

a) 63 ? 65 b) 76 : 73 c) 52 ? 53 ? 5

12 Calculate the integer square root and remainder for the following numbers.

a) 35 c) 64 e) 315

b) 50 d) 136 f ) 462

Combined operations

13 Compute the following operations.

a) ? ?( )6 2 49 5 1+ +

b) : ( )4 16 9 4 52 0- + +

14 Every weekend Louis receives 6 € and spends 4 €. How many weeks does he need in order to save 18 €?

15 There are 1 752 pines planted in a greenhouse ready for reforestation.

a) If they are sold in groups of 12 for 4 € per group, how much money will they earn?

b) How many more pines would they need to make a total of 600 €?

SELF EVALUATION

SUMMARY

11

Natural numbers 1

ES0000000126360 146700_Unidad01_100696.indd 11 16/04/2020 9:37:56

Divisibility1

Multiples of a number2

A number D is divisible by another number d when the division D : d is exact; that is, the remainder is 0.

• D is divisible by d.

• D contains an exact number c of times d.

If D is divisible by d, we say there is a divisibility relation between D and d.

D d0 c

EXAMPLE

1. Is there a divisibility relation between 104 and 8?

104 824 130

The division is exact, as the remainder is 0.

Dividend = divisor ? quotient

104 = 8 · 13

We say:

• 104 is divisible by 8.

• 104 contains 8, 13 times.

• As 104 is divisible by 8, there is a divisibility relation between 104 and 8.

A number b is a multiple of another number a if the division b : a is exact.

Multiples of a number are obtained by multiplying it by consecutive natural numbers.

EXAMPLES

2. Is 35 a multiple of 5?

35 50 7

The division 35 : 5 is exact.

35 is a multiple of 5 " 35 is divisible by 5.

3. Calculate the multiples of 5.

5o = {5 · 1, 5 · 2, 5 · 3, 5 · 4, 5 · 5…} = {5, 10, 15, 20, 25…}.

ACTIVITIES

1 Is 144 divisible by 2? And by 10?

2 Calculate the first six multiples of 12.

2 Divisibility

Dividend divisorremainder quotient

ao "   Represents the set of all multiples of a.

3o "   All multiples of 3.

12o "   All multiples of 12.

12

ES0000000126360 146700_Unidad02_98640.indd 12 16/04/2020 9:37:10

Divisors of a number3

A number a is a divisor of a number b if the division b : a is exact.

The divisors of a number are obtained by dividing it by consecutive natural numbers until the quotient of the division is less than the divisor.

EXAMPLE

4. Is 3 a divisor of 72?

72 30 24

The division 72 : 3 is exact.

3 is a divisor of 72 " 72 is divisible by 3.

Prime and composite numbers4

ACTIVITIES

3 For which of these numbers is 8 a divisor?

a) 144 b) 18 c) 120

4 Calculate all the divisors of these numbers.

a) 10 b) 26 c) 33

5 Is 101 a prime or a composite number? What about 121?

EXAMPLE

5. Is 45 a prime or a composite number?

Calculate its divisors and check how many there are.

Div (45) = {1, 3, 5, 9, 15, 45} " 45 has more than 2 divisors.45 is a composite number.

Divisible by Divisibility Criteria

2 The last digit is either 0 or even.

3 The sum of its digits is divisible by 3.

5 The last digit is either 0 or 5.

10 The last digit is 0.

11The sum of the digits in an even position minus the sum of the digits in an odd position is either 0 or divisible by 11.

24 is divisible by 3.

3 is a divisor of 24.

24 is a multiple of 3.

GF

G

F

GF

2 3 5 7 11

13 17 19 23 29

31 37 41 43 47

53 59 61 67 71

73 79 83 89 97

• A number is prime if it has only two divisors: itself and the unit.

• A number is composite if it has more than two divisors.

• The number 1 is neither prime nor composite.

The divisibility criteria are rules that allow us to recognise, without dividing, whether a number is divisible by another.

Prime numbers up to 100

13

Divisibility 2

ES0000000126360 146700_Unidad02_98640.indd 13 16/04/2020 9:37:12

Decomposition of a number into factors5

Decomposing a number into factors is expressing it as a product of various factors.

• A prime number only has one possible decomposition into factors, the product of itself with the unit.

• A composite number has multiple different decompositions into factors.

EXAMPLE

6. Decompose 13 and 24 into factors.

13 is a prime number " Div (13) = {1, 13}.

13 ? 1 is the only decomposition into factors 13 has.

24 is a composite number " Div (24) = {1, 2, 3, 4, 6, 8, 12, 24}.

24 = 24 ? 1 = 2 ? 12 = 3 ? 8 = 4 ? 6 …

24 ? 1, 2 ? 12, 3 ? 8 and 4 ? 6 are all decompositions into factors of 24.

Factorising a number is decomposing it into its prime factors. That is, expressing it as a product of its prime divisors.

Factorise a number

Decompose 630 as a product of its prime factors.

Follow these steps

1. Divide the number by successive prime numbers (2, 3, 5, 7, 11…) as many times as possible until obtaining the unit.

• 630 is divisible by 2.

• 315 is not divisible by 2.315 is divisible by 3.

• 105 is not divisible by 2.105 is divisible by 3.

• 35 is not divisible by 2 or 3.35 is divisible by 5.

• 7 is a prime number.

2. Express the number as the product of its prime factors obtained above. Repeated prime factors will be expressed as a power.

The prime factorisation of 630 is:

630 = 2 ? 3 ? 3 ? 5 ? 7 = 2 ? 32 ? 5 ? 7

KNOW HOW TO

The decomposition ends when we arrive at a prime number. Dividing it by itself will give us the unit.

630 2 630 : 2 " 315 3

315 : 3 " 105 3

105 : 3 " 35 5

35 : 5 " 7 7

7 : 7 " 1

ACTIVITY

6 Decompose the following numbers into their prime factors.

a) 270 d) 405

b) 2 470 e) 675

c) 400 f ) 943

14

ES0000000126360 146700_Unidad02_98640.indd 14 16/04/2020 9:37:14

Greatest common divisor6

Lowest common multiple7

ACTIVITIES

7 Calculate the greatest common divisor.

a) 8 and 10 b) 15 and 20

8 Calculate the lowest common multiple.

a) 8 and 10 b) 15 and 25

The lowest common multiple of two or more numbers is the smallest of their multiples in common.

The lowest common multiple of two or more numbers a, b, c… is expressed as LCM (a, b, c…).

EXAMPLE

7. Calculate the greatest common divisor of 24, 82 and 132.

Firstly, we decompose the numbers into their prime factors.

24 2 84 2 132 212 2 42 2 66 2 6 2 21 3 33 3 3 3 7 7 11 11 1 1 1

24 = 23 ? 3 84 = 22 ? 3 ? 7 132 = 22 ? 3 ? 11

The prime factors that are common to the three of them are 2 and 3.

Taking their powers to the lowest exponent: 22 and 3.

The greatest common divisor is:

GCD (24, 84,132) = 22 ? 3 = 12

If a and b have no common divisors, then:

GCD (a, b) = 1

And a and b are prime with respect to each other.

If LCM (a, b) = a ? b, a and b have no common divisors.

The greatest common divisor of two or more numbers is the biggest of their common divisors.

The greatest common divisor of two or more numbers a, b, c… is expressed as GCD (a, b, c…).

EXAMPLE

8. Calculate the lowest common multiple of 135, 315 and 175.

Firstly, we decompose the numbers into their prime factors.

135 = 33 ? 5 315 = 32 ? 5 ? 7 175 = 52 ? 7

The prime factors appearing between the three of them are 3, 5 and 7.

Taking their powers to the greatest exponent: 33, 52 and 7.

The lowest common multiple is: LCM (135, 315, 175) = 33 ? 52 ? 7 = 4 725.

15

Divisibility 2

ES0000000126360 146700_Unidad02_98640.indd 15 16/04/2020 9:37:15

9 Is there a divisibility relation between the following pairs of numbers?

a) 135 and 45 b) 238 and 16

10 Which of these series is formed by multiples of 4? And by multiples of 5?

a) 1, 4, 9, 16, 25… d) 4, 8, 16, 24, 32, 40…

b) 5, 10, 15, 20… e) 1, 5, 10, 20, 30…

c) 8, 10, 12, 14, 16… f ) 20, 40, 60, 80…

11 Find three numbers that are multiples of both 6 and 5. Are they multiples of 10 too?

12 Given the relation 104 = 4 ? 26, which of the following statements are true?

a) 104 is divisible by 4. c) 26 is a divisor of 104.

b) 104 is a multiple of 4. d) 104 is divisible by 26.

13 The number a is divisible by 4. Calculate a if the quotient of the division is 29.

14 Are all numbers divisible by 2 also divisible by 4? And vice versa?

15 Copy and complete the table in your notebook.

Number Divisors Prime/Composite

17

29

58

72

97

113

16 If the division a : 4 is exact, is a a prime or a composite number?

17 Decompose the following numbers into their prime factors.

a) 560 b) 2 700 c) 616 d) 784 e) 378 f ) 405

18 Ben and Julie have factorised the number 2 250, obtaining the following results.

Ben: 2 ? 32 ? 53 Julie: 32 ? 52 ? 10

Are their results correct?

19 Group up factors and write down the following factorial decompositions correctly.

a) 22 ? 3 ? 2 ? 33 c) 32 ? 5 ? 5 ? 32

b) 52 ? 7 ? 54 ? 73 ? 7 d) 22 ? 7 ? 2 ? 72 ? 2

20 Calculate the greatest common divisor and the lowest common multiple of the following groups of numbers.

a) 10, 20 and 100 b) 5, 9 and 45 c) 4, 30 and 50

21 There are 18 000 plates in a warehouse. The company decides to put them in boxes. Each box contains a dozen plates.

a) How many boxes will they need?

b) If the number of plates in the warehouse was tripled, how many boxes they need?

c) If they could only fit half a dozen plates in each box, how many boxes would they need?

22 A stationery shop sells pencils in boxes of 8, 10 or 15. How many boxes of each size can it sell if there are 270 pencils and all of the boxes are of the same size? Will it sell all the pencils in each case?

23 Tania had a bag containing 35 sweets. She ate some of them, but does not remember how many. However, she knows she can group the ones she has left into bags of 2, 3 and 5 sweets without any remaining.

a) How many sweets does she have left?

b) If she groups them into bags of 2, how many bags will she need?

c) What about if she groups them into bags of 3? And into bags of 5?

24 Alice wants to put 45 books in a bookcase. She would like each shelf to contain the same number of books.

a) How many books can she put on each shelf?

b) How many shelves will she need in each case?

25 Helen and Martha have a collection of stickers. Helen counts them 7 by 7 and Martha 4 by 4. What is the minimum number of stickers their collection can have?

26 Rachel and David both go horse riding. Rachel goes every 3 days and David every 4 days. If they meet on the 24th February, when will they next meet? How many times will each of them have gone horse riding before?

FINAL ACTIVITIES

16

ES0000000126360 146700_Unidad02_98640.indd 16 16/04/2020 9:37:17

Divisibility

8 : 2 is an exact division.

F F

8 is divisible by 2.

8 is a multiple of 2 G F 2 is a divisor of 8.

Divisibility

Divisible by Criteria

2 The last digit is 0 or even.

3 The sum of its digits is a multiple of 3.

5 The last digit is a 5 or a 0.

10 The last digit is a 0.

11

The sum of the digits in an even position

minus the sum of the digits in an odd

position is either 0 or divisible by 11.

Prime number

Div (7) = {1, 7}

Div (11) = {1, 11}

Composite number

Div (10) = {1, 2, 5, 10}

Div (12) = {1, 2, 3, 4, 6, 12}

Greatest common divisor

12 = 22 ? 3

18 = 2 ? 32

GCD (12, 18) = 2 ? 3 = 6

Lowest common multiple

20 = 22 ? 5

24 = 23 ? 3

LCM (20, 24) = 23 ? 3 ? 5 = 120

Divisibility

1 Is there a divisibility relation between the following pairs of numbers?

a) 4 and 18 c) 3 and 4

b) 5 and 30 d) 7 and 91

2 Which of the following numbers are divisible by both 8 and 12?

a) 288 c) 576 e) 480

b) 364 d) 1 248 f ) 672

3 Calculate the divisors of these numbers.

a) 75 c) 81 e) 121

b) 77 d) 96 f ) 113

4 Lucas has made 45 cakes and wants to store them in boxes. He wants each box to contain the same number of cakes. In how many different ways can he store them so there are none remaining?

Prime and composite numbers

5 Which of the following numbers are prime and which are composite? Write down three divisors of the composite numbers.

a) 133 c) 179 e) 210

b) 153 d) 184 f ) 301

Decomposition into factors

6 Decompose the following numbers into their factors.

a) 240 b) 345 c) 99 d) 5 700

7 To which numbers do these prime decompositions correspond?

a) 2 ? 32 ? 5 b) 24 ? 5 c) 3 ? 52 d) 2 ? 32 ? 7

Greatest common divisor and lowest common multiple

8 Calculate the greatest common divisor of these numbers.

a) 45 and 75 c) 24, 66 and 84

b) 16 and 24 d) 72, 108 and 144

9 Calculate the lowest common multiple of these numbers.

a) 18 and 24 c) 12, 18 and 60

b) 15 and 25 d) 15, 25 and 45

10 Carol has 40 European stamps and 56 Asian stamps. She wants to make the smallest possible number of equal sets of stamps. She does not want to mix any European stamps with Asian stamps or have any left over. How many sets will she make? How many stamps will each set have?

SELF EVALUATION

SUMMARY

G

F

G

F

17

Divisibility 2

ES0000000126360 146700_Unidad02_98640.indd 17 16/04/2020 9:37:18

Integers1

Within the set of integers Z we can distinguish:

• Positive integers: +1, +2, +3, +4…, which are the natural numbers.

• The number 0.

• Negative integers: -1, -2, -3, -4…

Integers can be represented, in order, using the number line.

Negative integers Positive integers

0… …-7 -6 -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 +6 +764444444444744444444448 64444444444744444444448

1.1. Absolute value of an integer

The absolute value of an integer is the distance, in units, that separates it from the number zero in the number line. It is written as ; ;, and is equal to the number without its sign.

;+b; = b ;-a; = a

The absolute value of 0 is 0, ;0; = 0.

1.2. Opposite of an integer

We say two different integers are opposites of each other if they are the same distance from 0.

Op (+a) = -a Op (-a) = +a

3 Integers

The number 0 is the only integer that is neither a positive nor a negative integer.

ACTIVITIES

1 Calculate the absolute value of the following numbers.

a) ;+3;

b) ;-2;

c) ;-5;

2 What are the opposite integers of these numbers?

a) Op (+2)

b) Op (-6)

c) Op (-8)

3 Write down the following integers from largest to smallest.

-6, +7, -10, -4, +8

Ordering integers2

• Any positive integer is larger than any negative integer.

• When there are two positive integers, the largest is the one with a larger absolute value.

• When there are two negative integers, the largest is the one with a smaller absolute value.

18

ES0000000126360 146700_Unidad03_100697.indd 18 16/04/2020 9:37:25

Adding or subtracting 0 to or from any number will give you the same number.

(+7) + 0 = +7

(+6) - 0 = +6

(-5) + 0 = -5

(-4) - 0 = -4

Addition and subtraction of two integers3

When adding two integers together, we can encounter two different cases:

• Adding two integers with the same sign. In that case:1. We add their absolute values.2. We give our result the same sign as the original integers.

• Adding two integers with different signs. In that case:1. We subtract their absolute values (subtract the smallest

from the largest).2. We give our result the same sign as the original integer

with the largest absolute value.

EXAMPLE

1. Compute these sums.

a) (-2) + (-4) = -6

2 2

2 4 6; ;- =

+ ="4 4; ;- =

4

To subtract two integers from each other we add the opposite of the second one to the first one.

EXAMPLE

2. Compute these subtractions.

a) (+4) - (+7) = (+4) + Op (+7) = (+4) + (-7) = -3

b) (+2) - (-6) = (+2) + Op (-6) = (+2) + (+6) = +8

c) (-5) - (-9) = (-5) + Op (-9) = (-5) + (+9) = +4

FF F

F

Same sign We keep

the same sign

Sum of their

absolute values

b) (-8) +  (+2) = -6

8 8

8 2 6; ;- =

- ="2 2; ;+ =

4

FFDifferent

signWe keep the sign of the

number with the largest

absolute value

4 Compute the following sums.

a) (+5) + (-2) b) (-5) + (-2)

5 Compute the following subtractions.

a) (+3) - (+6) b) (+3) - (-6) c) (-3) - (+6)

6 Complete in your notebook.

a) 4 + (+4) = +9 c) 4 - (-9) = +2

b) 4 - (-9) = +16 d) 4 + (+4) = -1

7 Can the sum of two negative integers be positive?

ACTIVITIES

F

19

Integers 3

ES0000000126360 146700_Unidad03_100697.indd 19 16/04/2020 9:37:28

Addition and subtraction of multiple integers

4

To simplify the addition and subtraction of multiple integers we follow these rules:

Rule number 1: We remove the brackets from the first addend and if its sign is positive, we rewrite it without a sign.

(+3) + (-7) = 3 + (-7) (-9) + (-1) = -9 + (-1)

Rule number 2: When removing brackets preceded by a plus sign (+), we maintain the sign of the number.

(-5) + (+1) = -5 + 1 (-4) + (-8) = -4 - 8

Rule number 3: When removing brackets preceded by a minus sign (-), we write the sign of the number opposite to it.

(-7) - (+6) = -7 - 6 (+3) - (-4) = 3 + 4

After applying these rules, we say the expression is rewritten in its abbreviated form.

EXAMPLE

3. Write down this expression in its abbreviated form.

a) (+6) + (-4) - (+8) = Rule 1 " F

= 6 + (-4) - (+8) = Rule 2 " F

= 6 - 4 - (+8) = Rule 3 " F

= 6 - 4 - 8

b) (+8) - (+4) + (+5) + (-7) - (-3) = Rule 1 " F

= 8 - (+4) + (+5) + (-7) - (-3) = Rule 2 " F F

= 8 - (+4) + 5 - 7 - (-3) = Rule 3 " F F

= 8 - 4 + 5 - 7 + 3

In practice:

+(+a) = +a

+(-a) = -a

-(+a) = -a

-(-a) = +a

ACTIVITIES

8 Write down the following expressions in their abbreviated form.

a) (+3) - (+4) + (+8)

b) (-14) + (-13) - (-10)

c) (+20) - (+18) - (-9) + (-3)

d) (-16) + (-4) + (+12) - (+1)

e) (-21) - (-12) + (+9) + (-4)

f ) (+15) + (-6) - (+8) - (+14)

9 Transform these operations written down in abbreviated form into their original operations using brackets.

a) -3 + 7 - 2 c) 4 + 7 - 9 + 4

b) 7 + 8 - 7 d) -2 - 6 + 3 - 5

10 Describe a real-life situation in which addition and subtraction of integers is used.

20

ES0000000126360 146700_Unidad03_100697.indd 20 16/04/2020 9:37:29

Add and subtract multiple integers

Compute the following operation: (-7) - (+5) + (+3) + (+9) - (-4) + (-2).

Follow these steps

1. Write down the operation in its abbreviated form.

(-7) - (+5) + (+3) + (+9) - (-4) + (-2) =

= -7 - (+5) + (+3) + (+9) - (-4) + (-2) =

= -7 - (+5) + 3 + 9 - (-4) - 2 =

= -7 - 5 + 3 + 9 + 4 - 2

2. Add the numbers with the sign +.

Integers with the sign +

" 3 + 9 + 4 = 16

3. Add the numbers with the sign -.

Integers with the sign -

" 7 + 5 + 2 = 14

4. Subtract the result in step 3 from the result in step 2.

16 - 14 = 2

(-7) - (+5) + (+3) + (+9) - (-4) + (-2) = -7 - 5 + 3 + 9 + 4 - 2 = = 16 - 14 = 2

Remember that a number with no sign is equal to a positive integer, 16 = +16.

KNOW HOW TO

11 Calculate.

a) (-5) + (+18) - (-3) - (-5)

b) (+7) - (-19) - (+17) + (+6)

c) (+16) + (-8) - (+8) - (-4)

d) (-14) - (-2) - (+13) - (-7)

12 Compute the following operations.

a) 9 + (4 - 1 - 8) - 5

b) -13 + (-5 - 7 - 9) + (-12)

c) 18 - (-8 - 3) + 22

d) -26 + (-8 - 4 + 2) - (-5)

ACTIVITIES

KNOW HOW TO

Compute additions and subtractions including brackets

Compute the following expression: 5 + (-7 + 2 - 1) - (8 - 4 + 3 - 2) - 4.

Follow these steps

1. First compute the result of the expression inside the brackets.

5 + (-7 + 2 - 1) - (8 - 4 + 3 - 2) - 4 = F F

= 5 + (-5 - 1) - (4 + 3 - 2) - 4 = F F

= 5 + (-6) - (7 - 2) - 4 = F

= 5 + (-6) - (5) - 4 =

2. Use the rules above in the remaining expression.

= 5 - 6 - 5 - 4 = 5 - 15 = -10

When solving operations inside brackets, remember that the result is also left inside the brackets.

6 - (-3 + 2) = = 6 - (-1) = 6 + 1

21

Integers 3

ES0000000126360 146700_Unidad03_100697.indd 21 16/04/2020 9:37:30

Multiplication and division of integers5

5.1. Multiplication of integers

To multiply two integers:

1. Multiply their absolute values.

2. Give the result a positive sign when multiplying two numbers that have the same sign, and a negative sign when the two numbers have different signs.

EXAMPLE

4. Compute these multiplications.

a) (-3) ? (+5) = -15 b) (-7) ? (-2) = +14

F

Different sign F

Same sign

5.2. Division of integers

To divide two integers:

1. Divide their absolute values.

2. Give the result a positive sign if they have the same sign, and a negative sign if their signs are different.

EXAMPLE

5. Compute these divisions.

a) (-12) : (-4) = +3 b) (-9) : (+9) = -1

F

Same sign F

Different sign

Multiply and divide multiple integers

Compute the value of the following expression: (-15) : (+5) ? (+2).

Follow these steps

1. Calculate the sign of the result of the operation. (-15) : (+5) ? (+2) = -

2. Multiply or divide, from left to right, the absolute values of the numbers. Then add the sign calculated above.

;-15; : ;+5; ? ;+2; = 3 ? ;+2; = 6(-15) : (+5) ? (+2) = -6

Perform the operations in the order they are given, from left to right.

Sign rules

+ ? + = +

- ? - = +

+ ? - = -

- ? + = -

+ : + = +

- : - = +

+ : - = -

- : + = -

ACTIVITIES

13 Compute.

a) (-6) ? (-8)

b) (-64) : (-8)

14 Calculate.

a) (+36) : (-9) ? (-3)

b) (-12) ? (+5) : (-3)

KNOW HOW TO

22

ES0000000126360 146700_Unidad03_100697.indd 22 16/04/2020 9:37:33

ACTIVITIES

15 Perform the following operations.

a) (-10) : (+5) + (+2) ? (+3)

b) (+4) : (-2) - (+6) : (+3)

16 Calculate.

a) [(+4) - (+16)] : (-4)

b) (+2) ? [(-5) - (+4)]

Combined operations6

When dealing with addition, subtraction, multiplication and division of integers in one expression, we must perform these operations in the following order:

1. Start by computing multiplications and divisions from left to right.

2. Compute the remaining additions and subtractions from left to right.

EXAMPLES

6. Perform the following operation.

(-10) : (+2) - (-4) ? (+1) =F F Multiplications and divisions

= (-5) - (-4) =F Additions and subtractions

= -1

7. Compute.

a) -9 - 5 ? (-3) - 6 = -9 - (-15) - 6 = 6 - 6 = 0

b) (+10) : (-2) ? (+3) + 9 = (-5) ? (+3) + 9 = -15 + 9 = -6

KNOW HOW TO

Perform combined operations including brackets

Perform the following operation.

(-4) - [(-8) - (+2)] : (-5) + (-6) : [(+1) - (-2)]

Follow these steps

1. Perform the operations inside the brackets.

2. Compute the multiplications and divisions from left to right.

3. Compute the remaining additions and subtractions from left to right.

Once multiplications and divisions have been computed, we can write the result in its abbreviated form.

(-4) - (+2) + (-2) = = -4 - 2 - 2 = -8

(-4) - [(-8) - (+2)] : (-5) + (-6) : [(+1) - (-2)] =

F F

= (-4) - (-10) : (-5) + (-6) : (+3) =

= (-4) - (+2) + (-2) =

= (-6) + (-2) =

= -8

F

F

F F

23

Integers 3

ES0000000126360 146700_Unidad03_100697.indd 23 16/04/2020 9:37:35

17 Write down the integer corresponding to each of the points marked in the number line.

a) A B C D

0 1

b) A B C D

0 1

18 Order the following numbers from smallest to largest.

4 -6 -7 2 -9 -11 12 -8 16 -19

Order their opposites and absolute values too.

19 Eliminate the brackets and compute the value of the following.

a) -(-8) + 17 e) -(-9) + 15

b) -(-5) - 23 f ) -(-12) - (-20)

c) -(-30) + (-12) g) -(-24) + (-19)

d) -(-6) - (-18) h) -(-16) - (-14)

20 Compute the following additions and subtractions of integers.

a) (-3 + 8) - (-2) e) (5 - 9) + (-3)

b) (-2 + 4) - (+4) f ) (7 - 10) + (-8)

c) (1 - 3) + (-4) g) (-2 - 8) - (+6)

d) (6 - 3) + (-2) h) (-1 + 4) - (-7)

21 Perform the following operations.

a) –1 – [–3 – 2 + (–4)]

b) (2 – 8) + [1 – (–9) – 3]

c) (–5 + 3) – [7 – (–8)]

d) –6 – [5 – 10 + (–3)]

e) –4 + [–6 + (–2) – (–5)]

f ) 12 + (–9) – [(–7) – (+5)]

g) –[5 + (–18) + (–6 – 12) – 3]

h) 4 + [–6 + (–9) – 12]

22 Calculate.

a) (+12) : (+3) d) (+6) ? (-8)

b) (+15) : (-3) e) (-12) ? (-3)

c) (-28) : (-7) f) (-7) ? (+10)

23 Calculate.

a) (7 - 10) ? (1 - 6) e) (-3 + 9) ? (4 - 2)

b) (5 - 12) ? (-3 + 5) f ) (-1 - 3) ? (9 - 7)

c) (-15 + 3) : (-7 + 4) g) (9 - 18) : (6 - 3)

d) (-12 - 6) : (-1 - 2) h) (-8 + 16) : (-4 + 6)

24 An aeroplane flies at a height of 7 950 metres above sea level and a submarine is directly beneath it, at 275 m below sea level. How many metres are there between them?

25 A mountain climber reaches the peak of a mountain at a height of 2 532 m. A miner is digging directly beneath him at a depth of 180 m.

a) Express these measurements as integers.

b) How many metres are there between them? Compute this using an operation between integer numbers.

26 The temperature in a room increases by 30 °C and then decreases by 42 °C. If the final temperature is -6 °C, what was the initial temperature?

27 Peter and Louise have a savings account where they receive their salaries and pay their expenses. These are the latest movements in their account.

Movement Balance Description

-120 200 Electricity bill

1 500 Peter’s salary

1 400 Gas bill

-1 470 Mortgage

730 Louise’s salary

a) What was their balance before paying the electricity bill?

b) What is their balance after receiving Peter’s salary?

c) How much was the gas bill?

d) What is their balance after paying their mortgage?

e) How much is Louise’s salary?

28 Alexander works on the 23rd floor of a building. When he parks his car in the company’s carpark, he must go up 27 floors to get to his office. On what floor does he park his car?

29 Euclid, a famous mathematician, died in the year 265 BC and lived for 60 years.

a) What year was he born?

b) How much older than you is Euclid?

c) In which year was a person who is two years older than Euclid born? EUCLIDES

FINAL ACTIVITIES

24

ES0000000126360 146700_Unidad03_100697.indd 24 16/04/2020 9:37:40

Integers. Ordering

1 Answer the following, providing an explanation.

a) How many numbers are there between -50 and +128?

b) What about between -48 and 48?

c) What integer has 9 as its opposite?

d) What integers have the same absolute value?

2 Write down the sets corresponding to the following.

a) Numbers greater than -7 and smaller than -2.

b) Numbers greater than -4 and smaller than +2.

c) Integers which are at a distance of 7 units from 3.

d) Integers whose absolute value is less than 6.

3 Order the following integers from smallest to largest.

Op (+5) -8 Op (-3) ;-4; +6

Operations with integers

4 Perform the following operations.

a) 6 + (-4 + 2) - (-3 - 1)

b) 7 - (4 - 3) + (-1 - 2)

c) 3 + (2 - 3) - (1 - 5 - 7)

d) -8 + (1 + 4) + (-7 - 9)

e) 10 - (8 - 7) + (-9 - 3)

f ) 7 - (4 + 3) + (-1 + 2)

5 Compute the following.

a) 5 - (-3) + 7 + (-9) - 14

b) 12 : (-5 + 3) - 4 ? (4 - 9) ? (-1)

c) 28 - 3 ? [(4 - 6 + 7) + (-5) ? (-4)]

6 Calculate the following.

a) 8 - 6 ? 3 + 12 : 2 - 5

b) 8 - 6 ? 3 + 12 : (2 - 5)

c) 8 - 6 ? (3 + 12 : 2) - 5

d) 8 - 6 ? (3 + 12 : 2 - 5)

e) (8 - 6 ? 3 + 12) : 2 - 5

7 Complete the following in your notebook.

a) 13 ? (4 - 8) = -26

b) 15 - 4 ? 3 = 33

c) 7 + (9 - 4 - 10) ? (-2) = 1

8 In a biology lab, biologists are studying the resistance of a microorganism to changes in temperature. They have a sample at -3 °C. They raise its temperature to 40 °C, then decrease it by 50 °C and finally raise it by 12 °C. What is the final temperature of the sample?

9 Xavier owed his brother 24 €. As he only had 15 €, he asked his friend Lucy for a loan. Lucy loaned him some money. With that money, Xavier paid back his brother and then bought 3 notebooks costing 2 € each. After that, he had 5 € remaining. Which of the following expressions can be used to find out how much money Lucy loaned Xavier?

a) 15 + 24 + 2 ? 3 +4? = 5

b) 15 - 24 -4? = 2 ? 3 + 5

c) 24 - 15 =4? - 2 ? 3 - 5

d) 15 - 24 +4? = 2 ? 3 + 5

e) 15 - 24 - 2 ? 3 =4? - 5

SELF‑EVALUATION

Integers

• Positive integers:+1, +2, +3, +4, +5, +6, +7…

• The number 0.

• Negative integers:-1, -2, -3, -4, -5, -6, -7…

Absolute value

;+a; = a ;-a; = a ;0; = 0

Opposite of an integer

Op (+a) = -a Op (-a) = +a

Op (0) = 0

Sign rules

(+) ? (+) = + (+) : (+) = +

(-) ? (-) = + (-) : (-) = +

(+) ? (-) = - (+) : (-) = -

(-) ? (+) = - (-) : (+) = -

SUMMARY

25

Integers 3

ES0000000126360 146700_Unidad03_100697.indd 25 16/04/2020 9:37:41

ES0000000126360 146700_Unidad03_100697.indd 26 16/04/2020 9:37:42

ES0000000126360 146700_Unidad03_100697.indd 27 16/04/2020 9:37:42

ES0000000126360 146700_Unidad03_100697.indd 28 16/04/2020 9:37:43

ES0000000126360 146700_Unidad03_100697.indd 29 16/04/2020 9:37:44

ES0000000126360 146700_Unidad03_100697.indd 30 16/04/2020 9:37:44

ES0000000126360 146700_Unidad03_100697.indd 31 16/04/2020 9:37:45

© 2020 by Santillana Educación, S. L. / RichmondAvda. de los Artesanos, 6. Tres Cantos 28760 Madrid

Richmond is an imprintof Santillana Educación, S. L.

Printed in Spain

Richmond58 St Aldate'sOxford 0X1 1STUnited Kingdom

CP: 146700

All rights reserved. No part of this book may be reproduced, stored in retrieval systems or transmitted in any form, electronic, mechanical, photocopying or otherwise without the prior permission in writing of the copyright holders. Any infraction of the rights mentioned would be considered a violation of the intellectual property. If you need to photocopy or scan any fragment of this work, contact CEDRO (Centro Español de Derechos Reprográficos, www.cedro.org).

ES0000000126360 146700_Unidad03_100697.indd 32 16/04/2020 9:37:45