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NCV 2 MATHEMATICS WORKBOOK This material was written by Gayle Staegermann And edited by Jackie Scheiber RADMASTE Centre University of the Witwatersrand Telephone (011) 717-6070 Fax (011) 339-1054 Copyright of the material remains with the authors. However, this material may be photocopied and used for educational purposes. ii CONTENTS Page TOPIC 1: NUMBERS Chapter 1 Natural Numbers, Whole Numbers and Integers ………………... 1 Chapter 2 Fractions and Decimals ………………………………………….. 20 Chapter 3 Rational Numbers and Real Numbers …………………………… 39 TOPIC 2: FUNCTIONS Chapter 4 Working with Algebraic Expressions …………………………… 58 Chapter 5 Linear Equations and Linear Functions …………………………. 75 Chapter 6 Quadratic Equations and Functions, Hyperbolic functions and Exponential Equations and Functions …………………………… 100 TOPIC 3: SPACE. SHAPE AND ORIENTATION Chapter 7 Properties of Shapes ……………………………………………... 128 Chapter 8 Analytical and Transformation Geometry ……………………….. 144 Chapter 9 Trigonometry …………………………………………………….. 153 TOPIC 4: STATISTICAL AND PROBABILITY MODELS Chapter 10 Data Handling …………………………………………………… 170 TOPIC 5: FINANCIAL MATHEMATICS Chapter 11 Financial Mathematics …………………………………………... iii 190 NCV2 – Chapter 1: Natural numbers, whole numbers and integers CHAPTER 1 Natural numbers, whole numbers and integers In this chapter you will: • Use the order of operations to do calculations • Calculate factors of a number • Classify numbers as either prime or composite • Calculate prime factors • Calculate the Highest Common Factor (HCF) • Calculate multiples of a number • Calculate the Lowest Common Multiple (LCM) • Use laws of exponents to simplify expressions • Calculate square roots and cube roots • Distinguish between sets of natural numbers and whole numbers • Add, subtract, multiply and divide using a calculator • Find squares, cubes and higher powers on a calculator • Find square roots, cube roots and higher roots on a calculator • Estimate answers • Add and subtract integers using a number line • Add and subtract integers without using a number line • Multiply and divide integers • Distinguish between integers, whole numbers and natural numbers. This chapter covers material from Topic 1: Numbers SUBJECT OUTCOME 1.1: Use Computational Tools and Strategies and Make Estimates and Approximations Use a scientific calculator competently and efficiently Learning Outcome 1: Execute algorithms appropriately in calculations Learning Outcome 2: 1 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.1 1.1 ORDER OF OPERATIONS If a calculation has more than one operation, we do them in the following fixed order: 1. ( ) Brackets 2. Exponents 3. × ÷ Multiplication and Division, working from left to right 4. + – Addition and Subtraction, working from left to right BEMDAS will help you to remember the order in which you must work. Unless you follow this order, you will get wrong answers. EXAMPLES: Calculate the answers without using a calculator: 1) 5+2–7 2) 24 × 2 ÷ 8 ÷ 3 3) 9–4 × 2 4) 7 + 4 × (3 + 2) 5) 22 + 4 × 3 6) 8–2 × 6+5 7) 6 + 32 ÷ 3 8) (6 + 9) ÷ 3 9) 2(3 + 1) – 4 ÷ 2 SOLUTIONS 5 + 2 – 7 = 7 – 7 = 0 ….. Work from left to right 24 × 2 ÷ 8 ÷ 3 = 48 ÷ 8 ÷ 3 = 6 ÷ 3 = 2 ….. Work from left to right 9 – 4 × 2 = 9 – (4 × 2) = 9 – 8 = 1 ….. Multiply before subtracting 7 + 4 × (3 + 2) = 7 + (4 × 5) = 7 + 20 = 27 …..Simplify brackets, multiply, then add 22 + 4 × 3 = 4 + (4 × 3) = 4 + 12 = 16 ..… Simplify the exponent, multiply, then add 8 – 2 × 6 + 5 = 8 – (2 × 6 ) + 5 = 8 – 12 + 5 = 1 ..… Multiply, subtract, then add and subtract 6 + 32 ÷ 3 = 6 + (9 ÷ 3) = 6 + 3 = 9 ..…Simplify the exponent, divide, then add (6 + 9) ÷ 3 = 15 ÷ 3 = 5 ..… Simplify brackets, then divide 2(3 + 1) – 4 ÷ 2 = 2(4) – (4 ÷ 2) = 8 – 2 = 6 ..… Simplify brackets, divide, then subtract Exercise 1.1 Calculate the answers without using a calculator: 1) 9 – 8 + 1 = ……………………………... 2) 13 – 10 – 3 = ……………………………. 3) 8 + 5 – 2 + 3 = …………………………. 4) 12 + 5 – 2 = ……………………………... 5) 12 ÷ 6 ÷ 2 = ……………………………. 6) 4 × 6 – 3 = ………………………………. 7) 15 – 3 × 4 = ……………………………. 8) (15 – 3) × 4 = ……………………………. 9) 12 ÷ 4 + 2 = …………………………… 10) 12 ÷ (4 + 2) = ……………………………. 11) 3 × (2 + 1) – 3 ÷ 3 = ………………………………………………………………………………. 12) 7 – 2 × 3 + 4 ÷ 4 = ………………………………………………………………………………… 13) 19 – 4(5 – 3) × 2 = ………………………………………………………………………………… 14) 9 × 105 × 28 × (3 – 3) = …………………………………………………………………………. 15) 23 – 15 + 4 – 12 + 9 – 3 = …………………………………………………………………………. 16) 6 + 2 × 8 – 8 – 5 = ………………………………………………………………………………… 2 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.2 1.2 FACTORS A factor of a number divides into that number exactly, without a remainder. 3 is a factor of 12, since 12 ÷ 3 = 4. 3 divides exactly into 12. 3 is not a factor of 16, since 16 ÷ 3 = 5 remainder 1. There are other factors of 12. To find them, divide by 1, 2, 3, ….. in this order. Those that divide into 12 exactly are factors. Note: • F stands for factor and F12 means "the set of factors of 12". EXAMPLE SOLUTION 1) Is 7 a factor of 42? Why? 7 is a factor of 42 since 42 ÷ 7 = 6 2) 3) Is 7 a factor of 41? Why? List all the factors of 12. 4) List all the factors of 30. 7 is not a factor of 41, since 41 ÷ 7 = 5 remainder 6 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1. So F12 = {1, 2, 3, 4, 6, 12}. There are 6 factors of 12. 1 is the least factor and 12 is the greatest factor. F30 = {1, 2, 3, 5, 6, 10, 15, 30} Note: • The least factor of any number is always 1, since every number can be divided by 1. • The greatest factor of any number is the number itself. Exercise 1.2 1) Is 6 a factor of 54? Why? ………………………………………………………………………… 2) Is 6 a factor of 46? Why? ………………………………………………………………………… 3) a) List all the factors of 18. F18 = {……………………………………………………………. b) How many factors does 18 have? …………………………………………………………… c) What is the least factor of 18? ……………………………………………………………… d) What is the greatest factor of 18? …………………………………………………………… 4) List all the factors of 24. F24 = { ………………………………………………………………… 5) List all the factors of 36. F36 = { ………………………………………………………………… 6) List all the factors of 45. …………………………………………………………………………. 7) What are the greatest and the least factors of 179? Greatest factor = ……......…………………………………………… Least factor = ………………………………………………………… 3 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.3 1.3 PRIME PRIME AND COMPOSITE NUMBERS A prime number has only 2 factors, itself and 1. A composite number has more than 2 factors. NOTE: • 7 is prime, since its only factors are 1 and 7. It has only 2 factors. • 6 is not prime, since its factors are 1, 2, 3, and 6. 6 is a composite number since it has more than 2 factors. • 1 is not prime. The only factor of 1 is 1, it does not have 2 factors. So 1 is neither prime nor composite. EXAMPLE 1) Is 1 740 prime or composite? Give a reason for your answer. 2) Is 365 prime? Why? SOLUTION 1, 10, 174 and 1 740 are some of the factors of 1 740. It is composite since it has more than 2 factors. No, 365 is NOT prime. 1, 5, and 365 are factors of 365, so it has more than 2 factors. NOTE: • Numbers that end in 0 have a factor of 10 • Numbers that end in 5 have a factor of 5 • The set of even numbers = {2; 4; 6; 8; 10; ………………..}. In other words, numbers that end in an even number have a factor of 2. Exercise 1.3 1) Are the following numbers prime or composite? Give a reason for your answer. a) 8 ……………………………………………………………………………………….............. b) 11 ……………………………………………………………………………………………… c) 14 ……………………………………………………………………………………………… d) 19 ……………………………………………………………………………………………… e) 1 ……………………………………………………………………………………….............. f) 23 ………………………………………………………………………………………………. g) 445 ……………………………………………………………………………………............... h) 986 ……………………………………………………………………………………............... 2) List all the prime numbers up to and including 20 . …………………………………………….. ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… 3) List all the composite numbers up to and including 20 ……………………………………………. ……………………………………………………………………………………………………… ………………………………………………………………………………………………………. 4 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.4 1.4 PRIME FACTORS A prime factor is a factor that is a prime number. EXAMPLE 1) We said that F12 = {1, 2, 3, 4, 6, 12} List the prime factors of 12 2) Which factors of 12 are composite? 3) 4) SOLUTION The prime factors are 2 and 3 The composite factors are 4; 6; 12 Why is 1 not a prime factor of 12? Factorise 36 into prime factors 1 has only one factor, so it is not prime. Prime Other factors factors 2 36 After dividing, we can say 36 = 2 × 2 × 3 × 3 = 2 2 × 32 2 18 3 9 3 3 Note: We do not include 1 because it is not a prime number 1 Exercise 1.4 1) a) List all the factors of 15: F15 = { ; ; ; } b) List the prime factors of 15 ……………………………………………………………………. c) Which factors of 15 are composite? …………………………………………………………… 4) a) Factorise 60, 72, and 90 into prime factors. Prime factors Prime factors Other factors 60 Other factors 72 Prime factors Other factors 75 b) List the prime factors of 60 …………………………………………………………………… c) List the prime factors of 72 …………………………………………………………………… d) List the prime factors of 75 …………………………………………………………………… e) Is 1 a factor of 60? Give a reason for your answer …………………………………………………………………………………………………. f) Is 1 a prime factor of 60? Give a reason for your answer …………………………………………………………………………………………………. 5 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.5 1.5 HIGHEST COMMON FACTOR HCF stands for Highest Common Factor. It is the greatest factor common to two or more numbers. EXAMPLE 1) Find the HCF of 36 and 24 by listing all their factors. 2) SOLUTION F24 = {1, 2, 3, 4, 6, 8, 12, 24} F36 = {1, 2, 3, 4, 6, 9, 12, 18, 36} The HCF of 36 and 24 = 12, since this is the highest factor common to both. Find the HCF of 36 and 24 by breaking each number into its prime factors. Prime Other Prime Other factors factors factors factors 2 24 2 36 2 12 2 18 2 6 3 9 3 3 3 3 1 1 Look for the factors that are common to both numbers 24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3 The HCF of 36 and 24 = 2 × 2 × 3 = 12. Exercise 1.5 1) Find the HCF of 120 and 90 by breaking each number into its prime factors. Prime factors Other factors 120 Prime factors Other factors 90 120 = ………………………………………………………………………. 90 = ………………………………………………………………………… HCF = ………………………………………………………………………. 2) Find the HCF of 125 and 75 by listing all their factors. F125 = { ; F75 = { ; ; ; ; ; } ; ; } HCF = ………………………………………………………………………. 3) Find the HCF of 180 and 150 by listing all their factors. F180 = {………………………………………………………………………. F150 = {………………………………………………………………………. HCF = …………………………………………………………………….... 6 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.6 1.6 MULTIPLES A multiple of a whole number is found by multiplying it by any whole number. Examples: The multiples of 5 are 5 × 1, 5 × 2, 5 × 3, 5 × 4, 5 × 5, …… = 5, 10, 15, 20, 25, ……….. The multiples of 1 are 1, 2, 3, 4, 5, ………… Note that M stands for multiple and M24 means "the set of multiples of 24" We cannot list all the multiples of a number since there are an infinite number of them, so we use dots to show that they are ongoing EXAMPLE 1) a) List the factors of 12 b) List the multiples of 12 2) a) Write down the factors of 18 b) Write down the multiples of 18. 3) Which of the numbers 11, 22, 41, 77 are multiples of 11? SOLUTION a) F12 = {1; 2; 3; 4; 6; 12} b) Multiples of 12 are 12 × 1; 12 × 2; 12 × 3; …. M12 = {12; 24; 36; 48; 60; …………..} a) F18 = {1; 2; 3; 6; 9; 18} b) The multiples of 18 are 18 × 1 = 18, 18 × 2 = 36, 18 × 3 = 54 etc. So M18 = {18; 36; 54; ……} 11 × 1 = 11, 11 × 2 = 22 and 11 × 7 = 77, so 11, 22 and 77 are multiples of 11 Note: • factors of a number are the number itself and smaller • multiples of a number are the number itself and greater Exercise 1.6 1) a) List the factors of 15 ………………………………………………………………………….. b) List the multiples of 15 ……………………………………………………………………….. c) Which number is both a factor and a multiple of 15? ………………………………………… 2) a) List the multiples of 20 ……………………………………………………………………….. b) List the factors of 20 ………………………………………………………………………….. c) Which multiple of 20 is also a factor of 20? ………………………………………………….. 3) Which of the following numbers are multiples of 9: 9, 18, 36, 109? Why? ………………………………………………………………………………………………………. ……………………………………………………………………………………………………… 4) Write down the first twelve multiples of 8 ………………………………………………………………………………………………………. ………………………………………………………………………………………………………. 7 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.7 1.7 LCM LCM stands for Lowest Common Multiple. It is the lowest number into which two or more numbers divide. EXAMPLE 1) Find the Lowest Common Multiple (LCM) of 36 and 24 by listing the multiples of each. SOLUTION M36 = {36, 72, 108, …} M24 = {24, 48, 72, …} The LCM = 72. This is the lowest multiple which is common to both 36 and 24. 2) Use prime factors to find the HCF and LCM of 504 and 180. Prime factors 2 2 2 3 3 7 Other factors 504 252 126 63 21 7 1 Prime Other factors factors 2 180 2 90 3 45 3 15 5 5 1 504 = 2 × 2 × 2 × 3 × 3 × 7 180 = 2 × 2 × 3 × 3 × 5 HCF = 2 × 2 × 3 × 3 = 36 To find the LCM, multiply the HCF by those factors that are not common. LCM = 36 × 2 × 5 × 7 = 2 520 Exercise 1.7 1) Find the LCM of 15 and 20 by listing the multiples of each. ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… 2) Find the HCF of 105 and 140 by listing the factors of each. ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… …………………………………………………………………………………………………….. 3) Find the LCM of 105 and 140 by listing the multiples of each. ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… 8 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.8 1.8 EXPONENTS, SQUARES AND CUBES In the expression 37 , 3 is called the base and 7 is called the exponent, index or power. The exponent gives the number of times the base is multiplied by itself. 37 expanded means 3 × 3 × 3 × 3 × 3 × 3 × 3 37 is said to be in index form or exponent form or written as a power of the base. EXAMPLE 1) Expand the following: a) 24 b) x 3 c) (a + b)2 2) Write the following in exponential form: a) 6 × 6 × 6 b) x × x c) p × p × p × p × p d) (d + g)(d + g) SOLUTION 24 = 2 × 2 × 2 × 2 x3 = x × x × x (a + b)2 = (a + b)(a + b) 6 × 6 × 6 = 63 x × x = x2 p × p × p × p × p = p5 (d + g)(d + g) = (d + g)2 Exercise 1.8 1) Expand the following: a) 54 = ……………………. b) x3 = ……………………. c) ( x + y ) 2 = ……………………. 2) Write the following in index form: a) 8 × 8 × 8 × 8 = ………... b) b × b × b = ………… c) (p + q)(p + q)(p + q) = …………… The square of any number is the number multiplied by itself. The cube of any number is the number multiplied by itself and by itself again. EXAMPLE Evaluate: 1) 72 2) 23 3) (4 + 2)2 4) 42 + 22 SOLUTION 72 = 7 × 7 = 49 23 = 2 × 2 × 2 = 8 (4 + 2)2 = 62 = 36 42 + 22 = (4 × 4) + (2 × 2) = 16 + 4 = 20 Note: • 72 is read "seven squared" and 23 is read "two cubed" • 49 = 72 is called a "perfect square" and 8 = 23 is called a "perfect cube". 3) Calculate: a) 52 + 12 = …………..……. b) (5 + 1)2 = …………….…… c) 14 + 41 = ……………..…. d) (4 – 3)2 = ………………. e) 43 – 32 = …………………. g) (1 + 2)3 = …………...….. h) (2 + 5 – 3)2 = .......................................................................... 4) a) 36 is the square of …….... f) 25 – 2 = …………………. b) 64 is the cube of .…….. and the square of ……........... c) Twelve squared = …….... d) Ten squared = …….…….. f) …..….. is the square of 11 g) The square of 9 = ................... h) Twenty squared = ............. 9 e) Five cubed = ……........…. NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.9 1.9 SQUARE ROOTS AND CUBE ROOTS Note: is the symbol for square root and means 3 is the symbol for cube root 16 is read "the square root of 16" 3 2 where the 2 is assumed 27 is read "the cube root of 27" The square root of a number multiplied by itself gives the number. 3 × 3 = 3× 3 = 3 Take one from every pair of identical numbers under the square root sign as the answer. 5 × 5 = 3 × 5 = 15 3× 3 × 3× 5× 3× 5 = So The cube root of a number multiplied by itself and by itself again gives the number. 3 2 × 3 2 × 3 2 = 3 2× 2× 2 = 2 Take one from every three identical numbers under the cube root sign as the answer. So 3 EXAMPLE Evaluate 1) 16 2) 3 3) 4) 5) 27 7) 8) 1 59 × 59 × 59 1 10) 1 = 1× 1× 1 = 1 25 = 5×5 = 5 78 × 78 = 78 3 59 × 59 × 59 = 59 9 + 16 = 7 9 1 8 × 18 Exercise 1.9 Calculate 1) 81 = ………. 0×0 = 0 3 9 + 16 = 9 + 16 9) 3× 3× 3 = 2 × 3 = 6 27 = 3 3 × 3 × 3 = 3 0 = 3 78 × 78 3 3 16 = 4 × 4 = 4 3 9 + 16 6) 2× 2× 2 × SOLUTION 0 3 3 2 × 3× 3× 2 × 3× 2 = 7 = 9 16 = 9 8 × 18 = 2) 3× 3 + 4× 4 4 = 3 3× 3 (2 × 2 × 2) × (2 × 3 × 3) = 2 × 2 × 3 = 12 144 = …..….. 4 = ………………..…… 4×4 = 3 + 4 = 7 3) 3 27 = …….... 4) 3 8 = …..….. 5) 16 – 7) 169 – 16 = …………………… 8) 9) 25 − 16 = …………………….. 10) 36 + 3 125 = ……………………… 12) 3 16 = ………….. 25 11) 27 = …………… 125 15) 3 19 × 19 × 19 = ……………. 13) 3 6) 169 − 25 = ..………… 3 1 + 81 = ……..………………… 100 14) 16) 10 64 5 3 = ………………….. 4 = ………………………….….. 9 9 × 4 × 6 = …………………………………. NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.10 1.10 THE SET OF NATURAL NUMBERS AND WHOLE NUMBERS Numbers belong in number sets which have names. Some numbers belong to more than one number set. So far, all the numbers we have worked with have been either natural numbers or whole numbers. Here are the first two sets. Remember, the dots in the set brackets mean that the numbers continue in the same way without end. The set of Natural numbers (also called counting numbers): N = {1; 2; 3; 4; …………..} The set of Whole numbers: W or N0 = {0; 1; 2; 3; …………….} These numbers can be shown on a number line. The arrow on the end means that the numbers continue in the same way without end. 1 2 3 4 5 6 0 1 2 3 4 5 N = {1; 2; 3; 4; …….….….} 6 W = {0; 1; 2; 3; ……….} EXAMPLE: 1) 0 is a Whole number (W) 2) 1 is a Natural numbers (N) and a Whole numbers (W) 3) 245 is a Natural numbers (N) and a Whole numbers (W) Exercise 1.10 1) Write down the letter that stands for: a) the set of whole numbers ……………………………………………………………… b) the set of natural numbers ……………………………………………………………... 2) What is the smallest natural number? ………………………………………………………. 3) What is the greatest natural number? ……………………………….………………………. 4) What is the smallest whole number? ……………………………….……………………….. 5) Write down a number that is a whole number but not a natural number. ..……………….… 6) Are the following numbers whole numbers and/or natural numbers? a) 4 ………………………………………………………………………………………… b) 56 ……………………………………………………………………………………….. c) 0 …………………………………………………………………………………………. d) 249 ………………………………………………………………………………………. 11 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.11 1.11 ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION USING A CALCULATOR Remember BIMDAS, the correct order of operations. Your scientific calculator follows this order automatically. EXAMPLE Calculate 1) 8− 2 3 2) (8 – 2) ÷ 3 3) 8− 2÷3 4) 2 8– 3 SOLUTION Key sequence Display ÷ 3 = 8 – 2 = ( 8 – 2 8 – 2 ÷ 3 = ) 2 ÷ 3 = 2 7.33… Answer 8− 2 =2 3 (8 – 2) ÷ 3 = 2 8− 2÷3 = 8 – 2 = 7,33… 3 Note: For 1), you have to enter = after 8 – 2 before dividing by 4. This is because the entire numerator is divided by 3. For 2), you use brackets instead of = . Questions 1) and 2) are identical. For 3) and 4) the answer is not 2. The calculator first divides 2 by 3 before subtracting the answer from 8. Exercise 1.11 Use your calculator to calculate the following: 1) 100 – 84 ÷ 2 = ………. 2) (100 – 84) ÷ 2 = ………. 3) 310 × 54 – 22 = ………. 4) 310 × (54 – 22) = ………. 5) 840 ÷ 21 + 3 = ………. 6) 840 ÷ (21 + 3) = ………. 7) 321 + 12 ÷ 3 = ………. 8) (321 + 12) ÷ 3 = ………. 9) 102 – 34 × 5 + 96 = ………. 10) 102 – 34 × (5 + 96) = ………. 11) (102 – 34) × 5 + 96 = ………. 12) (102 – 34) × (5 + 96) = ………. 13) 126 + 18 ÷ 9 = ………. 14) (126 + 18) ÷ 9 = ………. 15) 145 × 23 ÷ 5 = ………. 16) 114 ÷ 19 ÷ 2 = ………. 17) 459 ÷ 3 × 9 = ………. 18) 459 ÷ (3 × 9) = ………. 19) 154 – 6 ÷ 2 = ………. 20) (154 – 6) ÷ 2 = ………. 21) 119 ÷ 17 – 7 = ………. 22) 119 ÷ (17 – 7) = ………. 12 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.12 1.12 SQUARES, CUBES AND HIGHER POWERS ON A CALCULATOR On a CASIO fx-82ES scientific calculator, x 2 is used for squaring a number x3 is used for cubing a number x is used for raising a base to any exponent Note: On some calculators the key x is marked EXAMPLE Use your calculator to calculate: ^ or xy or yx SOLUTION Key Sequence 1) 1982 198 x2 2) 183 18 x3 3) 145 14 x Answer 1982 = 39 204 = 183 = 5 832 = 5 145 = 537 824 = 4) 25 × 163 25 × 16 x3 5) 2 3792 − 453 2 379 x2 – 6) (23 + 19)3 ( 23 + 19 7) (172 – 132)4 ( 17 x2 25 × 163 = 102 400 = 45 x3 = 1232 - 453 = 5 568 516 ) x3 = (23 + 19)3 = 74 088 – 13 x2 x ) Note: • On some calculators you may not have to enter the 4 = = (172 – 132)4 = 207 360 000 key for squares and cubes. Exercise 1.12 Use your calculator to evaluate: 1) 542 = ………. 2) 433 = ………. 3) 184 = ………. 4) 153 – 143 = ………. 5) 172 – 162 + 52 = ………. 6) (222 – 192)3 = ………. 7) 182 – 82 = ………. 8) (18 – 8)2 = ………. 9) (43 – 19)4 = ………. 10) 434 – 194 = ………. 11) 153 – 142 + 162 = ………. 12) 144 – (132 + 123) = ………. 13) 1232 – 133 + 124 = ………. 14) (252 – 202 – 52)2 = ………. 15) (233 + 124)2 = ………. 16) (56 – 23 + 45)4 = ………. 17) 12 × 152 × 132 = ………. 18) (32 + 47)3 = ………. 13 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.13 1.13 SQUARE ROOTS, CUBE ROOTS AND HIGHER ROOTS ON A CALCULATOR On your CASIO fx-82ES scientific calculator, the key: is used for finding the square root 3 is used for finding the cube root is used for finding other roots greater than 3 Note: 1 On some calculators the key EXAMPLES Use your calculator to calculate: 1) 5 3) 841 5 9 + 16 ( 3 9 261 – 5) 4 2 825 761 3 361 x or Key sequence 1889 568 4) yx or or x y SOLUTION 841 2) 1 xy is marked Answer 841 = 29 = 1 889 568 9 9 261 4 + 16 = ) 5 9 + 16 = 5 = – 361 2 825 761 1889 568 = 18 = = 3 9 261 – 361 = 2 4 2 825 761 = 41 Note: • In 3) the brackets around the '9 + 16' are essential to get the right answer If you forget the brackets in 3), your calculator will calculate • 9 + 16 = 19 Exercise 1.13 Use your calculator to calculate: 1) 5 537 824 = ………. 2) 145 = ………. 3) 3 12 167 = ………. 4) 233 = ………. 14 400 = ………. 6) 729 + 3 3 375 = ………. 62 748 517 = ………. 8) 961 – 4 28 561 = ………. 252 − 242 = ………. 10) 323 – 5) 7) 7 9) 11) ( 4 130 321 + 13) 3 15) 289 )2 = ………. 361 – 3 1 331 = ………. 12) ( 256 – 225 )2 = ………. 35 937 − 256 = ………. 14) 4 130 000 + 321 = ………. 100 + 100 + 100 + 100 = ………………. 16) 3 19 × 19 × 19 = …………………… 14 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.14 1.14 ESTIMATING AN ANSWER Rounding off or correcting to 1 significant figure: As you read a number from left to right, the first figure you come to that is not zero is called the first significant figure. Numbers greater than 1 1st significant figure 325 1 456 2,789 The same is true for numbers less than 1 1st significant figure 0,0345 0,0092 0,000 549 An easy way to approximate a number is to round it off to the first significant figure. Procedure for rounding to 1 significant figure: i) Underline the first significant figure ii) Look at the number to the right: • If it is less than 5, the first significant figure remains the same • If it is 5 or more, the first significant figure is increased by 1 iii) If the number is more than 1, fill in zeros to keep the place value correct EXAMPLE 1) Round the following numbers off to 1 SOLUTION significant figure: a) 634 810 b) 38 126 c) 0,0346 d) 0,006 825 13 2) a) Calculate 12 012 × 45 ÷ 462 using a calculator b) As a check, estimate the answer to a) by rounding each number off to 1 significant figure 634 810 ≈ 600 000 38 126 ≈ 40 000 0,0346 ≈ 0,03 0,006 825 13 ≈ 0,007 Calculator: 12 012 × 45 ÷ 462 = 1 170 Estimate: 10 000 × 50 ÷ 500 = 500 000 ÷ 500 = 1 000 The answers to a) and b) are nearly the same, so the answer to a) is probably correct Exercise 1.14 1) Round each of the following numbers to 1 significant figure: a) 5 499 ≈ ………………………………………………… b) 491 ≈ …………………………………………………… c) 35 ≈ …………………………………………………….. d) 0,005 53 ≈ ………………………………………………. 2) a) Use a calculator to calculate: 924 × 28 ÷ 87 ……………………………………………….. b) Check your answer, by rounding each number to 1 significant figure ………………………………………………………………………………………………… 3) Round each number to 1 significant figure, to estimate the answer to 5059 × 99 × 17 ……………………………………………………………………………………………………... 15 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.15 1.15 ADDITION AND SUBTRACTION OF INTEGERS USING A NUMBER LINE The set of integers is the set of numbers {….; –5; –4; –3; –2; –1; 0; 1; 2; 3; 4; 5; …….} We can show these integers on a number line like this: ● ● ● ● ● ● ● ● ● ● -4 -3 -2 -1 0 1 2 3 4 5 The set of integers consists of the set of whole numbers as well as the negatives of the whole numbers. When we add integers, we move a number of digits to the right. When we subtract integers , we move a number of digits to the left. Note: • If there is no sign in front of a digit, it is assumed to be + • The sign before an integer belongs to that integer Use the number line for the following: –4 EXAMPLE 1. Add 2 and 3 2. 3. 4. 5. –3 –2 –1 0 1 2 3 4 5 SOLUTION Start at 2 and move 3 digits to the right. Your answer is 2 + 3 = 5. Add −3 and 4 Start at −3 and move 4 digits to the right. Your answer is −3 + 4 = 1. Add –1 and −3 A tricky one! Start at −1 and move –3 digits to the right, which is the same as moving 3 digits to the left. Your answer is –1 + (–3) = –4. Subtract 4 from 3 Start at 3 and move 4 digits to the left. Your answer is 3 – 4 = –1. Subtract −3 from −2 Another tricky one! Start at –2 and move –3 to the left, which is the same as moving +3 to the right. Your answer is –2 – (–3) = –2 + 3 = 1. Exercise 1.15 1) Use the number line above to help you to calculate the answers to each question. a) –2 + 7 = …… b) 5 – 9 = ……… c) 1 – (–2) = ……… d) 1 – 5 = ……… e) –3 – (–2) = ……... f) –1 – 3 = …….. g) –2 + (–1) = …….. h) –2 – (–3) = …….. 2) Use the centimetre marks along a full length ruler if the number line above is too small. a) –4 + 2 = …….. b) –1 – 2 = …….. c) –3 + (–1) = …….. d) –5 – (–2) = …….. e) 2 – 3 = …….. f) 2 – (–2) = …….. g) –2 – 1 = …….. h) –7 + 12 = …….. i) –3 – 6 = …….. j) –1 + (–2) = …….. k) 4 + (–4) = …….. l) –11 – 1 = …….. m) 4 – 6 = …….. n) –3 + 5 = …….. o) 1 – (–2) = …….. p) –10 – (–1) = ……. 16 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.1 1.16 ADDITION AND SUBTRACTION OF INTEGERS WITHOUT USING A NUMBER LINE As a short cut, remember: • + × +=+ • + × –=– • – × +=– • – × –=+ When multiplying: • if the signs are the same, the answer is + • if the signs are different, the answer is – EXAMPLE Evaluate 1) +5 + (+4) 2) –3 + (–5) 3) –8 – (+3) = –11 4) –2 – (–4) 5) –2 – (+6) + (+1) + (–3) – (–5) 6) 12 + (–6) – (–3) – (+2) – (–1) 7) –15 + (–18) + (+6) – (–2) + (–5) – (+7) 8) –11 – (–6) + (–3) + 4 – 2 SOLUTION +5 + (+4) = 5 + 4 = 9 –3 + (–5) = – 3 – 8 = – 8 –8 – (+3) = – 8 – 3 = –11 –2 – (–4) = –2 + 4 = 2 –2 – (+6) + (+1) + (–3) – (–5) = –2 – 6 + 1 – 3 + 5 = – 5 12 + (–6) – (–3) – (+2) – (–1) = 12 – 6 + 3 – 2 + 1 = 8 –15 + (–18) + (+6) – (–2) + (–5) – (+7) = –15 – 18 + 6 + 2 – 5 – 7 = – 37 –11 – (–6) + (–3) + 4 – 2 = – 11 + 6 – 3 + 4 – 2 = – 6 Exercise 1.16 Evaluate without the use of a calculator. 1) – (+4) = …………………………………… 2) – (–5) = …………………………………. 3) –7 – (+15) = ……………………………… 4) 13 – (–9) = ……………………………… 5) +14 + (+6) = ……………………………… 6) –9 + (–5) = ……………………………… 7) –9 + 0 = …………………………………… 8) 19 – (–7) = ………………………………. 9) –13 + (–5) = ……………………………… 10) –13 – 0 = ………………………………… 11) 23 – (–4) = ………………………………... 12) –25 – (+12) = …………………………… 13) 0 – (+13) = ………………………………... 14) 0 – (–23) = ………………………………. 15) –3 – 4 + 2 + (–7) + 6 = ………………………………………………………………………….. 16) 14 – 17 + 4 + 3 = ………………………………………………………………………………… 17) –4 – (–3 + –5) = …………………………………………………………………………………. 18) 11 + 9 + (–6) – 2 = ………………………………………………………………………………. 19) 24 – 7 – 5 – (+3) + 1 = ………………………………………………………………………….. 20) –19 +(–21) – (–32) + 5 = ……………………………………………………………………….. 21) –23 – 12 – 34 – 15 = ……………………………………………………………………………. 17 NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.1 1.17 MULTIPLICATION AND DIVISION OF INTEGERS INTEGERS Remember: • + ÷ +=+ • + ÷ –=– • – ÷ +=– • – ÷ –=+ EXAMPLE 1) Evaluate a) 7 × 11 b) 5 × –3 c) –4 × 7 d) –6 × –2 e) 18 ÷ 6 When dividing, as for multiplying: • if the signs are the same, the answer is + • if the signs are different, the answer is – SOLUTION 7 × 11 = 77 5 × -3 = –15 –4 × 7 = –28 –6 × –2 = 12 18 ÷ 6 = 3 EXAMPLE 2) Evaluate a) 24 ÷ –3 b) –15 ÷ 3 c) –49 ÷ –7 d) –2 × 5 × –6 × 4 e) (–2 + –3)(9 – 11) SOLUTION 24 ÷ –3 = –8 –15 ÷ 3 = –5 –49 ÷ –7 = 7 –2 × 5 × –6 × 4 = 240 (–2 + –3)(9 – 11) = –5 × –2 = 10 Note: An integer multiplied by zero gives an answer of zero, for example –6 × 0 = –6 Zero divided by any integer other than zero, is zero, for example 0 ÷ 9 = 0 Multiplying an integer by 1 does not change the integer, for example –8 × 1 = –8 Division by zero is undefined, for example 3 ÷ 0 = undefined Exercise 1.17 Calculate: 1) 0 × 24 = ……………… 2) –6 × 1 = …………… 3) 1 × –6 = …………...... 4) –2 × 7 = ……………… 5) 5 × –4 = ……………. 6) –4 × –1 = …………… 7) –5 × 0 = ……………… 8) 15 × 1 = ……………. 9) 0 ÷ 5 = …………….. 10) –24 × –10 = ………….. 11) –9 ÷ 0 = ……………. 12) –1 × 4 = …………… 13) –12 ÷ –4 = …….. 14) –8 ÷ –2 = …………... 15) –320 ÷ 10 = ………… 16) –2 × –3 × –1 × 5 17) –3 × 4 × –2 × 5 18) 4 × –5 × 7 × – 2 = ………………………. 19) (–2 + –1)(–4 +–5) = ………………………. 22) 56 ÷ –8 –4 = ……………………… 25) (–5 × 2) × 3 × –2 = ……………………… = …………………… 20) 28 ÷ –7 × –2 = ……………………. 23) –24 ÷ (7 – 4) = ……………………. 26) 7 × (–3 – 2) = …………………….. 18 = ……………………. 21) –4(–1 – 6) = ……………………. 24) 12 ÷ –4 + 2 = ……………………. 27) 24 ÷ –3 × 4 ÷ –2 = ……………………. NCV2 – Chapter 1: Natural numbers, whole numbers and integers § 1.1 1.18 THE SET OF INTEGERS So far we have worked with three sets of numbers. The set of integers: Z = {……; -3; -2; -1; 0; 1; 2; 3; ………} The set of whole numbers: W = {0; 1; 2; 3; …………….} The set of natural numbers: N = {1; 2; 3; 4; ……………..} Exercise 1.18 1) Answer the following: a) In which set/s of numbers do negative numbers occur? ……………………………… b) Is there a smallest integer? …………………………………………………………..... c) Is there a greatest integer? ……………………………………………………………… 2) In which set/s of number do you find each of the following numbers? a) 3 …………………………………………………………………… b) 1 …………………………………………………………………… c) –9 ………………………………………………………………….. d) 0 …………………………………………………………………… e) 27 ………………………………………………………………….. f) 14 ………………………………………………………………….. g) –41 ………………………………………………………………… h) 102 ………………………………………………………………… i) –15 ………………………………………………………………… j) 200 ………………………………………………………………… 3) Answer True or False: a) The set of natural numbers includes zero ……………………………………………… b) There are negative numbers in N0 ……………………………………………………… c) Zero is an integer but not a natural number …………………………………………… d) Zero is a whole number but not an integer ……………………………………………. e) All integers are whole numbers ……………………………………………………….. f) All natural numbers are whole numbers ………………………………………………. g) All natural numbers are integers ………………………………………………………. h) All whole numbers are integers ……………………………………………………….. 19 NCV2 – Chapter2: Fractions and Decimals CHAPTER 2 Fractions and decimals In this chapter you will: • Define fractions • Find equivalent fractions • Compare fractions • Convert between mixed numbers and improper fractions • Add and subtract fractions with the same denominators • Add and subtract fractions with different denominators • Multiply fraction • Find the reciprocal of a number • Divide fractions • Convert fractions to decimals • Compare the sizes of decimals • Multiply decimals • Divide decimals • Round off to the nearest whole number and to 1 or 2 decimal places • Round off and estimate answers • Work with fractions and decimals on the scientific calculator • Add and subtract fractions on a scientific calculator • Use a scientific calculator to convert from recurring decimals to fractions This chapter covers material from Topic 1: Numbers SUBJECT OUTCOME 1.1: Use Computational Tools and Strategies and Make Estimates and Approximations Use a scientific calculator competently and efficiently Learning Outcome 1: Execute algorithms appropriately in calculations Learning Outcome 2: SUBJECT OUTCOME 1.2: Demonstrate understanding of numbers and relationships among numbers and number systems and represent numbers in different ways Identify rational numbers and convert between terminating or recurring decimals like Learning Outcome 1: a ; a; b ∈ Z; b ≠ 0 b 20 NCV2 – Chapter2: Fractions and Decimals § 2.1 2.1 FRACTIONS 1 4 1 1 , -1 , -3,6 and 4,5. 2 4 Examples of fractions and decimals are 3 , The position of these numbers can be shown on a number line: –4 –3 –2 -1 0 2 3 1 2 -1 1 -3,6 1 4 4 5 31 4,5 4 Note: • the arrows on the ends of the number line show that it extends in both directions • there is an infinite number of positive and negative numbers along the number line This new set of numbers is called the set of rational numbers. Between any two integers there are an infinite number of rational numbers, so it is impossible to list them all. Note: • In the fraction 2 , 2 is called the numerator and 3 is called the denominator. 3 4 is a proper fraction, since the numerator is less than the denominator 5 12 • is an improper fraction, since the numerator is greater than the denominator 5 3 • 2 is a mixed number, since it is made up of a whole number and a proper fraction 4 • 1 5 1 5 1 5 1 5 1 5 o The rectangle has 5 equal parts. o Each piece is 1 part of the 5 equal parts. o Each piece is 1 1 of the rectangle: 5 × = 1 5 5 o The shaded area is 3 parts of the 5 equal parts, so it is written as 3 of the rectangle. 5 Exercise 2.1 For each of the figures below, answer the following questions: A B C A 1) How many equal parts does the figure have? 2) How many of these parts are shaded? 3) What fraction is the shaded area of the whole figure? 21 B C NCV2 – Chapter2: Fractions and Decimals § 2.2 EQUIVALENT FRACTIONS FRACTIONS We call 1 2 3 4 1 2 3 4 , , and equivalent fractions. This is because = = = . 12 12 3 6 9 3 6 9 We can find equivalent fractions by: • multiplying the numerator and the denominator by the same number • dividing the numerator and the denominator by the same number Examples: 1) 2) 3) 2 4 = , by multiplying both the numerator and the denominator by 2 3 6 2 6 = , by multiplying both the numerator and denominator by 3 3 9 12 6 = , dividing both the numerator and the denominator by 2 24 12 4) 6 3 = , dividing both the numerator and the denominator by 2 12 6 5) 3 1 = , dividing both the numerator and the denominator by 3 6 2 2 4 6 = = 3 6 9 12 6 3 = = = 24 12 6 Exercise 2.2 1) Fill in the missing numbers to make the fractions equivalent: b) 2 c) 2 10 a) 1 = = = e) i) m) q) 2 8 4 = 2 8 3 21 = 7 3 9 12 2 = 18 12 = 4 16 f) j) n) 10 = 3 15 21 7 = 8 36 3 = 48 24 6 = 7 r) g) k) d) 5 3 9 = 4 3 = 5 45 o) 81 = 3 = 4 8 2 6 = 5 5 = 8 16 h) l) p) 5 9 4 = 2 9 2) Write down the first five equivalent fractions starting with the number given: a) c) e) g) 1 5 2 3 5 8 3 4 = = = = b) = = = = d) = = = = f) = = = = h) 22 4 = 7 4 = 9 11 = 12 2 = 5 = = = = = = = = = = = = NCV2 – Chapter2: Fractions and Decimals § 2.3 COMPARING FRACTIONS 1) When the denominators are the same: Is 4 3 greater than or less than ? 5 5 4 1 5 5 3 5 1 5 From the diagrams we can see that 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 4 3 4 3 is greater than . We write > . 5 5 5 5 2) When numerators are the same: Which is the smaller, 1 8 1 6 5 5 or ? 8 6 1 1 8 8 1 8 1 6 1 8 1 6 From the diagram we can see that 1 8 1 6 1 8 1 8 1 6 1 6 5 8 5 6 5 5 5 5 is less than . So < . 8 6 8 6 3) When numerators and denominators are different: Compare 4 5 with . Which is greater? 5 6 Choose the lowest multiple (LCM) of 5 and 6, which is 30. Then write each fraction as an equivalent fraction having a denominator of 30. 4 4 6 24 = × = 5 5 6 30 5 5 5 25 = × = 6 6 5 30 But 25 24 5 4 > , so > . 30 30 6 5 Exercise 2.3 1) Which fraction is greater? Show all calculations. a) b) c) d) 4 7 5 7 7 9 2 3 6 ………………………………………………………………………………..... 7 5 or …………………………………………………………………………………. 9 20 or ……………………………………………………………………………..…. 27 5 or …………………………………………………………………………..……... 8 or …………………………………………………………………………………. 2) Which fraction is smallest, 4 3 5 or or ? Show all calculations. 4 5 6 ………………………………………………………………………………………….…… …………………..…………………………………………………………………………... 23 NCV2 – Chapter2: Fractions and Decimals § 2.4 CONVERTING FRACTIONS 1. Write fractions in simplest form: A fraction is in simplest form when its numerator and denominator are as small as possible. They cannot be divided by any further number and still remain whole numbers. Examples: 24 15 , b) in simplest form (simplify) 30 45 24 24 ÷ 6 4 15 15 ÷ 5 3 3÷3 1 Solution: a) = = b) = = = = 30 30 ÷ 6 5 45 45 ÷ 5 9 9÷3 3 Write a) 2. Convert improper fractions to mixed numbers: Examples: 6 2 2 2 7 3 3 1 1 1 = + + =1+1+1=3 2) = + + = 1 + 1 + = 2 2 2 2 2 3 3 3 3 3 3 9 5 4 4 9 4 3) = + = 1 As a shortcut: = 1 remainder 4 = 1 5 5 5 5 5 5 1) 3. Convert mixed numbers into improper fractions: Examples: 1 1 7 1 8 3 3 4 4 3 11 =1+ = + = 2) 2 = 1 + 1 + = + + = 4 4 4 4 4 4 7 7 7 7 7 3 2× 4 + 3 11 As a shortcut: 2 = = 4 4 4 1) 1 Exercise 2.4 1) Write each fraction in simplest form / simplify: 4 = …………………... 10 12 d) = …………………... 27 a) 14 = …………………… 28 16 e) = …………………… 28 b) 25 = ……………… 40 33 f) = ……………… 55 c) 2) Convert these improper fractions to mixed numbers in simplest form: 12 48 = ………………………………………… b) = …………………..……………. 5 7 64 38 c) = ………………………………………… d) = …..…………………………..…. 4 9 75 44 e) = …….………………………………….. f) = …………………………….…… 9 8 a) 3) Convert these mixed numbers to improper fractions: 1 3 = ……………………………………….... b) 2 = ………………………………… 4 5 5 2 c) 3 = ………………………………………… d) 4 = ………………………………… 8 7 a) 1 24 NCV2 – Chapter2: Fractions and Decimals § 2.5 ADDING AND SUBTRACTING FRACTIONS WITH THE SAME DENOMINATORS To add or subtract fractions: Make sure their denominators are the same – use equivalent fractions if they are not Add or subtract the numerators Simplify the answer EXAMPLES 1) 1 8 2) 3) 1 8 1 8 1 8 1 8 1 8 7 3 4 4÷4 1 – = = = 2 8 8 8 8÷4 1 8 1 16 1 15 7 2 – = 2( ) – = – = =1 8 8 8 8 8 8 8 1 8 1 8 Note: 1 = 5 2 7 + = 8 8 8 2 3 4 5 = = = etc. 2 4 3 5 Exercise 2.5 Calculate and simplify your answers: 2 = …………………………………………………………………………………… 7 3 2) + = …………………………………………………………………………………… 5 4 3) – = ……………………………………………………………………………………. 9 2 4) – = …………………………………………………………………………………… 6 1 5) 1 – = …………………………………………………………………………………….. 5 3 6) 1 – = ………………………..…………………………………………………………… 4 1 3 2 7) + + = ……………………………………………………………………………… 8 8 8 2 1 8) 1 + – = ……………………………………………………………………………….. 9 9 2 1 9) 1 – – = ……………………………………………………………………………….. 7 7 14 11 2 10) – + = …………………………………………………………………………… 15 15 15 4 1 11) 2 – + =.......................................................................................................................... 5 5 3 1 12) 1 + = ………………………………………………………………………………….. 8 8 7 3 1 13) – + = …………………………………………………………………………... 20 20 20 4 14) 3 – = …………………………………………………………………………………….. 5 1) 3 7 1 5 7 9 5 6 + 25 NCV2 – Chapter2: Fractions and Decimals § 2.6 ADDING AND SUBTRACTING FRACTIONS WITH DIFFERENT DENOMINATORS To add or subtract fractions like 1 1 and they must each be written as equivalent 2 3 fractions with the same denominator. EXAMPLES Calculate: 1) 1 1 + 2 3 2) 5 1 – 12 3 3) SOLUTION 1) The LCM of 2 and 3 is 6. 3) 1 1× 3 3 1 1× 2 2 = = and = = 2 2×3 6 3 3× 2 6 1 1 3 2 5 = + = So + 2 3 6 6 6 3 2 + The LCM of 4 and 5 is 20. 4 5 3× 5 2× 4 = + 4×5 5× 4 15 8 = + 20 20 23 3 = =1 20 20 3 2 + 4 5 4) 3 1 3 –1 4 8 2) The LCM of 12 and 3 is 12 5 1 5 1× 4 5 4 1 – = – = – = 12 12 12 12 12 3 3× 4 4) 3 1 – 1 3 The LCM of 8 and 4 is 8 4 8 25 7 = – 4 8 25 7×2 = – 4× 2 8 25 14 11 3 = – = =1 8 8 8 8 Exercise 2.6 Calculate without a calculator. Write your answers as mixed numbers where possible. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 1 1 + = …………………………………………………………………………………. 2 8 3 5 + = …………………………………………………………………………………. 4 8 4 1 – = ………………………………………………………………………………….. 5 3 5 2 3 + + = …………………………………………………………………………… 12 3 8 5 2 1 – = ……………………………………………………………………………….. 6 3 5 2 3 – 1 = ………………………………………………………………………………. 8 3 1 5 1 1 – + = …………………………………………………………………………… 3 6 6 1 3 2 – 1 = ………………………………………………………………………………. 2 4 7 5 2 – 1 = ………………………………………………………………………………. 8 6 7 2 4 – 2 = …………………………………………………………………………......... 9 3 26 NCV2 – Chapter2: Fractions and Decimals § 2.7 MULTIPLICATION OF FRACTIONS EXAMPLE Find the following products: 1) 2 4 × 5 9 2) 3 3 × 9 14 3 5 3) 2 × 1 SOLUTION 1) 2 × 4 = 2 × 4 = 8 5 9 5× 9 2 13 4) 3 3 1 × 2 7 2) 45 3 3 3× 3 1 × = = 9 14 9 × 14 4 3 is a factor of 3 and 9, so cancel 3) 3 2 13 15 2 ×1 = × 5 13 5 13 13 ×15 cancel numerator and denominator by 13 and 5 = 5 × 13 =3 4) 3 3 1 24 1 24 × 1 × = × = 2 2 7× 2 7 7 = 12 5 =1 7 7 2 is a factor of 24 and 2, so you can cancel by 2 The answer is given as a mixed number Exercise 2.7 Find the following products. Whenever possible, write your answers as mixed numbers. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 9 3 × = ………………………………………………………………………………. 2 10 2 3 × = ………………………………………………………………………………... 7 5 5 9 × = ……………………………………………………………………………… 6 10 13 4 × = ……………………………………………………………………………… 16 5 2 3 × 3 = ……………………………………………………………………………… 4 5 1 2 3 × = ………………………………………………………………………………. 4 9 1 2 2 × 12 = …………………………………………………………………………… 19 3 3 1 1 × 2 = ……………………………………………………………………………… 2 5 3 5 8 × × = …………………………………………………………………………. 4 9 25 3 2 1 × 2 × 3 = ……………………………………………………………………….. 4 4 3 1 2 3 4 × × × = ……………………………………………………………………. 2 4 3 5 8 2 2 × 1 = …………………………………………………………………………….. 9 13 27 NCV2 – Chapter2: Fractions and Decimals § 2.8 THE RECIPROCAL RECIPROCAL OF A NUMBER When we multiply any number by its reciprocal, we get 1 as an answer. 1 1 , because 2 × = 1. 2 2 1 is 6, because × 6=1 6 3 2 3 is , because × =1 2 2 3 The reciprocal of 2 is 1 6 2 The reciprocal of 3 The reciprocal of EXAMPLES What is the reciprocal of 1) 3? 2) 3 ? 4 SOLUTIONS The reciprocal of 3 is The reciprocal of 1 1 , since 3 × = 1 3 3 3 4 3 4 is , since × =1 4 3 4 3 1 5 3) –5 The reciprocal of –5 is − , since -5 × − 1 8 3 5) 2 ? 8 1 =1 5 1 1 is –8, since − × -8 = 1 8 8 3 8 3 19 19 8 The reciprocal of 2 is , since 2 = and × =1 8 19 8 8 8 19 4) − ? The reciprocal of − Exercise 2.8 1) The reciprocal of 6 is ………… since ………………………………………………… 2) The reciprocal of 2 is ………… since ………………………………………………... 3 3) The reciprocal of −4 is ………… since ……………………………………………….. 3 4) The reciprocal of –2 is ………… since ………………………………………………… 5) The reciprocal of 5 is ………… since …………………………………………………. 9 6) The reciprocal of −6 is ………… since ………………………………………………… 7 7) The reciprocal of 3 8) The reciprocal of –9 is ………… since …………………………………………………. 9) The reciprocal of 2 1 is ………… since ………………………………………………… 2 3 is ………… since ……………………………………………….. 4 28 NCV2 – Chapter2: Fractions and Decimals § 2.9 DIVISION OF FRACTIONS 1) If we are asked to calculate 3 ÷ 1 , we are actually being asked "how many sixths are 6 there in 3 wholes". 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 There are 18 sixths in 3 wholes, so 3 ÷ 2) 6 ÷ 2 = 3 but 6 × 1 6 1 6 1 = 18 6 1 6 1 6 OR 1 6 3 ÷ 1 6 1 6 1 6 1 6 1 6 1 3 3× 6 18 = = = = 18 1 1 1 6 ×6 6 6 1 1 = 3, so division by 2 is the same as multiplying by , its reciprocal. 2 2 As a shortcut, when you divide by a fraction, you multiply by its reciprocal. EXAMPLES Calculate SOLUTIONS 1) 2 1 ÷ 3 3 2 1 2 3 2 3/ ÷ = × = × =2 3 1 3 3 3/ 1 2) −6 3 ÷ 14 7 −6 3 −6 14 −2 6/ 14 ÷ = × = × = –4 14 7 7 3 7/ 3/ 9 9 1 1 ÷ 18 = × = 10 10 18 20 1 1 7 7 7 2 2 2 ÷3 = ÷ = × = 3 2 2 3 3 7 3 9 ÷ 18 10 1 1 4) 2 ÷ 3 3 2 3) Exercise 2.9 Find the following quotients and simplify your answers: 1) 2) 3) 4) 5) 6) 7) 8) 9) – 5 5 ÷ = …………………………………………………………………………… 12 6 2 ÷ 6 = ……………………………………………………………………………… 3 1 45 ÷ 7 = …………………………………………………………………………… 2 3 1 4 ÷ = ……………………………………………………………………………. 4 4 3 –3 ÷ = …………………………………………………………………………….. 4 5 1 2 ÷ 3 = …………………………………………………………………………… 2 8 4 1 4 ÷ 1 = …………………………………………………………………………… 5 5 4 1 3 ÷1 = ………………………………………………………………………….. 14 7 2 5 1 ÷ ÷ = ………………………………………………………………………. 7 7 10 29 NCV2 – Chapter2: Fractions and Decimals § 2.10 CONVERTING FRACTIONS TO DECIMALS To convert a fraction to a decimal, divide the denominator into the numerator. Note: We can write any number of • 2 = 2,0 = 2,00 = 2,0000…. zeros after the decimal comma. • 3,5 = 3,50 = 3,500 = 3,5000 = 3,50000…. EXAMPLES Convert the following fractions to decimals 1) SOLUTIONS 1 2 1 1, 0 = = 0,5 2 2 3 7 7, 000 1 = = = 1,75 4 4 4 19 = 0,019 1000 3 4 19 3) 1000 2) 1 Exercise 2.10 1) Convert these fractions to decimals: a) 2 b) 3 7 = ……….. 10 c) e) f) 5 = ………….. 8 g) 1 = ……….. 2 7 = ………... 80 7 = ………….. 8 10 = …………. 8 d) h) 2 = ………. 100 45 = ………... 150 Recurring decimal notation: Some answers will be a recurring or repeating decimal. In a recurring decimal, a digit or group of digits, which occur after the decimal comma, are repeated. The numbers which recur are written under a line, or under and between dots. EXAMPLES Write as a decimal: 1) SOLUTIONS • 1 = 0,333333.... = 0,3 or 1,3 3 1 3 2) 4 23 99 4 3) 9 152 999 9 4) 3 97 666 • • 23 = 4,23232323..... = 4, 23 or 4, 23 99 • • 152 = 9,152 152 152 ..... = 9,152 999 97 3 = 3,1 456 456 456…. = 3,1456 , since the 1 does not recur 666 2) Write as a recurring decimal: a) 3,88888……… = ………………….. c) 5,238238238238 …….. = …………… e) 1 2 = ……………………………….. 3 b) 12,45454545……….. = ………………. d) 3,417171717…… = ………………….. f) 30 5 = …………………………………… 6 NCV2 – Chapter2: Fractions and Decimals § 2.11 COMPARING AND CONVERTING Comparing the size of decimal numbers 1) Compare the signs: 1,2 > – 6,7 since a positive number is always greater than a negative number. – 1,2 > – 3,8 since on the number line, the greater negative number is further from 0. 2) Compare the whole numbers: 3,49 is greater than 2,9 since 3 > 2. 3) Compare the numbers in the first decimal place: If the integer parts are the same, then we compare the numbers in the first decimal place. e.g. 4,69 is greater than 4,298 since in the first decimal place, 6 > 2. 4) Compare the second decimal place: If the numbers in first decimal place are the same, compare the numbers in the second decimal place. e.g. 2,369 9 is less than 2,391 97 since 6 < 9. We can continue in this way. 5) Compare recurring numbers: • 1, 3 > 1,3 since 1,3333… is greater than 1,3 Exercise 2.11 1) Insert = or > or <: a) 3,1659 …... 3,1700 b) 0,00879 …... 0,00891 c) 5,1903 …... 5,1903000 d) 1,001 …... 0,009 e) –3,6 …… 0,1 f) –2,78 …… –2,99 h) –0,019 …… 0 i) 11,009 …… 11,0019 • g) 2,5 …… 2, 5 Converting from a decimal to a fraction We write one zero in the denominator for each decimal place EXAMPLES Convert to a fraction and simplify: 1) 0,5 SOLUTION 5 10 1) 0,5 = 4) 3,79 = 3 79 379 = 100 100 2) 0,34 34 100 2) 0,34 = 5 0,005 = 3) 0,056 4) 3,79 3) 0,056 = 5 1 000 2) Convert to a fraction and simplify: a) 3,65 = ……………….. b) c) 7,5 = ………………... d) 2,04 = ……………….. e) 0,045 = ……………….. 31 5) 0,005 0,1 = ……………….. 56 1 000 NCV2 – Chapter2: Fractions and Decimals § 2.12 MULTIPLYING DECIMALS EXAMPLES Calculate: 1) 1,2 × 0,3 SOLUTIONS 1,2 × 0,3 = 12 3 36 × = = 0,36 10 10 100 Shortcut: Multiply the numbers ignoring the decimal comma. 3 × 12 = 36. There is 1 decimal place in each number, so there will be 1 + 1 = 2 decimal places in the answer. 2) 0,04 × 200 4 200 × = 8,00 1 100 0,04 × 200 = Shortcut: 4 × 200 = 800. There are 2 decimal places in the answer 13 1 × = 0,130 10 10 3) 1,30 × 0,1 1,30 × 0,1 = 4) 0,0042 × 1,2 Shortcut: 130 × 1 = 130. There are 2 + 1 = 3 decimal places in the answer. 0,0042 × 1,2 = 0,00504 Shortcut: 42 × 12 = 504. There are 4 + 1 = 5 decimal places in the answer. Add zeros to the left of 504 in order to get the correct number of decimal places Exercise 2.12 Simplify without using a calculator: 1) 0,25 × 4 = ………………………………………………………………………………….. 2) 0,004 × 1,6 = ………………………………………………………………………………. 3) 14,5 × 1,1 = ………………………………………………………………………………... 4) 1,25 × 0,4 = ………………………………………………………………………………... 5) 21,5 × 0,02 = ………………………………………………………………………………. 6) 1,004 × 10 = ……………………………………………………………………………….. 7) 0,02 × 500 = ……………………………………………………………………………….. 8) 1,300 × 0,40 = ……………………………………………………………………………... 9) 0,005 × 1,02 = ……………………………………………………………………………... 10) 1,000 × 0,09 = ……………………………………………………………………………... 11) 250 × 4,0 = ………………………………………………………………………………… 12) 2,5 × 0,002 = ………………………………………………………………………………. 32 NCV2 – Chapter2: Fractions and Decimals § 2.13 DIVIDING DECIMALS To divide decimals: Count the number of decimal places in the denominator. Make the denominator a whole number by multiplying both the numerator and the denominator by the same multiple of 10 as the number of decimal places in the denominator. EXAMPLE Calculate 1) 4,2 ÷ 0,02 SOLUTION 4,2 ÷ 0,02 = 4, 2 4, 2 ×100 420 = = = 210. 2 0, 02 0, 02 × 100 Note: There are two decimal places in the denominator, so multiply both numerator and denominator by 100 2) 32 ÷ 0,005 32 ÷ 0,005 = 32 32 ×1 000 32 000 = = = 64 000 0, 005 ×1 000 0, 005 5 Note: There are 3 decimal places in the denominator, so multiply both numerator and denominator by 1 000 Exercise 2.13 1) Calculate the following quotients without using a calculator: a) 350 ÷ 100 = ………………………... b) 37,4 ÷ 11 = ………………………… c) 12,65 ÷ 0,05 = ……………………... d) 300,05 ÷ 0,005 = …………………... e) 1,004 ÷ 0,02 = ……………………... f) g) 3,411 ÷ 0,9 = ………………………. h) 45,550 ÷ 0,5 = ……………………... i) 39,3 ÷ 0,3 = ………………………... j) 284,6 ÷ 1,2 = ………………………. 1,44 ÷ 1,2 = ………………………... 2) Calculate the following products and quotients without using a calculator: a) 600 ÷ 0,003 = ………………………………………………………………………… b) 0,002 × 60 = ………………………………………………………………………….. c) 120 × 0,4 = …………………………………………………………………………… d) 36 ÷ 1,2 = …………………………………………………………………………….. e) 0,2 × 0,3 × 200 = …………………………………………………………………….. f) 4 500 ÷ 0,5 ÷ 1 000 = ………………………………………………………………... g) 2,4 ÷ 0,02 × 5 = ……………………………………………………………………… h) 900 ÷ 0,03 × 100 = …………………………………………………………………... i) 1,08 ÷ 9 ÷ 0,4 = ……………………………………………………………………… j) 12,1 ÷ 0,11 × 0,002 = ………………………………………………………………… 33 NCV2 – Chapter2: Fractions and Decimals § 2.14 ROUNDING OFF TO THE NEAREST WHOLE NUMBER AND TO 1 OR 2 DECIMAL PLACES 1. Rounding off to the nearest whole number: Underline the number in the units position. If the number to the right is 5 or greater, round up to the next whole number. If it is less than 5, round down to the given whole number. EXAMPLE Round the following numbers to the nearest whole number 1) 4,276 2) 12,53 3) 0,199 SOLUTION 4,276 ≈ 4, since 2 < 5 12,53 ≈ 13, since 5 ≥ 5 0,199 ≈ 0, since 1 < 5 Exercise 2.14 1) Round off to the nearest whole number: a) 3,29 ≈ ………. b) 21,51 ≈ ……... c) 0,482 ≈ …….. d) 0,59 ≈ …….. e) 109,51 ≈ ……. f) 120,49 ≈ …… g) 2 000,82 ≈ … h) 199,67 ≈ ….. 2. Rounding off or correcting to 1 decimal place: Underline the number in the first decimal place. If the number to the right is 5 or greater, round up the number in the first decimal place. If it is less than 5, round down. Note: To round off or correct to any number of decimal places, look at the next decimal place to see whether it is 5 or greater. If so, round up. If not, round down. EXAMPLE Round the following numbers to 1 decimal place: 1) 3,551 SOLUTION 3,551 ≈ 3,6 since 5 ≥ 5 2) 11,949 11,949 ≈ 11,9 since 4 < 5 3) 3,991 3,991 ≈ 4,0 since 9 ≥ 5 2) Round off to 1 decimal place: a) 23,451 = …….. b) 36,623 = …….. c) 0,025 = …….. d) 329,95 = …….. e) 145,459 = …… f) 6,019 = …….. g) 0,009 = …….. h) 26,951 = …….. c) 99,995 = ……. d) 0,0051 = …….. 3) Round off to 2 decimal places: a) 3,5608 = …….. b) 12,915 = …….. 34 NCV2 – Chapter2: Fractions and Decimals § 2.15 ROUNDING OFF AND ESTIMATING ANSWERS EXAMPLE 1) Use a calculator to calculate 6,179 × 180,5 correct to 2 decimal places. 2) Check your answer by rounding each number to the nearest whole number. 3) Check your answer by rounding each number to 1 significant figure SOLUTION 6,179 × 180,5 = 1 115,3095… = 1 115,31 correct to 2 decimal places. 6,179 × 180,5 ≈ 6 × 181 = 1 086 correct to the nearest whole number. 6,179 × 180,5 = 6 × 200 = 1 200 Exercise 2.15 1) a) Calculate 1, 26 × 25,3 using a calculator correct to the nearest whole number. 0,995 6 × 10, 729 b) Estimate the answer by rounding each number to one significant figure. c) Does your answer seem to be correct? 2) a) Use a calculator to calculate 207,8 × 15,995 correct to 1 decimal place. b) Check this answer by rounding each number to one significant figure. c) Estimate the answer by rounding each number to the nearest whole number d) Do these estimates show that your answer is reasonably correct? 35 NCV2 – Chapter2: Fractions and Decimals § 2.16 FRACTIONS AND DECIMALS ON THE SCIENTIFIC CALCULATOR Fraction keys on the calculator enable you to: Simplify fractions (i.e. write them in their simplest form) Convert between mixed numbers, improper fractions and decimals Do calculations involving fractions The CASIO fx-82ES PLUS Scientific Calculator: has fraction keys marked and . is shown above the key and is used by pressing shift and then The arrow keys move the cursor around the blocks on the display. . Note: • First get the calculator in the correct mode by entering the following keys: MODE: Select 1: COMP SETUP: Select 1: Mth IO • Repeated entering of S ⇔ D will enable you to go back and forth between a decimal and an improper fraction, i.e. you toggle from one to the other. EXAMPLE 1) Use your calculator to: a) Simplify b) Write SOLUTION KEY SEQUENCE 18 27 18 18 as a decimal 27 2) Use your calculator to write 2 ANSWER 27 = 2 3 0,6666 S⇔ D 5 9 a) as an improper fraction 2 b) as a decimal 5 9 = S⇔ D 23 9 2,5555.. 3) Use your calculator to: a) Simplify 36 15 36 36 as a mixed number 15 36 c) Write as a decimal 15 b) Write shift 15 = S⇔ D S⇔ D 12 5 2 2 5 2,4 Exercise 2.16 Use your calculator to: 24 ……………………………. 18 24 as a decimal … 18 1) Simplify 3) Write 5) Write 2,54 as a mixed number …………………………………………………….. 24 as a mixed number ………….. 18 2) Write 4) Write 2,54 as an improper fraction ……. 36 NCV2 – Chapter2: Fractions and Decimals § 2.17 ADDING AND SUBTRACTING FRACTIONS ON A SCIENTIFIC CALCULATOR CALCULATOR EXAMPLES 1) Calculate 4 2 + , and write the answer as a) an improper fraction, b) a decimal and 5 3 c) a proper fraction. Key Sequence 4 5 + 2) Calculate 1 2 3 = (a) KS (b) KS 22 15 S⇔ D 1,466... shift S ⇔ D 5 3) Calculate 7 + 3 7 15 Display KS Display 69 28 shift S ⇔ D 2 13 Display KS Display 36 5 shift S ⇔ D 4 = 28 3 as a proper fraction 2 1 − 3 4 Key sequence 3 1 5 3 + . Write your answer as a proper fraction. 4 7 Key sequence 1 (c) 2 3 – 1 4 = 7 1 5 Exercise 2.17 12 as a decimal (correct to 1 decimal place) ................……….… 13 23 2) Use a calculator to write as 14 1) Use a calculator to write a) a mixed number …………………………………. b) a decimal (correct to 1 decimal place) ………………………………………… 3) Use a calculator to write each answer as i) a decimal (correct to 2 decimal places) and ii) a mixed number. a) 3 4 + = ….…………………………………………………………………………… 4 7 b) 7 11 7 – 5 = …………………………………………………………………………… 12 8 c) 1 6 5 + 4 = ……………………………………………………………………………. 7 8 d) 3 5 1 – 2 = ……………………………………………………………………………. 4 8 37 NCV2 – Chapter2: Fractions and Decimals § 2.18 USING A SCIENTIFIC CALCULATOR TO CONVERT FROM RECURRING DECIMALS DECIMALS TO FRACTIONS EXAMPLE Use your calculator to: • 5 1) Prove that 3,5 = 3 9 • SOLUTION KEY SEQUENCE 3,55555…… (until the display is full) = shift S ⇔ D • ANSWER 5 9 3 145 999 2) Convert 1, 145 to a mixed number 1,145145145 …… = shift S⇔ D 1 3) Convert 2,3 45 to a mixed number 2,34545454545… = shift S⇔ D 2 19 55 Note: • In all three examples it is necessary to fill the screen with the recurring decimals • In example 3, the 3 in the first decimal place does not recur Exercise 2.18 Use a calculator to write each recurring decimal as a fraction. Where possible, write as a mixed number. • 1) 0,3 = ……………………………………… 2) 2,8 = ……………………………………… • 3) 3,5 = ………………………………………. 4) 12, 46 = ……………………………………. 5) 1,444 ….. = ……………………………….. • • 6) 0, 35 = …………………………………….. • • 7) 5, 153 = ……………………………………. • 8) 1,2 4 = ……………………………………... 38 NCV2 – Chapter 3: Rational Numbers and Real Numbers CHAPTER 3 Rational numbers and real numbers In this chapter you will: • Find ratios of two or more quantities • Find equivalent ratios • Divide a quantity in a given ratio • Calculate rates • Work with ratios in direct proportion • Work with ratios in indirect proportion • Calculate percentages • Multiply and divide exponents • Work with zero exponents and negative exponents • Simplify products and quotients raised to a power • Raise an exponent to a power • Work with number sequences • Find the value of missing terms in a sequence • Decide whether numbers are rational numbers or not • Decide whether numbers are irrational numbers or not • Simplify surds • Decide whether numbers are real or not • Use the real number diagram to answer questions This chapter covers material from Topic 1: Numbers SUBJECT OUTCOME 1.1: Use Computational Tools and Strategies and Make Estimates and Approximations Use a scientific calculator competently and efficiently Learning Outcome 1: Execute algorithms appropriately in calculations Learning Outcome 2: SUBJECT OUTCOME 1.2: Demonstrate understanding of numbers and relationships among numbers and number systems and represent numbers in different ways Identify rational numbers and convert between terminating or recurring decimals like Learning Outcome 1: a ; a; b ∈ Z; b ≠ 0 b Learning Outcome 2: Learning Outcome 3: Learning Outcome 4: Know, understand and apply the laws of exponents Convert surds into rational forms Identify and work with arithmetic progressions, sequences and series 39 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.1 RATIO A ratio compares two or more quantities. In the nine identical rectangles below, three are shaded and six are unshaded. We say that: • the ratio of the shaded to the unshaded rectangles is 3 to 6 which we write 3 : 6 • the ratio of the unshaded to the shaded rectangles is 6 to 3 which we write 6 : 3 • the ratio of the shaded rectangles to all the rectangles is 3 to 9 which we write 3 : 9 • the ratio of the unshaded rectangles to all the rectangles is 6 to 9 which we write 6 : 9 Note: • The ratios remain the same no matter which three of the nine rectangles are shaded • The order in which the ratio is written is important • The ratio is written without units, since we are comparing rectangles with rectangles • The ratio 3 : 6 can be written as the fraction 3 1 = =1:2 2 6 • 3 : 6 = 1 : 2 are equivalent ratios EXAMPLE Fruit juice is made by mixing 1 cup of concentrate with 3 cups of water. Find the ratio of the number of cups of: 1) concentrate to water 2) concentrate to mixture 3) mixture to water 4) water to mixture 5) water to concentrate 6) mixture to concentrate SOLUTION 1:3 1:4 4:3 3:4 3:1 4:1 Note: • the unit here is the cup • A ratio is always written without units Exercise 3.1 In a class of 30 learners, 12 are boys. Find the following ratios and write in simplest form: 1) number of boys : number of girls ………………………………………… 2) number of girls : number in class …………………………………………. 3) number of boys : number in class …………………………………………. 4) number of girls : number of boys ………………………………………….. 5) number in class : number of boys ………………………………………….. 6) number in class : number of girls ………………………………………….. 7) number of boys and girls : number in class ………………………………… 40 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.2 EQUIVALENT RATIOS EXAMPLE SOLUTION 1 1) A recipe uses 1 cups of 2 Flour : sugar : milk flour, 1 cup of sugar and 3 4 of a cup of milk. Express the ratio of flour to sugar to milk in its simplest form. 2) Palesa receives R 90 pocket money per month. The ratio of the pocket money received by Palesa and Thabo is 3 : 4. How much does Thabo receive per month? 1 3 :1: 2 4 3 3 = :1: The LCM of 2 and 4 = 4, so multiply by 4. 2 4 =1 =6:4:3 Note: • The quantities to be compared have the same unit (in this case cups) • A ratio has no units Let Thabo receive Rx. P : T = 3 : 4 = 90 : x 3 90 = 4 x 3 × 4 x 90 × 4 x = (The LCM of 4 and x is 4x) 4 x 3x = 4 × 90 = 360 x = 360 3 = R 120 Thabo receives R 120. Exercise 3.2 1) Mortar (for bricklaying) is made by mixing 3 wheelbarrows of cement with 2 wheelbarrows of sand and 1 wheelbarrow of water. a) Find the ratio of cement : sand : water ……………………………… b) How much sand will I need to order to mix with 9 wheelbarrows of cement? …………………………………………………………………………………………. c) How much cement and water will I need to order to mix with 8 wheelbarrows of sand? …………………………………………………………………………………………. d) How much cement and sand will I need to order to mix with 6 wheelbarrows of water? ………………………………………………………………………………………….. 2) John, who is 16 years old and Thandi, who is 11 years old, receive pocket money in the ratio of their ages. John is given R 80 per month. How much is Thandi given? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 41 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.3 DIVIDING A QUANTITY IN A GIVEN RATIO To divide something in a given ratio, find a) The total number of parts from the ratio b) What one part is EXAMPLE Divide 66 sweets between Jim and Busi in the ratio of 6 : 5. SOLUTION Suppose that the sweets are divided into 6 + 5 = 11 equal shares. 6 × 66 = 36 sweets. 11 5 Busi gets × 66 = 30 sweets. 11 Jim gets Note: • As a calculation check, 36 + 30 = 66 sweets Exercise 3.3 1) Divide R650 between Susan and James in the ratio of 7 : 3. 2) The perimeter of a triangle is 612 cm. The sides of the triangle are in the ratio 3 : 4 : 5. What is the length of the longest side? 3) A green paint is mixed from blue and yellow paint in the ration 3 : 5. How much of each colour is needed to make 40 litres of this green paint? 42 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.4 RATE A rate links two quantities. It tells us how one quantity compares or changes with another. The unit of a rate is usually given as “somethings” per “something” e.g. kilometres per hour, cents per litre, revolutions per minute. “Per” means “for each” or “for every”. It is often shortened to p or /. EXAMPLE 1) A box of 15 naartjies costs R16,65. A bag of 8 naartjies of the same size costs R9,10. Which is the best deal? 2) A motorist travels 56 km in 29 minutes. What is his average speed in km/hour? SOLUTION 16,65 = R1,11 15 9,10 BAG: 8 naartjies cost R9,10 so 1 naartjie costs R = R1,14 8 BOX: 15 naartjies cost R16,65 so 1 naartjie costs R The bag is the best deal. The larger quantity is more expensive. Note: In each case you calculated the price per 1 naartjie or the unit price, which is a rate. In 29 minutes, 56 km is travelled. So in 1 minute, 56 ÷ 29 = 1,931….km are travelled (do not round off) In 60 minutes 60 × 2,115….= 115,862… km are travelled. The average speed ≈ 116 km per hour. Note: Average speed = distance time Exercise 3.4 1) A 500g tin of jam is marked R15,98. The 300g tin of the same jam costs R9,50. Which size is the better buy? 2) A car travels at a speed of 120 km per hour. How long will a trip of 2 520 km take if he travels at the same speed all the way? 3) My car has a petrol consumption of 8,5 litres per 100 km. a) How many litres of petrol will I need to travel 1 km? b) How many litres of petrol will I need to travel 420 km? c) How far, to the nearest km, can I travel on 20 litres? 43 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.5 DIRECT (or linear) PROPORTION When two ratios are equal to each other, we say that they are in proportion. When quantities are in direct proportion, one quantity increases in the same ratio as the other quantity increases, or one quantity decreases in the same ratio as the other quantity decreases. EXAMPLE 1) The price of a sweet is R3,45. What is the price of two sweets? SOLUTION Two sweets will cost 2 × R3,45 = R6,90. 2) Andile can run 5 km in 20 minutes. How far can he run at the same speed in 15 minutes? In 20 minutes Andile runs 5 km Note: The number of sweets and the money both doubled – they increased in the same ratio, so they are in direct (linear) proportion 5 km (less distance) 20 5 In 15 minutes (more time) he runs × 15 = 3,75 km 20 In 1 minute (less time) he runs ….. (more km) Note: The distance (km) decreased in the same ratio as the time (minutes) decreased, so the distance and the time are in direct (linear) proportion Exercise 3.5 1) Ben’s computer can print 1 800 words in 5 minutes. Working at the same rate, how many words can he print in a) 1 minute? ………………………………………………………………………….. ………………………………………………………………………….. b) 7 minutes? ………………………………………………………………………….. …………………………………………………………………………. 2) My diesel motorcar has a petrol consumption of 6,5 litres per 100 km. a) How many litres of petrol will I need to travel 1 km? ……………………………….… ………………………………………………………………………………………….. b) How many litres of petrol will I need to travel 350 km? ………………………………. ………………………………………………………………………………………….. c) How far can I travel on 10 litres (correct to 1 decimal place)? ………………………… ………………………………………………………………………………………….. 3) A factory produces 600 washers every 3 minutes. How many washers will it produce in 8 minutes? ………………………………………………………………………………….. ……………………………………………………………………………………………… ……………………………………………………………………………………………... 44 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.6 INDIRECT (inverse) PROPORTION When quantities are in indirect proportion, one quantity decreases in the same ratio as the other quantity increases. EXAMPLE Betty has enough sandwiches to give 5 children, 4 sandwiches each. However, 10 children arrive. How many sandwiches will each child receive, if they are shared equally? SOLUTION The total number of sandwiches available is 5 × 4 = 20. The number of sandwiches per child will be 20 ÷ 10 = 2. Note: The number of children has doubled and the number of sandwiches has halved, so the number of children and the number of sandwiches are in indirect (inverse) proportion. Exercise 3.6 1) 3 campers have enough food to last them for 15 days. 2 others join them. How many days will they take to finish the food if all eat the same quantity? 2) Seven builders complete a building project in 4 weeks. a) How long will 14 builders, working at the same rate, take to complete the project? b) How long will 2 builders, working at the same rate, take to complete the project? 3) It is estimated that a group of 6 plasterers can plaster a house in 9 days. Only 4 arrive for work. How long will it take these plasterers to finish the job? 45 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.7 PERCENTAGES “Percent” means “per hundred”, and is written %. We can convert a percentage to a fraction or a percentage to a decimal number. EXAMPLE 1) Convert 6% to a fraction SOLUTION 6% means 6 equal parts per 100 equal parts, so 6% = 6 3 = . 100 50 2) Convert to a decimal number a) 9% 9% means b) 4,5% 4,5% = 9 100 or 9 ÷ 100 = 0,09. 4,5 45 = = 0,045 100 1000 It is useful to know the following percentage to decimal conversions off by heart: 1% = 0,01 10% = 0,1 25% = 0,25 100% = 1 5% = 0,05 50% = 0,5 75% = 0,75 12,5% = 0,125 We can also convert any fraction to a percentage, even those whose denominators do not divide exactly into 100. EXAMPLE SOLUTION 1 1) Express as a percentage 4 25 1 1 25 = × = = 25% 4 4 25 100 2) Use a calculator to write the fraction 15 as a percentage, 43 rounded to 1 decimal place. This was easy since 100 is exactly divisible by 4 Key Sequence Display Answer 38.888…… 38,9 % 15 43 % = or 15 ÷ 43 × 100 = Exercise 3.7 1) Convert 24% to a fraction …………………………………………………………………. 2) Convert 46,5% to a fraction ……………………………………………………………….. 3) Convert 45% to a decimal …………………………………………………………………. 4) Convert 32,9% to a decimal ……………………………………………………………….. 5) Convert 3 to a percentage ……………………………………………………………….. 25 6) Convert 34 to a percentage correct to 2 decimal places ………………………………….. 77 7) Convert 2 5 to a percentage correct to 1 decimal place …………………………………… 9 46 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.8 EXPONENTS 3 In 2 , 2 is the base and 3 is the exponent 24 = 2 × 2 × 2 × 2 . There are 4 factors. We have expanded 24. If we contract 3 × 3 × 3 …. 7 factors, we have 37. Multiplication: 2 3 × 2 4 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 7 . But 3 + 4 = 7. We can add the exponents rather than expand, provided the bases are the same. Note: • 4 = 41 EXAMPLE Simplify 1) 23 × 32 × 24 × 3 2) a3 × a4 SOLUTION 3) 2 b × 2 3 b × 2 4 × 2 2 b × 2 3 b × 2 4 × 2 = 2 b + 3b + 4 + 1 = 2 4 b + 5 4) x −4 × x5 × x 0 × x x −4 × x 5 × x 0 × x = x − 4 + 5 + 0 + 1 = x 2 23 × 32 × 24 × 31 = 23+4 × 32+1 = 27 × 33 a 3 × a 4 = a 3+ 4 = a 7 Division: You know that 25 2× 2× 2× 2× 2 = = 2 2 . But 5 – 3 = 2. We can subtract the exponents 23 2× 2× 2 rather than expand, provided the bases are the same. EXAMPLE 8 SOLUTION EXAMPLE 8 SOLUTION 1) 5 56 5 = 58 – 6 = 52 6 5 2) a a2 a6 = a 6− 2 = a 4 2 a 3) 23 × 28 2 4× 2 23 × 28 = 2 3+8− 4 −1 = 26 2 4× 2 4) 23 25 23 1 1 = 5 −3 = 2 5 2 2 2 9x3 ÷ 3x2 × x = 3x3 – 2 + 1 = 3x2 6) d 3c 4 e c d 2e2 d 3c 4 e d 3 − 2 c 4 − 1 d c 3 = = e c d 2 e2 e 2−1 5) 9x3 ÷ 3x2 × x 6 Exercise 3.8 Simplify: 1) 52 × 54 ÷ 53 = ……………………………………………………………………………….. 2) 2c2 × 4c4 ÷ 8c5 = ………………………………………………………………………….. 3) 42 × 4 5 = …………………………………………………………………………………… 48 4) a 5 × 5) a2 ÷ a = ……………………………………………………………………………….. a4 24 2 × × 3 = ………………………………………………………………………………… 32 23 6) 32 × 34 ÷ 33 × 36 × 3 ÷ 37 = ………………………………………………………………. 47 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.9 ZERO EXPONENTS AND NEGATIVE EXPONENTS You know that 42 42 2−2 0 = 4 = 4 . But = 1 , since any non-zero number divided by itself is 42 42 1, so 4 0 = 1 . Note: • 0 0 is undefined. The base of a zero exponent may not be zero • Any number other than zero raised to the power of zero is 1 • Zero raised to any other power is zero, e.g. 03 = 0 24 24 2× 2× 2× 2 1 1 1 = . But = 2 4 − 7 = 2 − 3 . So 2 − 3 = 3 = = 7 7 3 2 2× 2× 2× 2× 2× 2× 2 2× 2× 2 2 2 2 In this way we can convert negative exponents to positive exponents. EXAMPLE Write with positive exponents: 1) 3 − 2 SOLUTION 3) p − a 5) x2 x5 EXAMPLE SOLUTION 3− 2 = 2) x − 3 x− 3 = p−a 4) ( − 4) −2 ( − 4) −2 = 6) 1 c −1 1 1 c = 1 ÷ = 1× = c −1 c 1 c 1 32 1 = a p x2 1 = x −3 = 3 5 x x 1 x3 1 1 = 2 ( − 4 ) 16 Exercise 3.9 Simplify, giving your answers with positive exponents: 3) 32 = ………………… 36 6) –30 = ………………… 8) ( −3) − 2 = ………………. 9) −( 3) − 2 = ……………. 11) 1 = ………………….. 3−2 12) 1 1 + −1 = …………. −1 x y ( a + b ) −1 = ……………. 14) 23 = ………………….. 23 15) a2 b3 c = …………… a4 b2 c0 17) 5-4 × 52 × 5 = ………… 18) 2−3 × 22 = ……………. 2−4 1) 03 = ………………….. 2) 4) c3 × c = ……………. c2 × c6 5) x−4 7) 3 4 1 = ………………… −1 = ………………. 10) 45 a − a = ………………. 13) 0 ( −7 ) = ……………….. 16) 4– 3 × 4 3 = …………… 48 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.10 PRUDUCTS AND QUOTIENTS RAISED TO A POWER You know that ( 5 × 3) 2 = ( 5 × 3) × ( 5 × 3) = 5 2 × 3 2 . In the same way, ( a × b ) 3 = ( a × b ) × ( a × b ) × ( a × b ) = a 3 × b 3 2 4 2 2 2 2 24 Also = × × × = 4 3 3 3 3 3 3 EXAMPLE 1) Write without brackets 2 a) ( 3.4 ) 3 b) (x × y) c) (x 2 y 3z x d) y ) 2 y 3 z1 ) 4 = x 2 × 4 y 3 × 4 z 1 × 4 = x 8 y 12 z 4 3 x x3 = 3 y y 2 2 2x 2 2 x 2 4x 2 = = 2 2 2 9y 3y 3 y 2) Write in brackets a) a 2 × b 2 b) 3 4 × 2 4 c) (x + y) × (x + y) d) a6 × b4 × c8 3 2 3 e) × 3 4 3 2 f) 2 ( 3.4 ) = 32.42 3 ( x × y ) = x3 × y 3 (x 4 3 2x e) 3y SOLUTION 3a 2 a 3 × 2b b a 2 × b 2 = (a b) 2 3 4 × 2 4 = (3 × 2 ) 4 (x + y) × (x + y) = (x + y)2 a6 × b4 × c8 = (a3 b2 c4)2 3 3 3 2 3 2×3 1 3 × 4 = 3× 4 = 2 2 2 2 2 3 3a 2 a 3 3a 2 a 3 3a 5 × × = = b 2b 2 2b b 2b 2 Exercise 3.10 1) Remove the brackets and simplify where possible: a) (4 + 5)2 = ………………………………. b) c) (a3 b2 c)3 = ……………………………... d) e) (2p4 × 6q2)2 = …………………………. f) 3 g) 3y2 3 = ……………………………….. 4x i) 2x2 3 = ……………………………….. 4x (2 × 3)2 = …………………………….. (p × q)3 = …………………………….. – (–4a3b)2 = …………………………… ( 5a ) 2 h) 2 2 = ………………………………. 5a 2 3 j) 10 z 4 2 = ……………………………… 5z 2) Contract to 1 bracket and simplify where possible: a) 23 × 53 = ……………………………… c) 2 3 3 × 2 = ………………………… 4 4 49 b) 24 × 54 × 54 × 24 = ………………….. 2 72 2 d) × 2 = ……………………. 7 25 25 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.11 POWER OF AN EXPONENT 4 From the exponent definition, ( 3 2 ) = (3 × 3) × (3 × 3) × (3 × 3) × (3 × 3) = 3 8 As a shortcut, we can multiply the exponents, keeping the existing base. Note: • 3 2 × 3 4 = 3 6 , the bases are both 3, so we add the exponents, but (3 ) 2 4 = 3 8 , the base of the 2 is 3 and the base of the 4 is 3 squared, so the bases are different and we cannot add the exponents. • 2 ( x × y ) = x 2 × y 2 So (2 × 3)2 = 62 = 36 = 4 × 9 = 22 × 32 2 But ( x + y ) ≠ x 2 + y 2 So (2 + 3)2 = 52 = 25, but 22 + 32 = 4 + 9 = 13 We can factorise the base of an exponent and write is as the product of prime factors For example: 63 = (2 × 3)3 = 23 × 33 Check: 63 = 216 and 23 × 33 = 8 × 27 = 216 EXAMPLE SOLUTION Remove the brackets and simplify: 1) (2 ) 2) (x ) 4 2 3 (2 ) 4 3 (a ) b 2 3 2×3 = x6 x2 × x3 = x5 c 24 2 3 = 2 4 × 3 = 2 12 (x ) = x 3) x2 × x3 4) 3 (a ) b c = ab×c = abc 5 2 4 2 4 × 5 2 20 2 = 2 × 5 = 10 3 3 3 5 3 3a 2 3 5b 5) 6) 3a 2 3 5b 7) 32 × 2 62 3 = 27a 6 33 a 2×3 = 125b9 53 b3×3 32 × 2 32 × 2 1 = 2 2 = 2 2 3 ×2 6 Exercise 3.11 Simplify the following. Where necessary write the base as the product of prime factors. 1) 4) 7) 10) 11) 12) 2 (3 x ) = …………….. (8 ) = ………………. 5 0 5 3a 2 2b (s 5) (-43)2 = ……………… 2 ×t5 ) 3 2) = …………. 3c) (22 + 32)2 = …………… (( 2 ) ) 9) 52 b 4 5 5b 4 = …………….. 8) 3 (4x – x) = …………… 2 3 6) 2 = ……………… 3 = ………………. 122 = ……………………………….…………………………………………………….. 82 63 = …………………………….…………………………………………………….. 27 × 42 182 × 2 = …………………………………………………………………………………. 42 × 33 50 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.12 NUMBER SEQUENCES Look at the number sequence: 1; 4; 7; 10; ….. ; ……. ; … Each number MINUS the preceding number gives three: 4 – 1 = 3, 7 – 4 = 3, 10 – 7 = 3, OR, each number PLUS 3 gives us the next number in the sequence. Sequences like the one above, where adding the same quantity each time gives us the terms of the sequence, are called arithmetic sequences. EXAMPLE SOLUTION 1) Is the following an arithmetic To find out whether this is an arithmetic sequence, sequence: 13; 8; 3; –2; –7; … ? calculate Term 2 – Term 1 and then calculate Term 3 – Term 2. If both answers are the same, the sequence is an arithmetic sequence. Term 2 – Term 1 = 8 – 13 = –5 Term 3 – Term 2 = 3 – 8 = –5 So this is an arithmetic sequence. 2) If so, write down the next three We subtract 5 in order to find the next three terms in numbers in the sequence. the sequence. –7 – 5 = – 12; –12 – 5 = –17 and – 17 – 5 = –22 So the next three terms are –12; –17 and –22. Exercise 3.12 Determine whether the following are arithmetic sequences. If they are, write down the next three numbers in the sequence. 1) 2; 7; 12; …………………………………………………………………………………… ………………………………………………………………………………………… 2) 3; 6; 12; ..…………………………………………………………………………………. ........................................................................................................................................ 3) 0; -4; -8; …………………………………………………………………………………… ………………………………………………………………………………………… 4) 1,2; 2,3; 3,4; ……………………………………………………………………………… ………………………………………………………………………………………… 5) 1 1 ; ; 0; …………………………………………………………………………………… 2 4 ………………………………………………………………………………………… 6) 1; 4; 9; 16; …………………………………………………………………………………. ………………………………………………………………………………………… 51 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.13 TERMS OF A SEQUENCE Each number in a sequence is called a term. T1 is the first term of the sequence, T2 is the second term and so on. In the sequence 1; 4; 7; 10; …….. T1 = 1, T2 = 4, T3 = 7 and T4 = 10. The dots show that there are an infinite number of terms. Sometimes a general term (Tx) of a sequence is given. You have to use it to find a specific term or the number of a term. EXAMPLE 1) Find the 1st, 2nd, 4th, 8th and 12th terms of a sequence with the general term Tx = 3x – 1 T1 Term T x = 3x – 1 T2 T4 T8 T12 17 29 2) Which terms have the values 17 and 29? 3) Fill the missing values on the table SOLUTION T2 = 3(2) – 1 T4 = 3(4) – 1 T8 = 3(8) – 1 1) T1 = 3(1) – 1 =2 =5 = 11 = 23 Tx = 3 x – 1 Tx = 3 x – 1 2) 17 = 3x – 1 29 = 3x – 1 17 + 1 = 3x – 1 + 1 29 + 1 = 3x – 1 + 1 18 = 3x 30 = 3x 30 3 x = 3 3 18 3x = 3 3 10 = x 6=x 3) T12 = 3(12) – 1 = 35 Term T1 T2 T4 T6 T8 T10 T12 T x = 3x - 1 2 5 11 17 23 29 35 Exercise 3.13 Complete the following two tables: 1) n 1 2 7 10 13 T n = 2n + 3 27 2) x T x = 3x – 4 1 4 5 5 10 20 52 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.14 THE SET OF RATIONAL RATIONAL NUMBERS The letter that stands for the set of rational numbers is Q. We define rational numbers in the following way: All rational number can be written in the form a where a and b are integers and b b ≠ 0. Things to remember: • Division of a number by zero is undefined in any number system • Every rational number can be written as an integer divided by a non-zero integer. • If you can't write the number in this form, it is not rational. EXAMPLE If possible, write each of the following numbers in the form a where a and b are integers and b b ≠ 0. Hence say whether the number is rational or not. Write in the form Number 1) 2 3 4 2) 1,5 3) – 12 5 4) 25 5) 3 −27 6) – 7 a b 3 11 = 4 4 15 3 = 1,5 = 10 2 12 −12 12 – = or 5 5 −5 5 25 = 5 = 1 −3 3 3 −27 = –3 = = 1 −1 −7 –7= 1 2 a, b ∈ Z, b ≠ 0 yes/no Rational yes/no Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Exercise 3.14 Use the table to help you decide whether the following numbers are rational numbers. Number Write number in the form a b 1) – 2,5 2) 22 7 3) – 49 4) 1 5) 0,05 6) 4⅔ 7) 0 8) – 53 9) 8 53 Are a, b ∈ Z, b ≠ 0? Rational? NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.15 IRRATIONAL NUMBERS Real numbers that are not rational are called irrational. We use the rational number definition to decide whether numbers are rational or irrational. EXAMPLE Decide whether the following numbers are rational or irrational: Number Test for a rational number 1) 49 49 = 7 = 2) 7 7 = 3 3) 64 3 7 1 7 but 1 4 64 = 4 = 1 Rational / Irrational Rational 7 is not an integer Irrational Rational Other examples of irrational numbers are 3 , 11 , 3 24 etc. They are the square roots of non-perfect squares or the cube roots of non-perfect cubes. We call these irrational numbers are surds. Find 5 on your calculator. 5 = 2,236 067 978 …….. On your calculator, the digits after the decimal comma seem to come to an end. This is because your screen can only hold a certain number of digits. In actual fact, these digits never end and do not recur. These surds are irrational numbers. Note: • π = 3,141 592 654 ……… is irrational and is a non-terminating, non-recurring decimal. 22 is an approximation for 7 • π , and is rational. π = 3,142 857 142 … and is a recurring decimal. Exercise 3.15 Are the following numbers rational or irrational? Number Test for a rational number • 1) 9,8 2) 5 2 3) 4 4) 7,148 148 148 ... 5) 1,54 6) 2,14 7) π 8) 22 7 9) 4 16 54 Rational / irrational NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.16 SURDS Each irrational number lies between two rational numbers, and so can be shown approximately on a number line. You know that 16 < 18 < 25. By taking the square root of these numbers, 4 < 18 < 5. Check this by finding 18 on your calculator. 18 = 4,242 640……. EXAMPLE 1) Simplify 20 a) b) ( 7 )3 c) 1,21 2) Simplify a) 7 + 7 b) 3 2 – 2 2 c) 7 × 7 d) 2 × 3 e) –2 5 × 3 5 f) 6 × 12 g) SOLUTION 20 = 5×4 = 2 5 3 ( 7) = 7 × 7 × 7 =7 7 1,21 = 1,1× 1,1 = 1,1 7+ 7 =2 7 3 2 –2 2 = 2 7 × 7 = 49 = 7 2 × 3 = 6 –2 5 × 3 5 = – 6 × 5 = – 30 6 × 12 = 6 × 6 × 2 = 6 2 8 8 32 32 = 2 2 4 2 = 1 2 Rationalisation of the denominator: If the denominator of a faction is a surd, we can make the denominator a rational number. Example: 4 = 2 = 4 2 × 2 2 Multiply numerator and denominator by the same surd 4 2 =2 2 2 Exercise 3.16 1) Simplify: a) 8 × 8 = ……………………………. c) 4 3 – 3 = ……………………………. e) 4 3 × 3 3 = …………………………. g) 18 6 2 = ………………………………….. b) 2 3 × 3 2 = ……………………… d) 3 5 + 4 5 = ………………………… f) h) 6 × 27 = …………………………. 625 ÷ 3 125 = …………………….. 2) Rationalise the denominators: a) 6 3 = …………………………………… b) 14 7 = …………………………………. 3) Between which two integers does 7 lie? ……………………………………………….. 4) Between which two integers does 3 lie? ……………………………………………….. 55 NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.17 THE SET OF REAL NUMBERS All the rational numbers together with all the irrational numbers form the set of Real numbers, R. Here are some of them on a number line. - 5 -3,5 -4 -3 -1 -2 3 4 1 2 • - 0, 3 -1 0 11 3 2 1 2 3 4 Note: • −4 is not a real number, since 2 × 2 = 4 and –2 × –2 = 4. It is a complex number. EXAMPLE On the table, mark with an X the number system/s to which each number belongs: SOLUTION Natural Whole Rational Irrational Real Number Integer Number Number Number Number Number –56 1) 0 2) 13⅛ 3) 5 4) 3 5) 27 6) – 12 5 7) π 8) 22 7 Exercise 3.17 On the table, mark with an X the number system/s to which each number belongs: Natural Whole Rational Irrational Number Integer Number Number Number Number • 1) 19, 2 2) 3) 4) 5) π –3 2 3 2 22 7 6) 1,148796… 7) 4 16 8) 15,78 9) 0 10) 144 56 Real Number NCV2 – Chapter 3: Rational Numbers and Real Numbers § 3.18 THE REAL NUMBER DIAGRAM Here is a diagram to help you remember what you have learned about the real number system. Real numbers Rational numbers Integers Whole numbers Irrational numbers Fractions Negative integers 0 Natural numbers Exercise 3.18 Use the number sets diagram to help you answer these questions: Statement 1) 25 is rational. 2) 13 is rational. 3) 4) Any recurring decimal number is rational. 3 −27 is irrational 56 99 5) 4,56 = 4 6) 0,6 = 0,6 7) 3 4 is rational because 4 is a perfect square 8) All surds are rational 9) 3,11111.... = 3,1 • 10) 8,2 = 8 2 10 11) All rational numbers are real numbers 12) All real numbers are rational numbers 13) All whole numbers are rational numbers 14) All irrational numbers are real numbers. 15) All real numbers are whole numbers. 16) Zero is a real number 17) Some real numbers are irrational 57 True (T) / False (F)