2REAL NUMBERS Applications of this chapter We all learn how to count, add and subtract from a very young age. Real numbers help us calculate and measure airspeed, rainfall, wind speed and distance. Of course, people who work with medical devices, insurance policies, accounts, and other finance-related jobs may use real numbers more often, but people in retail, purchasing, catering, and even publishing must also use real numbers. For example, real numbers are used to display stock price charts on NYMEX web pages. Why the positive or negative numbers is important in stock market? 36 ©Praxis Publishing_Focus On Maths
Rational Numbers Irrational Numbers Real Numbers Concept Map • Integers • Rational number • Irrational number • Terminating decimal • Recurring decimal • Real numbers • Fractions • Percentages Key Terms Learning Outcomes Learning Outcomes • Identify rational numbers and irrational numbers. • Change fractions to decimals and vice versa. • Use dot notation to write recurring decimals. • Extend the concept of integers to fractions to solve problems. • Extend the concept of integers to decimals to solve problems. • Perform computations involving directed numbers (integers, fractions and decimals). • Understand and use the symbols , , , , = and ≠. Decimals Natural Numbers Whole Numbers Fractions Decimals Terminating Non-terminating Positive Decimals Positive Fractions Negative Decimals Negative Fractions Non-integers Integers Pythagoras is a Greek mathematician who invented rational numbers in the sixth century. The core belief of Pythagoras and his followers is that everything is numbers. He stated that, numbers can be expressed as a ratio or fraction between two integers. Maths History 37 ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 38 Flashback 1. Find the value of each of the following. (a) –10 + 6 (b) –2 – (–7) (c) (– 4) × 3 (d) (–16) ÷ (–2) 2. Find the value of each of the following. (a) 25 (b) 0.04 (c) 9 16 3. Change each of the following fractions to a decimal. (a) 1 4 (b) 5 2 (c) 8 5 Based on the stock market report above, write the subtraction statements using rational numbers. Thinking The table shows part of a stock market reports for February 5, 2020 for some ASEAN companies. Company Stock price at the end of the day ($) Stock price at the start of the day ($) Difference in prices of the day ($) P 3.670 3.710 +0.04 Q 40.630 41.330 + 0.7 R 64.840 65.970 +1.13 S 142.580 142.15 – 0.43 (a) What does it mean when the difference in prices is positive or negative? (b) Draw a number line to show each subtraction. (c) Arrange the amounts from the greatest loss to the greatest gain. 1 ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 39 The label on a bag of frozen mixed berries says that it must be stored at a temperature between negative 18ºC and negative 22ºC. (a) Name some possible temperatures. (b) How could these temperatures be shown on a number line? 2.1 Rational Numbers (a) Explain why a rational number p q is subjected to the condition q ≠ 0. (b) All integers are rational numbers. Are all rational numbers integers? 2 Any number that can be written in fractional form a b such that a and b are integers, where b ≠ 0, are known as rational numbers. For example, 2 3 , – 9 14 , 6 5 and 8 1 . Rational numbers can be illustrated on the number line. 9 4 – 1 2 – 1 2 5 2 10 3 –4 –3 –2 –1 0 1 2 3 4 Objective: To identify and describe rational numbers. Instruction: Do this activity in groups of four. 1. You are given some number cards as follows. 0.5 –1.75 1 1 4 5 –9 – 3 4 6 2. Observe all the number cards given. Discuss with your group members how would you write these numbers in the form of a b such that a and b are integers. 3. Present your findings in class. 1 Critical Thinking ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 40 EXAMPLE 2 Find the rational number which is in the middle of (a) –6 and –7, (b) 5 12 and 3 4 . Solution: (a) –6 –7 –12 2 –14 2 –13 2 lies between –6 and –7. \ –13 2 is in the middle of –6 and –7. (b) 5 12 3 4 5 12 9 12 6 12 , 7 12 and 8 12 lie between 5 12 and 3 4 . \ 7 12 is in the middle of 5 12 and 3 4 . EXAMPLE 3 Determine whether the following numbers are rational numbers. Explain your answer. (a) 1 2 3 (b) – 0.24 (c) 4 EXAMPLE 1 State all the possible rational numbers in the form n 4 between 2 and 3, where n is an integer. Solution: 2 3 8 4 12 4 9 4 , 10 4 and 11 4 lie between 2 and 3. \ All the possible rational numbers in the form n 4 between 2 and 3 are 9 4 , 10 4 and 11 4 . 10 4 = 5 2 The set of rational numbers is denoted by Q. Since Z = {... , –3, –2, –1, 0, 1, 2, 3, ...} and Q = {all rational numbers}, hence Q ⊂ Z. ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 41 2.2 Irrational Numbers An irrational number is a number that cannot be expressed in the form a b , where a and b are integers and b ≠ 0. For example, π = 3.14159… cannot be expressed as a ratio of a b . – 0.24 can be expressed in the fraction form by using a scientific calculator. Press (–) 0 . 2 4 = SHIFT d /c Practice 2.1 Basic Intermediate Advanced 1 State all the possible rational numbers (a) between 3 and 6 and in the form of n 2 , where n is an integer, (b) between 4 and 5 and in the form of n 5 , where n is an integer. 2 Find the rational number which is in the middle of (a) 0 and –1, (b) 1 5 and 1, (c) 1 3 and 2 3 , (d) 1 4 and 5 8 . 3 Determine whether the numbers – 8, 3.15, 11 5 and – 2 3 are rational numbers. Give your reason. 4 Calculate each of the following. Give your answer in decimal form. (a) – 5.2 + 2 1 2 × 1– 4 5 2 (b) [6 + (–2.13)] × 2 3 (c) – 2 1 2 ÷ [5.2 – (– 2.8)] (d) – 2.53 + 1– 1 8 – 3 4 ÷ 0.62 5 Puan Hasnah had 4.5 kg of sugar. She used 1 4 of the sugar to bake a cake. Calculate the mass, in kg, of the remaining sugar. Give the answer in fraction form. 6 The initial depth of a pond was 2.52 m. When the surrounding temperature increased, the water level of the pond descended 1 3 of the initial level. Then, water was pumped into the pond until the levels increased by 12.5 cm. Calculate the current height, in m, of the water level of the pond. Give the answer in decimal form. Discuss with your classmates. Give three examples of irrational numbers. INTERACTIVE ZONE Solution: (a) 1 2 3 = 5 3 Express the number in improper fraction. Thus, 1 2 3 is a rational number. (b) –0.24 = – 24 100 Express in the fraction of hundredths. = – 6 25 In the lowest term. Thus, – 0.24 is a rational number. (c) 4 = 4 1 Thus, 4 is a rational number. ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 42 Practice 2.2 Basic Intermediate Advanced 1 Determine whether each of the following is a rational number or an irrational number. (a) 0.2 (b) 1.04 (c) 0.333… (d) 9 (e) 3 (f) 6 2 Circle the irrational numbers in the list given below. 0.666 3 8 10 0.838383… 11 1 2 2.3 Fractions A Understanding fractions A fraction is a number that represents a part of the whole. For example, If a square is divided into 4 equal parts, each equal part is called a fraction of the square. A fraction is written in the form of a b where a is called numerator and b is called denominator. 2 cannot be represented as a fraction of two integers. \ 2 is an irrational number. 9 is not an irrational number as it can be simplified to 3, which is rational. The square root of a prime number is an irrational number. There is an irrational number between any two rational numbers. 3 (b) 0.06 = 0.06 1 = 6 100 = 6 100 = 3 50 50 \ 0.06 is a rational number. (c) 4 = 2 \ 4 is a rational number. (d) 2 = 1.14142135… \ 2 is an irrational number. 1 4 Numerator Denominator EXAMPLE 4 Determine whether the following is a rational number or an irrational number. (a) 1.25 (b) 0.06 (c) 4 (d) 2 Solution: 5 (a) 1.25 = 1.25 1 = 125 100 = 125 100 = 5 4 4 \ 1.25 is a rational number. π cannot be expressed as a fraction because π ≠ 22 7 , π ≈ 22 7 . ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 43 (I) Representing fractions with diagrams We follow the steps below to represent a fraction with a diagram. Draw a suitable diagram (square, rectangle or circle). 2 Divide the diagram into equal parts shown by the denominator. 3 Shade the number of equal parts shown by the numerator. For example, 1 3 3 4 2 5 Divide a circle into 3 equal parts. (Denominator = 3) Shade 1 out of the equal parts. (Numerator = 1) Divide a square into 4 equal parts. (Denominator = 4) Shade 3 out of the equal parts. (Numerator = 3) Divide a rectangle into 5 equal parts. (Denominator = 5) Shade 2 out of the equal parts. (Numerator = 2) (II) Writing fractions for given diagrams We follow the steps below to determine the denominator and numerator from a given diagram. Count the number of equal parts (denominator). 2 Count the number of shaded parts (numerator). For example, 5 6 4 9 There are 6 equal parts and 5 parts are shaded. There are 9 equal parts and 4 parts are shaded. For example, 1 out of the 4 equal parts can be written as 1 4 . The denominator tells us the total number the unit is divided into equal parts. The numerator tells us how many of the parts in the unit are to be taken. 1 4 is read as ‘one over four’ or ‘one quarter’. ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 44 EXAMPLE 5 Write each of the following fractions as a decimal. (a) 3 10 (b) 17 100 (c) 506 1000 (d) 8215 1000 (e) 2 4 100 Solution: (a) 3 10 = 0.3 (b) 17 100 = 0.17 (c) 506 1000 = 0.506 (d) 8215 1000 = 8.215 (e) 2 4 100 = 2 + 4 100 = 2 + 0.04 = 2.04 EXAMPLE 6 Change each of the following fractions to a decimal. (a) 3 4 (b) 7 2 5 Solution: (a) 3 4 = 3 ÷ 4 = 0.75 or 3 4 = 3 × 25 4 × 25 = 75 100 = 0.75 (b) 7 2 5 = 37 5 = 7.4 or 7 2 5 = 7 + 2 5 = 7 + 4 10 = 7.4 (III) Representing fractions as decimals Decimals are numbers representing fractions with denominators that are powers of 10 such as 10, 100, 1000 and so on. For example, 1 10 = 0.1 1 100 = 0.01 1 1000 = 0.001 Read as ‘zero point one’. Read as ‘zero point zero one’. Read as ‘zero point zero zero one’. 3.45 is read as 'three point four five', and not 'three point forty-five'. We divide the numerator by the denominator to change a fraction to a decimal. Press 7 a b/c 2 a b/c 5 = a b/c 7.4 Change to an equivalent fraction with denominator that is power of 10. × 2 2 5 = 4 10 × 2 ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 45 EXAMPLE 7 Write each of the following decimals as a fraction or mixed number in its lowest terms. (a) 0.25 (b) 4.625 Solution: (a) 0.25 = 25 100 = 1 4 (b) 4.625 = 4 625 1000 = 4 5 8 We express the decimal as a fraction with denominator that is a power of 10 such as 10, 100 or 1000 and so on to change a decimal to a fraction. (IV) Representing fractions as decimals Percentage is a fraction with a denominator of 100. For example, 4 100 = 4%, is read as ‘four per cent’. We multiply the fraction or the decimal by 100% to change a fraction or a decimal to a percentage. For example, change 4 5 to percentage. 4 5 = 4 5 × 100% = 80% Maths LINK History When the New York Stock Exchange was established in 1792, it was modelled after the Spanish system. The Spanish dollar is divided into eight parts, so when the value of a stock increases or decreases, the change is expressed in eighths. Decreases are represented by negative fractions, while increases are expressed as positive fractions. In 2000, the New York Stock Exchange migrated to its current system, which displays changes in stock values as decimals. Here are the values of 5 different stocks changed on a specific day: 1 2 5 , – 3 16 , 3 4 , –1 7 16, –1 1 2 Arrange the fractions from least to greatest. Write each fraction as a decimal to show the change in dollars. Press 4 . 6 2 5 = a b/c 4 5/8 ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 46 Solution: (a) 4 5 = 4 51 × 1 20 00% = 80% (b) 11 4 = 5 41 × 1 25 00% = 125% EXAMPLE 9 Change each of the following decimals to a percentage. (a) 0.23 (b) 3.1 Solution: (a) 0.23 = 0.23 × 100% = 23% (b) 3.1 = 3.1 × 100% = 310% EXAMPLE 8 Change each of the following fractions to a percentage. (a) 4 5 (b) 11 4 Conversely, we divide the percentage by 100% to change a percentage to a fraction or a decimal. EXAMPLE 10 Change each of the following percentages to a fraction in its lowest terms. (a) 60% (b) 150% Solution: (a) 60% = 60 100 (b) 150% = 150 100 = 3 5 = 3 2 = 11 2 To multiply a number by 100, alternatively, we can also move the decimal point two places to the right. For example, 0. 23 = 23. Press 5 ÷ 4 SHIFT = 125 ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 47 From the findings in Activity 2, it is found that representing the positive and the negative fractions on a number line is the same as integers. 4 –– 3 3 –– 3 2 –– 3 1 –– 3 1 – 3 1 –– 3 1 –– 3 1 –– 3 1 –– 3 1 + – 3 1 + – 3 1 + – 3 1 + – 3 2 – 3 3 – 3 4 – 3 0 Negative fractions Zero Positive fractions The value is decreasing The value is increasing Note that the positive fractions are located on the right hand side of zero whereas the negative fractions are located on the left hand side of zero. Objective: To explore positive and negative fractions on a number line. Instruction: Do this activity in groups of four. 1. Open the file Fraction number line using GeoGebra. 2. Click and drag the ‘Numerator’ and ‘Denominator’ slider to define a fraction on the number line. 3. Tick at the checkbox ‘Change sign’ to change between positive and negative fraction. 4. Observe the position of the fractions defined on the number line. 5. Discuss with your group member and determine the best method to identify the position of fractions on number line. 6. Present your finding to the class. 2 B Positive and negative fractions (I) Represent positive and negative fractions on number lines The fractions can also be represented by using vertical number line. Scan or click the above QR code to download this activity file. GeoGebra ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 48 (II) Compare and arrange fractions in order On a number line, the fractions to the right is larger than the fractions to the left. Based on the position of fractions on a number line, the fractions can either be arranged in ascending order or in descending order. EXAMPLE 12 Determine the largest fraction. (a) – 1 2 or 1 3 (b) – 3 7 8 or –1 3 4 (c) –1 2 3 or –1 5 6 Solution: (a) 1 3 is larger than – 1 2 . A positive fraction is always larger than a negative fraction. (b) – 3 7 8 or –1 3 4 Compare the whole number of the two mixed numbers: –1 is larger than – 3. Thus, –1 3 4 is larger than –3 7 8 . EXAMPLE 11 Represent the following fractions on number lines. (a) 1 5 , – 2 5 , – 4 5 , 3 5 (b) – 5 6 , 1 3 , – 2 3 , – 1 6 Solution: (a) 4 –– 5 3 –– 5 2 –– 5 1 –– 5 1 – 5 2 – 5 3 – 5 0 (b) 5 –– 6 –– = – – 2 –– 3 1 –– 6 3 –– 6 2 –– 6 1 – 6 1 – 3 0 4 6 2 3 – = – 2 6 1 3 ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 49 (c) –1 2 3 or –1 5 6 ➞ –1 4 6 or –1 5 6 ➞ –10 6 or –11 6 Compare the values of numerators for the two negative fractions with the same denominator: –10 is larger than –11. Thus, – 10 6 is larger than – 11 6 . Thus, –1 2 3 is larger than –1 5 6 . EXAMPLE 13 (a) Arrange – 1 5 , 1 10, – 1 2 , 1 5 and – 3 10 in ascending order. (b) Arrange 1 4 , – 1 8 , – 1 4 , 3 8 and – 1 2 in descending order. Solution: (a) – 1 5 , 1 10 , – 1 2 , 1 5 , – 3 10 ↓ ↓ ↓ ↓ ↓ – 2 10 , 1 10 , – 5 10, 2 10 , – 3 10 Draw a number line to represent the given fractions. 5 –– 10 3 –– 10 2 –– 10 1 –– 2 1 –– 5 1 – 10 1 – 5 3 –– 10 2 – 10 1 – 10 Original fractions Thus, the fractions arranged in ascending order is – 1 2 , – 3 10, – 1 5 , 1 10, 1 5 . (List the fractions on a number line from left to right.) Equalise the denominator Compare the numerators with the same denominators. Arrange the fractions based on the values of numerators, from the smallest value to the largest value. – 5 10 , – 3 10 , – 2 10 , 1 10 , 2 10 = – 1 2 , – 3 10 , – 1 5 , 1 10 , 1 5 ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 50 (III) Perform computations involving combined basic arithmetic operations of positive and negative fractions EXAMPLE 14 Solve each of the following. (a) –1 1 8 + 1 2 × 3 4 (b) 1 1 8 ÷ 1–1 1 2 – 3 4 2 (c) –1 1 3 ÷ 4 9 + 1 4 × 2 1 2 Bracket ↓ × or ÷ from left to right ↓ + or – from left to right Amisha wants to make mixed fruit punch. Answer the following questions. • How many cups of punch does the recipe make? • Suppose Amisha makes the punch, then pours herself 3 4 cup of punch. How much punch does she have left? • Suppose Amisha only has 1 3 cup of pineapple juice. How much of each of the other ingredients does she need to keep the flavour the same? • Suppose Amisha decides to make one-third of the recipe. How much soda will she need? 3 cup orange juice 7 – 8 cup pineapple juice 2 – 3 cup lemon lime soda 3 – 4 cup cranberry juice 1 – 4 cup ice cubes 1 – 3 FRUIT PUNCH The order of operations involving addition, subtraction, multiplication, division and brackets for fractions is the same as the order of operations involving integers. (b) 1 4 , – 1 8 , – 1 4 , 3 8 , – 1 2 ↓ ↓ ↓ ↓ ↓ 2 8 , – 1 8 , – 2 8 , 3 8 , – 4 8 Draw a number line to represent the given fractions. 4 –– 8 2 –– 8 1 –– 8 2 – 8 3 – 8 Original fractions 1 –– 2 1 –– 8 1 – 4 3 – 8 1 –– 4 Thus, the fractions arranged in descending order are 3 8 , 1 4 , – 1 8 , – 1 4 , – 1 2 . (List the fractions on a number line from right to left.) Equalise the denominators Compare the numerators that have the same denominators. Arrange the fractions based on the values of denominators, from the largest value to the smallest value. 3 8 , 2 8 , – 1 8 , – 2 8 , – 4 8 = 3 8 , 1 4 , – 1 8 , – 1 4 , – 1 2 ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 51 Solution: (a) –1 1 8 + 1 2 × 3 4 = – 9 8 + 1 2 × 3 4 = – 9 8 + 1 1 2 × 3 4 2 = – 9 8 + 3 8 = – 6 8 = – 3 4 The lowest term (b) 1 1 8 ÷ 1–1 1 2 – 3 4 2 = 9 8 ÷ 1– 3 2 – 3 4 2 = 9 8 ÷ 1– 6 4 – 3 4 2 = 9 8 ÷ 1– 9 4 2 = 9 8 × 1– 4 9 2 Change ÷ into × and inverse the fraction – 9 4 . = – 1 2 (c) –1 1 3 ÷ 4 9 + 1 4 × 2 1 2 = – 4 3 ÷ 4 9 + 1 4 × 5 2 = 1– 4 3 ÷ 4 9 2 + 1 1 4 × 5 2 2 × / ÷ from left to right = 1– 4 3 × 9 4 2 + 1 1 4 × 5 2 2 = –3 + 5 8 = –2 3 8 × / ÷ from left to right Change to improper fraction. Equalise the denominators for the calculation in the bracket. Change to improper fraction. (IV) Solve problems involving fractions EXAMPLE 15 A mathematics quiz contains 20 questions. A score of 2 marks is awarded for every correct answer and a score of – 1 2 mark is given for every incorrect answer. Mei Ling participated in the quiz and answered all the questions. Her score for incorrect answers was – 4. What was the total score Mei Ling obtained in the quiz? Do the operations inside the brackets first. Change the mixed number to improper fraction. ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 52 Solution: Stage 1: Understand the problem List the facts and the question. Facts: Score for every correct answer = 2 marks Score for every incorrect answer = – 1 2 Mei Ling's score for incorrect answers = – 4 Question: Find the total score obtained. Stage 2: Think of a plan • +2 represents the score for every correct answer. • – 1 2 represents the score for every incorrect answer. • Find the total number of incorrect answers using division. • Find the total score using multiplication and addition. Stage 3: Carry out the plan Number of incorrect answers = – 4 ÷ 1– 1 2 2 = 8 Total score obtained = (20 – 8) × 2 + (– 4) = 12 × 2 – 4 = 24 – 4 = 20 Stage 4: Look back Total score for correct answers = 12 × 2 = 24 Total score obtained = 24 + (– 4) = 20 Total score for incorrect answers = 84 × 1– 1 2 1 2 = – 2 There were some sweets in a jar. Jolin was given 7 12 of the sweets. Selena was given 3 5 of the remaining sweets. There were 72 sweets left in the jar. Calculate the number of sweets given to Jolin. ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 53 Practice 2.3 Basic Intermediate Advanced 1 Represent each of the following fractions with a diagram. (a) 2 4 (b) 1 6 (c) 3 5 (d) 5 9 2 Write the fraction represented by the shaded parts in each of the following diagrams. (a) (b) (c) (d) 3 Express each of the following fractions as a percentage. (a) 3 100 (b) 47 100 (c) 201 100 4 Change each of the following percentages to a fraction with 100 as its denominator. (a) 7% (b) 29% (c) 121% 5 Express each of the following decimals as a percentage. (a) 0.6 (b) 0.28 (c) 1.204 6 Express each of the following percentages as a fraction in its lowest terms. (a) 2% (b) 32% (c) 160% 7 Express each of the following percentages as a decimal. (a) 0.8% (b) 96% (c) 175% 8 Represent each of the following on number lines. (a) 1 7 , – 2 7 , 3 7 , – 4 7 (b) 1 3 , – 2 3 , 0, –1 1 3 (c) 1 6 , – 1 2 , – 1 3 , 1 3 (d) – 1 4 , 1 8 , 1 2 , – 3 8 9 Determine the larger fraction. (a) – 2 3 , 4 5 (b) –5 1 2 , –1 1 2 (c) – 2 3 5 , –2 7 10 (d) –3 7 8, –2 1 9, –2 1 6 Arrange each of the following in (i) ascending order, (ii) descending order. (a) 1 2 , – 3 8 , – 1 2 , 5 8 , – 1 8 (b) – 2 5 , 1 10, 1 2 , 3 5 , – 7 10 (c) –1 1 3 , – 5 6 , 4 9 , –1 1 6 , 7 9 (d) 1 1 4 , – 5 8 , –1 1 2 , –1 7 8 , 3 2 Calculate each of the following. (a) 1 2 + 1–2 1 4 2 – 1– 3 8 2 (b) –2 1 2 × 4 5 ÷ 1– 2 3 2 (c) 1 1 9 × 1– 1 5 2 – 5 6 (d) – 7 8 + 3 10 × 5 12 (e) –2 1 2 ÷ 5 8 + 3 4 × 1– 1 6 2 A submarine was submerged as deep as 3 5 of 250 m per minute. Find the new position, in m, of the submarine after 1 4 hour. Normala buys 6 packets of sugar with mass of 3 10 kg each. She needs 2 1 2 kg of sugar to bake a cake. Is Normala’s sugar enough for her to bake the cake? Give your reason. The table below shows the profit/loss of a company in three consecutive years. Year Profit / Loss 2020 Profit of $1 1 3 million 2021 Loss of $2 1 4 million 2022 2 times of the profit in the year 2020 Calculate the profit or loss, in million, of the company in the period of three years. Give your answer in fraction form. ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 54 EXAMPLE 16 Change each of the following fractions into decimals. (a) 3 8 (b) 7 16 Solution: (a) 0.375 8 3.000 0 3 0 2 4 60 56 40 40 (b) 0.4375 16 7.0000 0 7 0 6 4 60 48 120 112 80 80 The set of rational numbers includes integers, fractions, decimals and percentages. Every rational number can be written as either a terminating decimal or a recurring decimal. B Recurring decimals A recurring decimal has an infinite number of decimal places, with one or more digits repeating themselves over and over. For example, 0.343434… If one or two digits recur, a dot is placed over these digits. For example, 3 12 = 0.272727… = 0.2· 7 · . If three or more digits recur, a dot is placed over the first and last digits. For example, 124 999 = 0.124124124… = 0.1· 24· . 2.4 Decimals The number system is a base 10 or decimal system. From right to left, the place value of each column increases by a factor of 10. The decimal point is used to separate the whole part of a number from the fractional part. This table shows some of the place columns in the decimal system. Thousands Hundreds Tens Ones • Tenths Hundredths Thousandths 1000 100 10 1 • 1 10 1 100 1 1000 A Terminating decimals A decimal number that has digits that do not go on forever is called a terminating decimal. A terminating decimal has a finite number of decimal places. For example: 3.125, which has three decimal digits. A fraction can be changed to decimal by dividing the numerator by the denominator. For example, 5 4 = 5 ÷ 4 = 1.25. ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 55 A recurring decimal is also called as a repeating decimal. EXAMPLE 17 Express each of the following recurring decimals using the dot notation. (a) 1.333… (b) 0.454545… (c) 0.142857 142857 142857… Solution: (a) 1.333… = 1.3· (b) 0.454545… = 0.4· 5 · (c) 0.142857 142857 142857… = 0.1 · 42857· EXAMPLE 18 Express each of the following recurring decimals in the ordinary notation. (a) 0.8· 1 · (b) 0.2· 9 8· (c) 3.7· 14285· Solution: (a) 0.8· 1 · = 0.818181… (b) 0.2· 98 · = 0.298298298… (c) 3.7· 14285· = 3.714285 714285 714285… EXAMPLE 19 Change the following fractions into decimals. (a) 5 9 (b) 7 11 Solution: (a) 0.555 9 5 0 5 0 4 5 50 45 50 45 5 (b) 0.6363 11 7.0 6.6 40 33 70 66 40 33 7 \ 5 9 = 0.555… = 0.5· \ 7 11 = 0.6363… = 0.6· 3 · ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 56 Note that the positive decimals are located on the right hand side of zero whereas the negative decimals are located on the left hand side of zero. –0.1 –0.1 –0.1 –0.1 +0.1 +0.1 +0.1 +0.1 0 Negative decimals Positive decimals Zero The value is decreasing The value is increasing –0.4 –0.3 –0.2 –0.1 0.1 0.2 0.3 0.4 EXAMPLE 20 Represent the following decimals on number lines. (a) 1.0, –1.5, –0.5, 1.5, –2.0 (b) –1.2, 1.2, –4.8, 3.6, –2.4 Solution: (a) –2.0 –0.5 –1.0 0 –1.5 0.5 1.0 1.5 (b) –4.8 –2.4 –1.2 –3.6 0 1.2 2.4 3.6 C Positive and negative decimals (I) Represent positive and negative decimals on number lines Objective: To explore positive and negative decimals on a number line. Instruction: Do this activity in groups of four. 1. Open the file Decimal number line using GeoGebra. 2. Click and drag the ‘Tenth’ and ‘Hundredth’ slider to define a decimal on the number line. 3. Tick at the ‘Change sign’ to change between positive and negative decimals. 4. Observe the position of the decimal defined on the number line. 5. Discuss with your group members about the position of a decimal on number line. 6. Present your finding to the class. 3 From the findings in Activity 3, it is found that representing the decimals on a number line is the same as integers. Scan or click the above QR code to download this activity file. GeoGebra ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 57 (II) Compare and arrange decimals On a number line, the decimals to the right is larger than the decimals to the left. Based on the position of decimals on the number line, the decimals can either be arranged in ascending order or in descending order. EXAMPLE 21 Determine the largest decimal. (a) 1.8 or –4.5 (b) –5.35 or –8.6 (c) –3.62, –4.82 or –3.67 Solution: (a) 1.8 is larger than –4.5. A positive decimal is always larger than a negative decimal. (b) –5.35 –5 is larger than –8. – 8.6 –5.35 is larger than –8.6. (c) –3.62, –4.82 or –3.67 Compare the value of first digit, –3 is larger than –4. Thus, –4.82 is the smallest. For –3.62 and –3.67, compare the value of the last digit because the values of the first digit and the second digit are the same. Since, –0.02 is larger than –0.07, –3.62 is larger than –3.67. Therefore, –3.62 is the largest. EXAMPLE 22 (a) Arrange –1.6, 1.4, –3.8, –2.5 and 2.35 in ascending order. (b) Arrange 3.28, –4.1, –1.03, 2.2 and –2.3 in descending order. Solution: (a) Positive numbers: 1.4 and 2.35 Compare the values of the first digits, 1 and 2. 2 is larger than 1. Thus, 2.35 is larger than 1.4. Arrange in ascending order: 1.4, 2.35 Negative numbers: –1.6, –3.8 and –2.5 ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 58 Compare the values of the first digits and arrange the decimals in ascending order based on the values of the first digits, that is –3.8, –2.5, –1.6. Thus, the decimals arranged in ascending order are –3.8, –2.5, –1.6, 1.4, 2.35. (b) Positive numbers: 3.28, 2.2 Compare the values of the first digits, 3 and 2. 3 is larger than 2. Thus, 3.28 is larger than 2.2. Arrange in descending order: 3.28, 2.2 Negative numbers: –4.1, –1.03 and –2.3 Compare the values of the first digits and arrange the decimals in descending order based on the values of the first digits, that is –1.03, –2.3, –4.1. Thus, the decimals arranged in descending order are 3.28, 2.2, –1.03, –2.3, –4.1. (III) Perform computations involving combined basic arithmetic operations of positive and negative decimals The Lucas family and the Nabila family have similar homes. The Lucas family sets its thermostat to 20ºC during the winter months. Its monthly heating bills were: $171.23, $134.35, and $123.21. The Nabila family used a programmable thermostat to lower the temperature at night, and during the day when the family was out. The Nabila family's monthly heating bills were: $134.25, $103.27, and $98.66. (a) How much money did each family pay to heat its home during the winter months? (b) How much money did the Lucas family pay? Estimate to check your answer is reasonable. (c) What other things could a family do to reduce its heating costs? 4 The order of operation involving addition, subtraction, multiplication, division and brackets for decimals is the same as integers. Bracket ↓ × or ÷ from left to right ↓ + or – from left to right ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 59 Discuss with your classmates. When you multiply 2 decimals, how do you know where to place the decimal point in the product? Use example to explain. INTERACTIVE ZONE EXAMPLE 23 Solve each of the following. (a) 4.2 + (–1.25) × 8.2 (b) 8.91 ÷ (–0.02 – 1.6) (c) (–5.2 + 1.48) – 3.12 × 2.5 Solution: (a) 4.2 + (–1.25) × 8.2 × / ÷ from left to right = 4.2 + (–10.25) = 4.2 – 10.25 = – 6.05 (b) 8.91 ÷ (–0.02 – 1.6) = 8.91 ÷ (–1.62) = – (8.91 ÷ 1.62) = – 5.5 (c) (–5.2 + 1.48) – 3.12 × 2.5 = –3.72 – 3.12 × 2.5 = –3.72 – 7.8 = –11.52 Perform computation in the bracket first. Perform computation in the bracket first. × / ÷ from left to right (IV) Solve problems involving decimals EXAMPLE 24 The price of the stock of a company was $2.05 at a certain time. The price hiked by $0.32, then subsequently dropped $0.28 every hour for the next three hours. Calculate the final price of the stock. ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 60 Solution: Stage 1: Understand the problem List the facts and the question. Facts: Price of stock = $2.05 Price of stock hiked = $0.32 Price of stock dropped every hour for the next three hours = $0.28 Question: Calculate the final price of the stock. Stage 2: Think of a plan • Increase in price is written as +0.32. • Decrease in price is written as – 0.28. • Use multiplication and addition. Stage 3: Carry out the plan The final price of the stock= 2.05 + 0.32 + 3 × (– 0.28) = 2.37 + (– 0.84) = 2.37 – 0.84 = 1.53 The final price of the stock was $1.53. Stage 4: Look back Work backwards to check. $1.53 + $0.84 = $2.37 $2.37 – $0.32 = $2.05 (V) Perform computations involving combined basic arithmetic operations of rational numbers A positive rational number is multiplied by a negative rational number. Is it possible that the product is closer to 0 than either of the numbers being multiplied? Explain. ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 61 EXAMPLE 25 Calculate each of the following. (a) 4.7 – 1–1 5 8 2 × 1 5 , give the answer as a decimal. (b) 1–0.75 + 1 8 ÷ 1 2 2 × 0.8, give the answer as a fraction. Solution: (a) 4.7 – 1–1 5 8 2 × 1 5 = 4.7 + 1 5 8 × 1 5 = 4.7 + 1.625 × 0.2 = 4.7 + 0.325 = 5.025 (b) 1– 0.75 + 1 8 ÷ 1 2 2 × 0.8 = 1– 75 100 + 1 8 ÷ 1 2 2 × 8 10 Change into fraction. = 1– 3 4 + 1 8 × 2 2 × 4 5 = 1– 3 4 + 1 4 2 × 4 5 = – 2 4 × 4 5 = – 2 5 EXAMPLE 26 A submarine was located 650 m below the sea level. The submarine rose 20.5 m per minute for 20 minutes and then descended 135 2 5 m. Find the final position, in m, of the submarine. Solution: Final position of the submarine = –650 + 20 × 20.5 + 1–135 2 5 2 = –650 + 20 × 20.5 – 135 2 5 = –650 + 410 – 135.4 = –240 – 135.4 = –375.4 Thus, the final position of the submarine was 375.4 m below the sea level. Change into decimals. ©Praxis Publishing_Focus On Maths
62 CHAPTER 2 Real Numbers 1 Arrange these decimal numbers in ascending order. (a) 5.4, 5.29, 5.38, 5.8 (b) 0.63, 0.6, 0.56, 0.7 (c) 2.12, 2.22, 2.02, 2.21 (d) 18.162, 1.816, 18.216, 1.826 2 Change the following fractions into decimals. (a) 24 5 (b) 18 25 (c) 13 8 (d) 5 16 (e) 17 32 (f) 9 64 3 Express each of the following recurring decimals using the dot notation. (a) 3.333… (b) 2.727272… (c) 0.298 298 298… (d) 8.571428 571428 571428… 4 Express each of the following recurring decimals in the ordinary notation. (a) 0.6· (b) 0.0· 9 · (c) 0.4· 01· (d) 0.0· 123456789· 5 Change the following fractions into recurring decimals. (a) 1 9 (b) 63 99 (c) 41 333 (d) 1 15 (e) 46 55 (f) 7 22 6 Represent the following decimals on number lines. (a) 0.2, –0.1, 0.3, –0.4, –0.2 (b) –1.3, –3.9, 2.6, –5.2, 3.9 7 Determine the largest decimal. (a) –2.3 or 2.9 (b) –5.6 or –7.3 (c) –1.12, –3.52 or –1.11 8 Arrange each of the following in (i) ascending order, (ii) descending order. (a) 3.87, –1.4, –6.7, 4.5, –3.31 (b) –0.4, 0.9, –5.2, 1.4, –3.0 (c) 2.33, –3.22, –4.11, 1.55, –1.44 (d) –5.42, –5.44, 0.3, –2.9, 0.03 9 Calculate each of the following. (a) 0.245 + (–2.3) × (–3.8) (b) (–1.75 + 4.85) × (–2.15) (c) –2.25 – 1.125 × 3.8 ÷ 0.25 (d) –0.5 ÷ 0.8 + 1.55 × (–3.24) (e) –4.8 × 1.4 – (–2.28) ÷ 0.625 In an experiment, the initial temperature of a metal rod was 35.3°C. The temperature of the metal rod dropped by 45.5°C and then increased by 6.2°C. Find the final temperature, in °C, of the metal rod. A turtle was 10.28 m below the sea level. The vertical distance between a fish and the turtle is 3.85 m. State the possible positions, in m, of the fish. The estimated fuel consumption of Roy's car is: City: 21.2 km/l Highway: 23.3 km/l The car's gas tank holds 40.2 l of fuel. (a) How far could Roy drive on a full tank of gas on the highway before he runs out of fuel? (b) How far could he drive on a full tank of gas in the city? What assumptions did you make? In a Mathematics Quiz, each contestant is given 10 questions. Each correct answer will be awarded 3 marks, each wrong answer will be deducted 1.5 marks and each unanswered question will be deducted 1 2 marks. The table shows the number of questions answered by 3 contestants. Contestant Number of questions answered correctly Number of questions answered wrongly Emilia 5 4 Ker Er 5 5 Tharishini 4 0 Among the three contestants, who scored the highest marks? Explain your answer. Practice 2.4 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 63 EXAMPLE 27 Draw a number line from –4 to 4. Show the approximate position of each of the following. (a) – 15 4 (b) –2.5 (c) e (d) π Solution: –4 –3 –2 –1 0 1 2 3 4 15 4 – –2.5 e π e = 2.718281… π = 3.1421592… Discuss with your classmate, is 3.141592654... a rational number? Explain. INTERACTIVE ZONE 2.5 Set of Real Numbers The set of real numbers includes whole numbers, integers, fractions, decimals, rational and irrational numbers. For example, 2, 5.4, – 0.2681, 3 4 , 3, 209. The set of real numbers is called R. Real numbers are made up of rational numbers and irrational numbers. The relationship between the sets of numbers that we have learnt can be represented by the diagram. EXAMPLE 28 Find the values of the following. (a) 6 + 3 4 – 1.2 (b) ( 2 × 2) – 5 2 Solution: (a) 6 + 3 4 – 1.2 (b) ( 2 × 2) – 5 2 = 2 – 5 2 = 2 – 2 1 2 = – 1 2 = 6 + 0.75 – 1.2 = 5.55 Real numbers Rational numbers Integers Whole numbers Positive integers (e.g. 2, 8) Zero Negative integers (e.g. – 4, –7) Non-integer rational numbers (e.g. , –5 ) Irrational numbers (e.g. π, √7, –3 √5) 1 3 1 8 ©Praxis Publishing_Focus On Maths
CHAPTER 2 Real Numbers 64 2.6 Use of the Symbol <, >, ≤, ≥, =, ≠ A The meaning of the basic symbols The table below shows some of the mathematical symbols and its meaning. Symbol Symbol Name Meaning Example = Equal sign Equal to 4 = 1 + 3 ≠ Not equal sign Not equal to 2 ≠ 3 , Strict inequality Less than 3 , 4 . Strict inequality Greater than 7 . 5 < Inequality Less than or equal to x < 1 > Inequality Greater than or equal to x > 2 If a . b, then it can be written as b , a, for all values of a and b. For example, 3 . 2, then 2 , 3. The direction of the inequality sign will not be affected by adding or subtracting a number on both sides. For example, If 5 . 3, then 5 + 1 . 3 + 1 and if 3 , 5, then 3 – 2 , 5 – 2. The direction of the inequality sign will not be affected by multiplying or dividing a positive number on both sides. For example, If 3 . 2, then 3 × 4 . 2 × 4 and if 4 , 6, then 4 2 , 6 2 . The direction of the inequality sign need to be reversed when multiplying or dividing a negative number on both sides. For example, If 3 . 2, then 3 × (–2) , 2 × (–2) and if 2 , 6, then 2 –2 . 6 –2. EXAMPLE 29 Insert one of the symbols, ,, ., or = in the box provided for each of the following. (a) 42 (–4)2 (b) 1 3 4 3 2 (c) 64 32 Solution: (a) 42 = (–4)2 (b) 1 3 4 . 3 2 (c) 64 , 32 Practice 2.5 Basic Intermediate Advanced 1 Draw a number line from –5 to 4. Show the approximate position of each of the following. (a) –4.8 (b) 2 (c) 20 9 (d) – π 2 2 Find the value of each of the following. (a) 2.8 – 2 5 + 3 (b) ( 3 × 3) – 5 1 4 (c) 7 2 – 3 2 (d) – 4 7 ÷ 5 14 × 10 3 Give three examples of non-real numbers. ©Praxis Publishing_Focus On Maths
Real Numbers CHAPTER 2 65 EXAMPLE 30 Rewrite each of the following statements using ,, ., <, >, = or ≠. (a) 10 is greater than 7. (b) 4 is equal to (2 × 2). (c) 1 2 is not equal to 1 1 3 ÷ 3 2 2 . (d) y is greater than or equal to k. (e) $3.50 is less than 400¢. (f) The number of apples in the box is less than or equal to 24. Solution: (a) 10 . 7 (b) 4 = (2 × 2) (c) 1 2 ≠ 1 1 3 ÷ 3 22 (d) y > k (e) $3.50 , 400¢ (f) Number of apples in the box < 24 EXAMPLE 31 Write each of the following statements using ,, ., < or >. (a) Najah has $15. She goes to her school canteen for lunch. How much will she spend? (b) A 12-cm string is cut into two pieces. How long is the (i) longer piece? (ii) shorter piece? Solution: (a) $0 < Najah's expenditure < $15 (b) (i) 6 cm , the longer piece , 12 cm (ii) 0 cm , the shorter piece , 6 cm Practice 2.6 Basic Intermediate Advanced 1 Insert ,, ., or = to make a true statement. (a) 10 12 (b) –5 –4 (c) (–3)2 9 (d) 4 5 3 4 (e) 2 2 3 2.8 (f) 49 22 (g) 250¢ $2.6 (h) –0.2 0.1 B Rewrite each of these statements using ,, ., <, >, = or ≠. (a) p is not equal to –4. (b) The length of a rope is less than or equal to 2.6 m. (c) 1 4 is greater than 0.2. (d) w is less than 3 2 . (e) The number of students in the class is greater than or equal to 25. C Rewrite the following statements using the inequalities symbols. (a) Jonah’s daily pocket money is at least $500. (b) Rana spends not more than 2 hours on computer games. D List down all the whole numbers which satisfy the following inequalities. (a) x < 7 (b) x . –4 (c) 5 < x < 1 (d) –1 , x , 8 ©Praxis Publishing_Focus On Maths
66 CHAPTER 2 Real Numbers Summary Summary Summary Rational number • A number that can be expressed in the form a b , where a and b are integers and b ≠ 0. • For example, 2 3 , – 9 14, 6 5 , 8 1 . Integers Real Numbers Integers Irrational numbers Integers Positive fractions Integers Positive decimals Integers Negative fractions Integers Negative decimals Integers Non-integers Integers Integers Non-terminating • A whole number that has a positive sign (+) or negative sign (–), including zero. • For example, ..., –3, –2, –1, 0, 1, 2, 3,... Terminating • A terminating decimal has a finite number of decimal places. • For example, 3.125 (3 d.p) 2 Section A 1. Which of the following shows ascending order? A 2, 7 2 , 0.5, –2, –1 B –12, –14, –16, –18, –20 C –0.25, 1 4 , 3 5 , 4.2, 14 3 D 0.9, 0.6, 0.3, 0, –0.3 2. 2 7 + 2 7 + 2 7 + 3 7 = × 2 7 + 5 7 The missing value in is A 2 C 4 B 3 D 5 3. Which of the following is the correct calculation for 6(–2 + 8) × 4.8 ÷ 3 5 ? A –36 × 8 C 36 × 4.8 × 5 3 B 36 × 4.8 × 3 5 D 6(–6) × 8 4. The diagram below shows a number line. P and Q are decimal numbers. –1.7 P Q –0.8 –0.5 Determine the values of P and Q. A P = –1.4, Q = –1.3 B P = –1.1, Q = –1.4 C P = –1.3, Q = –1.4 D P = –1.4, Q = –1.1 5. 0.219 K 0.223 0.224 The diagram shows part of a number line. Find the value of K. A 221 100 C 221 1000 B 222 100 D 222 1000 6. 3.4257 7.0346 4.8625 2.6458 P Q R S The diagram shows four decimals written on cards P, Q, R and S respectively. Which card has the smallest value for the digit 4? A P C R B Q D S Section B 1. (a) The diagram below shows 4 number cards. –10 11 –13 5 Fill in the blanks with the suitable ©Praxis Publishing_Focus On Maths
67 Real Numbers CHAPTER 2 number from the diagram to form an arrangement of integer in descending order. 8 , , – 8 , , –12 (b) Mark (3) against the rational number and (7) against the irrational number. (i) π (ii) −3.5 2. (a) Fill in the blanks with ‘+’, ‘-’, ‘×’, ‘÷’. (i) −9 (−7) = −2 × [6 (−2)] (ii) 6 + (−8) 4 = −4 (−1) (b) Fill in the blanks with ‘+’ or ‘−’ to obtain the smallest value. (i) 1 4 (−6) 5.3 (ii) −3.5 7 5 (−8) 3. (a) Cindy has 2480 pieces of stamps. The number of Ahmad’s stamp is 1 8 of Cindy’s stamp, while Muthu has 4 times the number of stamps that Ahmad has. All the stamps are collected and arrange into 10 albums. Calculate the number of stamps in each album. (b) Given that 3 4 of Alvin’s money is equal to 2 3 of Daniel’s money. If Daniel has $360, calculate the total amount of their money. 4. (a) Sam wanted to simplify 20 9 . The following are Sam’s steps of working. 20 9 = 20 4 + 5 Step 1 = 20 4 + 20 5 Step 2 = 5 + 4 Step 3 = 9 Step 4 Sam’s friend, Daniel told him that his working was incorrect. Which step of Sam’s working is incorrect? What is the actual answer? (b) Given that 7.5 ÷ 2 5 − 3.51 = x × 0.5. Find the value x. 5. Mr. Foo bought 2000 units of shares of Company A & B at the price of $2.48 per unit. On the next day, the shares decreased by 5 cents per unit and Mr. Foo bought another 3000 units of shares. If Mr. Foo sold all his shares at the price of $2.46 per unit, determine whether Mr. Foo gained profit. Hence, give your opinion on why Mr. Foo bought the shares when the price of shares decreased. 6. The thermostat on a freezer is set at –18ºC. The compressor on the freezer turns on and cools down the freezer when the rises to –15.5ºC. The compressor turns off when the temperature drops to –19.5ºC. (a) Draw a thermometer and mark the 3 freezer temperatures. (b) A package of meat must remain below –18ºC. Should this freezer be used? Explain. 7. In Asia, the lowest point on land is the shore of the Dead Sea, which is 417.5 m below sea level. The highest point is the peak of Mount Everest, which 8844.43 m above the sea level. (a) Write each measurement above as a rational number. (b) Write a substraction statement that represents the distance between the highest point and the lowest point. What is this distance? 8. Three hikers are returning to base camp after a mountain climbing expedition. Hiker A is 26.4 m above base camp, Hiker B is 37.2 m below base camp, and Hiker C is 15.7 m below base camp. (a) Represent each hiker's distance above or below base camp as a rational number. (b) Draw and label a vertical number line to show the base camp and the positions of the hikers. (c) Which hiker is closest to base camp? Explain your reason. (d) Which hiker has the lowest altitude? How do you know? ©Praxis Publishing_Focus On Maths
RATIOS, RATES AND 3PROPORTIONS Applications of this chapter Ratios and proportions are widely used in fields where concepts need to be compared. Astronomers use the concept of ratios to measure distances in the solar system. They measure the distance between the Earth and the Sun to determine the distances between Earth and other planets in the solar system. Other common examples: using a standard cement and sand ratios in constructions of houses and buildings that we live in, comparing prices per ounce while grocery shopping, calculating the proper amounts for ingredients in recipes and determining how long a car trip might take. What is the distance between the Earth and the Sun? 68 ©Praxis Publishing_Focus On Maths
Concept Map Learning Outcomes • Compare two quantities of ratio. • Compare three quantities of ratio. • Write a ratio in its simplest form and as fraction. • Divide quantities in a given ratio. • Increase and decrease a value in a given ratio. • Solve problems involving ratios. • Convert measurements on maps, plans and scale drawings to actual measurements and vice versa. • Determine whether two quantities are in proportion. • Solve practical problems on direct and inverse proportions. • Ratio • Equivalent • Simplest form • Simplify • Rate • Scale • Lowest term • Proportion • Direct proportions • Inverse proportions Key Terms Ratios, Rates and Proportions Direct Proportions Simplifying Ratios Increase and Decrease in Ratios Ratios as Fractions Two Quantities of Same Unit Two Quantities of Different Unit Equivalent Ratios Inverse Proportions Ratio Rate Scales Proportions Fibonacci, also known as Leonardo Bonacci was an Italian mathematician that invented the Fibonacci sequence. Fibonacci ratios were derived from the Fibonacci sequence. He wrote Liber abaci which means ‘Book of the Abacus’, which introduced the Hindu-Arabic numeral systems to Europe. Maths History 69 ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 70 Flashback 1. Complete each of the following. (a) 2 km = m (b) 1.6 m = cm (c) 0.5 kg = g (d) $36 = cents 2. Find the HCF of each of the following. (a) 6, 9, 15 (b) 4, 8, 20 3. Find the LCM of each of the following. (a) 2, 5, 10 (b) 2, 3, 4 Explain how to compare these numbers more effectively. Thinking The advertisements show there are different ways to compare numbers. (a) How are the numbers in each advertisement compared? (b) Which advertisement is most effective? (c) Why do you think so? 1 PEOPLE SURVEYED PREFERRED 2 out of 3 tasty HOTDOGS PEOPLE SURVEYED PREFERRED 7023 out of 10559 tasty HOTDOGS MANY PEOPLE PREFERRED TWICE AS tasty HOTDOGS MORE PEOPLE PREFERRED 3516 tasty HOTDOGS ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 71 Sham saved 15 coins from school pocket money in January. (a) How many 5 cents does he have? (b) How many 10 cents does he have? (c) How many 20 cents does he have? (d) What fraction of the coins are 5 cents? (e) What fraction of the coins are 10 cents? ( f) What fraction of the coins are 20 cents? 2 3.1 Ratios A Comparing two quantities of same unit A ratio is a term used to compare two quantities of the same unit in a specified order. The ratio of the quantities can be written as a : b. ‘:’ or ‘to’ is used to separate the values. For example, the ratio of 3 boys and 2 girls standing in a row is written as 3 : 2 or 3 to 2. The two quantities of the ratio must have the same unit. EXAMPLE 1 A plastic bag contains 2 red sweets and 5 yellow sweets. Write each of the following as a ratio. (a) Number of red sweets : Number of yellow sweets (b) Number of yellow sweets : Total number of sweets Solution: (a) Number of red sweets : Number of yellow sweets = 2 : 5 (b) Number of yellow sweets : Total number of sweets = 5 : (2 + 5) = 5 : 7 ‘:’ is read as ‘to’. Discuss with your classmates. What is the difference between a part-to-whole ratio and a part-to-part ratio? INTERACTIVE ZONE ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 72 EXAMPLE 2 Write each of the following as a ratio by using ‘:’. (a) $8 to $3. (b) 1 green apple to 4 red apples. (c) 2 kg of big fish to 7 kg of small fish. Solution: (a) $8 to $3 (b) 1 green apple to 4 red apples = 8 : 3 = 1 : 4 (c) 2 kg of big fish to 7 kg of small fish = 2 : 7 Two quantities are expressed as a ratio without the units. B Comparing two quantities of different unit Ratios can be used to compare measurements. However, when comparing two quantities, we must ensure both the quantities are in the same unit before expressing them in ratio form. EXAMPLE 3 Write each of the following ratios in the form a : b. (a) 200 cm to 5 m (b) 2 hours to 60 minutes (c) 1 year to 5 months Solution: (a) 200 cm to 5 m 1 m = 100 cm = 2 m to 5 m = 2 : 5 (b) 2 hours to 60 minutes 1 hour = 60 minutes = 2 hours to 1 hour = 2 : 1 (c) 1 year to 5 months 1 year = 12 months = 12 months to 5 months = 12 : 5 Before determining the ratio, convert both quantities of different units to the same unit. Ratio has no unit. To convert a larger unit (eg. kg) to a smaller unit (eg. g), we multiply. To convert a smaller unit to a larger unit, we divide. 1 km = 1000 m 1 m = 100 cm 1 kg = 1000 g 1 l = 1000 ml 1 h = 60 min = 3600 s $1 = 100 cents C Simplifying ratios to the simplest form A ratio is in its simplest form when both sides are whole numbers and there is no whole number which both sides can be divided by. A ratio can be simplified to the simplest form by multiplying or dividing both quantities by the same whole numbers. ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 73 EXAMPLE 4 Simplify each of the following ratios in its simplest form. (a) 4 : 8 (b) 3.5 : 10 (c) 2 : 3 4 (d) 1500 g : 2 kg Solution: (a) 4 : 8 = 4 4 : 8 4 Divide both quantities by 4. = 1 : 2 (b) 3.5 : 10 = 3.5 × 10 : 10 × 10 Multiply both quantities by 10. = 35 : 100 = 35 5 : 100 5 Divide both quantities by 5. = 7 : 20 (c) 2 : 3 4 = 2 × 4 : 3 4 × 4 Multiply both quantities by 4. = 8 : 3 (d) 1500 g : 2 kg = 1500 g : 2000 g 1 kg = 1000 g = 1500 : 2000 = 15 5 : 20 5 Divide both quantities by 5. = 3 : 4 Simplify a ratio by dividing both quantities by their HCF or multiplying both quantities by their LCM. Simplest form is also called ‘lowest terms’. Thinking The diagram shows three photos of different sizes. How do you represent the relationship of the sizes of these three photos in a ratio? 5 cm 7 cm 10 cm 14 cm 15 cm 21 cm ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 74 D Equivalent ratios If two ratios have the same value, then they are equivalent. Equivalent ratios are basically the same as equivalent fractions. We can find equivalent ratios by multiplying or dividing both quantities of a ratio by the same number. For example, 2 : 6 = 1 : 3. Observe the diagram, the equivalent fractions are represented by the shaded parts in each circle. 1 2 2 4 6 12 The areas of the shaded parts are always of the same size. Mona makes her cocoa drink with 2 scoops of cocoa powder to 5 cups of water. Jess makes her cocoa drink with 3 scoops of cocoa powder to 7 cups of water. (a) How much water is used for 1 scoop of cocoa powder? (b) Whose cocoa drink is stronger? Critical Thinking EXAMPLE 5 Determine whether the following pairs of ratios are equivalent. (a) 2 : 8 and 1 : 4 (b) 4 : 3 and 12 : 9 (c) 3 : 5 and 15 : 20 (d) 20 : 36 and 5 : 9 Solution: (a) 2 : 8 = 2 2 : 8 2 Divide both quantities by 2. = 1 : 4 Therefore, 2 : 8 and 1 : 4 are equivalent. (b) 4 : 3 = 4 × 3 : 3 × 3 Multiply both quantities by 3. = 12 : 9 Therefore, 4 : 3 and 12 : 9 are equivalent. (c) 3 : 5 = 3 × 5 : 5 × 5 Multiply both quantities by 5. = 15 : 25 Therefore, 3 : 5 and 15 : 20 are not equivalent. Ratio a : b ≠ b : a. For example, 2 : 3 ≠ 3 : 2. The equivalent ratios of 1 4 and 2 8 can be represented by the following diagrams. Case I: comparing the shaded part to the whole diagram 1 : 4 2 : 8 ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 75 Write two ratios that are equivalent. Explain how you know thay are equivalent. Write two ratios that are not equivalent. Explain how you know they are not equivalent. INTERACTIVE ZONE E Ratios of three or more quantities EXAMPLE 6 Simplify each of the following ratios in its simplest form. (a) 6 : 8 : 12 (b) 2 3 : 1 2 : 5 6 (c) 1.4 km : 0.7 km : 350 m (d) 150 minutes : 90 minutes : 3 hours Ratios of two quantities a : b can be extended to three or more quantities. A ratio a : b : c compares three quantities in the same unit. Nora prepared some Kuih for tea time with her family members. (a) How many Kuih Lapis did she prepare? (b) How many Karipap did she prepare? (c) How many Kuih Seri Muka did she prepare? (d) How many Kuih Nona Manis did she prepare? (e) Represent the relationship of the number of Kuih Lapis, Kuih Seri Muka and Kuih Nona Manis in the form of a : b : c. Express your answer in its simplest form. ( f) Write the ratio of the number of Kuih Seri Muka to the number of Karipap to the total number of Kuih in the form of a : b : c. Express your answer in its simplest form. 3 Kuih Nona Manis Kuih Seri Muka Karipap Kuih Lapis Case II: comparing the shaded parts to the unshaded parts of the diagram 1 : 4 2 : 8 (d) 20 : 36 = 20 4 : 36 4 Divide both quantities by 4. = 5 : 9 Therefore, 20 : 36 and 5 : 9 are equivalent. ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 76 Solution: (a) 6 : 8 : 12 = 6 2 : 8 2 : 12 2 Divide by 2, the HCF of 6, 8 and 12. = 3 : 4 : 6 (b) 2 3 : 1 2 : 5 6 = 2 3 × 6 : 1 2 × 6 : 5 6 × 6 Multiply by 6, the LCM of 3, 2 and 6. = 4 : 3 : 5 (c) 1.4 km : 0.7 km : 350 m = 1400 m : 700 m : 350 m 1 km = 1000 m = 4 : 2 : 1 Divide by 350. (d) 150 minutes : 90 minutes : 3 hours = 150 minutes : 90 minutes : 180 minutes 1 h = 60 min = 5 : 3 : 6 Divide by 30. Work with your team members. Trail Mix Recipe 3 scoops raisins 9 scoops nuts 6 scoops dried apple 3 scoops sunflower seeds Write as many ratios as you can for the trail mix recipe. Explain what each ratio compares. team work F Increase and decrease in ratios When a number is increased in the ratio p : q, where p . q, the resulting value is obtained by multiplying the value by the multiplying factor, p q . Similarly when a number is decreased in the ratio p : q, where p , q, the resulting value is obtained by multiplying the value by the multiplying factor, p q . EXAMPLE 7 (a) Increase 32 in the ratio 7 : 4. (b) Decrease 64 in the ratio 5 : 8. When the multiplying factor . 1, the value increases. When the multiplying factor , 1, the value decreases. ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 77 Solution: (a) Multiplying factor = 7 4 New value = 32 × 7 4 = 56 (b) Multiplying factor = 5 8 New value = 64 × 5 8 = 40 EXAMPLE 8 The price of a printer has been increased from $160 to $192. In what ratio has the price increased? Solution: Increased price : Original price 192 : 160 = 192 32 : 160 32 = 6 : 5 \ The price has been increased in the ratio 6 : 5. EXAMPLE 9 In a sale, the price of a television is decreased in the ratio 3 : 4. If the television costs $1500 after the decrease, what is its original price? Solution: New price = Original price × 3 4 Original price × 3 4 = 1500 Original price = 1500 × 4 3 = 2000 \ The original price is $2000. Maths LINK Your World In a supermarket, the tag on the shelf with the barcode of a product shows the price of 1 g of the product. With this information, you can compare the prices of packages of the product in different sizes. ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 78 G Ratio as fractions A ratio a : b can also be expressed as the simplest fraction form, a b . For example, 2 : 5 = 2 5 . The order in which a ratio is written is important. The quantity which is mentioned first in the ratio is the numerator of the fraction. What is the difference between the concepts of ratio and fraction? Critical Thinking EXAMPLE 10 Write each of the following ratios as a fraction. (a) 4 to 7 (b) 5 to 3 (c) 6 : 27 (d) 16 : 12 Solution: (a) 4 to 7 = 4 7 (b) 5 to 3 = 5 3 (c) 6 : 27 = 6 27 = 2 9 (d) 16 : 12 = 16 12 = 4 3 EXAMPLE 11 Express each of the following ratios as a fraction. Give your answer in its simplest. (a) 10 cm : 35 cm (b) $200 : $500 (c) 45 minutes : 2 hours (d) 4 hours : 1 day Solution: (a) 10 cm : 35 cm (b) $200 : $500 = 10 35 = 2 7 = 200 500 = 2 5 (c) 45 minutes : 2 hours = 45 minutes : 120 minutes = 45 120 = 9 24 = 3 8 • A ratio a : b is equivalent to the fraction a b . • The first part of a ratio is the numerator and the second part is the denominator. A ratio 5 3 (improper fraction) is not changed to a mixed fraction (1 2 3 ). 1 hour = 60 minutes 1 day = 24 hours (d) 4 hours : 1 day = 4 hours : 24 hours = 4 24 = 1 6 ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 79 EXAMPLE 12 A person may have either single eyelids or double eyelids due to genetics. In a class of 40 students, there are 16 students with single eyelids. Express each of the following ratios as a fraction. Give your answer in its simplest. (a) Number of students with double eyelids to total number of students. (b) Number of students with single eyelids to number of students with double eyelids. Solution: (a) Number of students with double eyelids Total number of students = 40 – 16 40 = 24 40 = 3 5 (b) Number of students with single eyelids Number of students with double eyelids = 16 24 = 2 3 H Solving problems involving ratios EXAMPLE 13 Laura wants to decorate the walls of her kitchen with blue and white tiles in the ratio 15 : 6. She requires 200 white tiles. Find the number of blue tiles that she requires. Solution: Let the number of white tiles = w the number of blue tiles = r r : w = 15 : 6 r : 200 = 15 : 6 r 200 = 15 6 r × 6 = 200 × 15 r = 15 6 × 200 = 3000 6 = 500 Therefore, she requires 500 blue tiles. ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 80 EXAMPLE 14 In the year 2021, the ratio of rainy days to dry days in Town A was 23 : 50. How many rainy days did the town have in that year? Solution: Let the number of rainy days = r Given that rainy days : dry days = 23 : 50 r : 365 = 23 : (23 + 50) r 365 = 23 73 r × 73 = 365 × 23 r = 23 73 × 365 r = 115 Therefore, the town had 115 rainy days in the year 2021. EXAMPLE 15 A piece of 40 cm long wire is cut into two pieces in the ratio 3 : 5. (a) What is the length of the longer wire? (b) Find the length of the shorter wire. Solution: (a) Ratio = 3 : 5 Number of parts = 3 + 5 = 8 Each part = 40 cm 8 = 5 cm \ Length of the longer wire = 5 × 5 cm = 25 cm (b) Length of the shorter wire = 3 × 5 cm = 15 cm Total number of equal part = 3 + 5 = 8 Length of the longer wire = 5 8 of the whole wire = 5 8 × 40 cm = 25 cm Length of the shorter wire = 3 8 of the whole wire = 3 8 × 40 cm = 15 cm ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 81 EXAMPLE 16 Sparkling punch is made from a mixture of Fruit juice P and Fruit juice Q. Fruit juice P and Fruit juice Q are mixed in the ratio of 4 : 5. If 1 l of Fruit juice P costs $15.50 and 1 l of Fruit juice Q costs $20, what is the cost of 1 l of the sparkling punch? Solution: Fruit juice P : Fruit juice Q = 4 : 5 9 l of the sparkling punch contains 4 l of Fruit juice P and 5 l of Fruit juice Q. Cost of 9 l of the sparkling punch = (4 × $15.50) + (5 × $20) = $162 \ Cost of 1 l of sparkling punch = $162 9 = $18. Practice 3.1 Basic Intermediate Advanced A Write each of the following ratios in the form a : b. (a) 100 cm to 4 m (b) 120 minutes to 5 hours (c) 2 years to 36 months (d) 3 l to 1000 ml B Determine whether each of the following pairs of ratios is equivalent. (a) 2 : 4 and 1 : 2 (b) 3 : 2 and 12 : 10 (c) 5 : 8 and 15 : 24 (d) 1 : 6 and 18 : 3 C Simplify each of the following ratios. (a) 6 : 9 : 15 (b) 2 5 : 1 2 : 3 10 (c) 2.8 l : 1.2 l : 800 ml (d) 240 min : 90 min : 2.5 h D In a sale at a boutique, the price of a gown has been decreased from $450 to $400. In what ratio has the price been decreased? E 3 cm 6 cm P 8 cm 12 cm Q The diagram shows two rectangles, P and Q. Express the following relationships as ratios in its simplest form. (a) Length of rectangle P to length of rectangle Q. (b) Width of rectangle P to width of rectangle Q. Maths LINK Your World Contrast ratio is associated with televisions and computer monitors. It is a measure of the difference between the brightest and darkest colours displayed on a screen. A high contrast ratio, such as 800 : 1, provides a better image than a low contrast ratio, such as 150 : 1. ©Praxis Publishing_Focus On Maths
82 CHAPTER 3 Ratios, Rates and Proportions (c) Perimeter of rectangle Q to perimeter of rectangle P. (d) Area of rectangle Q to area of rectangle P. F 3 cm 2 cm 7 cm A B C A 12 cm long ribbon is cut into three parts, A, B and C. Express each of the following ratios as a fraction. Give your answer in its simplest form. (a) Length of Part B to length of Part C. (b) Length of Part A to total length of the ribbon. G The difference in temperature between a cup of hot coffee and hot cooking oil is 110°C. If the ratio of their temperatures is 17 : 6, find the sum of their temperatures. H Koko chocolate powder is a mixture of Grade A and Grade B chocolate powder in the ratio of 3 : 4. If 1 kg of Grade A chocolate powder costs $350 and 1 kg of Grade B chocolate powder costs $420, what is the cost of 1 kg of Koko chocolate powder? 9 Lina, Phang and Gary put a total of $9000 into a business in the ratio 5 : 4 : 3. (a) How much did each person contribute? (b) If the business made a profit of $22 560, how should the profit be divided among Lina, Phang and Gary? Catherine has diabetes. For each meal, she must estimate the mass in grams of carbohydrates she plans to eat, then inject the appropriate amount of insulin. Catherine needs 1 unit of insulin for 15 grams of carbohydrates. Catherine's lunch contains 60 grams of carbohydrates. How many units of insulin should Catherine inject? 3.2 Rate A Determining the relationship between ratios and rates Objective : To determine the relationship between ratios and rates. Instruction : Do this activity in pairs. 1. In the table below, write the ratio of two quantities for the measurements involved in each of the situations given. 2. Write the quantities involved and also their units of measurement. Situation Ratio in the form a b Quantities involved Units of measurement A lorry travels 375 km in 5 hours. 375 km 5 hours Distance and time km and hour A tree grows 272 cm in 2 months. A baby monkey's mass increases by 3.5 kg in 60 days. Zhang's family paid $48 for 320 kWh of electricity. 50 l of water flows from the tap in 4 minutes. 1 ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 83 In Activity 1, we compare two quantities measured in different units. For example, in the ratio 375 km 5 hours , we compare the distance travelled in km with the time taken in hours. The ratio 375 km 5 hours is known as a rate. Rates show how two quantities with different units are related to each other. Rate describes the change of one quantity with respect to the change of another quantity. When the value of the second quantity in a rate is 1, the rate is called a unit rate. Rate Ratio Unit Speed Distance to time km/h or m/s Acceleration Speed to time km/h2 or m/s2 Density Mass to volume kg/m3 or g/cm3 Pressure Force to area N/m2 or N/cm2 B Identifying the two related quantities in determining the rates EXAMPLE 17 Siva finished reading 3 story books in 1 week. Determine Siva’s rate of reading and identify the two quantities involved (including the units). Solution: Rate = 3 books 1 week The quantities involved are number of books and time (week). C Calculate rate EXAMPLE 18 Item Quantity Price Milk 1 tin $2 Fish 2 kg $14 Egg 10 units $4 Based on the price list above, state the price rate for each item. Unit is number per week or number/week. Rate is a special ratio that compares two quantities with different units of measurement. ©Praxis Publishing_Focus On Maths
CHAPTER 3 Ratios, Rates and Proportions 84 Solution: Price rate for milk = $2 1 tin = $2 per tin Price rate for fish = $14 2 kg = $7 1 kg = $7 per kg Price rate for egg = $4 10 eggs = $0.40 1 egg = $0.40 per egg Ratio as a fraction. Unit rate. Simplifying. Unit rate. Divide. Unit rate. EXAMPLE 19 (a) The temperature of a jug of water dropped constantly from 90°C to 69°C in 15 minutes. Find the cooling rate of the water, in °C/min. (b) Water flow into a tank at a constant rate. If the height of water increases from 20 cm to 56 cm in half an hour, what is the flow rate of the water, in cm/min. Solution: (a) Cooling rate = Temperature dropped Time taken = 90°C – 69°C 15 min = 1.4°C/min (b) The flow rate = Water level increased Time taken = 56 cm – 20 cm 30 min = 1.2 cm/min A unit rate is a rate with a denominator of 1. A rate of change can be specified per unit of time / area / volume / capacity / mass etc. ©Praxis Publishing_Focus On Maths
Ratios, Rates and Proportions CHAPTER 3 85 D Conversion of units of rates (I) Conversion from one compound unit to another compound unit In this section, we will learn how to change a measurement from one compound unit to another compound unit. For example, convert km/h to m/s. EXAMPLE 20 Convert each of the following. (a) $90/h to cents/min (b) 16 km/l to m/ml (c) 42 ml/kg to l/ton (d) 65 cents/cm2 to $/m2 Solution: (a) $90/h = 90 × 100 cents / 60 min = 9000 cents 60 min = 150 cents/min ∴ $90/h = 150 cents/min (b) 16 km/l = 16 × 1000 m / 1000 ml = 16 000 m 1000 ml ∴ 16 km/l = 16 m/ml • km ×1000 ÷1000 m ×100 ÷100 cm ×10 ÷10 mm • ton ×1000 ÷1000 kg ×1000 ÷1000 g • l ×1000 ÷1000 ml • $ ×100 ÷100 cents • h ×60 ÷60 min ×60 ÷60 sec • m2 ×10 000 ÷10 000 cm2 (c) 42 ml/kg = 42 1000 l / 1 1000 ton = 42 1000 × 1000 l / 1 1000 × 1000 ton = 42 l/ton ∴ 42 ml/kg = 42 l/ton (d) 65 cents/cm2 = $ 65 100 / 1 10 000 m2 = $ 65 100 × 10 000 / 1 10 000 × 10 000 m2 = $650/m2 ∴ 6.5 cents/cm2 = $650/m2 ©Praxis Publishing_Focus On Maths