G8 U1-3 Rational Numbers PDF

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Members of the AAEB management, have to be commended for their painstaking e orts in addressing L WD W HD L H I D H JH H LHZL J D G DIW D G S LGL J L HGLDWH IHHGED N :LW W W HL H H HG GH WL W H WL H D G H I HD L DWL IW L JH Z N Z G W D H EHH S LE H H /D W GHHSH W D N Z HGJH H W HHG W J W W H S L LSD I D ZL J W H WH WE N Z LWH W EH I HH I W HL HJ D M E I WHD L J D G W I W H DWH LD S HSD DWL H H W HL HD H W D G L WH W D SS W GH H H SH LD Z G I S DL H  0DW HPDWLFV UDGH 7 ( 2) 217(176 1.1. The concept of Rational numbers.................................................................. 2 1.1.1. Representation of Rational Numbers on a Number line......................... 3 1.1.2. Relationship among W, ℤ and ℚ............................................................ 8 1.1.3. Absolute value of Rational numbers....................................................... 9 1.2. Comparing and Ordering Rational numbers................................................ 13 1.2.1. Comparing Rational numbers............................................................... 13 1.3. Operation and properties of Rational Numbers........................................... 20 1.3.1. Addition of rational numbers................................................................ 20 1.3.2. Subtraction of rational numbers............................................................ 26 1.3.3. Multiplication of rational numbers....................................................... 28 1.3.4. Division of Rational Numbers.............................................................. 33 1.4. Real life applications of rational numbers................................................... 35 1.4.1. Application in sharing something among friends................................. 35 1.4.2. Application in calculating interest and loans........................................ 36 SUMMARY FOR UNIT 1.................................................................................. 38 REVIEW EXERCISE FOR UNIT 1................................................................... 39 2.1. Squares and Square roots............................................................................. 42 2.1.1. Square of a rational number.................................................................. 42 2.1.2.Use of table values and scientific calculator to find squares................ 48 of rational numbers......................................................................................... 48 2.1.3. Square Roots of a Rational number...................................................... 51 2.1.4. Use of table values and scientific calculator to find square roots of rational numbers.............................................................................................. 55 2.2. Cubes and Cube roots.................................................................................. 56 2.2.1. Cube of a rational number..................................................................... 56 2.2.2. Cube Root of a rational number............................................................ 60 0DW HPDWLFV UDGH 2.3. Applications on squares, square roots, cubes and cube roots...................... 61 SUMMARY FOR UNIT 2.................................................................................. 63 REVIEW EXERCISE FOR UNIT 2................................................................... 65 3.1. Revision of Cartesian coordinate system................................................. 68 3.2: Graphs of Linear Equations......................................................................... 71 3.3. Solving Linear Inequalities.......................................................................... 83 3.4. Applications of Linear Equations and Inequalities...................................... 90 SUMMARY FOR UNIT 3.................................................................................. 98 REVIEW EXERCISE FOR UNIT 3................................................................. 100 4.1. Similar Plane Figures................................................................................. 104 4.1.1. Definition and Illustration of Similar Figures..................................... 105 4.1.2. Similar Triangles................................................................................. 108 4.1.3. Tests for similarity of triangles [AA, SSS, and SAS]......................... 112.. Perimeter and Area of Similar Triangles.................................................. 120 SUMMARY FOR UNIT 4................................................................................ 125 REVIEW EXERCISE FOR UNIT 4................................................................. 126 5.1 The three angles of a triangle add up to 180°............................................. 130 5.2. The exterior angle of a triangle equals to the sum of two remote interior angles 136 5.3. Theorems on the right angled triangle....................................................... 140 5.3.1. s Theorem and its Converse.................................................... 140 5.3.2. The Pythagoras' Theorem and its converse......................................... 145 SUMMARY FOR UNIT 5................................................................................ 154 REVIEW EXERCISE FOR UNIT 5................................................................. 156  0DW HPDWLFV UDGH REVIEW EXERCISE FOR UNIT 5................................................................. 156 6.1. Circles........................................................................................................ 159 6.1.1. Lines and circles................................................................................ 159 6.1.2. Central angle and inscribed angle....................................................... 166 6.1.3. Angles formed by two intersecting chords..................................... 174 6.2. Applications of Circle................................................................................ 177 SUMMARY FOR UNIT 6................................................................................ 179 REVIEW EXERCISE FOR UNIT 6................................................................. 181 7.1. Solid Figures.............................................................................................. 185 7.1.1 Prisms and Cylinders........................................................................... 185 7.1.2. Pyramids and Cones............................................................................ 191 7.2. Surface Area and Volume of Solid Figures........................................... 194 7.2.1. Surface area of Prisms and Cylinders................................................. 194 7.2.2. Volume of Prisms and Cylinders........................................................ 203 7.3. Applications on Solid Figures and Measurements..................................... 209 SUMMARY FOR UNIT 7................................................................................ 215 REVIEW EXERCISE FOR UNIT 7................................................................. 217 8.1. The concept of probability......................................................................... 221 8.2. Probability of Simple events...................................................................... 227 8.3. Applications on Business, Climate, Road Transport,................................ 234 Accidents and Drug Effects.............................................................................. 234 SUMMARY FOR UNIT 8................................................................................ 239 REVIEW EXERCISE FOR UNIT 8................................................................. 240 0DW HPDWLFV UDGH DS H UNIT RATIONAL NUMBERS HDUQLQ 2XWFRPHV 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. 0DLQ RQWHQWV 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 DS H 0DW HPDWLFV UDGH ,1752'8 7,21 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. 7 H FRQFHSW RI 5DWLRQDO QXPEHUV RPSHWHQFLHV At the end of this sub-topic, students should:  Describe the concept of Rational Numbers practically.  Express Rational Numbers as fractions. URXS ZRUN De 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. 0DW HPDWLFV UDGH DS H 5HSUHVHQWDWLRQ RI 5DWLRQDO 1XPEHUV RQ D 1XPEHU OLQH Competency: At the end this sub-topic, students should:  Represent rational numbers as a set of fractions on a number line. 5HYLVLRQ RQ )UDFWLRQV 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 8 of the DVD one part is one-ninth, 9 of the 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. 6ROXWLRQ Figure 1.3 Green area Numerator 3 Denominator 5  DS H 0DW HPDWLFV UDGH In grade 6 and 7 you have discussed the important ideas about fractions and integers. i. 3URSHU IUDFWLRQ A fraction in which the numerator is less than the denominator. ii. ,PSURSHU IUDFWLRQ 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 on the number line above? You observe that the number is greater than 2 but less than 3. So, it belongs to the interval between 2 and 3. Thus 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: 'HILQLWLRQ A number that can be written in the form of where and are integers and ≠ 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. 0DW HPDWLFV UDGH DS H 1RWH 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. 5 b. 2 c. − 4 d. − 2 6ROXWLRQ 2 a) Since > 0, and proper, so it lies on the right side of 0 and on the left side 5 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 on number line. 5 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. DS H 0DW HPDWLFV UDGH 3 Thus, = 1. The fraction lies between 1 and 2 at 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 − < 0 and improper, first change in to mixed fraction. That is, 5 − = −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 −. 1RWH 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 and − are opposites. 0DW HPDWLFV UDGH DS H Figure 1.8 ([HUFLVH 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) b) c) − d) − 7 5 5 5 3. Change the following improper fractions to mixed fractions. 3 7 7 a) b) − c) 5 10 3 4. Represent the following rational numbers on a number line. 5 3 5 8 a) b) c) − d) − e) 2 6 5 6 5 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 3 4 a) 5 b) − 3 c) 2 5 d) −3 7  DS H 0DW HPDWLFV UDGH 5HODWLRQV LS DPRQ : ℤ DQG 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 0DW HPDWLFV UDGH DS H Example 1.5: Classify the following numbers by naming the set or sets to which it belongs. 1 5 a. −13 b. c. − d. 10 7 76 6ROXWLRQ 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. 6ROXWLRQ No, because a whole number is an integer. ([HUFLVH 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 belongs to negative integers. 7 EVROXWH YDOXH RI 5DWLRQDO QXPEHUV RPSHWHQF At the end of this section, students should:  Determine the absolute value of a rational number. FWLYLW 1. What is the distance between 0 and 5 on the number line? Between 0 and −5 on the number line?  DS H 0DW HPDWLFV UDGH 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. 'HILQLWLRQ ’, denoted by │ │, is defined as: , ≥0 │ │ ={ − , 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 (TXDWLRQ LQYROYLQ DEVROXWH YDOXH 'HILQLWLRQ : An equation of the form │ │= for any rational number is called an absolute value equation. Geometrically the equation │ │ = 8 means that the point with coordinate 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 │ │ = 8 has two solutions = 8 and = −8. 0DW HPDWLFV UDGH DS H Opposite -8 0 8 Figure 1.12 1RWH The solution of the equation │ │= for any rational number , has i. Two solutions = and = − if > 0. ii. One solution, = 0 if = 0 and iii. No solution, if < 0. Example 1.9: Solve the following absolute value equations. a. │ │ =13 b. │ │ = 0 c. │ │ = −6 6ROXWLRQ a. │ │ =13 Since, 13 > 0, │ │ = 13 has two solutions: = 13 and = −13 b. │ │ = 0 If │ │ =0 , then = 0 c. │ │ = −6 Since, −6 < 0 , │ │ = −6 has no solution.  DS H 0DW HPDWLFV UDGH ([HUFLVH 1. Complete the following table. 7 │ │ 0 2 2. Find all rational numbers whose absolute values are given below a. 3.5 b. c. d. 3 3. Evaluate each of the following expressions. a. │ 5│+│5│ b. │ 13│ │ 8│+│7│ 1 c. │0│+23 d. │ 8+5│ 1 e. │ 13 │+│ 15│ 15 4. Evaluate each of the following expressions for the given values of and. a. 5 │ 3│, = 5 b. │ │ 9, =3 c. │ │ │ │, = 3 and =6 d. │ │+│ │, =5 and = 10 e. 3│ 6│, = 5 │x│−│ y│ f. , =4 and =8 │x + y│ 5. Solve the following absolute value equations. 3 a. │ │=8 b. │ │= D H H S R HP 6. Solve the following absolute value equations. a. │ 4│=10 d. │5 3│= 2 2 b. 4│ 3│=123 e. │ 5│=3 3 c. 3 2│ 5│ = 9 0DW HPDWLFV UDGH DS H RPSDULQ DQG 2UGHULQ 5DWLRQDO QXPEHUV Competency: At the end of this sub-topic students should:  Compare and order Rational numbers. RPSDULQ 5DWLRQDO QXPEHUV 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. FWLYLW 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. ------- 3 2 RPSDULQ 'HFLPDOV A rational number can be expressed as a decimal number by dividing the numerator by the denominator. 1RWH 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  DS H 0DW HPDWLFV UDGH 6ROXWLRQ 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 RPSDULQ )UDFWLRQV RPSDULQ IUDFWLRQV ZLW W H VDPH GHQRPLQDWRU If the denominators of two rational numbers are the same, then the number with the greater numerator is the greater number. That is and are a given rational numbers and > if and only if >. Example 1.11: 15 13 a. > because 15 >13 7 7 10 15 b. < because 10 < 15 3 3 Fractions that represent the same point on a number line are called Equivalent fractions. For any fraction and is a rational number different from 0 ( ≠ 0), then = ×. 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: 0HW RG Change the fractions to equivalent fractions with the same denominators. 6WHS. Determine the LCM of the positive denominators. 6WHS Write down the given rational numbers with the same denominators. 6WHS. Compare the numerators of the obtained rational numbers. 0DW HPDWLFV UDGH DS H Example 1.12: Compare the following pairs of rational numbers. 3 1 11 7 a. 5 and 2 b. 16 and 8 6ROXWLRQ 3 1 a. To compare and 5 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 = × 2 =10 and 2 × 5 = 10 5 5 6 5 iii. Since 6 > 5, 10 > 10. 3 1 Therefore, >. 5 2 11 7 b. To compare and 16 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 0HW RG URVV SURGXFW PHW RG Suppose and are two rational numbers with positive denominators. Then I. < , if and only if < II. > , if and only if > III. = , if and only if =  DS H 0DW HPDWLFV UDGH Example 1.13: 5 a. < because 5 × 3 =15 < 7 × 8 =56 2 b. − < because −12 × 7 = −84 < 9 × 6 = 54 9 9 c. > because 9 × 7 =63 > 5 × 11 = 55. 5 RPSDULQ 5DWLRQDO QXPEHUV XVLQ QXPEHU OLQH 1RWH 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 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│. 0DW HPDWLFV UDGH DS H ([HUFLVH 1. Which of the following statements are true and which are false? 3 2 3 3 12 10 a. −0.15 < 1.5 b. 2 >3 c. │− │< d. = 5 5 5 5 8 15 5 21 5 7 4 25 e. 3 > f. > g. 6.53 < 6.053 h. 3 = 7 6 12 18 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. , c. 3 , 2 8 7 5 7 13 1 d. −9 , e. − , −1.5 5 3 4. , , , , , are natural numbers represented on a number line as follows: Figure 1. 14 Compare the numbers using > or 32 > −15 > −23 0DW HPDWLFV UDGH DS H b. To order these rational numbers, first change the fractions with the 70 84 75 same denominator. thus , , , 30 30 30 30 Now compare only the numerators, 84 > 75 > 70 >16 4 5 7 8 Therefore, 2 > > > 5 2 3 5 c. 5.17 > 4.75 > 4.5 > 3.75 ([HUFLVH 1. Arrange the following rational numbers in ascending order. 4 3 2 23 a. ,` , , −5 , 9 25 7 3 5 b. 5.24, 8.13, 6.75, 12.42, −12.51 5 c. 3.92 , 3 , 47 , 4.73, 9 2. Arrange the following rational numbers in descending order. a. 13.72, 23.86, 15.02, 13.05 2 3 7 9 4 b. , , 2 , , 2 2 9 7 9 5 8 c. 3 , 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 a. Which more? b. Order the differences from lowest to highest. Difference from Average Plant Growth Student Rahel Kassa Alemitu Munir Difference 3 −2.2 1.7 −1 7 4 0  DS H 0DW HPDWLFV UDGH 2SHUDWLRQ DQG SURSHUWLHV RI 5DWLRQDO 1XPEHUV Competencies: At the end of this sub-unit students should:  Add rational numbers.  Subtract rational numbers.  Multiply rational numbers  Divide rational numbers. GGLWLRQ RI UDWLRQDO QXPEHUV FWLYLW 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 6 and 6 using fraction bar. GGLQ UDWLRQDO QXPEHUV ZLW VDPH GHQRPLQDWRUV To add two or more rational numbers with the same denominators, we add all the numerators and write the common denominator. + For any two rational numbers and , + = 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 0DW HPDWLFV UDGH DS H 6ROXWLRQ 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. + = = =2 4 4 4 4 1 4 7 4 3 7+4+3 14 d. 2 + +1 = + + = = 3 3 3 3 3 3 3 GGLQ UDWLRQDO QXPEHUV ZLW GLIIHUHQW GHQRPLQDWRUV 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 6ROXWLRQ a. The LCM of 7 and 4 is 28. Then, write the fraction as a common denominator 28. 3 4 12 5 7 35 × = and × = 7 4 28 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  DS H 0DW HPDWLFV UDGH 11 5 3 44 30 27 44 + 30 + 27 101 9 + 6 +4= 36 + + = 36 36 36 = 36 1RWH Always reduce your final answer to its lowest term. Now you are going to discover some efficient rules for adding any two rational numbers. 5XOH 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 6ROXWLRQ 8 11 −6 (− 8 ) i. Sign (−) 8 11 8 11 65 ii. │− │+│− │= + = 6 8 6 8 24 8 11 65 Therefore, − (− ) = − 6 8 24 5XOH 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. 0DW HPDWLFV UDGH DS H Example 1.18: Perform the following operation 9 5 a. − + 4 4 5 3 b. − + 6 4 6ROXWLRQ 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 │ − │−│ │ = − = =1 4 4 4 4 4 9 5 4 Therefore, − + = − = −1 4 4 4 5 3 b. − + 6 4 5 3 i. Sign(−) because │ − │ > │ │ 6 4 ii. Take the difference of the absolute values: 5 3 5 3 │− │−│ │= − = 6 4 6 4 2 5 3 Therefore, − + = − 6 4 2 Example 1.19: 5 2 Find the sum of and using fraction bar. 6ROXWLRQ 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  DS H 0DW HPDWLFV UDGH 5 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 7 This shows that + = 8 8 8 Example 1.20: Find the sum of and using fraction bar. 6ROXWLRQ Divide the fraction bar into 6 equal parts and shade one of them, which represents 3. Similarly, Shade the other three parts in different colors which represents =. 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 3 4 implies that + = + = = 3 3 RSH H R R R 5D R D 1 P H For any rational numbers , and the following properties of addition holds true: a. RPPXWDWLYH + = + b. VVRFLDWLYH + ( + ) = ( + ) + c. 3URSHUWLHV RI + 0 = = 0 + d. 3URSHUWLHV RI RSSRVLWHV : + (- ) = 0 Example 1.21: 3 7 Using properties of addition find the sum: + + 3 5 0DW HPDWLFV UDGH DS H 6ROXWLRQ 1 3 7 1 3 7 = ( )+ -------- Associative property 3 5 6 3 5 6 5 + 9 7 =( ) + 15 6 14 7 = + 15 6 28 + 35 63 = = 30 30 21 = 10 1 3 7 1 3 7 Similarly, = ( ) ------ Associative property 3 5 6 3 5 6 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 ([HUFLVH 1. Find the sum: 13 21 a. + 5 5 5 3 b. + 6 8 3 3 c. +2 5 5 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 3.23 oc more than that of Monday. What was the temperature on Thursday? DS H 0DW HPDWLFV UDGH 6XEWUDFWLRQ RI UDWLRQDO QXPEHUV FWLYLW 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: That is, for two rational numbers and , subtracting from means adding the negative of to. Thus = ( ) Example 1.22: Compute the following difference. 7 4 3 5 1 9 a. 9 3 b. 7 ( 9 ) c. 12 8 6ROXWLRQ 7 4 7 4 3 5 3 5 a. = ( ) b. –( ) = 9 3 9 3 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 1RWH i. The difference of two rational numbers is always a rational number. ii. Addition and subtraction are inverse operations of each other. 0DW HPDWLFV UDGH DS H Example 1.23: 7 2 Find the difference of − using fraction bar. 3 6ROXWLRQ 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 2 represent of the fraction bar. 3 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 −3 = − = ([HUFLVH 1. Find the difference of each of the following 5 3 a. 4 −2 6 4 b. −5.3 −3.45 6 7 c. −│− │ 3 3 5 3 d. │− │−│ │ 7 4 e. −32.24 − │−32.24│ 2 3 f. −3 − 2 5 7 2. Evaluate the following expressions: 2 a. y−( + 5), when y = 7 4 4 b. 15−(−y − ), when y = 7 2 7  DS H 4 0DW HPDWLFV UDGH −(− − = 2 3. From a rope 23 m long, two pieces of lengths m and 7 7 4 m are cut off. What is the length of the remaining rope? 4. A basket contains three types of fruits, apples, oranges and bananas, 58 18 weighing kg in all. If kg be apples, 3 7 11 kg be oranges and the rest are bananas. What is the weight of the 9 bananas in the basket? 0XOWLSOLFDWLRQ RI UDWLRQDO QXPEHUV FWLYLW 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. 7 ×5 d. 35 × 27 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) × b) × 2 8 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. 0DW HPDWLFV UDGH DS H 6ROXWLRQ 1 3 a. model by shading half of a b. use a different color to shade 2 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 3 4 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 of , or × 4 2 4 2 3 1 3 Therefore, × = 4 2 8 DS H 0DW HPDWLFV UDGH 1RWH The product of two rational numbers with different signs can be determine in three steps 1 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 6ROXWLRQ 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 2 =2+ = , then 7 7 7 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 1RWH The product of two negative rational numbers is a positive rational number. Example 1.27: 2 7 Find the product: − × (− ) 5 3 0DW HPDWLFV UDGH DS H 2 7 2 7 14 6ROXWLRQ − 5 × (− 3) = 5 × 3 = 15 The following table 1.1 summarizes the facts about product of rational numbers. 7 H WZR IDFWRUV 7 H SURGXFW ([DPSOH RW SRVLWLYH 3RVLWLYH 2 × 3 RW QH DWLYH 3RVLWLYH − × − 2I RSSRVLWH VL Q 1H DWLYH − × = − 2QH RU ERW =HUR − × Table 1.1 3 RSH H R P S D R R D R D P H For any rational numbers , and ,the following properties of multiplication holds true: a. Commutative: × = × b. Associative: ×( × )= ( × )× c. Distributive: ×( + ) = × + × d. Property of 0: ×0 = 0 = 0 × e. Property of 1: ×1 = = 1× Example 1.28: Using the properties of multiplication, find the following products. 3 6 2 4 5 2 3 4 8 5 3 D × E × × F ×( + ) G ( × ) × 0 H 2 ×1 5 7 3 5 2 5 7 3 9 4 5 6ROXWLRQ 3 6 6 3 18 a. × = × = 5 7 7 5 35 2 4 5 2 4 5 b.( × )× property ×( × ) 3 5 2 3 5 2 8 5 2 20 = × = × 15 2 3 10 40 40 = = 30 30 DS H 0DW HPDWLFV UDGH 4 4 =3 = 3 2 3 4 c. × ( + ) 5 7 3 2 3 2 4 = × + × distributive property 5 7 5 3 6 8 = + 35 15 74 = 105 8 5 d. ( × ) × 0 = property of 0 9 4 3 3 e. 2 ×1= 2 5 5 Example 1.29: 35 162 Find the cost of 9 m of cloth, if the cost of a cloth per meter is Birr 4. 6ROXWLRQ 35 162 5670 315 Total cost = 9 × 4 = 36 = Birr 2 = Birr 157.5 1RWH 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. ([HUFLVH 1. Determine the product 5 3 3 3 8 3 a. × b. × (− ) c. − × (− ) 8 4 7 4 7 5 3 2 2 5 3 d. 2 × 4 e. − × × 5 3 3 4 2 3 f. - 3(5 +5), when = 4 2. An airplane covers 1250 km in an hour. How much distance will it cover in 23 hours? 6 0DW HPDWLFV UDGH DS H 1 2 3. Find the product of 3 and 5 using grid model. 'LYLVLRQ RI 5DWLRQDO 1XPEHUV FWLYLW 3 1. How many groups of are in 12? 2 3 2. How many groups of 5 are in 35? 1 2 3. Use grids to model 3 ÷. 3 3 Example 1.30: 1 2 Find the quotient by dividing 4 by using grid model. 3 3 6ROXWLRQ 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 ÷ = 6 + = 3 3 2 2 1RWH  ÷ is read as is divided by.  In ÷ = , is called the quotient, is called the dividend and is called the divisor.  DS H 0DW HPDWLFV UDGH  The quotient ÷ is also denoted by.  If , and are integers, ≠ 0 and ÷ = , if and only if = × 5 H R R R 5D R D P H 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 1RWH For any two rational numbers and ÷ = × = (where ≠ 0) Example 1.32: Determine the quotient: 6 3 9 5 a. ÷ b. 3÷ c. ÷ 4 14 7 15 7 6ROXWLRQ 6 3 6 7 6 × 7 42 a. 14 ÷ 7 = 14 × 3 = 14 = =1 × 3 42 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 1RWH For any rational number where ≠ 0; 0DW HPDWLFV UDGH DS H ([HUFLVH 1. Determine the quotient 5 3 a. ÷ 8 4 3 6 b. − 5 ÷ 7 8 5 c. − ÷ (− ) 9 3 3 4 d. 25 ÷ (− 3) 3 3 2. Rahel made of a pound of trail mix. If she puts of a pound into each bag, 4 8 how many bags can Rahel fill? × = = = 1, Then is called the reciprocal of. 5HDO OLIH DSSOLFDWLRQV RI UDWLRQDO QXPEHUV Rational numbers used to express many day to- day real life activities. For instance, to share something among friends, to calculate interest rates on loans and mortgages, to calculate interest on saving accounts, to determine shopping discounts, to calculate prices, to prepare budgets, etc. So, in this sub-topic we will discuss some of them. SSOLFDWLRQ LQ V DULQ VRPHW LQ DPRQ IULHQGV Rational numbers are used in sharing and distributing something among a group of friends. Example 1.33: There are four friends and they want to divide a cake equally among themselves. Then, the amount of cake each friend will get is one fourth of the total cake. Example 1.34: DS H 0DW HPDWLFV UDGH 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? 6ROXWLRQ 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 ÷ 3, which is. so each brother gets: 5 15 1 1 1 4 + = + = 5 15 15 15 15 SSOLFDWLRQ LQ FDOFXODWLQ LQWHUHVW DQG ORDQV 6LPSOH LQWHUHVW 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 VLPSOH LQWHUHVW. 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? 6ROXWLRQ 0DW HPDWLFV UDGH DS H Given Required P = irr 21100 I=? R = 15% I = PRT I = Birr 21100 × 15% × , Where, T = 5 months = I = Birr 21100 × 0.15 × I = Birr 3165 × I = Birr 1318.75 Therefore, Abebe pay an additional 1318.75 Birr of simple interest as per their agreement. Example 1.36: What principal would give Birr 250 interest in 2 years at a rate of 10%? Solution: Given Required I = Birr 250 P =? R = 10% T = 2 years = 2.5 years I = PRT = = % ×. =. ×. = Birr1000 ([HUFLVH 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. DS H 0DW HPDWLFV UDGH 6800 5 )25 81,7 1. A rational number is a number that can be written as where and are integers and ≠ 0.The set of rational numbers is denoted by ℚ. 2. Rational numbers can represent on a number line. 3. The absolute v denoted by │ │, is defined as: , >0 │ │ ={ 0, =0 − , c. │2 │= │ 2 │ 9 b. 3 > > d. 3 0", " + ≥ 0" where and are rational numbers and ≠ 0 0DW HPDWLFV UDGH DS H 6. Two linear inequalities are said to be equivalent if and only if they have the same solution set. 7. [Rules of transformation] For any rational numbers , , and I. If < , then + < + II. If < , then − < − III. If < and > 0, then < IV) If < and > 0, then < V) If < and < 0, then > VI) If < and < 0, then >  DS H 0DW HPDWLFV UDGH 5(9,(: ( (5 ,6( )25 81,7 1. Write true for correct statement and false for the incorrect one. a) The graph of the line = −2 passes through the II and III quadrants. b) The graph of the equation = , ∈ ℚ , and > 0 is a vertical line that lies to the right of the y-axis. c) A horizontal line has the equation − = 0. d) If < and = −3, then > e) The inequality 8 ≥ 5 , ∈ W has a finite solution set. 2. Plot the following points in a Cartesian coordinate plane: (0, 6), (2, −3), (−4, 5) and (−4, −5). Which point lies in neither of the quadrant? 3. Refer Figure 3.12 and answer the following questions. a) Name the coordinates of the point D, E and F. b) Which point has the coordinates (−2, 1)? c) Which coordinate of B is 3? d) In which quadrant does the point C, E lie? Figure 3.12 0DW HPDWLFV UDGH DS H 4. Fill in the blank with an appropriate inequality sign. a) If x ≥ 3, then −4 ------- −12 b) If 4 < 2 , then ------ 2 c) If < −2, then 1 – ------ 3 5. Determine whether the given points are on the graph of the equation −2 −1 = 0 a) (0, 0) b) (1, 0) c) (−1, −1) d) (2, 1) 6. Sketch the graph of the following equations. a) = −7 b) − 11 = 0 c) 3 = −7 6 d) −7=0 7. Find and , if the points P(3, 1) and Q(0,2) lie on the graph of − =6 8. Consider the equations = 1 and = 1−. a) Determine the values of y for each equation when the values of x are −1, 0 and 1. b) Plot the ordered pairs on the Cartesian coordinate plane. c) Which point is a common point, called intersection point? 9. Solve the following inequalities and graph the solution set. a) 3x + 11 ≤ 6x + 8 c) 4−3 ≤ − (2 8 ) b) 6−2 > 9 10. Solve the following inequalities a) ≥ − b) 2( − ) < 3(1 ) 5 c) − −1 ≥0 + d) ( − 3) < e) 7( 1) – > 2(3 4) f) −4( − 1) 3 ≥ 1− DS H 0DW HPDWLFV UDGH 11. Solve the following inequalities in the given domain. a) < 4(4 − ), ∈ ℤ+ b) − ≥2 , ∈ W 6 2(3 − 7) − 14 ≥ 3(2 − 11), ∈ ℚ 12. Translate the following sentences in to mathematical expressions. a) A year ago a b) Three fourth of a number is greater than 12. c) The average mark of Saron is not smaller than 89. 13. A board with 2.5 m in length must be cut so that one piece is 30 cm more than the other piece. Find the length of each piece. Figure 3.13 14. Rodas is 25 years old and her brother Mathanya is 10 years old. After how many years will Rodas be exactly twice as old as Mathanya. 15. Yoseph wants to surprise his wife with a birthday party at her favourite restaurant. It will cost Birr 56.50 per person for dinner, including tip and tax. His budget for the party is Birr 735. What is the maximum number of people Yoseph can have at the party? 16. Getaneh and Salhedin play in the same soccer team. Last Saturday Salhedin scored 3 more goals than Getaneh, but together they scored less than 7 goals. What are the possible number of goals Salhedin scored? 0DW HPDWLFV UDGH DS H UNIT SIMILARITY OF FIGURES Learning Outcomes: At the end of this unit, learners will able to:  Know the concept of similar figures and related terminologies.  Understand the condition for triangles being similar.  Apply tests to check whether two given triangles are similar or not.  Apply real-life situations in solving geometric problems. 0DLQ RQWHQWV 0DLQ RQWHQWV 4.1 Similar plane figures 4.2 Perimeter and Area of Similar Triangles Summary Review Exercise  DS H 0DW HPDWLFV UDGH 6LPLODU 3ODQH )L XUHV Competencies: At the end of this sub-topic, students should:  Identify figures that are similar to each other.  Apply the definition of similarity of two triangles to solve related problems.  Determine the similarity of two triangles. ,1752'8 7,21 This unit focused on similarity of plane figures. In common language the word similar can have many meanings but in mathematics, the word similar and similarity have very specific meanings. In mathematics, we say that two objects are similar if they have the same shape, but are not necessarily the same size. For instance, an overhead projector forms an image on the screen which has the same shape as the image on the transparency but with the size altered. Two figures that have the same shape but not necessarily the same size are called similar figures. FWLYLW 1. Which of the following figures are similar? A B C D E F G H Figure 4.1 0DW HPDWLFV UDGH DS H 2. Are any two equilateral triangles always congruent? 3. Let ABCD be a square and be its diagonal. Are ∆ ABC ≅ ∆ ADC? 4. In the triangles, ∆ ABC and ∆ PQR, if ∠ ≅ ∠ , and ∠ ≅ ∠ , then what is the relation between each ∠ and ∠ ? 5. Decide whether the polygons are similar. a. Rectangle and parallelogram b. Square and trapezium 'HILQLWLRQ DQG ,OOXVWUDWLRQ RI 6LPLODU )L XUHV In this section, we will discuss how to compare the size and shape of two given figures. Recall that, one way to determine whether two geometric figures are congruent is. If two figures have the same shape, but not necessarily the same size (That is, one figure is an exact scale model of the other figure), then we say that the two geometric figures are similar. The symbol ~ the term similar. Example 4.1: A tree and its shadow are similar. Example 4.2: The two photograph of the same size of the same person, one at the age of 5 years and the other at the age of 50 years are not similar. Because similar figures differ only in size, there is a test we can perform to make sure that our shapes are really similar. DS H 0DW HPDWLFV UDGH 6LPLODU 3RO RQV 'HILQLWLRQ Two polygons are said to be similar if there is a one to one correspondence between their vertices such that: i) all pairs of corresponding angles are congruent ii) the ratio of the lengths of all pairs of corresponding sides are equal. In the diagram, ABCD is similar to EFGH. That is, ABCD ~ EFGH, then i)∠ ≅ ∠ , ∠ ≅ ∠ , ∠ ≅ ∠ , ∠ ≅∠ ii) = = = Figure 4.2 Example 4.3: The following pairs of figures are always similar a) Any two squares. b) Any two equilateral triangles. Example 4.4: Consider the following polygons. Which of these are similar? Rectangle Square Rhombus 0DW HPDWLFV UDGH DS H 6ROXWLRQ The square and the rectangle are not similar. Because their corresponding angles are congruent but their corresponding sides are not proportional. The square and the rhombus are not similar. Because their corresponding sides are proportional but their corresponding angles are not congruent. The rectangle and the rhombus are not similar. Because their corresponding angles are not congruent and corresponding sides are not proportional. 'HILQLWLRQ. The ratio of two corresponding sides of similar polygons is called the scale factor or constant of proportionality ( ). Example 4.5: The sides of a quadrilateral are 2cm, 5cm, 6cm and 8cm. Find the sides of a similar quadrilateral whose shortest side is 3cm. 6ROXWLRQ Let the corresponding sides of a quadrilateral are 3, x, y, and z. Since the corresponding sides of similar polygons are proportional, 2 5 2 6 2 8 = , = , = 3 3 3 15 = , =9, z = 12 2 Therefore, the sides of the second quadrilateral are 3cm, 7.5cm, 9cm, and 12cm. Example 4.6: Decide whether the polygons are similar. If so find the scale factor of Figure A to Figure B. 18 8 Figure A Figure B 36 16 Figure 4.3