Geometry 51 1.12 Problems on Sum of Angles of Quadrilateral Read, Think and Learn The figure KLMN is a quadrilateral. N K 85° 70° 92° ? L M What is the sum of angles of the quadrilateral KLMN? If, in the quadrilateral KLMN, ∠K = 85°, ∠L = 92° and ∠M = 70°, then ∠N = ? For this, we know that ∠K + ∠L + ∠M + ∠N = 360° or, 85° + 92° + 70° + ∠N = 360° or, 247° + ∠N = 360° or, ∠N = 360° – 247° = 113° CLASSWORK EXAMPLES Example 1: In the adjoining quadrilateral HIRA, ∠H = 3a + 10°, ∠I = 2a, ∠R =100° and ∠A = 95°. (a) What is the sum of angles of a quadrilateral. (b) Find the value of a. (c) Find the measure of ∠H. Solution: (a) The sum of angles of a quadrilateral is 360°. 95° 2a 100° 3a + 10° A H I R
52 The Leading Mathematics - 5 (b) Here, in the given quadrilateral HIRA, ∠H = 3a + 10°, ∠I = 2a, ∠R = 100° and ∠A = 95°. Now, we know that, ∠H + ∠I + ∠R + ∠A = 360° [ Sum of angles of the quad. HIRA] or, 3a + 10° + 2a + 100° + 95° = 360° or, 5a + 205° = 360° or, 5a = 360° – 205° or, a = 155° 5 or, a = 31° (c) ∠H = 3a + 10° = 3 × 31° + 10° = 103° PRACTICE 1.12 Your mastery depends on practice. Practice as you play. 1. In the quadrilateral ABCD, (a) If ∠A = 60°, ∠B = 85°, ∠C = 110°, find the measure of ∠D. (b) If ∠D = 75°, ∠C = 115°, ∠B = 98°, find the measure of ∠A. 2. Study the following quadrilaterals: i. P a S 82° Q 97° R 80° ii. M 75° 115° a 82° N K L iii. D C a 87° 98° 105° B A iv. 3a + 8° 90° a + 50° 80° S R A M v. 3a + 8° 90° a + 50° 80° S R A M vi. 3a + 5° 50° 2a + 10° 110° O N L M
Geometry 53 (b) Write the sum of angles of a quadrilateral. (c) Find the value of 'x' in each quadrilateral. (d) Find the measures of the unknown angles in each quadrilateral. 3. Observe the following quadrilaterals: i. P S R Q 5p 118° 92° 85° ii. R S P Q 95° 125° 88° 3p + 10° iii. 75° S Q P R 90° 3p + 10° 5p – 15° iv. 3b + 27° 4b – 30° 85° P Q R S 2b + 35° v. 3c 4c S P Q R 3c 2c vi. 6a 5a 5a 4a R S Q P (a) How many triangles are formed in a quadrilateral by one of its diagonals? (b) Find the value of 'p' in the quadrilaterals (i), (ii) and (iii). (c) Find the value of 'b' in the quadrilateral (iv). (d) Find the value of 'c' in the quadrilateral (v). (e) Find the value of 'a' in the quadrilateral (vi). (f) Find the measure of ∠PQR in each of the quadrilateral PQRS. (g) Find the measures of their unknown angles in each quadrilateral.
54 The Leading Mathematics - 5 1.13 Drawing Perpendicular and Parallel Lines At the end of this topic, the students will be able to: ¾ draw the perpendicular and parallel lines on graph. Learning Objectives Read, Think and Learn Perpendicular Lines Take two points A and B on the same straight horizontal line of the given square grid and join them. Take another point C just above the point B and join them. Now, measure the angle ABC. What is the measure of ∠ABC? Here the angle between the lines AB and BC is 90°. These types of lines are called perpendicular lines. Here, CB is perpendicular to BA at B. It is denoted by CB⊥BA or CB⊥BA. The adjoining lines m or PQ and l or RS intersect in right angle at O. So, they are also perpendicular lines. Discuss the objects in the perpendicular lines at your home, around surroundings and in your school. C B A P m l O Q R S
Geometry 55 Parallel Lines Take two points A and B on the same straight horizontal line of the given square grid and join them. Take another two points C and D on the another same straight horizontal line of the same square grid and join them. Now, measure the distance between the lines AB and CD at every units of AB. What do you find? What is the angle between them ? Can you measure the angle between them ? That means the angle between the lines AB and CD is 0° and the distance between them is always the same. These lines are called perpendicular lines. Here, AB is parallel to CD or CD is perpendicular to AB. It is denoted by AB //CD or CD // AB. The adjoining lines m or PQ and l or RS never intersect at any point. So, they are also parallel lines. Discuss the objects in the parallel lines at your home, around surroundings and in your school. C D A B Q m S l P R
56 The Leading Mathematics - 5 PRACTICE Your mastery depends on practice. Practice as you play. 1.13 1. Tick () the perpendicular lines, circle (O) the parallel lines and cross () the non of them in the boxes. (a) (b) (c) (d) (e) (f) 2. Tick () the perpendicular objects and cross () the parallel objects in the boxes. (a) (b) (c) (d) (e) (f) 90° m l
Geometry 57 3. Draw a pair of perpendicular lines on the given square grid and write their names. (a) (b) (c) (d) 4. Draw a pair of parallel lines on the given square grid and write their names. (a) (b) (c) (d)
58 The Leading Mathematics - 5 5. Observe a pair of two parallel lines AB and CD drawn in the given square grid. Copy this figure in your note copy and answer the following questions. (a) How many perpendicular lines can be drawn from C with the line AB? (b) Draw a perpendicular line DE on CD at D. (c) Write the relation between AC and DE. (d) Join B and C. Measure ∠ABC and ∠BCD by using a protractor. (e) Writer the relation between ∠ABC and ∠BCD. 6. Observe the adjoining figure. (a) What are perpendicular lines? Define. (b) Write the relation between AB and CD. (c) Draw a perpendicular line AE at A on AB. (d) Write the relation between AE and CD. (e) Join A and D. Measure ∠BDA and ∠DAE by using a protractor. (f) Write the relation between ∠BDA and ∠DAE. C A D B C A DB Search any ten objects in your home and surrounding that have the perpendicular and parallel lines. Prepare a report and present it in your classroom. PROJECT WORK
Geometry 59 CHAPTER 2 SOLID OBJECTS What is a solid ?Why is solid different from a plane surface ? What is the meaning of 3-dimensional figure? What are the shapes of your book, Rubik, duster, eraser and pencil ? What is the difference between the shapes of duster and Rubik ? What are the faces, edges and vertices of the solid ? What are the faces of duster ? What are the faces of Rubik ? How many faces, edges and vertices are in Rubik ? How many faces, edges and vertices are in duster ? WARM-UP Lesson Topics Pages 2.1 Review on Solid Object 60 2.2 Cube and Cuboid 63
60 The Leading Mathematics - 5 2.1 Review on Solid Objects Read, Think and Learn Observe the given solids: Which are the surfaces of the book, ball, cone and pencil ? Which are the edges of the book, ball, cone and pencil ? Which are the vertices of the book, ball, cone and pencil ? Book has six rectangular surfaces, eight vertices and twelve edges. Ball has only one round surface, no vertex and no edge. Cone has one circular surface, one curved surface, one circular edge and one vertex. Pencil has one circular surface, two curved surface, one circular edge and one vertex. PRACTICE Your mastery depends on practice. Practice as you play. 2.1 1. Observe the following solids and write the name of their surfaces.
Geometry 61 2. Observe the following solids and write the number of plane surfaces, curved surfaces, edges and corners. Solid No. of plane surfaces No. of plane surfaces No. of curved surfaces No. of edges No. of corners Cube box Cuboid box Traffic Cone Stick Brick
62 The Leading Mathematics - 5 Football Glue stick Triangular pyramid Pencil Collect any ten objects in your home and write the number of plane surfaces, curved surfaces, edges and corners in tabular form. Present it in your classroom. PROJECT WORK
Geometry 63 2.2 Cube and Cuboid At the end of this topic, the students will be able to: ¾ identify the cube and cuboid. Learning Objectives Read, Think and Learn Observe the following solids objects: Ö Which are the faces or surface of the duster ? Ö Which are the edges of the duster ? Ö Which are the vertices of the duster ? Ö Which are the geometric name of duster and dustbin? Name of faces in cuboid: ....................... Name of faces in cube: ....................... Name of edges in cuboid: ....................... Name of faces in cube: ....................... Name of vertices in cuboid: ....................... A B C Faces Edges Vertices E D F G H I
64 The Leading Mathematics - 5 PRACTICE Your mastery depends on practice. Practice as you play. 2.2 1. Tick () the cubical objects and cross () the others. 2. Tick () the cuboid objects and cross () the others.
Geometry 65 3. Observe the following objects and write their name. 4. Name the faces, edges and vertices of the objects given below: R W X S V U Q T M L I P K N O J F C B A H D E G Collect any 10 cuboid and cubical objects in your home and surroundings. Draw their geometric shapes and level them. Identify and name the vertices, edges and faces. Prepare a table and present in your classroom. PROJECT WORK
66 The Leading Mathematics - 5 Read, Understand, Think and Do 1. The hands of a clock make the angles. (a) What type of angle can be at 9 o'clock? (b) Find the angle made by second hand and hour hand in the given clock. (c) Is the angle made by second hand and minute hand 120°? 2. Two angles ∠PQS and ∠SQR are made by 35° and a°. (a) What types of angle is ∠ PQR? (b) ∠RQS and ∠PQS both are acute angles. Justify it. (c) What should be the value of a? 3. A triangle ABC is shown with BC = 4 cm. (a) How many sides are there in the triangle? (b) Prove that the length of AC is greater than AB. (c) Which angles among ∠A, ∠B and ∠C is greater? 4. A quadrilateral ABCD is shown in the figure. (a) How many angles are there in a quadrilateral? (b) Which angles among ∠A, ∠B, ∠C and ∠D is greater? Measure by a protractor and write it. (d) Prove that the sum of angles ∠A, ∠B, ∠C and ∠D is 360°. P B C D A B C MIXED PRACTICE–I Q R a 35° S P
Geometry 67 5. In the given square grid, two parallel lines AB and CD are drawn. Copy this figure in your copy and answer the following. (a) How many perpendicular lines can be drawn on the line AB? (b) Draw a perpendicular line on CD. (c) Is the distance between two parallel lines always the same everywhere? (d) Join A and D. Measure the ∠ DAB by using a protractor. 6. The figure given alongside is a cubical die. (a) What is the geometric name of the die. (b) Count the vertices, edges and faces of the cube and fill in the table. Number of Vertices Number of Edges Number of Faces (c) Justify V – E + F = 2 7. The figure given alongside is a shape of cuboid match box. (a) Write any two differences between cube and cuboid. (b) Count the vertices, edges and faces of the cuboidal match box and plot in the table. Number of Vertices Number of Edges Number of Faces (c) Justify V – E + F = 2 8. Four small cubes are joined to make a new solid as shown in the figure. (a) Write the name of the solid made by 4 cubes. (b) Is there any changes in the number of faces, edges and vertices? (c) Write a common property of cube and cuboid. C A D B
68 The Leading Mathematics - 5 FM : 20 CONFIDENCE LEVEL TEST GEOMETRY Time : 40 Min. I 1. Given figure is ∠ABC. (a) Measure the ∠ABC using protractor. [1] (b) Construct an angle of 30° at point C. (c) Such that it cuts the line AB at D. [1] (d) Find the measure of ∠BDC without using protractor. [1] 2. Study the given figure. (a) What is the measure of ∠AMR? [1] (b) Measure the angel PMR and find the value of a. [2] (c) Find the measure of ∠AMP without using protractor. [2] 3. In the given figure, ∠TON = 45°, SON = 3a and ∠ROS is a right angle. (a) What is the measure of ∠ROS ? Write it. [1] (b) Find the value of 'a'. [2] 4. Observe the given plane figure. (a) Name the quadrilateral. [1] (b) Name the sides and vertices. [1] (c) Measure ∠RUP by using protractor. [1] 5. Observe the given figure and answer the following questions. (a) Write the name of given solid figure. [1] (b) How many vertices are there in this figure. [1] (c) Write the name of a pair of parallel faces. [1] (d) Verify that V – E + F = 2. [1] B A C A M P B O R 45° 3a N S U R A P A B C F D E H G Attempt all the questions.
Arithmetic 69 ARITHMETIC UNIT II COMPETENCY Count, read and write the numerals in digits and words in Devanagari and Hindu Arabic numeral systems upto crore. Solving mathematical problems of daily life including addition, subtraction, multiplication and division. Solving the simple behaviour problems related to fraction, decimal and percentage. CHAPTERS 3 Number Sense 4. Simplification 5. Fractions 6. Decimal 7. Percentage LEARNING OUTCOMES After completion of this content area, the learner is expected to be able to: count the numbers more than crore in Hindu-Arabic numerals (numbers and number names) and tell and write the place. read and write the numbers up to million in English. identify the prime and composite numbers from 1 to 100. compute the prime factors of the 3-digit numbers by factorization. round off of the numbers in the given place. compute the square numbers from 1 to 10 and cube numbers from 1 to 5 and their roots (square and cube).
70 The Leading Mathematics - 5 CHAPTER 3 NUMBERS SENSE Why do we use numbers in our daily life? Which is the starting number of counting? Which is the starting number of whole numbers? Count the scales of the ruler in cm and inches. What is the area of Nepal now ? Which numbers divide 12 ? Which numbers divide 60 ? Which numbers divide 1 ? What are the factors of 6 ? What are the factors of 1 ? WARM-UP Lesson Topics Pages 3.1 Concept of Numbers 71 3.2 Counting Numbers up to Crore 75 3.3 Place Value 78 3.4 Numbers up to Million 79 3.5 Rounding off the Nearest to Hundred 82 3.6 Rounding off the Nearest to Thousand 85 3.7 Factors 88 3.8 Prime Number 91 3.9 Prime Factorization 94
Arithmetic 71 3.1 Concept of Numbers At the end of this topic, the students will be able to: ¾ read and write the numerals up to lakhs Learning Objectives Read, Think and Learn What short of number symbols are there in the wall clock? They are Roman numerals (numbers.) The numerals used in wrist watch are Hindu-Arabic numerals. The basic number symbols used in Roman system are : I V X L C D M One Five Ten Fifty Hundred Five hundred Thousand VI = 5 + 1 = 6 IX = 10 – 1 = 9 XVI = 10 + 5 + 1 = 16 XL = 50 – 10 = 40 etc. The basic number symbols used in Hindu Arabic-System are: 1 2 3 4 5 6 7 8 9 0 One Two Three Four Five Six Seven Eight Nine Zero Lets us see the two numerals (numbers): Roman : III - (Represent three) Hindu-Arabic : 111 - (Represent One hundred eleven) (eleven is one ten and one)
72 The Leading Mathematics - 5 The basic symbols used in Devanagari system are : ! @ # $ % ^ & * ( ) Ps b'O{ tLg rf/ kfFr 5 ;ft cf7 gf} z"Go In the Hindu-Arabic number system, which digit is the greatest and which digit is the smallest among the digits 1, 2, 3 ,4, 5, 6, 7, 8, 9 and 0? What is the greatest number formed by 3 digits and which is the smallest number formed by 3 digits? 999 is the greatest number and 000 is the smallest. 000 is not a number. So, 1 is increased in 99, it becomes 100, which is the smallest number of 3 digits. Table of the smallest and the greatest numbers of specific digits No. of Digits Smallest Number Greatest Number 1 1 9 2 10 99 3 100 999 4 1000 9999 5 10000 99999 6 100000 999999 7 1000000 9999999 CLASSWORK EXAMPLES Example 1: Write the 3-digit numbers formed by the digits 2, 4 and 5. Which is the greatest and which is the smallest among them? Write them. Solution: The 3-digit numbers formed by the digits 2, 4 and 5 are given below: 245, 254, 425, 452, 524 and 542 The smallest number is 245. The greatest number is 542.
Arithmetic 73 Example 2: Find the sum of the greatest number and the smallest number formed by the digits 2, 0, 7 and 9. Solution: Here, the given digits are 2, 0, 7 and 9. If 0 is used in thousand place, it is meaningless and it becomes a three-digit number. The greatest number is 9720. The smallest number is 2079. ∴ The required sum is 11799. PRACTICE 3.1 Your mastery depends on practice. Practice as you play. 1. Write the smallest number and the greatest number of 5 digits, 7 digits and 9 digits. 2. Find the sum and difference of the greatest number and the smallest number formed by 6 digits. 3. Find the sum and difference of the greatest number and the smallest number formed by 9 digits. 4. Write the 3-digit numbers formed by 1, 7 and 8. Identify the greatest and the smallest numbers. 5. Write the 4-digit numbers formed by 3, 0, 5 and 9. Identify the greatest and the smallest numbers. 6. Find the sum and difference of the greatest and the smallest number of 5 digits formed by the digits 4, 5, 0, 8 and 9. 7. sdfsf] k|of]u u/L cIf/df n]Vg'xf];\ M (a) @&%$# (b) ^%$#^) (c) %#@!)( (d) #@*)(&^ (e) @!)#$^& (f) *^$@#)& (g) %!!@&%$ (h) #@!*&%$#^ (i) #@!)!($!# (j) ($!#!@)#!
74 The Leading Mathematics - 5 8. sdfsf] k|of]u u/L cª\sdf n]Vg'xf];\ M (a) gAa] nfv pgfGc;L xhf/ 5 ;o . (b) c7f;L nfv pgGtL; xhf/ rf/ ;o afx| . (c) afpGg nfv b'O{ xhf/ tLg ;o gf}F . (d) PsfgAa] nfv pgGrfnL; xhf/ kfFr ;o ;q . (e) gf} nfv gf} xhf/ gf} ;o gf} . (f) krxQ/ nfv rfln; xhf/ kfFr ;o lqkGg . (g) cG7fgAa] nfv 5};l¶ xhf/ cf7 ;o alQ; . (h) ;txQ/ nfv b'O{ ;o ;ft . 9. Write the following Roman numerals in Hindu Arabic numerals: (a) XVII (b) XXXIX (c) LXXX (d) CM (e) MIX (f) MLI (g) DCLIX (h) MMXL (i) MMMXLVII 10. Write the following Hindu-Arabic numerals in Roman numerals: (a) 24 (b) 38 (c) 74 (d) 96 (e) 129 (f) 520 (g) 778 (h) 3449 (i) 3638 (j) 1763 (k) 2338 (l) 1483 (m) 2087 (n) 1242 (o) 2691 Search the population of five metropolitan city of Nepal in internet. Write them in Devanagari and Hindu-Arabic numeral systems by using comma. PROJECT WORK
Arithmetic 75 3.2 Counting the Numbers up to Crore At the end of this topic, the students will be able to: ¾ read and write the numerals up to crore Learning Objectives Read, Think and Learn In Hindu-Arabic numeration system, numerals have different value in different places, but in Roman system they have the same value in different places. The place value and number names in our national system are given below. National Numbers National Numbers Ones 1 Crore 10000000 Tens 10 Ten crore 100000000 Hundreds 100 Arab 1000000000 Thousands 1000 Ten Arab 10000000000 Ten Thousands 10000 Kharab 100000000000 Lakhs 100000 Ten Kharab 1000000000000 Ten lakhs 1000000 Neel 10000000000000 To ease the process of counting and writing numerals, we put the digits in place value chart then we write it by using commas to indicate the periods as in the following. Place Value Tablie in National system of Numeration Crores Lakhs Thousands Ones Numeral with period T O T O T O H T O 3 1 4 2 3 1 0 5 3 31, 42, 31, 053
76 The Leading Mathematics - 5 Note In national system, we use comma (,) before three digits counting from the right to left and other commas are put before every two digits. We read it as, Thirty one crore forty two lakh, thirty one thousand and fifty three. b]jgfu/L ;ª\Vof b; s/f]8 s/f]8 b; nfv nfv b; xhf/ xhf/ ;o b; Ps # ! $ @ # ! ) % # The periods are the same as National number system. The number #!,$@,#!,)%# reads as, Pslt; s/f]8 aofln; nfv Pslt; xhf/ lqkGg . CLASSWORK EXAMPLES Example 1: Write the number of the following and put the comma (,) in national system. Forty-nine crore three lakh twenty four thousand five hundred twenty-six Solution: Forty-nine crore three lakh twenty four thousand five hundred and twenty six = 490324526 Using comma (,) = 49,03,24,526 PRACTICE 3.2 Your mastery depends on practice. Practice as you play. 1. Use the comma (,) in National system and write the number names of the population of the given countries below: Country Population Country Population Maldives 394999 Bhutan 787424 Sri Lanka 20238000 Nepal 29164578
Arithmetic 77 Country Population Country Population Afghanistan 33369945 Japan 126323715 Bangladesh 186987563 Brazil 209567920 USA 324118787 India 1210193422 China 1382323332 Pakistan 235824862 2. Write the above population of countries mentioned in question 1 in Devanagari numerals and Nepali. 3. Write the numbers of the following and put the comma (,) in National system. (a) Three crore seventy two lakh twenty seven thousand seven hundred and forty six (b) Eight crore forty four lakh three thousand and seven (c) Twelve crore seven lakh five thousand and seven (d) Thirty eight crore three lakh seventy thousand and fifty one (e) Forty nine crore forty three lakh two thousand and ten (f) Fifty eight crore five lakh two hundred and three (g) Sixty two crore thirteen lakh five thousands and twenty seven (h) Seventy two crore ninety three lakh seventy eight thousand and one (i) Ninety three crore four lakh two thousands and three hundred 4. Write the total population of Nepal according to the National Census - 2078 and answers the following questions. (a) Use the comma (,) according to national system. (b) Show the total population of Nepal in the place value table. (c) How many lakhs of people are there in Nepal? (d) How many thousands of people are there in Nepal?
78 The Leading Mathematics - 5 3.3 Place Value Read, Think and Learn Value of a digit in a numeral What are the place and place value of 5 in each of the following numerals? Ö 532 : 5 is in the hundreds place. The place of 5 is hundreds. The place value of 5 is (5 × 100 = 500) five hundred. Ö 3205 : 5 is in ones place. The place of 5 is ones. The place value of 5 is (5 × 1 = 5) five. Ö 25345023 5 is in ten lakhs place. The place of 5 is ten lakh. The place value of 5 is (5 × 1000000) 50 lakh. Ö 5200000123451 The 5 is in ten kharbs place. The place of 5 is ten kharabs. The place value of 5 is (5 × 1000000000000 = 5 × 1012= 5000000000000) fifty kharab. PRACTICE 3.3 Your mastery depends on practice. Practice as you play. 1. In which place is 7 in the following numbers? (a) 2378 (b) 22307 (c) 44289782 (d) 720081810 2. What is the place value of 6 in each of the following? (a) 612 (b) 5430630 (c) 262140002143 (d) 611212110082 3. What is the place and numerical value of 3 in each of the followings? (a) 3252 (b) 32000122 (c) 3001122520 (d) 300000278001 4. What is the value of 9 in each of the following? (a) 39004003 (b) 8920418435 (c) 2239040041673 (d) 9246047802535
Arithmetic 79 3.4 Numbers up to Million At the end of this topic, the students will be able to: ¾ read and write the numerals up to million Learning Objectives Read, Think and Learn The Hindu-Arabic numeration system is also called decimal system. Decimal system has base ten. This means each place has a value ten times the place to its right. Thus, a standard numeral. 42316 = 4 × 104 + 2 × 103 + 3 × 102 + 9 ×101 + 6 × 100 = 4 × 10000 + 2 × 1000 + 3 × 100 + 1 × 10 + 6 × 1 = 4 ten thousands + 2 thousands + 3 hundreds + 1 tens and 6 ones If there is zero in any place other than units place, we do not write but in units place, we write zero as 0 ones in expanded form. Example : 420 = 4 hundreds + 2 tens + 0 ones 402 = 4 hundreds + 2 ones Word names To write numbers in words we make groups of 3, 2, 2, but the last group may contain one digit. We say ones, thousands, lakhs, crores, arabs and kharab etc. in National system. We group digits of 3, whereas in international system ones, thousands, millions and billions. We say that the number is written as in place value table below. Billions Millions Thousands Ones kharabs arabs crores lakhs thousands Hundred units/ones International system National system
80 The Leading Mathematics - 5 We show the grouping for 1234567891 also in words as follows: 1,23,45,67,82,291 and 123,456,782,291 in national and international systems respectively. Number in Hindu Arabic words One kharab twenty three arab forty five crore sixty seven lakh eighty two thousand two hundred and ninety one Number in words in international system One hundred twenty three billion four hundred fifty six million seven hundred eighty two thousand two hundred and ninety one CLASSWORK EXAMPLES Example 1: Write the number 288543320 by using comma (,) in place value table. Also, write its number name in international system. Solution: The given number is written by using comma (,) as 378,543,210. Writing the number in place value table given below: Millions Thousands Ones H T O H T O H T O 2 8 8 5 4 3 3 2 0 In words, 288,543, 320 = Two hundred eighty eight million five hundred forty three thousand three hundred and twenty PRACTICE 3.4 Your mastery depends on practice. Practice as you play. 1. Insert the comma (,) in the following numbers and write the numbers in words in international system by using place value table. (a) 43231 (b) 343102 (c) 56123456 (d) 32345345 (e) 787924021 (f) 102005304 (g) 327800201 (h) 300201023 (i) 501003570
Arithmetic 81 2. Write the numbers in words in place value table and in figures by using comma (,). (a) Eight hundred forty nine thousand two hundred and two (b) Six million two hundred twenty four thousand five hundred and seventy seven (c) Five million fifty eight thousand and ninety (d) Thirteen million four hundred seventy one thousand two hundred and thirty four (e) Sixty seven million three hundred three thousand and eight (f) Two hundred ninety five million seven hundred ten thousand four hundred and seventy three (g) Five hundred five million sixty thousand and ninety seven (h) Six hundred seventy million and five (i) Seven hundred three million five hundred and six (j) Nine hundred fifty million three thousand and nineteen Search the population of any ten counties in internet and show them in place value table and abacus. Also, write in numerals in digits and words in national and international numeral systems by using comma (,) and express them in expanded forms. PROJECT WORK
82 The Leading Mathematics - 5 3.5 Rounding off the Nearest to Hundreds At the end of this topic, the students will be able to: ¾ round off the given numbers Learning Objectives Read, Think and Learn Which house fly is nearer from the frog? Which housefly will be quickly caught by the frog? 0 10 20 30 40 50 60 70 80 90 100 far near A B Housefly B is nearer from the frog than housefly A. So, housefly B will be quickly caught by the frog. 60 is nearer to 100 and far from 0. So, the rounding off 60 to the nearest to 100 is 100. Ram is very hungry. Which fruits basket will he take quickly? 200 210 220 230 240 250 260 270 280 290 300 near far B F 230 is near to 200 than that 300. So, the rounding off 230 to the near to 100 is 200.
Arithmetic 83 John is thirsty. Which juice of glass will he take first? 800 810 820 830 840 850 860 870 880 890 900 same same G J Here, 850 is in the same distance from 800 and 900. In this case, we round off the greater hundreds. So, the rounding off 850 to the nearer 100 is 900. Remember Oh! To round off the nearest 100, we observe the second last digit of the number. If the second last digit is equal or greater than 5, the rounding off the given number is upper hundreds, other than it is lower hundreds. But in the case of the rounding off the nearest to tens if the last digit of the number is 1, 2, 3 or 4 round off the lower tens. If the last digit of the number is 5, 6, 7, 8 or 9, round off the upper tens. PRACTICE 3.5 Your mastery depends on practice. Practice as you play. 1. Round off the following height of the students in the class 5 to the nearest 10. Name Gita Hari Sophal Muna Ram John Rejhon Height 45 38 45 38 35 42 52 Round off
84 The Leading Mathematics - 5 2. Round off the following length of the highways of Nepal to the nearest to hundred. Highway Siddhartha Tribhuvan Prithvi Kodari Mechi Karnali Mahendra Lenght (km) 182 158 174 113 224 232 1028 Round off 3. Round off the following height of mountains of the world to the nearest 100. Mountain Mt. Everest Mt. K2 Kanchanjunga Lhotse Makalu Height (m) 8849 8611 8586 8516 8485 Round off Mountain Cho Oyu Dhaulagiri Manaslu Nanga parbat Annapurna Height (m) 8188 8848 8611 8586 8516 Round off 4. Round off the following numbers as stated within brackets. (a) 24 (to the nearest 10) (b) 456 (to the nearest 10) (c) 5697 (to the nearest 10) (d) 4789 (to the nearest 100) (e) 9563 (to the nearest 100) (f) 7896 (to the nearest 100) (g) 34786 (to the nearest 100) (h) 78964 (to the nearest 10) Ask the weights of your family members and list them. Round off these in to nearer to ten. PROJECT WORK
Arithmetic 85 3.6 Rounding off Nearest to 1000 At the end of this topic, the students will be able to: ¾ round off the given numbers neatest to 1000 Learning Objectives Read, Think and Learn Tiger is very hungry. Which animal does it attack? Why? 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 G D Tiger is nearer to goat than that of deer. So, the tiger goes to goat. The rounding off 1300 to the nearest thousand is 1000. Where does Hari go to take Rs 1000 note toward 2000 or 3000 ? 2000 2100 2200 2300 2400 2500 2600 2700 2800 2780 2900 3000 T H Hari is the nearest to Rs. 1000 note toward 3000. So, he goes towards 3000. 2780 is rounded off 3000 under the nearest thousands. Remember Oh! To round off the nearest 1000, we observe the third last digit of the number. If the third last digit is equal or greater than 5, the rounding off the given number is upper hundreds, other than it is lower hundreds.
86 The Leading Mathematics - 5 PRACTICE 3.6 Your mastery depends on practice. Practice as you play. 1. Round off the given numbers the nearest 1000 by using arrow head. 1552 4348 13850 17232 38570 59849 2. Round off the following length of rivers in the World to the nearest 1000. River Amazon Nile Yangtze Mississippi Yenisei Haanytle Length (km) 6992 6852 6300 6275 5539 5464 Rounding off 3. Round off the following height of pass (bhanjyang) as in the nearest 1000. Pass Lhola Nangpa Lipulekh Ganja Larke Thorang Frenchla Height (m) 6992 6852 6300 6275 5539 5464 5410 Rounding off 4. Round off the following numbers as stated within brackets. (a) 219 (to the nearest 10) (b) 345 (to the nearest 10) (c) 429 (to the nearest 100) (d) 156 (to the nearest 100)
Arithmetic 87 (e) 1596 (to the nearest 1000) (f) 2272 (to the nearest 1000) (g) 3753 (to the nearest 1000) (h) 5453 (to the nearest 1000) (i) 27148 (to the nearest 1000) (j) 57812 (to the nearest 1000) (k) 875700 (to the nearest 1000) (l) 95409(to the nearest 1000) 5. Round off the following numbers for the circled digit. (a) 4 9 (b) 2 5 8 (c) 327 3 (d) 2 8 19 (e) 4 8 97 (f) 4 5 024 (g) 23 5 73 (h) 949 9 4 (i) 3 4 786 (j) 4 8 497 (k) 95 2 78 (l) 8 7 827 (m) 15 3 72 (n) 865 2 3 (o) 2 9 763 (p) 4 5 237 (q) 47 8 97 (r) 7 8 532 Circled last digit means rounding off the nearest 10 and circled second last digit means rounding off the nearest. Search the male, female and total population of your local level and round off them to the nearest 10, 100 and 1000. Prepare a table and present in your classroom. PROJECT WORK
88 The Leading Mathematics - 5 3.7 Factors At the end of this topic, the students will be able to: ¾ find the factors of the given number Learning Objectives Read, Think and Learn Look at the multiplication table given below. Take a number say 12. This number is obtained by multiplying 2 by 6. i.e., 2 × 6 = 12. We call the number 2 and 6 are the factors of 12 and 12 is the product or multiple of 2 and 6. We see that a factor of a number divides the number exactly with no remainder. e.g., 2 6 12 12 × 6 2 12 12 × and ∴ 12 = 2 × 6 ∴ 12 = 6 × 2 A factor of a number is a perfect divisor of the numbers. But, 2 is not a factor of 13. Since, 2 6 13 12 1 With remainder What are the factors of 12? Circle the number 12 in the multiplication table. The first circled number 12 shows that 2 and 6 are the factors of 12. Similarly, by second circled 12 shows 3 and 4 are the factors of 12. ∴ 12 = 3 × 4 As we defined a factor of a number is a divisor of the number. Thus, 1 and the number 12 itself divide 12. × 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36
Arithmetic 89 ∴ 12 = 1 × 12 Thus, 1 and 12 are also the factors of 12. They are called trivial factors of 12. The largest factor of a number is the number itself. Hence, the factors of 12 are 1, 3, 4, 6 and 12. Note No number greater than the given number can be the factor of the number. CLASSWORK EXAMPLES Example 1: List all the factors of the following numbers and write down the set of factors of a. 9 b. 15 c. Write down the set of common factors of 9 and 15. Solution: Since, 1 × 9 = 9 3 × 3 = 9 The factors of 9 = F(9) = 1, 3, 9 Alternatively, 9 ÷ 1 = 9 9 ÷ 9 = 1 9 ÷ 3 = 3 ∴ The factors of 9 are 1, 3, 9. F(9) = 1, 3, 9 Since, 1 × 15 = 15 3 × 5 = 15 The factors of 15 = F(15) = 1, 3, 5, 15 Alternatively, 15 ÷ 1 = 15 15 ÷ 15 = 1 15 ÷ 3 = 5 15 ÷ 5 = 3 ∴ The factors of 15 are 1, 3, 5, 15. F(15) = 1, 3, 5, 15 Set of common factors of 9 and 15 = 1, 3
90 The Leading Mathematics - 5 Example 2: Write down the factors of 6. Show it in jellyfish pattern. Solution: Since, 1 × 6 = 6, 2 × 3 = 6, So, the factors of 6 are 1, 2, 3 and 6. These factors are shown in the jellyfish pattern as. PRACTICE 3.7 Your mastery depends on practice. Practice as you play. 1. Find the set of factors of the following numbers. (a) 4 (b) 6 (c) 8 (d) 12 (e) 15 (f) 16 (g) 19 (h) 18 (i) 20 (j) 25 (k) 30 (l) 35 (m) 19 (n) 7 (o) 40 (p) 50 (q) 55 (r) 60 (s) 70 (t) 23 (u) 77 (v) 23 (w) 11 (x) 17 (y) 18 (z) 36 2. Use the set of factors of question 1 and write down the set of common factors of : (a) 4 and 6 (b) 6 and 8 (c) 8 and 12 (d) 12 and 18 (e) 30 and 50 (f) 25 and 40 3. Write the jellyfish pattern for the factors of the following numbers: (a) (b) (c) 1 2 3 6 6 12 21 28
Arithmetic 91 3.8 Prime Number At the end of this topic, the students will be able to: ¾ find the prime of the given number Learning Objectives Read, Think and Learn What are the factors in each of the following numbers? 1, 2, 3, 4, 5, 6, 7, 8, 9 From the multiplication, we see the different factors of these numbers are: Numbers 1 2 3 4 5 6 7 8 9 Test 1 × 1 1 × 2 1× 3 1 × 4 2 × 2 1 × 5 1 × 6 2 × 3 1 × 7 1 × 8 2 × 4 1 × 9 3 × 3 Factors 1 1, 2 1, 3 1, 2, 4 1, 5 1,2, 3, 6 1, 7 1, 2, 4, 8 1, 3, 9 We see from the above facts, 1 has only one factor and 2, 3, 5, 7 have only two factors and 4, 6, 8, 9 have more than 2 different factors. A whole number which has exactly two different factors is called a prime number. A whole number which has more than two different factors is called a composite number. The number 1 is neither prime nor composite. It is called the identity number. Finding prime numbers There is a familiar method of finding prime numbers called method of Eratosthenes (Er-uh-tos-then-neez), Eratoshthenes is a Greek mathematician. Write the whole numbers from 1 to 50 in rows, each row contains ten whole numbers.
92 The Leading Mathematics - 5 Since, 1 is not a prime number, so cross it. 2 is a prime number circle it. Cross the multiplies of 2. The first crossed number is second from the circled one, the other crossed numbers are the second from the crossed one. The numbers are multiples of 2. Circle 3 and cross the multiples of 3. The crossed number is third from the circled one and the remaining crossed numbers are third from the last crossed number. Circle the numbers 5 and 7 and do similar process as we did for 2 and 3. Continue the process until when you circled 23, we will have just one multiple 46 to cross it. After that we do not have to cross the numbers in the list but we just circle them. The circled numbers are the prime numbers less than 50. Thus, they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 What is the smallest prime? What is the largest prime between 1 and 50? How many even prime numbers are there? PRACTICE 3.8 Your mastery depends on practice. Practice as you play. 1. Table below gives the numbers from 1 to 100. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
Arithmetic 93 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Now carry out the following activities. Is 1 prime number? Is 1 composite number? Circle 1. It is neither prime nor composite because it has only one factor. Circle all the numbers divisible by 2 but not 2 itself. Circle all the numbers divisible by 3 but not 3 itself. Circle all the numbers divisible by 5 but not 5 itself. Circle all the numbers divisible by 7 but not 7 itself and so on. The numbers which are not circled are the prime numbers and the numbers which are circled except 1 are composite numbers based on the above activity. Answer the following questions. (a) How many even prime numbers are there between 1 to 10? (b) How many prime numbers are there between 10 to 30? (c) Write down the prime numbers between 20 to 40. (d) How many prime numbers are there between 30 to 50? (e) Write down the prime numbers from 1 to 100. (f) List all the prime numbers between 1 to 100. (g) Which is the greatest prime number of one digit? (h) Which is the smallest prime number of two digits? (i) Write down the composite numbers between 1 to 100. (j) How many composite numbers are there between 1 to 75?
94 The Leading Mathematics - 5 3.9 Prime Factorization At the end of this topic, the students will be able to: ¾ find the prime factors of the given number. Learning Objectives Read, Think and Learn Express the composite number 12 as a product of prime numbers. 12 = 2 × 6 Is this a product of primes? No, can you express 6 into prime factors? 12 = 2 × 2 × 3 Is this a product of primes? Yes. Here, 12 is composite number. Every composite number have at least 3 different factors. To express a composite number as product of prime numbers is called the prime factorization of the number. Prime Factorization by Division Write prime factorization of 12. Step 1 : If you know a prime divisor or you divide the number (if not try from a small prime number not necessarily the smallest.) Step 2 : Repeat the step 1, for the quotient. Generally, we apply this method as follows ∴ 12 = 2 × 2 × 3 6 ÷ 2 = 3 or 6 4 3 2 2 2 12 6 3 2 2 12 4 2 3 2 i.e., 12 = 2 × 2 × 3 12 ÷ 2 = 6 or 12 12 6 4 2 3 i.e., 12 = 2 × 6 or 12 = 3 × 4 12 6 3 2 2
Arithmetic 95 Prime factorization by tree diagram We know that, 12 = 2 × 2 × 3 You can work out prime factors of 12 showing the factors in a diagram like a tree, called the factor tree diagram; as in the following. 12 2 × 6 2 × 2 × 3 12 is even, divide by 2. 6 is divisible by 2, divide by 2. 2 and 3 are prime stop dividing. Or, you can put the factors starting from the bottom as. 12 2 × 2 × 3 2 × 6 2 and 3 are prime, stop dividing. 6 is divisible by 2, divide 6 by 2. 12 is even, divide by 2. You can conclude the process as 12 6 3 2 2 Divide 12 by 2 Divide 6 by 2 3 is prime stop dividing Hence, 12 = 2 × 2 × 3 The prime factorization of the number is unique except for the order of the factors. 12 6 3 2 2
96 The Leading Mathematics - 5 CLASSWORK EXAMPLES Example 1: Express 126 as the product of prime factors, and show the factors in a tree diagram. Solution: 126 even divide by 2. 6 + 3 = 9, 9 is divisible by 3. 2 +1 = 3, 3 is divisible by 3. 7 is prime; stop dividing. 2 126 3 63 3 21 7 Therefore, 126 = 2 × 3 × 3 × 3 × 7 Showing the factors in a factor tree, we have, 126 2 × 63 × × × × × 2 3 21 2 3 3 7 ∴ The prime factors of 126 are 2, 3, 3 and 7. PRACTICE 3.9 Your mastery depends on practice. Practice as you play. 1. Express each of the following numbers as the product of their prime factors: (a) 6 (b) 20 (c) 28 (d) 35 (e) 50 (f) 64 (g) 72 (h) 99 (i) 125 (j) 360 2. Factorize into prime factors for each of the following: (a) 14 (b) 16 (c) 18 (d) 48 (e) 60 (f) 75 (g) 135 (h) 180
Arithmetic 97 (i) 270 (j) 625 (k) 575 (l) 668 (m) 590 (n) 620 (o) 725 3. Express each of the following numbers as the product of their prime factors and show the factors in the factor tree diagram. (a) 18 (b) 35 (c) 42 (d) 63 (e) 128 (f) 144 (g) 210 (h) 156 (i) 44 (j) 72 (k) 525 (l) 663 (m) 221 (n) 728 (o) 878 (p) 928 4. Divide as in the example (Express numerator and denominator into prime factors.) 72 18 = 2 × 2× 2 × 3 × 3 2 × 3 × 3 = 2 × 2 = 4 (a) 16 8 (b) 36 12 (c) 24 6 (d) 18 6 (e) 45 9 (f) 63 9 (g) 65 13 (h) 54 18 (i) 210 14 (j) 550 25 (k) 625 25 (l) 625 50 (m) 495 11 (n) 663 9 (o) 125 10 (p) 169 13 (q) 1155 105 (r) 1911 273 (s) 7497 153 (t) 1001 77 (u) 2015 75 (v) 4500 125 (w) 2144 67 (x) 2080 65
98 The Leading Mathematics - 5 CHAPTER 4 SIMPLIFICATION Add, subtract, multiply, and divide the numbers 8 and 4. The are 4 bananas and 2 bananas on two plates and three monkeys eat 1 banana each. How many bananas are left in the plate ? Which numbers divide 4, 12 and 16 separately ? Which numbers are divided by 2 ? Similarly, by 3. What is the sum of 8, 5 and 2 ? What is the sum of 12, 5 and 2 ? How many small squares are there in all faces of the given Rubik ? WARM-UP Lesson Topics Pages 4.1 Simplification Without Brackets 99 4.2 Verbal Problems on Without Brackets 102
Arithmetic 99 4.1 Simplification Without Brackets At the end of this topic, the students will be able to: ¾ simplify the given numeral expressions. Learning Objectives Read, Think and Learn While working with numbers and operations you may come across the problems that include some or all of the four basic operations of addition (+), subtraction (–), multiplication (×) and division (÷). This type of problems are called simplification problems. Look at the following examples: (a) 4 × 3 + 2 = 12 + 2 = 14 (b) 4 × 3 + 2 = 4 × 5 = 20 (c) 15 ÷ 5 × 3 = 15 ÷ 15 = 1 (d) 15 ÷ 5 × 3 = 3 × 3 = 9 But in some cases order of operations are not counted. (e) 20 + 15 – 10 = 35 – 10 = 25 (f) 20 + 15 – 10 = 20 + 5 = 25 Thus, we must have a definite order of working rule. If a simplification problem consists of all or some operations addition (+), subtraction (–) multiplication (×) and division (÷), we follow the order, DMAS. Where, D = Division M = Multiplication A = Addition and S = Subtraction. We perform two operations of division (D), multiplication (M), addition (A) and subtraction (S) in that order whenever these occur in a problem.
100 The Leading Mathematics - 5 CLASSWORK EXAMPLES Example 1: Simplify : 25 ÷ 5 × 7 + 3 – 12 Solution: 25 ÷ 5 × 7 + 3 – 12 = 5 × 7 + 3 – 12 (Division) = 35 + 3 – 12 (Multiplication) = 38 – 12 (Addition) = 26 (Subtraction) Example 2: Simplify : 49 + 64 ÷ 8 × 3 – 10 Solution: 49 + 64 ÷ 8 × 3 – 10 = 49 + 8 × 3 – 10 (Division) = 49 + 24 – 10 (Multiplication) = 73 – 10 (Addition) = 63 (Subtraction) Example 3: Simplify : 50 – 75 ÷ 15 + 11 × 5 Solution: 50 – 75 ÷ 15 + 11 × 5 = 50 – 5 + 11 × 5 (Division) = 50 – 5 + 55 (Multiplication) = 45 + 55 (Subtraction) = 100 (Addition) PRACTICE 4.1 Your mastery depends on practice. Practice as you play. 1. Simplify: (a) 7 + 8 – 12 (b) 6 – 2 + 8 (c) 8 + 9 – 7 (d) 37 – 25 + 12 (e) 16 + 8 – 10 (f) 42 + 43 – 48 (g) 5 × 4 + 2 (h) 6 × 3 + 7 (i) 3 + 4 × 7