Grade 8 Mathematics: Rational Numbers PDF

1 Mathematics: Grade 8 Chapter 1 UNIT RATIONAL UNIT 1 NUMBERS RATIONAL NUMBERS Learning Outcomes: At the end of...

1 Mathematics: Grade 8 Chapter 1 UNIT RATIONAL UNIT 1 NUMBERS RATIONAL NUMBERS Learning Outcomes: At the end of this unit, learners will able to:  Define and represent rational numbers as fractions  Show the relationship among ℕ, W, ℤ and ℚ.  Order rational numbers.  Solve problems involving addition, Subtraction, Multiplication and division of rational numbers  Apply Rational Numbers to solve practical problems.  Aware the four operations as they relate to Rational Numbers. Main Contents 1.1 The concept of Rational numbers 1.2 Comparing and Ordering Rational numbers 1.3 Operation and Properties of Rational numbers 1.4 Application of Rational numbers Summary Review Exercise 1 Chapter 1 Mathematics: Grade 8 INTRODUCTION Many times throughout your mathematics lessons, you will be manipulating specific kinds of numbers that are related to your real life activities. So, it is important to understand how mathematicians classify numbers and what kinds of major classifications exist. In the previous grades you have learnt about the set of Natural numbers, Whole numbers and Integers and their basic properties. In this unit you will learn about the set of numbers which contains the other set of numbers (i.e. ℕ, W, ℤ) called Rational numbers. And also you will learn about the basic properties, operations and real life applications of rational numbers. 1.1. The concept of Rational numbers Competencies: At the end of this sub-topic, students should:  Describe the concept of Rational Numbers practically.  Express Rational Numbers as fractions. Group work 1.1 DeVFULEH the following questions with your groups 1. Provide an example of each of the following numbers a. Natural numbers smaller than 10. b. Whole number that is not a natural number c. Integers that is not a whole number 2. Which of the following set of numbers include the other set of numbers? a. Whole numbers b. Integers c. Natural numbers 3. Define a rational number in your own words. 4. Solomon has 3 cats and 2 dogs. He wants to buy a toy for each of his pets. Solomon has 22 Birr to spend on pet toys. How much can he spend on each pet? Write your answer as a fraction and as an amount in Birr and Cents. 2 Mathematics: Grade 8 Chapter 1 1.1.1. Representation of Rational Numbers on a Number line Competency: At the end this sub-topic, students should:  Represent rational numbers as a set of fractions on a number line. Revision on Fractions A fraction represents the portion or part of the whole thing. For example, one- half, three-quarters. A fraction has two parts, namely numerator (the number on the top) and denominator (the number on the bottom). Example 1.1: Figure 1.1 DVD Figure 1.2 Rectangular fields 7 1 The shaded part is of the DVD one part is one-ninth, of the 8 9 rectangular field Example 1.2: If three-fifths of the green area be covered by indigenous plants, then find the numerator and denominator of the covered area. Solution: Figure 1.3 Green area  Numerator 3 Denominator 5 3 Chapter 1 Mathematics: Grade 8 In grade 6 and 7 you have discussed the important ideas about fractions and integers. i. Proper fraction: A fraction in which the numerator is less than the denominator. ii. Improper fraction: A fraction in which the numerator is greater than or equal to the denominator. If an improper fraction is expressed as a whole number and proper fraction, then it is called mixed fraction. Integers are represented on number line as shown below in figure 1.4. Figure 1.4 What number is represented by the marked letter 𝑥𝑥 on the number line above? You observe that the number 𝑥𝑥 is greater than 2 but less than 3. So, it belongs to the interval between 2 and 3. Thus 𝑥𝑥 is not a natural number, or a whole number, or an integer. What type of a number is? Using the above discussion, we define a rational number as follow: Definition 1.1 𝑎𝑎 A number that can be written in the form of where 𝑎𝑎 and 𝑏𝑏 are integers and 𝑏𝑏 ≠ 𝑏𝑏 0, is called a rational number. Example 1.3: 1 3 3 8 14 5 5 , 7 , 5 , −2 , − 4 and 9 are rational numbers. 4 Mathematics: Grade 8 Chapter 1 Note: The set of rational numbers is denoted by ℚ. How can we locate rational numbers on number line? Rational number can be represented on a number line by considering the following facts. I. Positive rational numbers are always represented on the right side of zero and negative rational numbers are always represented on the left side of zero on a number line. II. Positive proper fractions always exist between zero and one on number line. III. Improper fractions are represented on number line by first converting into mixed fraction and then represented on the number line. Example 1.4 Sketch a number line and mark the location of each rational numbers. 2 3 3 5 a. b. c. − d. − 5 2 4 2 Solution: 2 a) Since 5 > 0, and proper, so it lies on the right side of 0 and on the left side of 1. How can we locate? Divide the number line between 0 and 1 into 5 equal parts. Then the second 2 part of the fifth parts will be a representation of 5 on number line. Figure 1. 5 3 b) Since is an improper fraction, first convert to mixed fraction to find 2 between which whole numbers the fraction exists on the number line. 5 Chapter 1 Mathematics: Grade 8 3 1 1 Thus, 2 = 1 2. The fraction lies between 1 and 2 at 2 point. Now, divide the number line between 1 and 2 in two equal parts and then the 1st part of 2 parts will be the required rational number on the number line. Figure 1.6 3 c) Since −1< − < 0, the fraction will lie between −1 and 0.To represent on 4 the number line, divide the number line between −1 and 0 in to 4 equal 3 parts and the third part of the four parts will be − 4. Figure 1.7 5 d) Since − 2 < 0 and improper, first change in to mixed fraction. That is, 5 1 − 2 = −2 2. To represent on the number line divide the number line between −3 and −2 in to two equal parts and the first part of the two parts 5 is − 2. Note: Two rational numbers are said to be opposite, if they have the same distance from 0 but in different sides of 0. 3 3 For instance 2 and − 2 are opposites. 6 Mathematics: Grade 8 Chapter 1 Figure 1.8 Exercise 1.1 1. Consider the following number line. Figure 1.10 Select a reasonable value for point B. a) 0.5 b) 3.6 c) – 0.2 d) 2 2. Between what consecutive integers the following rational numbers exist? 3 8 3 9 a) 7 b) 5 c) −5 d) − 5 3. Change the following improper fractions to mixed fractions. 32 27 7 a) 5 b) − 10 c) 3 4. Represent the following rational numbers on a number line. 5 3 5 8 2 a) 6 b) 5 c) − 6 d) − 5 e) 2 5 5. If you plot the point −8.85 on a number line, would you place it to the left or right of −8.8? Explain. 6. Find the opposite of the following rational numbers. 4 2 3 4 a) b) − c) 2 d) −3 7 5 3 5 7  Chapter 1 Mathematics: Grade 8 1.1.2. Relationship among W, ℤ and ℚ Competency: At the end of this section, students should:  Describe the relationship among the sets ℕ, W, ℤ and ℚ. In the previous grades of mathematics lesson you have learnt about the sets of natural numbers (ℕ), whole numbers (W) and integers (ℤ).In this subsection you will discuss the relationship among these set of numbers with the other set of number which is rational numbers. Recall that:  A collection of items is called a set.  The items in a set are called elements and is denoted by ∈.  A Venn diagram uses intersecting circles to show relationships among sets of numbers. The Venn diagram below shows how the set of natural numbers, whole numbers, integers, and rational numbers are related to each other. When a set is contained within a larger set in a Venn diagram, the numbers in the smaller set are members of the larger set. When we classify a number, we can use Venn diagram to help figure out which other sets, if any, it belongs to. Figure 1.11. Venn diagram 8 Mathematics: Grade 8 Chapter 1 Example 1.5: Classify the following numbers by naming the set or sets to which it belongs. 1 5 a. −13 b. 7 c. − 76 d. 10 Solution: a. integer, rational number b. rational number c. rational number d. natural number, whole number, integer, rational number. Example 1.6: Is it possible for a number to be a rational number that is not an integer but is a whole number? Explain. Solution: No, because a whole number is an integer. Exercise 1.2 1. Solomon says the number 0 belongs only to the set of rational numbers. Explain his error. 2. Write true if the statement is correct and false if it is not. a) The set of numbers consisting of whole numbers and its opposites is called integers. b) Every natural number is a whole number. 2 c) The number −3 7 belongs to negative integers. 1.1.3. Absolute value of Rational numbers Competency: At the end of this section, students should:  Determine the absolute value of a rational number. Activity 1.1 1. What is the distance between 0 and 5 on the number line? Between 0 and −5 on the number line? 9 Chapter 1 Mathematics: Grade 8 The absolute value of a rational number describes the distance from zero that a number is on a number line without considering direction. For example, the absolute value of a number is 5 means the point is 5 units from zero on the number line. Definition 1.2: The absolute value of a rational number ‘𝑥𝑥’, denoted by │𝑥𝑥│, is defined as: 𝑥𝑥, 𝑖𝑖𝑖𝑖 𝑥𝑥 ≥ 0 │𝑥𝑥│ ={ −𝑥𝑥, 𝑖𝑖𝑖𝑖 𝑥𝑥 < 0 Example 1.7: a. │6│ = 6 b. │0│ = 0 c. │−15│ = −(−15) = 15 Example 1.8: Simplify each of the following absolute value expressions. a. │8 −3│ b. │– 25+13│ c. │0 − 10│ Solution: a. Since 8−3 = 5 and 5 > 0, we have │8 −3│ = │5│ = 5 b. Since −25+13 = −12 and −12 < 0, we have │ −25+13│ = │ −12│ = −(−12) = 12 c. Since 0 –10 = −10 and −10 < 0, we have │0 −10│ =│−10│ = −(−10) = 10 Equation involving absolute value Definition 1.3: An equation of the form │𝑥𝑥│=𝑎𝑎 for any rational number 𝑎𝑎 is called an absolute value equation. Geometrically the equation │𝑥𝑥│ = 8 means that the point with coordinate 𝑥𝑥 is 8 units from 0 on the number line. Obviously the number line contains two points that are 8 units from the origin, one to the right and the other to the left of the origin. Thus │𝑥𝑥│ = 8 has two solutions 𝑥𝑥 = 8 and 𝑥𝑥 = −8. 10 Mathematics: Grade 8 Chapter 1 Opposite -8 0 8 Figure 1.12 Note The solution of the equation │𝑥𝑥│= 𝑎𝑎 for any rational number 𝑎𝑎, has i. Two solutions 𝑥𝑥 = 𝑎𝑎 and 𝑥𝑥 = −𝑎𝑎 if 𝑎𝑎 > 0. ii. One solution, 𝑥𝑥 = 0 if 𝑎𝑎 = 0 and iii. No solution, if 𝑎𝑎 < 0. Example 1.9: Solve the following absolute value equations. a. │𝒙𝒙│ =13 b. │𝒙𝒙│ = 0 c. │𝒙𝒙│ = −6 Solution: a. │𝒙𝒙│ =13 Since, 13 > 0, │𝒙𝒙│ = 13 has two solutions:  𝑥𝑥 = 13 and 𝑥𝑥 = −13 b. │𝒙𝒙│ = 0 If │𝒙𝒙│ =0 , then 𝑥𝑥 = 0 c. │𝒙𝒙│ = −6 Since, −6 < 0 , │𝒙𝒙│ = −6 has no solution. 11 Chapter 1 Mathematics: Grade 8 Exercise 1.3 1. Complete the following table. 𝒙𝒙 −7 𝟑𝟑 𝟐𝟐 − 𝟑𝟑 𝟓𝟓 │𝒙𝒙│ 0 2𝟗𝟗 𝟓𝟓 2. Find all rational numbers whose absolute values are given below 𝟒𝟒 𝟏𝟏𝟏𝟏 𝟐𝟐 a. 3.5 b. 𝟕𝟕 c. 𝟔𝟔 d. 3𝟓𝟓 3. Evaluate each of the following expressions. a. │−5│+│5│ b. │−13│−│−8│+│7│ 1 c. │0│+2 3 d. │−8+5│ 1 e. │− 13 │+│−15│−15 4. Evaluate each of the following expressions for the given values of 𝑥𝑥 and 𝑦𝑦. a. 5𝑥𝑥 − │𝑥𝑥 − 3│, 𝑥𝑥 = − 5 b. │𝑥𝑥│−𝑥𝑥 +9, 𝑥𝑥 = 3 c. │𝑥𝑥 + 𝑦𝑦│−│𝑥𝑥│, 𝑥𝑥 = −3 and 𝑦𝑦 =6 d. │𝑥𝑥│+│𝑦𝑦│, 𝑥𝑥 =5 and 𝑦𝑦 = −10 e. −3│𝑥𝑥 +6│, 𝑥𝑥 = −5 │x│−│5y│ f. , 𝑥𝑥 =4 and 𝑦𝑦 =8 │x + y│ 5. Solve the following absolute value equations. 3 a. │𝑥𝑥│=8 b. │𝑥𝑥│= 5 Challenge problem 6. Solve the following absolute value equations. 5 a. │𝑥𝑥 +4│=10 d. │5𝑥𝑥 −3│= 2 2 b. 4│𝑥𝑥 +3│=123 e. │𝑥𝑥 −5│=33 c. 3−2│𝑥𝑥 −5│ = 9 12 Mathematics: Grade 8 Chapter 1 1.2. Comparing and Ordering Rational numbers Competency: At the end of this sub-topic students should:  Compare and order Rational numbers. 1.2.1. Comparing Rational numbers In day to day activity, there are problems where rational numbers have to be compared. For instance, win and loss in games; positive and negative in temperature; profit and loss in trading etc. Activity 1.2 Insert to express the corresponding relationship between the following pairs of numbers. a. 0 ------- 15 b. −3 ------- −5 c. 6.7 ------- 6.89 12 8 d. ------- 2 3 Comparing Decimals 𝑎𝑎 A rational number 𝑏𝑏 can be expressed as a decimal number by dividing the numerator 𝑎𝑎 by the denominator 𝑏𝑏. Note: Decimal numbers are compared in the same way as comparing other numbers: By comparing the different place values from left to right. That is, compare the integer part first and if they are equal, compare the digits in the tenths place, hundredths place and so on. Example 1.10: Compare the following decimal numbers. a. 4.25-----12.33 c. 45.667 ----- 45.684 b. 15.52-----15.05 13 Chapter 1 Mathematics: Grade 8 Solution: a. since 4 0, then 15.52 > 15.05 c. here the corresponding integer part and the tenth place numbers are equal, so we move to the hundredth place: 6 < 8. Thus 45.667 < 45.684 Comparing Fractions Comparing fractions with the same denominator If the denominators of two rational numbers are the same, then the number with the greater numerator is the greater number. That is 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑐𝑐 and 𝑏𝑏 are a given rational numbers and > 𝑏𝑏 if and only if 𝑎𝑎 > 𝑐𝑐. 𝑏𝑏 𝑏𝑏 Example 1.11: 15 13 a. 7 > 7 because 15 >13 10 15 b. 3 < 3 because 10 < 15 Fractions that represent the same point on a number line are called Equivalent 𝑎𝑎 fractions. For any fraction 𝑏𝑏 and 𝑚𝑚 is a rational number different from 0 ( 𝑚𝑚 ≠ 𝑎𝑎 𝑎𝑎 𝑚𝑚 0), then 𝑏𝑏 = 𝑏𝑏 × 𝑚𝑚. Comparing fractions with different denominators In order to compare any two rational numbers with different denominators, you can use either of the following two methods: Method 1: Change the fractions to equivalent fractions with the same denominators. Step 1. Determine the LCM of the positive denominators. Step 2. Write down the given rational numbers with the same denominators. Step 3. Compare the numerators of the obtained rational numbers. 14 Mathematics: Grade 8 Chapter 1 Example 1.12: Compare the following pairs of rational numbers. 3 1 11 7 a. 5 and 2 b. 16 and 8 Solution: 3 1 a. To compare 5 and 2 i. Find the LCM of 5 and 2 which is 10. ii. Express the rational numbers with the same denominator 10. 3 3 2 6 1 5 5 5 = 5 × 2 =10 and 2 × 5 = 10 6 5 iii. Since 6 > 5, 10 > 10. 3 1 Therefore, 5 >2. 11 7 b. To compare 16 and 8 i. Find the LCM of 16 and 8 which is 16. ii. Express the rational numbers with the same denominator 16. 11 11 1 11 7 7 2 14 16 = 16 × 1 = 16 and 8 = 8 × 2 = 16 11 14 Since 11 < 14 , < 16 16 11 7 Therefore, 16 < 8 Method 2. ( Cross- product method ) 𝑎𝑎 𝑐𝑐 Suppose 𝑏𝑏 and 𝑑𝑑 are two rational numbers with positive denominators. Then 𝑎𝑎 𝑐𝑐 I. < , if and only if 𝑎𝑎𝑎𝑎 < 𝑏𝑏𝑏𝑏 𝑏𝑏 𝑑𝑑 𝑎𝑎 𝑐𝑐 II. 𝑏𝑏 > 𝑑𝑑 , if and only if 𝑎𝑎𝑎𝑎 > 𝑏𝑏𝑏𝑏 𝑎𝑎 𝑐𝑐 III. = , if and only if 𝑎𝑎𝑎𝑎 = 𝑏𝑏𝑏𝑏 𝑏𝑏 𝑑𝑑 15 Chapter 1 Mathematics: Grade 8 Example 1.13: 5 8 a. 7 < 3 because 5 × 3 =15 < 7 × 8 =56 12 6 b. − 9 < 7 because −12 × 7 = −84 < 9 × 6 = 54 9 11 c. 5 > 7 because 9 × 7 =63 > 5 × 11 = 55. Comparing Rational numbers using number line Note: For any two different rational numbers whose corresponding points are marked on the number line, then the one located to the left is smaller. Figure 1.13 1 1 1 Thus, −1< − 2 , − 2 < 0, 0< 2 From the above fact, it follows that:  Every positive rational number is greater than zero.  Every negative rational number is less than zero.  Every positive rational number is always greater than every negative rational number.  Among two negative rational numbers, the one with the largest absolute value is smaller than the other. For instance, −45 < −23 because│−45│ > │−23│. 16 Mathematics: Grade 8 Chapter 1 Exercise 1.4 1. Which of the following statements are true and which are false? 3 2 3 3 12 10 a. −0.15 < 1.5 b. 25 > 35 c. │− 5 │< 5 d. 8 = 15 5 21 5 7 4 25 e. 37 > 6 f. 12 > 18 g. 6.53 < 6.053 h. 3 7 = 7 2. Insert ( >, = o r < ) to express the corresponding relationship between the following pairs of numbers. 15 18 3 3 a. 9 _____ 9 e. │− 10 │_____ 10 21 28 12 13 b. − 12 _____ − 16 f. 8 _____ 9 8 c. 20 _____ 0.35 5 7 d. 36 _____ 38 ` 3. From each pair of numbers, which number is to the left of the other on the number line? 6 5 2 5 a. 3.5 , 7 b. 8 , 7 c. 35 , 27 13 1 d. −9 , 5 e. − 3 , −1.5 4. 𝑘𝑘, 𝑛𝑛, 𝑚𝑚, 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 are natural numbers represented on a number line as follows: Figure 1. 14 Compare the numbers using > or 32 > −15 > −23 18 Mathematics: Grade 8 Chapter 1 b. To order these rational numbers, first change the fractions with the 70 84 75 16 same denominator. thus , , , 30 30 30 30 Now compare only the numerators, 84 > 75 > 70 >16 4 5 7 8 Therefore, 25 > 2 > 3 > 15 c. 5.17 > 4.75 > 4.5 > 3.75 Exercise 1.5 1. Arrange the following rational numbers in ascending order. 4 3 11 2 23 a. ,` 25 , , −53 , 15 9 7 b. 5.24, 8.13, 6.75, 12.42, −12.51 5 6 11 c. 3.92 , 13 , 47 , 4.73, 9 2. Arrange the following rational numbers in descending order. a. 13.72, 23.86, 15.02, 13.05 21 13 7 9 4 b. 12 , 16 , 29 , 7 , 29 5 18 c. 36 , 3.75 , 5 , 4.23, 3.21 3. Samuel’s science class is “growing plants under different conditions”. The average plant growth during a week was 5.5cm.The table shows the difference from the average for some students’ plants. a. Which student’s plant growing more? b. Order the differences from lowest to highest. Difference from Average Plant Growth Student Rahel Kassa Alemitu Munir Difference 3 1 −2.2 1.7 −1 7 4 10 19 Chapter 1 Mathematics: Grade 8 1.3. Operation and properties of Rational Numbers Competencies: At the end of this sub-unit students should:  Add rational numbers.  Subtract rational numbers.  Multiply rational numbers  Divide rational numbers. 1.3.1. Addition of rational numbers Activity 1.3 1. Add the following numbers using a number line and show using arrows. a. 6 + (−3) c. 2 +(−4) b. −9 + 5 d. 6 +(−6) 3 2 2. Find the sum of and 6 using fraction bar. 6 Adding rational numbers with same denominators To add two or more rational numbers with the same denominators, we add all the numerators and write the common denominator. 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑎𝑎 + 𝑐𝑐 For any two rational numbers 𝑏𝑏 and 𝑏𝑏 , + 𝑏𝑏 = 𝑏𝑏 𝑏𝑏 Example 1.15: Find the sum of the following rational numbers 3 6 a. +5 5 8 5 6 b. 7 + 7 + 7 3 11 c. − 4 + 4 1 4 d. 23 + 3 +1 20 Mathematics: Grade 8 Chapter 1 Solution: 3 6 3+6 9 a. 5 + 5 = 5 =5 8 5 6 8+5+6 19 b. 7 + 7 + 7 = 7 = 7 3 11 −3 + 11 8 c. − 4 + 4 = 4 =4=2 1 4 7 4 3 7+4+3 14 d. 23 + 3 +1 = 3 + 3 +3 = 3 = 3 Adding rational numbers with different denominators To find the sum of two or more rational numbers which do not have the same denominator, we follow the following steps: I. Make all the denominators positive. II. Find the LCM of the denominators of the given rational numbers. III. Find the equivalent rational numbers with common denominator. IV. Add the numerators and take the common denominator. Example 1.16: Find the sum of the following rational numbers. 3 5 a. 7 + 4 11 5 3 b. 9 + 6 +4 Solution: a. The LCM of 7 and 4 is 28. Then, write the fraction as a common denominator 28. 3 4 12 5 7 35 7 × 4 = 28 and 4 × 7 = 28 3 5 12 35 12 + 35 47 7 + 4 = 28 + 28 = 28 = 28 b. The LCM of 9 , 6 and 4 is 36. Then, write the fraction as a common denominator 36. 11 4 44 5 6 30 3 9 27 9 × 4 = 36 , 6 × 6 = 36 and 4 × 9 = 36 21 Chapter 1 Mathematics: Grade 8 11 5 3 44 30 27 44 + 30 + 27 101 9 + 6 +4= 36 + + = 36 36 36 = 36 Note: Always reduce your final answer to its lowest term. Now you are going to discover some efficient rules for adding any two rational numbers. Rule 1: To find the sum of two rational numbers where both are negatives: i) Sign : Negative (−) ii) Take the sum of the absolute values of the addends. iii) Put the sign in front of the sum. Example 1.17: Perform the following operation: 8 11 − + (− ) 6 8 Solution: 8 11 − 6 + (− 8 ) i. Sign (−) 8 11 8 11 65 ii. │− 6 │+│− 8 │= 6 + 8 = 24 8 11 65 Therefore, − 6 + (− 8 ) = − 24 Rule 2: To find the sum of two rational numbers, where the signs of the addends are different, are as follows: i) Take the sign of the addend with the greater absolute value. ii) Take the absolute values of both numbers and subtract the addend with smaller absolute value from the addend with greater absolute value. iii) Put the sign in front of the difference. 22 Mathematics: Grade 8 Chapter 1 Example 1.18: Perform the following operation 9 5 a. − 4 + 4 5 3 b. − 6 + 4 Solution: 9 5 a. − 4 + 4 9 5 i. Sign(−) because │ − 4 │ > │ 4 │ ii. Take the difference of the absolute values: 9 5 9 5 4 │ − 4 │−│4│ = 4 −4 = 4 =1 9 5 4 Therefore, − 4 + 4 = − 4 = −1 5 3 b. − 6 + 4 5 3 i. Sign(−) because │ − 6 │ > │ 4 │ ii. Take the difference of the absolute values: 5 3 5 3 1 │ − 6│ − │4│ = 6 − 4 = 12 5 3 1 Therefore, − 6 + 4 = − 12 Example 1.19: 5 2 Find the sum of 8 and 8 using fraction bar. Solution: Divide the fraction bar into 8 equal parts. Now shade 5 of them with gray color and 2 of them with green color as shown in the figure below: 1 1 1 1 1 1 1 8 8 8 8 8 8 8 Figure 1.15 23 Chapter 1 Mathematics: Grade 8 5 2 The shaded part within gray color represents 8 and within green color represents 8 of the whole fraction bar. The shaded part in both colors is 7 of eight equal parts. 5 2 7 This shows that 8 + 8 = 8 Example 1.20: 1 1 Find the sum of 6 and 2 using fraction bar. Solution: 1 Divide the fraction bar into 6 equal parts and shade one of them, which represents 6 3 1. Similarly, Shade the other three parts in different colors which represents 6 = 2. 1 1 1 1 6 6 6 6 1 Figure 1.16 2 How many parts of the fraction bar shaded? 4 parts of the 6 equal parts. This 1 1 1 3 4 2 implies that 6 + 2 = 6 + 6 = 6 = 3 Properties of Addition of Rational Numbers For any rational numbers 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 the following properties of addition holds true: a. Commutative: 𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎 b. Associative: 𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐 c. Properties of 0: 𝑎𝑎 + 0 = 𝑎𝑎 = 0 + 𝑎𝑎 d. Properties of opposites : 𝑎𝑎 + (-𝑎𝑎) = 0 Example 1.21: 1 3 7 Using properties of addition find the sum: 3 + 5 + 6 24 Mathematics: Grade 8 Chapter 1 Solution: 1 3 7 1 3 7 3 +5+6 = (3+5)+ 6 -------- Associative property 5 + 9 7 =( 15 ) + 6 14 7 = 15 + 6 28 + 35 63 = 30 = 30 21 = 10 1 3 7 1 3 7 Similarly, 3 + 5 + 6 = 3 + (5 + 6) ------ Associative property 1 18+35 =3+( 30 ) 1 53 = 3 + 30 10+53 = 30 63 21 = 30 = 10 1 3 7 1 3 7 Therefore, (3 + 5 )+ 6 = 3 +( + ) 5 6 Exercise 1.6 1. Find the sum: 13 21 a. 5 + 5 5 3 b. 6 +8 3 3 c. 5 +25 1 3 5 d. 23 + 8 + 36 2 3 4 2 e. │− 5 + 8│ +│7 + 7│ 2. In the city where I live, the temperature on an outdoor thermometer on Monday was 23.72oc. The temperature on Thursday was 3.23oc more than that of Monday. What was the temperature on Thursday? 25 Chapter 1 Mathematics: Grade 8 1.3.2. Subtraction of rational numbers Activity:1.4 1. Can you express subtraction of rational numbers in the form of addition? 2. Are the commutative and associative properties holds true in subtraction of rational numbers? The process of subtraction of rational numbers is the same as that of addition. Subtraction of any rational numbers can be explained as the inverse of addition: 𝑎𝑎 𝑐𝑐 𝑐𝑐 𝑎𝑎 That is, for two rational numbers and 𝑑𝑑, subtracting from 𝑏𝑏 means adding the 𝑏𝑏 𝑑𝑑 𝑐𝑐 𝑎𝑎 negative of 𝑑𝑑 to 𝑏𝑏. 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑐𝑐 Thus 𝑏𝑏 − 𝑑𝑑 = 𝑏𝑏 + (− 𝑑𝑑) Example 1.22: Compute the following difference. 7 4 3 5 1 9 a. 9 −3 b. 7 – (− 9) c. − 12 − 8 Solution: 7 4 7 4 3 5 3 5 a. 9 − 3 = 9 + (− 3) b. 7 – (− 9) = 7 + 9 7 + (−12) 27+35 = 9 = 63 5 62 = −9 = 63 1 9 1 9 c. − 12 − 8 = − 12 + (− 8) −2 + (−27) = 24 29 = − 24 Note: i. The difference of two rational numbers is always a rational number. ii. Addition and subtraction are inverse operations of each other. Example 1.23: 26  Note: Mathematics: i.Grade The8difference of two rational numbers is always a rational number. Chapter 1 ii. Addition and subtraction are inverse operations of each other. Example 1.23: 7 2 Find the difference of 9 − 3 using fraction bar. Solution: Divide the fraction bar in to 9 equal parts and shade 7 parts of 6 them. Out of the shaded, mark 6 of them with x. Now, 6 parts of the 9 equal parts 9 2 represent 3 of the fraction bar. 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 Figure 1.17 How many parts of the shaded part is unmarked? One part. This implies that 7 2 7 6 1 9 −3 =9−9 = 9 Exercise 1.7 1. Find the difference of each of the following 5 3 a. 46 − 24 b. −5.3 −3.45 6 7 c. 13 − │ − 13│ 5 3 d. │− 7│−│4│ e. −32.24 − │−32.24│ 2 3 f. −35 − 27 2. Evaluate the following expressions: 2 9 a. y−(7 + 5), when y = 4 4 b. 15−(−y − 9), when y = 7 12 3. From a rope 23 m long, two pieces of lengths m and 7 27  a. y−(7 + 5), when y = 4 Chapter 1 4 Mathematics: Grade 8 b. 15−(−y − 9), when y = 7 12 3. From a rope 23 m long, two pieces of lengths m and 7 7 m are cut off. What is the length of the remaining rope? 4 4. A basket contains three types of fruits, apples, oranges and bananas, 58 18 weighing 3 kg in all. If 7 kg be apples, 11 9 kg be oranges and the rest are bananas. What is the weight of the bananas in the basket? 1.3.3. Multiplication of rational numbers Activity 1.5 Multiply the following rational numbers 5 8 2 5 3 3 2 a. 7 × 11 c. 3 ( 9 × (− 5)) e) − 8 × (− 3) 9 4 2 4 b. − ×5 d. 35 × 27 7 To multiply two or more rational numbers, we simply multiply the numerator with the numerator and the denominator with the denominator. Finally reduce the final answer to its lowest term if it is. Example 1.25: Find the product 3 7 5 4 a) 2 ×8 b) − 6 × 3 Solution: 3 7 3×7 21 a) 2 ×8= 2×8 = 16 5 4 −5×4 −20 −10 b) − 6 × 3 = 6×3 = 18 = 9 Example 1.25: 1 3 Find the product of 2 and 4 using grids model. 28 Mathematics: Grade 8 Chapter 1 Solution: 1 3 a. model 2 by shading half of a b. use a different color to shade 4 grid of the same grid i) Divide the grid in to 2 i) Divide the grid in to 4 rows columns. ii) shade 3 rows to show 4 3 1 ii) Shade 1 column to show 2 Figure 1.18 Figure 1.19 c. Determine what fraction of the grid is shaded with both colors. There are 8 equal parts, and 3 of the part are shaded with both colors. The 3 fraction shaded with both colors is. 8 Figure 1.20 1 1 The section of the grid shaded with both colors shows 3 part of when is divided 2 2 3 1 3 1 into 4 equal parts. In other words 4 of 2 , or 4 ×2 3 1 3 Therefore, 4 × 2 = 8 29 Chapter 1 Mathematics: Grade 8 Note: The product of two rational numbers with different signs can be determine in three steps 1. Decide the sign of the product; it is “– “. 2. Take the product of the absolute value of the numbers. 3. Put the sign in front of the product. Example 1.26: Find the product 4 5 a. 9 ×(− ) 2 5 3 b. 2 7 × (− 2 ) Solution: 4 5 a. 9 ×(− 2) i. Sign (−) ii. Multiply the absolute value 4 5 4 5 20 10 │ │×│− │= × = = 9 2 9 2 18 9 4 5 20 10 Therefore, 9 ×(− 2) = − 18 = − 9 5 3 b. 2 7 × (− 2 ) First change the mixed number to improper fraction. 5 5 19 27=2+7= 7 , then i. Sign ( − ) ii. Multiply the absolute value 19 3 19 3 57 | 7 |×|−2|= 7 × 2 = 14 5 3 57 Therefore, 2 7 × (− 2 ) = − 14 Note: The product of two negative rational numbers is a positive rational number. Example 1.27: 2 7 Find the product: − 5 × (− 3) 30 Mathematics: Grade 8 Chapter 1 2 7 2 7 14 Solution: − 5 × (− 3) = 5 × 3 = 15 The following table 1.1 summarizes the facts about product of rational numbers. The two factors The product Example Both positive Positive 2 × 𝟓𝟓 = 𝟏𝟏𝟏𝟏 3 𝟑𝟑 Both negative Positive 𝟓𝟓 (− 𝟒𝟒) × (− 𝟑𝟑 ) = 𝟕𝟕 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏 Of opposite sign Negative −3 × 5 = −15 One or both 0 Zero 𝟕𝟕 − 𝟗𝟗 × 0 = 0 Table 1.1 Properties of multiplication of rational numbers For any rational numbers 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 ,the following properties of multiplication holds true: a. Commutative: 𝑎𝑎 × 𝑏𝑏 = 𝑏𝑏 × 𝑎𝑎 b. Associative: 𝑎𝑎 × (𝑏𝑏 × 𝑐𝑐) = ( 𝑎𝑎 × 𝑏𝑏) × 𝑐𝑐 c. Distributive: 𝑎𝑎 × (𝑏𝑏 + 𝑐𝑐) = 𝑎𝑎 × 𝑏𝑏 + 𝑎𝑎 × 𝑐𝑐 d. Property of 0: 𝑎𝑎 ×0 = 0 = 0 × 𝑎𝑎 e. Property of 1: 𝑎𝑎 ×1 = 𝑎𝑎 = 1× 𝑎𝑎 Example 1.28: Using the properties of multiplication, find the following products. 3 6 2 4 5 2 3 4 8 5 3 a. 5 × 7 b. 3 × 5 × 2 c. 5 ×( 7 + 3) d. (9 × 4) × 0 e. 25 ×1 Solution: 3 6 6 3 18 a. 5 × 7 = 7 × 5 = 35 2 4 5 2 4 5 b.( 3 × 5 )× 2 …..associative property … 3 ×(5×2) 8 5 2 20 = 15 ×2 = 3 × 10 40 40 = 30 = 30 31 Chapter 1 Mathematics: Grade 8 4 4 =3 = 3 2 3 4 c. 5 × ( 7 + 3) 2 3 2 4 = 5 × 7 + 5 × 3 ….. distributive property 6 8 = 35 + 15 74 = 105 8 5 d. ( × ) × 0 = 0……..property of 0 9 4 3 3 e. 25 ×1= 25 ………property of 1 Example 1.29: 35 162 Find the cost of 9 m of cloth, if the cost of a cloth per meter is Birr 4. Solution: 35 162 5670 315 Total cost = 9 × 4 = 36 = Birr 2 = Birr 157.5 Note: To get the product with three or more factors, we use the following properties: a. The product of an even number of negative factors is positive. b. The product of an odd number of negative factors is negative. c. The product of a rational number with at least one factor 0 is zero. Exercise 1.8 1. Determine the product 5 3 3 3 8 3 a. 8 × 4 b. 7 × (− 4 ) c. − 7 × (− 5) 3 2 2 5 3 d. 25 × 43 e. − 3 × 4 × 2 3 f. - 3(5𝑥𝑥 +5), when 𝑥𝑥 = 4 2. An airplane covers 1250 km in an hour. How much distance will it cover in 23 hours? 6 32 Mathematics: Grade 8 Chapter 1 1 2 3. Find the product of 3 and 5 using grid model. 1.3.4. Division of Rational Numbers Activity1.6 3 1. How many groups of 4 are in 12? 2 3 2. How many groups of 5 are in 35? 1 2 3. Use grids to model 33 ÷ 3. Example 1.30: 1 2 Find the quotient by dividing 4 3 by 3 using grid model. Solution: 1 1 Since 4 3 = 4 + 3, divide each 5 grids in to 3 equal parts. 1 1 Shade 4 grids and 3 of a fifth grid to represent 4 3 and Divide the shaded grids in to equal groups of 2. Figure 1.21 2 1 1 2 There are 6 groups of 3, with in 3 left over. This piece is 2 of a group of 3. 1 2 1 Thus, there are 6 + 2 groups of 3 in 4. 3 1 2 1 13 Therefore, 4 3 ÷ 3 = 6 + 2 = 2 Note:  𝑎𝑎 ÷ 𝑏𝑏 is read as 𝑎𝑎 is divided by 𝑏𝑏.  In 𝑎𝑎 ÷ 𝑏𝑏 = 𝑐𝑐, 𝑐𝑐 is called the quotient, 𝑎𝑎 is called the dividend and 𝑏𝑏 is called the divisor. 33 Chapter 1 Mathematics: Grade 8 𝑎𝑎  The quotient 𝑎𝑎 ÷ 𝑏𝑏 is also denoted by. 𝑏𝑏  If 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 are integers, 𝑏𝑏 ≠ 0 and 𝑎𝑎 ÷ 𝑏𝑏 = 𝑐𝑐, if and only if 𝑎𝑎 = 𝑐𝑐 × Rules for Division of Rational numbers When dividing rational numbers: 1. Determine the sign of the quotient: a) If the sign of the dividend and the divisor are the same, then sign of the quotient is (+). b) If the sign of the dividend and the divisor are different, the sign of the quotient is (−). 2. Determine the value of the quotient by dividing the absolute value of the dividend by the divisor. Example 1.31: −324 324 −18 = 18 = 18 𝑎𝑎 𝑐𝑐 Note: For any two rational numbers and 𝑏𝑏 𝑑𝑑 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑑𝑑 𝑎𝑎𝑎𝑎 ÷ 𝑑𝑑 = 𝑏𝑏 × 𝑐𝑐 = (where 𝑐𝑐 ≠ 0) 𝑏𝑏 𝑏𝑏𝑐𝑐 Example 1.32: Determine the quotient: 6 3 9 5 a. 14 ÷7 b. −3÷ 15 c. − 7 ÷ −4 Solution: 6 3 6 7 6 × 7 42 a. ÷ 7 = 14 × 3 = 14 = 42 = 1 14 ×3 9 3 9 3 15 45 b.−3÷ 15 = − 1 ÷ 15 = − 1 × 9 =− 9 = −5 5 5 1 5 c. − 7 ÷ −4 = − 7 × − 4 = 28 𝑎𝑎 Note: For any rational number 𝑏𝑏 where 𝑎𝑎 ≠ 0; 34 Mathematics: Grade 8 Chapter 1 Exercise:1.9 1. Determine the quotient Exercise:1.9 5 3 a. 8 ÷ 4 1. Determine the quotient 3 6 a. b.8 ÷ 5 −3 ÷ 45 7 8 5 b. c.− − ÷ ÷ (− 3) 3 6 5 9 7 3 4 c. d.− 295÷÷(− (−53)) 8 3 3 3 d. 2.25Rahel ÷ (− made of a pound of trail mix. If she puts 8 of a pound into each bag, 3 4 3 ) 4 how many3bags can Rahel fill? 3 2. Rahel made 4 of a pound of trail mix. If she puts 8 of a pound into each bag, 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 × 𝑎𝑎 = 𝑏𝑏𝑏𝑏 = 𝑎𝑎𝑎𝑎 = 1, Then 𝑎𝑎 is called the reciprocal of 𝑏𝑏. how 𝑏𝑏 many bags can Rahel fill? 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎 × 𝑎𝑎 = 𝑏𝑏𝑏𝑏 = 𝑎𝑎𝑎𝑎 = 1, Then 𝑎𝑎 is called the reciprocal of. 𝑏𝑏 𝑏𝑏 1.4. Real life applications of rational numbers 1.4. Real life Rational applications numbers of rational used to express many numbers day –to- day real life activities. For instance, to share something among friends, to calculate interest rates on loans and Rational numbers used to express many day –to- day real life activities. For mortgages, to calculate interest on saving accounts, to determine shopping instance, to share something among friends, to calculate interest rates on loans and discounts, to calculate prices, to prepare budgets, etc. So, in this sub-topic we will mortgages, to calculate interest on saving accounts, to determine shopping discuss some of them. discounts, to calculate prices, to prepare budgets, etc. So, in this sub-topic we will 1.4.1. some discuss Application of them.in sharing something among friends Rational 1.4.1. numbersinare Application used insomething sharing sharing and distributing among friendssomething among a group of friends. Rational numbers are used in sharing and distributing something among a group of Example 1.33: friends. There are four friends and they want to divide a cake equally among themselves. Example 1.33: Then, the amount of cake each friend will get is one fourth of the total cake. There are four friends and they want to divide a cake equally among themselves. Example 1.34: Then, the amount of cake each friend will get is one fourth of the total cake. Example 1.34: 35 Chapter 1 Mathematics: Grade 8 Three brothers buy sugarcane. Their mother says that she will take over a fifth of the sugarcane. The brothers share the remaining sugarcane equally. What fraction of the original sugarcane does each brother get? Solution: 1 Brother 1 Brother 2 Brother 3 2 Mother 3 Figure 1.22 1 Each brother gets a big piece 5 and divides one big piece in to three equal parts, 1 1 5 ÷ 3, which is 15. so each brother gets: 1 1 3 1 4 5 + 15 = 15 + 15 = 15 1.4.2. Application in calculating interest and loans Simple interest Interest is a payment for the use of money or interest is the profit return on investment. Interest can be paid on money that is borrowed or loaned. The borrower pays interest and the lender receives interest. The money that is borrowed or loaned is called the principal (P). The portion paid for the use of money is called the interest (I). The length of time that money is used or for which interest is paid is called time (T). The rate per period (expressed as percentage) is called rate of interest (R). The interest paid on the original principal during the whole interest periods is called simple interest. Interest can be calculated by: I = PRT Example 1.35: Abebe borrowed Birr 21100 from CBE five months ago. When he first borrowed the money, they agreed that he would pay to CBE 15% simple interest. If Abebe pays to it back today, how much interest does he owe to it? Solution: 36 Mathematics: Grade 8 Chapter 1 Given Required P =Birr 21100 I=? R = 15% I = PRT 5 5 I = Birr 21100 × 15% × 12, Where, T = 5 months = 12 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 5 I = Birr 21100 × 0.15 × 12 5 I = Birr 3165 × 12 I = Birr 1318.75 Therefore, Abebe pay an additional 1318.75 Birr of simple interest as per their agreement. Example 1.36: 1 What principal would give Birr 250 interest in 2 2 years at a rate of 10%? Solution: Given Required I = Birr 250 P =? R = 10% 1 T = 2 2 years = 2.5 years I = PRT I Birr 250 𝑃𝑃 = 𝑅𝑅𝑅𝑅 = 10% × 2.5 Birr 250 𝑃𝑃 = 0.1 × 2.5 𝑝𝑝 = Birr1000 Exercise 1.10 1. If Birr 1200 is invested at 10% simple interest per annum, then What is simple interest after 5 years? 2. What principal will bring Birr 637 interest at a rate of 7% in 2 years? 3. Find the simple interest rate for a loan where Birr 6000 is borrowed and the amount owned after 5 months is Birr 7500. 37 Chapter 1 Mathematics: Grade 8 SUMMARY FOR UNIT 1 𝑎𝑎 1. A rational number is a number that can be written as 𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are integers and 𝑏𝑏 ≠ 0.The set of rational numbers is denoted by ℚ. 2. Rational numbers can represent on a number line. 3. The absolute value of a rational number ‘𝑥𝑥’, denoted by │𝑥𝑥│, is defined as: 𝑥𝑥, 𝑖𝑖𝑖𝑖 𝑥𝑥 > 0 │𝑥𝑥│ ={ 0, 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 −𝑥𝑥, 𝑖𝑖𝑖𝑖 𝑥𝑥 < 0 4. Rational numbers can be compared i. By changing to decimal numbers ii. By making the same denominators iii. By using cross product 5. Subtraction of any rational numbers can be explained as the inverse of addition. 6. The sum of two opposite rational numbers is 0. 7. Rules of signs of Addition a. The sum of two negative rational numbers is negative. b. The sum of negative and positive rational numbers is takes the sign of the greater absolute value. 8. Rules of signs of Multiplication a. The product of two negative rational numbers is positive. b. The product of negative and positive rational numbers is negative. 9. Rules of signs for division: a. positive divided by negative equals negative. b. negative divided by positive equals negative. c. negative divided by negative equals positive. 38 Mathematics: Grade 8 Chapter 1 REVIEW EXERCISE FOR UNIT 1 1. Which of the following statements are true? 5 8 3 3 a. − 3 > − 11 c. │25│= │−25│ 5 23 19 5 4 2 b. 3 7 > 7 > 6 d. 3 6 < 3 5 < 7 2. Solve each of the following absolute value equations. a. │𝑥𝑥│ = 7 d. 2│3𝑥𝑥 − 4│= 6 b. │𝑥𝑥 − 5│= 4 e. 5+3│𝑥𝑥 − 3│= 2 c. │2𝑥𝑥 + 3│ = 5 3│x−5│−│4y│ 3. If 𝑥𝑥 = −8 and 𝑦𝑦 = 4, then fined │x+y│ 4. Find the sum 3 9 3 3 a. 4 + 7 + 2 5 c. 3.35 + 2 7 1 4 b. −2 3 + 1 7 5. Find the difference 2 3 6 13 a. 7 − 8 b. −2 7 − 9 6. Determine the product 3 1 3 2 3 a. 2 7 × 4 c. −3 5 × 1 3 × (− 4) 7 b. 2.34 × 6 7. Determine the quotient 3 2 3 a. ÷ (− ) c. 2 7 ÷ 2.3 4 7 5 3 b. −3 8 ÷ 2 10 39 Chapter 1 Mathematics: Grade 8 8. Simplify the following expressions. 1 1 4 3 2 4 3 ÷ ( + ) a. 5 + 7(5 + 2) c. 5 6 3 1 2 5 3 + ( ÷ ) 2 3 5 5 4 5 6 b. 3 ÷ (2 − 7 ) 9. Eleni baked a batch of 32 cupcakes and iced 24 of them. What fractions of cupcakes were iced? 10. How long will take Birr 500 to get Birr 50 simple interest at rate of 9.5%? 40