The Leading Maths - 7

i Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Ph. No.: 01-5388827 The Leading MATHS 7 Author Ashok Dangol M.Ed., Maths (TU) ALLIEDllied Prepared on New Curriculum Issued by CDC, Sanothimi, Bhaktapur, Nepal

Allied The Leading MATHS 7 Publisher Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Phone : 01-5378629, 5388827 Written by Ashok Dangol M.Ed., Maths (TU) Special Thanks Nabaraj Pathak Sanjay Maharjan Lalbabu Prasad Yadav Jit Bahadur Khanal Ashok Dangol Copyright All rights reserved with the copyright holder. Edition First - 2080 (5000 pcs.) Computer Icon Design House #9849098999 Printed in Nepal

Approved by CDC, Nepal PREFACE This Allied The Leading Mathematics - 7 is basically meant for making the teachers and taught active while teaching and learning mathematics. The contents and extent of the series are strictly contained and arranged in accordance with the new vision and mission of the latest New Curriculum 6-8 of Basic Level of CDC, Nepal. This series is basically an outcome of my untiring effort and patience. The long and dedicated service in teaching and popularization of mathematics has been a great asset in preparing this series. It has been designed as a textbook for English medium private and government school students with a new approach. This book provides maximum benefit to both teachers and students because of the following unique features: Unique Features of this Book Ö Arranged especially focusing on child psychology of teaching and learning mathematics which are based on the Areas of Basic Level Crriculum 6-8. Ö Prepared with the firm belief that “Mathematics begins at Home, grows in the Surroundings and takes shape in School (HSS Way)”, and sincere attempts have been made to make the learner, the teacher and reader feel “Mathematics is Fun, Mathematics is Easy and Mathematics is Everywhere (FEE Concept)”. Ö Written the focus of students’ activities and easily perform teaching-learning activities for teachers. Ö Well arranged the four colours of the whole book supports to find easily Units, Chapters, Lessons, Examples, Practices, and other topics from Content. Ö Every Theme begins with its estimated teaching hours (Theory + Practical), competency, learning outcomes, Warm-Up for pre-knowledge. Ö Highlighted the important terms, notes and key points. Ö Included sufficiently all types of Classwork Examples and Home Assessments from simple to complex with suitable figures and reasons. Ö Provided QR Code of Chapters and Geogebra to present from Projector and learn from Mobile set. Ö Included Project Works and sample format of Project Work as self-practice to the students at home for some days' activities to memories long time about the entire chapters. Ö Included Mixed Practice and Confidence Level Tests as self-evaluations for Success and Competent by themselves. Ö Available Individual Practical Evaluation Sheet at the end.

PREFACE It is very much hoped that with all the above features, this book will be found really fruitful by teachers and students alike. Thank Allied Publication Pvt. Ltd., Kathmandu, Nepal for taking responsibility for publishing this book, Nariswor Gautam for language editing, Dev Krishna Maharjan for an attractive art of pictures, and Binod Bhandari for an attractive design. I would like to extend my sincere gratitude to the persons whose ideas or creations are directly or indirectly incorporated into the text. I would like to extend my thanks to the teachers and the students who helped me to verify the answers and to check the manuscript of this book. Also, many thanks to the schools that applied this book and suggested it to me. Finally, I heartily welcome criticisms, feedbacks and suggestions from readers so that it may appear with revise from in the coming edition and will be gratefully and thankfully acknowledged and honored.

UNIT - I: SETS 7 Chapter 1: Sets 8 UNIT - II: ARITHMETIC 27 Chapter 2: Real Numbers 28 Chapter 3: Integers 51 Chapter 4: Rational Numbers 75 Chapter 5: Fraction and Decimal 82 Chapter 6: Ratio and Proportion 119 Chapter 7: Profit and Loss 132 Chapter 8: Unitary Method 140 UNIT - III: MENSURATION 151 Chapter 9: Perimeter, Area and Volume 152 UNIT - IV: ALGEBRA 174 Chapter 10: Indices 175 Chapter 11: Algebraic Expressions 183 Chapter 12: Equation, Inequality and Graph 212 UNIT - V: GEOMETRY 225 Chapter 13: Lines and Angles 226 Chapter 14: Plane Shapes 244 Chapter 15: Congruency 266 Chapter 16: Solid Objects 270 Chapter 17: Coordinates 278 Chapter 18: Symmetry and Tessellation 289 Chapter 19: Transformation 296 Chapter 20: Bearing and Scale Drawing 311 UNIT - VI: STATISTICS 323 Chapter 21: Presentation of Data 324 CONTENTS

Sets 7 COMPETENCY  Solution of behaviour problems related to set. CHAPTER 1. Sets LEARNING OUTCOMES  To introduce the null set, equivalent sets, equal sets, finite and infinite sets.  To identify the universal set and subsets. SETS UNIT I Family ko fig Class 6 ko Father and daughter + = Daughter and mother Father, daughter and mother

8 The Leading Mathematics - 7 CHAPTER 1 SETS Lesson Topics Pages 1.1 Review on Sets 9 1.2 Types of Sets and Venn Diagram 15 1.3 More Types of Sets and Venn Diagram 19 1. What is the meaning of set? 2. What is a well-defined object? 3. In which condition is not a set defined ? 4. What is the definition of set? 5. What is the member element of a set? 6. Which letters are denoted for set and its elements? 7. How many ways are there to represent a set? WARM-UP

Sets 9 1.1 Review on Sets At the end of this topic, the students will be able to: ¾ write the set and its members in notation. ¾ write the representation of sets in different forms. Learning Objectives Modern mathematics begins with the idea of a “set”. Man started to use this idea a long time ago. But, its mathematical use appeared some two hundred years ago only. Set and Elements of a Set There are objects all around us. The objects may be living or non-living. They may be abstract ideas. Look at each of the following pairs of sentences: Ordinary Expression Mathematical Expression “ A bunch of flowers ”, “ A set of flowers ”, “ A group of children ”, “ A set of children ”, “ A class of students ”, “ A set of students ”, “ A collection of stamps ”, “ A set of stamps ”, “ A cluster of points in a plane ”, “ A set of points in a plane ”, The word “set ” thus carries the same meaning as the words: “bunch ”, “group ”, “class ”, “collection”, “cluster ” in the above phrases. Each individual object in a set is called its member or element. We make no attempt to define a set. For all practical purposes, we simply accept that “A set is said to be known (or well-defined) if it is possible to say whether a given object ‘is in’ (or ‘belongs to’) the set or ‘not’.” The students of Class VII are well-defined but the beautiful students of class VII is not well-defined. We can decide whether a student is in Class VII or not; but we cannot pinpoint who is beautiful and who is not. Set Notation and Set Membership A set is denoted by capital or upper case letters such as A, B, C, ... ..., X, Y, Z, and the elements by the small or lower case letters such as a, b, c, ... ..., x, y, z.

10 The Leading Mathematics - 7 Consider a set, P = {2, 3, 5, 7} In symbols, we write 2 ∈ S (∈ is the Greek letter epsilon) for i. ‘2’ is in the set S’ or, ii. ‘2’ belongs to the set S’ or, iii. ‘2’ is an element of the set S’ or, iv. ‘2’ is a member of the set S’ In the same way, write 4 ∉ S for i. ‘4 is not in the set S’ or, ii. ‘4 does not belong the set S’ or, iii. ‘4 is not an element of the set S’ or, iv. ‘4 is not a member of the set S’ . Specification of a Set Once we have a set, we always need the phrase “ the set of ”. To visualize or write “ the set of ” any object or objects, we first denote the phrase “ the set of ” by a simple closed curve or a pair of curly brackets or braces “ { }”. Inside the simple closed curve or the braces, we may write i. the pictures of the objects or, ii. names of the objects or some description about the objects. or, iii. some description about the objects. For instance, let’s consider the set consisting of the three geographical regions of Nepal: (a) The scale (b) protractor (c) compass (d) set-square.

Sets 11 (a) By drawing pictures inside a closed curve: (b) By listing the names inside a closed curve: scale, protractor, compass, set-square (c) By drawing pictures inside curly brackets (braces) “{ }” and separated by commas: (d) By writing the names inside curly brackets or braces “{ }” and separated by commas {scale, protractor, compass, set square}. Sometimes, we may write a descriptive phrase or sentence within the braces as show below: {The letters of the English alphabet}. But listing is not always a suitable method of describing a set. It may sometimes be a very tedious job. Sometimes, it becomes impossible. Listing may become simple and obvious in some cases. A set of three dots is quite often used when it is possible to tell what could be the missing elements. For instance, in the listing, {a, b, c, d, e, f, g, h, i, j, k, l, m, ..., z} Here, the three dots mean that the letters continue till z. In listing, we do not worry about the order of occurrence of the elements. Besides, we do not allow any repetition.

12 The Leading Mathematics - 7 In short, a set can be represented: (a) By drawing the pictures or names of the elements inside a closed curve, i.e., in the roster form, (b) By listing the objects or elements inside braces and separating them by commas, i.e., by listing or tabulation, (c) By writing a descriptive phrase or sentence inside two braces { }, i.e., by description. Standard Description of Sets Small sets can be written in the roster form or by listing. This is difficult in the case of large sets. The set of counting numbers is very large. All of its elements cannot be listed. We can denote such a set as shown below: {1, 2, 3, 4, 5, ... }, the dots (...) conveying that we have to continue without end. A better way is to use (i) the braces { }, (ii) a letter such as x followed by a colon ‘ : ’ within { } and (iii) a descriptive phrase after ‘ : ’. We then write the set {1, 2, 3, 4, 5, ... } as{x : x is a counting number}. Here, the symbol “ { } ” is read “ the set of ”, the colon “ : ” is read “ such that ” ; and the whole symbol as “ the set of all x such that x is a counting number ”. In the above notation, the colon ‘ : ‘ is sometimes replaced by the vertical stroke or bar ‘ | ’. As a second example, we can consider the set V = {a, e, i, o, u}, of the vowels of the English alphabet. It may be written as V = {x : x is a vowel in the English alphabet}. It is read as “the set of all x such that x is a vowel of the English alphabet”. A specification of a set in the form {x : x satisfying some property} is known as the set-builder form. Here the letter or symbol x is known as a variable.

Sets 13 EXERCISE 1.1 Your mastery depends on practice. Practice like you play. 1. State whether the following statements are true or false: (a) The words “ set ”, “group ” and “collection ” carry the same meaning. (b) The phrase “ the set of ”, may be denoted by a simple closed curve. (c) The phrase “ the set of ”, is denoted by a pair of curly brackets { }. (d) The words “ element ” and “member ” carry the same meaning. (e) The phrases “ is an element of ” and “belongs to ” carry the same meaning. 2. Fill in the blanks. (a) A set consists of objects called ______________ . (b) We make no attempt to define a ______________ . (c) “A set is said to be known (or well-defined) if it is possible to say whether a given object ______________ the set or ‘not’.” (d) While listing the elements of a set may occur in any ______________ . (e) While listing of elements of a set no element is ______________ . 3. Tick mark (or write down) the correct answer: (a) A set is denoted by i. English small alphabet ii. English capital letter iii. ∈ iv. ∴ (b) An element of a set is denoted by i. English small alphabet ii. English capital letter iii. ∈ iv. ∈ (c) The set {1, 3, 5, 7, 9} is said to be specified by i. Listing ii. Description iii. Set-builder form iv. None of the above

14 The Leading Mathematics - 7 4. State whether the following are true or false: (a) 2 ∈ {1, 2} (b) 3 {1, 2} (c) {2} ∈{1, 2} (d) {1, 2} ∈ {1, 2} (e) a ∈ {a, l } (f) a {all } (g) {1, 2} ∈ {{1, 2}, {1, 2, 3}} 5. Rewrite the following using set notation: (a) 1 belongs to the set N of counting numbers. (b) 1 is not an element of the English alphabet A. (c) e is a member of the set V of vowels of the English alphabet. (d) + is in the set S of signs of four simple rules of arithmetic. 6. Write the following sets in the tabular form (by listing) (a) The set of even numbers of one digit. (b) The set of the number zero (c) The set of letters of the word ‘ vowel ’ (d) The letters of the word ‘ mathematics ’ (e) The multiples of 2 greater than 4 and less than 13. 7. Specify the following sets by description method. (a) {l, o, v. e} (b) {1, 2, 3} (c) {Kathmandu} (d) The set {2, 4, 6, 8} (e) The multiples of 2 greater than 4 and less than 13. 8. Write in the set-builder form. (a) The set of odd numbers of one digit. (b) The set of numbers non-zero numerals. (c) The set of the letters of the word ‘ empty ’ (d) The set {2, 4, 6, 8} (e) The multiples of 2 greater than 4 and less than 13. ANSWERS Consult with your teacher.

Sets 15 1.2 Types of Sets and Venn Diagram At the end of this topic, the student will be able to: ¾ write the special types of sets. ¾ use the Venn diagram in set notation. Learning Objectives Sets of various kinds occur and recur in our daily life. They may give rise to i. One object or element set, e.g., {Sun}, ii. Two elements set, e.g. {Boy, Girl}, iii. Three elements set, e.g. {Sun, Earth, Moon} iv. Unlimited elements set, e.g. {1,2,3,…} and so on. (a) Null Set One special type of set is the set having no element at all. A set having no element is known as the empty set or null set or void set. Such a set is denoted by the Greek letter f (phi) or simply by the empty braces “{ }”. For instances: M = {a male student in a girls’ school}. S = {the number zero in Roman Numerals} and V = { volcanoes in Nepal}. (b) Universal Set We talk or think about one or more objects. The totality of things or objects or ideas under consideration is called the universe or the universal set. It is denoted by U. To construct an English word, we need the set of English alphabet. So, the letters of the English alphabet are under consideration. It is the universal set for our purpose of constructing an English word. In ordinary arithmetic, the set of all numbers is the universal set. (c) Singleton Set A set having only one element is called a singleton set. For instances : E = {highest peak of the world} and S = {an even prime number}

16 The Leading Mathematics - 7 (d) Finite Set The empty set and any other set that has a fixed number (zero or whole number) of elements is said to be finite. i. The set of English alphabet {a, b, c, ... ..., x, y, z} ii. The set of digits {1, 2, 3, 4, 5, 6, 7, 8, 9} iii. The set of districts of Nepal (e) Infinite Set Sets that are not finite are called infinite sets. For examples, i. The set of points in a line ii. The set of counting numbers (f) Venn Diagrams Diagrams may be used to understand sets. A rectangular region is often used to denote a universal set U. Other sets are then denoted by circular or elliptical or a closed region inside the rectangle. Rectangular Circular Elliptical Points inside each of them are used to denote the elements of a set. Such a pictorial representation or diagram of a set is commonly known as a “ Venn diagram ”. Shading may be done, if we like to do so. Two Venn diagrams are shown below: B A U A A U U .4 4 .3 3 .5 5 .1 .2 1 2 Fig (i) Fig (ii) Fig (iii) In Fig. (i), A and B are in the set U. In Fig. (ii), U = {1, 2, 3, 4, 5} and A = {3, 4}. Here, the elements 1, 2, 3, 4 and 5 are denoted by the dots within them. The dots are not used in practice, used only the elements.

Sets 17 CLASSWORK EXAMPLES Example 1 State whether each of the following is true or false: (a) The set of letters of the word “ empty ” is not empty. (b) The set of points on a line is infinite. (c) The set of number “ zero” is not finite. Solution: (a) True (b) True (c) False Example 2 Fill in the blanks: The set of objects under consideration is called the … ... Solution: The set of objects under consideration is called the Universal set. Example 3 Write Venn diagrams for each of the following sets : (a) The singleton set {0} (b) The doubleton set {Kathmandu, Nepal} (c) The letters of the word ‘ mathematics ’ (d) The multiples of 2 greater than 4 and less than 13. Solution: (a) (b) (c) (d) Example 4 What do you understand by a Venn diagram? Solution: A set represented pictorially by means of point or points inside a closed region is called a Venn diagram. 0 U Kathmandu Nepal U N m a t h e i c s U M 6 8 10 12 U T

18 The Leading Mathematics - 7 EXERCISE 1.2 Your mastery depends on practice. Practice like you play. 1. State whether each of the following is true or false: (a) The set of letters of the word “ void ” is not empty. (b) The set of stars in the visible universe is infinite. (c) The set of counting numbers is not finite. 2. Fill in the blanks: (a) A closed region representing a set is called a ............ . (b) A null set has ........... membership. (c) A set that has uncountable number of elements is called ............. . 3. Tick mark (or write down) the correct answer: (a) A set that is not finite is called i. Empty ii. Infinite iii. Finite iv. None of above (b) The set having no elements is called i. Empty ii. Infinite iii. Finite iv. None of above 4. State whether the following are empty, finite or infinite: (a) The set of moons of the Earth (b) {f} (c) The set of mountains of Nepal (d) The set of points in a line (e) The set of whole numbers. 5. Write Venn diagrams for each of the following sets: (a) The singleton set {1} (b) The doubleton set (boy, girl} (c) The tripleton set {g, o, d} (d) The letters of the word ‘ mathematics ’ (e) The multiples of 3 greater than 6 and less than 16. 6. (a) What do you understand by a Universal set? (b) What is null set? ANSWERS Consult with your teacher.

Sets 19 1.3 More Types of Sets and Venn Diagram At the end of this topic, the student will be able to: ¾ define disjoint and overlapping sets. ¾ define subset, proper subset and improper subsets. Learning Objectives Sets may or may not be related to each other. We begin with a simple example. We consider a class C consisting of (a) the girl-students: Maya and Priti (b) the boy-students: Arun and Ravi (c) the teachers: Ravi and Arun We denote, The set of girl-students by G, The set of boy-students by B, The set of girl-students and the teacher by X. The set of boy-students and the teacher by Y. and The set of the class teacher only by T. Here, the universal set U is the set consisting of (a) the girl-students, G = {Maya, Priti } or {m, p} (b) the boy-students, B = {Arun, Ravi} or {a, r} (c) the teacher. T = {Ravi, Arun} or {r, a}. We easily note that: (a) How many members are there in each set? (b) What is the relation between the sets G and B or T, and B and T ?

20 The Leading Mathematics - 7 The sets G, B or T have equal number of members and the sets B and T have the same members. Here, the members of the sets G and B are one to one correspondence, but not equal. The sets B and T are the same sets. Here, we discuss the following types of sets. Equivalent and Equal Sets In the Venn Diagram of the sets G, B and T, we notice that (a) the sets G and B or T have equal number of members. This type of sets are called equivalent sets. (b) the sets B and T has the same members. This type of sets are called equal sets. The sets having equal number of members or elements are known as equivalent sets. It is denoted by ‘~’. If A and B are two equivalent sets, they are denoted by A ~ B. Furthermore, if A, B, C and D are equivalents sets, they are denoted by A ~ B ~ C ~ D. The sets having the same members or elements are known as equal sets. It is denoted by ‘=’. If A and B are two equivalent sets, they are denoted by A = B. Furthermore, if A, B, C and D are equivalents sets, they are denoted by A = B = C = D. Subsets We further note that: (a) the sets G = { m, p} and X = {m, p, b} are overlapping with m and p as common elements, and (b) every element of G is also an element of X. Thus, we see that G is wholly within X or G is contained in X. We also say that G is a subset of X. In symbol, we write G ⊆ X. In the same way, (a) the sets B = {a, r, s} and Y = { a, r, s, b} are overlapping with a, r and s as common elements and (b) every element of B is also an element of Y. G T B m p a r U G X m p b U G B > > a r m p G B > > a r m p

Sets 21 Thus, we see that B is wholly within Y or B is contained in Y. We also say that B is a subset of Y. In symbols, we write B ⊆ Y. But, the two sets X = {m, p, b} and Y = {a, r, s, b} have one common element ‘b’; and neither X is contained in Y nor Y is contained in X. A set A is said to be a subset of another set B if every element of A is also an element of B. In symbol, we write it as A ⊆ B. Sometimes, we call B a superset of A; B ⊇ A. In particular, { 1, 2, 3, 4, 5} ⊇ {1, 2, 3} Proper Subsets A subset that contains at least one element less than that of the super set is called a proper subset. If A is the proper subset of B, it is denoted by A ⊂ B. For example, If A = {1, 2, 3}, then the subsets{ } {1}, {2}, {3}, {1, 2}, {2, 3} and {1, 3} are proper subsets of A. Improper Subsets A subset that contains all the elements of the set itself is called an improper subset. If A is the improper subset of B, it is denoted A ⊂ B. For example, If A = {1, 2, 3} then the subset B = {1, 2, 3} is the improper subset of A. CLASSWORK EXAMPLES Example 1 State whether the following are equivalent or equal. (a) A = {123} and B = {321} (b) M = {1, 2, 3} and N = {3, 2, 1} (c) F = {father, son} and M = {mother, son} Solution: (a) A and B are equivalent sets. (b) M and N are equal sets. (c) F and M are equivalent sets.

22 The Leading Mathematics - 7 Example 2 Is A a subset of B? (a) The set A = {t, o, p} and the set B = {s, t, o, p} (b) The set A = {123} and the set B = {321} (c) The set A = {r, e, a, p} and B = {p, e, a, r} Solution: (a) Yes (b) No (c) Yes Example 3 Find the proper subsets of the set A = {a, b}. Solution: The proper subsets of the set A = {a, b} are {}, {a} and {b}. Example 4 Write the relation of each of the following pairs of sets symbolically: (a) M = {m, a, n} and W = {w, o, m, a, n} (b) S = {1, 2, 3} and T = {1, 2} (c) A = {a, e, i, o, u} and B = {i, o, u, e, a} Solution: (a) M ⊂ W (b) T ⊂ S (c) A ⊂ B and B ⊂ A or A ⊂ B EXERCISE 1.3 Your mastery depends on practice. Practice like you play. 1. State whether each of the following is true or false: (a) Two sets are equivalent if they have the same elements. (b) Two sets are equal if they have equal number of elements. (c) Equivalent sets have equal number of members. (d) Equal sets have different elements. (e) If A is the subset of B, then B is the super set of A. (f) If the set P whose elements are contained in another set Q, P is the subset of Q. 2. Fill in the blanks: (a) If every element of a set is an element of another set, the latter set is called a ............. of the former. (b) If every element of a set is an element of another set, the former set is called a ............. of the latter.

Sets 23 3. Tick mark (or write down) the correct answer: (a) If two sets have the same elements, they are said to be i. equivalent ii. equal iii. different iv. subset (b) If two sets have equal number of elements, they are said to be i. subset ii. equal iii. equivalent iv. proper subset 4. State whether the following are equivalent or equal: (a) A = {12} and B = {21} (b) B = {but} and T = {tub} (c) A = {2, 3} and B = {3, 4} (d) U = {b, u, t} and T = {t, u, b} (e) A = {x : x is a counting number less than 3.} and B = {x : x is one of the first two counting numbers.} (f) Two sides of an equilateral triangle. 5. Write down the common element or elements, if any of the following sets are: (a) The singleton set {1} and the set {12, 21} (b) The set of counting numbers less 10 and the set of counting numbers greater than 5. (c) The set R = {e, a, t} and P = {t, e, a} (d) A = {x : x is an odd number} and B = {x : x is an even number}. 6. Is A a subset of B? (a) The set A = {t, e, a} and the set B = {t, e, a, m} (b) The set A = {12} and the set B = {123} (c) The set A = {l, e, a, p} and B = {p, l, e, a} B U A a r t c a r t

24 The Leading Mathematics - 7 7. Write the relation of each of the following pairs of sets symbolically. (a) M = {a, r, t} and W = {c, a, r, t} (b) E = {2, 4, 6, 8} and D = {1, 2, 3, 4, 5, 6, 7, 8, 9}} (c) A = {a, b, c, …, x, y, z} and B = {i, o, u, e, a} 8. (a) Given the definition of a superset of a set. (b) Distinguish between comparable and non-comparable sets. (c) Define a subset of a set. (d) Distinguish between disjoint and intersecting sets. 9. Distinguish weather the given pairs of sets are equal or equivalent sets. (a) P = {2, 3, 5, 7} and Q = {5, 3, 2, 7} (b) A = {x : x is odd number less than 10} and B = {x : x is even number less than 12} (c) N = {x : x < 5, x ∈ N} and M = {x : – 2 < x ≤ 2, x ∈ Z} (d) X = {x : x is letters in word ‘hiram’} and Y = {x : x is letters in word ‘mahir’} ANSWERS Consult with your teacher. Project Work 1.3 List the different things around you and your school and home. Construct the set on the basis of their common properties. Separate their types and present it in your classroom.

Sets 25 Read, Understand, Think and Do 1. Sets P = {prime numbers less than 10}, Q = {2,3,5,7} and R = {natural number between 5 and 6} are given. (a) Define empty set. (b) Are set P and set Q equal sets? Justify (c) Is the set R the proper subset of set U? Write with reason. 2. Two sets A and B are given below. A = {prime number less than 10} B = {x:x is odd number less than 10} (a) How many numbers of elements are in set A.? (b) Write any one common proper subset of sets A and B. (c) How many subsets can be made from set B? 3. Two sets A and B are given below. A = {even number less than 10} B = {x:x is odd number less than 10} (a) Define proper subset with example. (b) Sets A and B are equivalent sets but not equal sets. Justify with reason. 4. Two sets A and B are given below. A is the set of letters in “ALLOY” B is the set of letters in “LOYAL” (a) Write an example of singleton set. (b) Justify that two sets A and B are equal and equivalent sets. 5. Sets A, B and C are given below. A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. (a) Define finite and infinite sets with example. (b) Write the universal set for all three sets. (c) Can we write B ⊂ C? Give reason. MIXED PRACTICE–I ANSWERS Consult with your teacher.

26 The Leading Mathematics - 7 FM : 15 Time : 40 Min. CONFIDENCE LEVEL TEST I SETS 1. Write the numbers from 1 to 20. The sets P, Q and R are defined as follows; P = {Prime numbers less than 2}, Q = {Composite numbers less than 14} and R = {Factors of 20} (a) Write the above sets in listing method. [1] (b) Which of the above sets is empty ? [1] (c) State with reason whether the sets Q and R are equivalent or equal sets. [1] 2. A set U and its subsets A and B are defined below. U = {letters of the word ‘education’}, A = {vowel letters of the word ‘mahendra’} and B = {vowel letters of the word ‘birendra’} (a) Identify which set is subset of other among the sets A and B. [1] (b) Show the above information in a Venn diagram. [2] 3. The sets L, M and N are defined below. L = {1, 3, 5, 7, ......}, M = {1, 2, 4, 8} and N = {2, 4, 8}. (a) Identify which of the given sets are finite and infinite. [1] (b) How many subsets are formed by the set N? Write its all subsets. [ 2] 4. If a set A = {numbers less than 15}, answer the following questions. (a) Write the elements of the set A in the listing method. [1] (b) If B = {single digit prime numbers}, is A ⊂ B? [1] (c) How many subsets can be formed from the set B? [1] 5. When the sets A and B are defined as follows; A = { x:x is a composite number up to 14} and B = {x:x is an even number less than 12} (a) List the elements of the sets A and B in the listing method. [1] (b) Show the elements of the sets A and B in the same Venn diagram. [2] Attempt all the questions.

Arithmetic 27 COMPETENCY  Solving the behaviour problems related to number system. CHAPTERS 2. Whole Numbers 3. Integers 4. Rational Numbers 5. Fraction and Decimal 6. Ratio and Proportion 7. Profit and Loss 8. Unitary Method LEARNING OUTCOMES  find the square and square root by prime factorization and division methods.  find the cube and cube root by prime factorization method.  find HCF and LCM up to three numbers.  simplify integers by using four simple rules.  identify the rational and irrational numbers.  convert the decimal number (only repeating) of terminating and non-terminating into fraction.  simplify the fractions involving four basic operations.  simplify the decimals involving four basic operations.  give the introduction to ratio and proportion.  solve the problems related to profit and loss including percentage.  solve the problems including two variables by using direct and indirect variations. ARITHMETIC UNIT II

28 The Leading Mathematics - 7 CHAPTER 2 Real Numbers Lesson Topics Pages 2.1 Square and Square Root of Numbers 29 2.2 Cube and Cubic Root of Numbers 39 2.3 Highest Common Factor (HCF) 42 2.4 Least Common Multiple (LCM) 46 1. How many total students are there in the morning prayer of a school in the above picture? 2. How many lines of students are there in the morning prayer? 3. How many students are there in each line? 4. How many small yellow rooms are there in the length and height of the above Rubik ? 5. How many total coloured rooms are there in total in the given Rubik? 6. The given two bells ring in different time of intervals. Do the both bells chance to ring at the same time ? WARM-UP

Arithmetic 29 2.1 Square and Square Root of Numbers At the end of this topic, the students will be able to: ¾ find the square and square root of numbers. Learning Objectives Counting Numbers and System of Numeration The history of numbers is interwoven with the history of humankind. In the prehistoric era, man did have no idea about numbers. Even then they had the idea of ‘ more than ‘ or ‘ less than’. The stronger used to lead the weaker. Man created numbers to describe the two phenomena. That is, man created number for counting and ordering purposes. Fractions and non-fractional numbers were invented some thousand years ago only. Putting things together or separating or taking away something gave rise to the rules for the addition and subtraction of numbers. Multiplication and division of numbers followed them. Addition, subtraction, multiplication and division of numbers are known as the four simple rules of arithmetic. They form the foundation of what is known as Arithmetic. Man is said to have created numbers for counting. Counting numbers have different names (number names) in different languages. Of many counting systems or systems of numerations, only a few are most widely used. Below we have listed some of them: (a) Roman numerals: I, V, X, C, … (b) Arabic numerals: ١, ۲, ۳, ٤,۵ , ٦, ٧,٨, ٩ (c) Devanagari numerals: !, @, #, $, %, ^, &, *, ( (d) Hindu-Arabic numerals: 1, 2, 3, 4, 5, 6, 7, 8, 9. The number zero or sifr or sunya (zero), was not known in ancient times. It was created around 650 AD by the Hindus. The symbol ' 0 ' is used to denote it. Roman numerals in Ishango bone

30 The Leading Mathematics - 7 The most well known and widely used system of numeration is the Hindu-Arabic system of numeration. It uses the following ten different symbols. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 In the long history of civilization, man simply used the counting numbers: 1, 2, 3, 4, … . Everything was done without the number zero, denoted by 0. It was nearly 1450 years ago, man made this unique creation. The counting numbers together with the number zero form the set of whole numbers. The unending chain of consecutive positive numbers: W : 0, 1, 2, ..., 10, 11, 12, ..., 100, 101, 102, ... form the set of whole numbers. Square Numbers Numbers occur and recur in various forms and orders. Let us revisit the following geometrical patterns and the corresponding numbers. 1 = 12 1 + 3 = 4 = 22 3 + 6 = 9 = 32 6 + 10 = 16 = 42 10 + 15 = 25 = 52 1 Squared =1 2 Squared = 4 3 Squared = 9 4 Squared = 16 5 Squared = 25 Multiplication of a number by itself gives a number. We have seen that some whole numbers are the products of two identical numbers or factors. For instances, 1 = 1 × 1, 4 = 2 × 2, 9 = 3 × 3, 16 = 4 × 4, 25 = 5 × 5 and so on. Numbers such as 1, 4, 9, 16, 25, … are called square whole numbers or perfect square whole numbers. 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25

Arithmetic 31 Square Root of Numbers In case a number has two equal factors, each factor is called a square root of the number. Finding the square root is called extraction of the square root. In particular, 9 has two equal factors 3 and 3. Here 3 is a square root of 9. In general, if a2 = a × a, a is called the square root of a2 . In symbols, we write a = a2 , (read root over a2 ) To find one of the two equal factors of a perfect square number is to extract the square root. Squares and square roots can be pictorially visualized as shown below: Square roots Square 1 2 3 4 5 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42 1 + 3 + 5 + 7 + 9 = 52 It is easy to note that the squares of the counting numbers are again counting numbers; but the square root of counting number is not necessarily a counting number. The following pattern exhibits some interesting properties of squares of some numbers. Number Square Number Notation Square root Read it as 1 1 12 1= 1 Square root of 1 2 4 22 2 = ? Square root of 2 3 9 32 3 = ? Square root of 3 4 16 42 4 = 2 Square root of 4 ... ... ... ... ... 11 121 112 11 = ? Square root of 11 111 12321 1112 111 = ? Square root of 111

32 The Leading Mathematics - 7 We list below some interesting properties related to perfect square counting numbers. i) A perfect square number always ends in 0 or 1 or 4 or 5 or 6 or 9. ii) If a perfect square number ends in 0, the number of zeros at the end is always even. iii) Squares of even numbers are even and those of odd numbers are odd. To find the square root of a perfect square counting number, we may write the number as sum of the odd numbers 1, 3, 5, 7, 9, …. Then, count the number of such numbers. The number of odd numbers gives the required square root. For instance, to find the square root of 49, we first write it as the following sum of the first 7 odd counting numbers: 1 + 3 + 5 + 7 + 9 +11 + 13 = 49. Hence, the square root of 49 is 7. Methods of Finding Square Root of Numbers Now, we briefly discuss how square roots are computed. (a) By Product of Factorization To write the given number as i) the product of pairs of equal prime factors, ii) take one factor from each pair and iii) find the product of such factors. For example, the square root of 324 is reckoned as shown below: 324 = (2 × 2) × (3 × 3) × (3 × 3) = 22 × 32 × 32 We have, 324 = 22 × 32 × 32 ∴ 324 = 2 × 3 × 3 = 18 "Alternatively" ∴ 324 = (2 × 3 × 3 ) × (2 × 3 × 3) = 18 × 18 = 182 = 18. 2 324 2 162 3 81 3 27 3 9 3

Arithmetic 33 (b) By Method of Long Division Consider the following facts, 12 = 1 A one or two-digit number has its square root in 1 digit. 92 = 81 102 = 100 A three or four-digit number has its square root in 2 digits. 992 = 9810 1002 = 10000 A five or six-digit number has its square root in 3 digits. 9992 = 998001 10002 = 1000000 A seven or eight-digit number has its square root in 4 digits. 99992 = 49980001 This helps us to estimate the number of digits in the square root of a number. For example: 15129 = 123 and 617796 = 786 Here both the number, 15129 and 617796 have their square root in 3 digits. We can approximate/estimate number of digits in the square not by pairing up numbers from left to right in the given number. 151 29 = 3 digits in square root (2 + 1) 61 77 96 = 3 digits in square root. Rules for Long Division Method The usual method of finding the square root of a perfect square number is to follow the long division method. Generally, this method is used for large numbers. The algorithm may be summarized as follows: Consider for example the square of 15. 152 = 225 or, (10 + 5)2 = 102 + (2 × 10 + 5)5

34 The Leading Mathematics - 7 Let’s express this schematically as 10 + 5 15 10 + 10 225 100 Directly as → 1 + 1 2 25 1 ↓ (20 + 5) +5 125 125 25 +5 125 125 30 × 30 × ∴ 225 = 15 Let’s explain the second one. To find the square root of 225. First estimate the number of digits in the square root by pairing it from left to right as 225. It has 2 digits in the square root. Find two identical factors whose product is smaller or equal to 2; with a smallest difference. Subtract and bring down the number 25 in pair you have 125. Find two identical factors whose product is equal or smaller than 125 with the smallest difference. If the reminder is zero, you have the square root 15 on the top. CLASSWORK EXAMPLES Example 1 Find the square root of 218089. Solution: 467 4 + 4 21 80 89 16 [It has three digits in the square root.] 86 + 6 580 – 516 927 + 7 6489 – 6489 934 × ∴ 218089 = 467

Arithmetic 35 Example 2 What smallest number must be subtracted so that 18236 will be a perfect square number? Solution: Try finding the square root. 135 1 + 1 1 82 36 – 1 [It will have three digits in the square root if it exists.] 23 + 3 82 69 265 + 5 1336 1325 270 11 To be a perfect square, there must not be the reminder. Therefore; the smallest number to be subtracted is 11; so that 18236 – 11 = 18225 will be a perfect square of 135. i.e. (135)2 = 18225 Example 3 Find the smallest number which when multiplies 675 so that the resulting number is a square number. Solution: A square number can be expressed as the product of two identical factors. Therefore, we have to find the smallest number which multiplies 675. So that it will have two identical factors. By factorization: 675 = 3 × 3 × 3 × 5 × 5 = 32 × 52 × 3× .... The number 3 is not a squared number. So, we multiply 675 by 3 so that it can be expressed as the product of two identical factors. Hence, the required number is 3. EXERCISE 2.1 Your mastery depends on practice. Practice like you play. 1. Identify which of the following numbers are square: (a) 1 (b) 3 (c) 4 (d) 16 (e) 18 (f) 25 (g) 32 (h) 36 (i) 42 (j) 49 (k) 55 (l) 64 3 675 3 225 3 75 5 25 5

36 The Leading Mathematics - 7 (m) 78 (n) 80 (o) 81 (p) 88 (q) 94 (r) 94 (s) 99 (t) 100 2. Find the square of the following numbers: (a) 1 (b) 2 (c) 14 (d) 18 (e) 25 (f) 36 (g) 42 (h) 55 (i) 76 (j) 89 (k) 92 (l) 99 (m) 100 (n) 179 (o) 214 (p) 325 3. Find the square root of the following numbers by long division method: (a) 1 (b) 4 (c) 9 (d) 16 (e) 25 (f) 36 (g) 49 (h) 64 (i) 81 (j) 100 (k) 121 (l) 144 (m) 169 (n) 196 (o) 225 (p) 256 4. Find the square roots of the following numbers by factorization: (a) 1 (b) 4 (c) 9 (d) 16 (e) 64 (f) 36 (g) 81 (h) 121 (i) 169 (j) 196 (k) 625 (l) 2025 (m) 400 (n) 961 (o) 1156 (p) 3136 5. Find the smallest number which multiplies the following numbers so that the number so obtained is a perfect square. (a) 128 (b) 72 (c) 243 (d) 605 (e) 38 (f) 588 (g) 4375 (h) 2156 6. Find the smallest number which divides the following numbers so that the number so obtained is a perfect square. (a) 80 (b) 183 (c) 72 (d) 363 (e) 338 (f) 980 (g) 1250 (h) 6075 7. First guess the number of digits in the square root and find the square roots of the numbers given below by division method. (a) 121 (b) 144 (c) 7056 (d) 4489 (e) 15129 (f) 103684 (g) 53361 (h) 133225 (i) 1042441 (j) 1745041 (k) 1550025 (l) 42562576

Arithmetic 37 8. What smallest number could be subtracted from each number so that the difference is a perfect square number? (a) 21046 (b) 427729 (c) 931250 (d) 616265 (e) 968300 (f) 471984 (g) 996058 (h) 99899 9. Find the square roots of the following numbers: (a) 25 36 (b) 81 100 (c) 225 169 (d) 289 529 (e) 576 841 (f) 961 3600 (g) 5625 7225 (h) 15129 22500 10. (a) The classroom whiteboard is square. If its side is 2 m long, find its area. (b) A garden is shaped like a square with a side length of 25 m. Find the garden area. (c) In a morning assembly, students are arranged in a square. If there are 56 students in one row, how many students are there altogether? Find it. (d) In a march past the commander arranged his soldiers in a square. If there are 101 soldiers in a row, how many soldiers are there in total? Find it. 11. (a) 625 cauliflowers are planted in a nursery field as a square. How many cauliflowers are planted on one side? (b) 17424 students are assembled as a square. How many students are there in each row? (c) A square garden measures 41616 square meters. What is the length of the garden side? (d) 636804 tiles of square shape are paved in the form of a square courtyard. How many tiles are there in each side?

38 The Leading Mathematics - 7 ANSWERS 1. (a), (c), (d), (f), (h), (j), (l), (o), (t) are square numbers. 2 (a) 1 (b) 4 (c) 196 (d) 324 (e) 625 (f) 1296 (g) 1764 (h) 3025 (i) 5776 (j) 7921 (k) 8464 (l) 9801 (m) 10000 (n) 32041 (o) 45796 (p) 105625 3. (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 10 (k) 11 (l) 12 (m) 13 (n) 14 (o) 15 (p) 16 4. (a) 1 (b) 2 (c) 3 (d) 4 (e) 8 (f) 6 (g) 9 (h) 11 (i) 13 (j) 14 (k) 25 (l) 45 (m) 20 (n) 31 (o) 34 (p) 56 5. (a) 2 (b) 2 (c) 3 (d) 5 (e) 38 (f) 3 (g) 7 (h) 11 6. (a) 5 (b) 183 (c) 2 (d) 3 (e) 2 (f) 5 (g) 2 (h) 3 7. (a) 11 (b) 12 (c) 84 (d) 67 (e) 123 (f) 322 (g) 231 (h) 365 (i) 1021 (j) 1321 (k) 1245 (l) 6524 8. (a) 21 (b) 13 (c) 25 (d) 40 (e) 44 (f) 15 (g) 54 (h) 43 9. (a) 5 6 (b) 9 10 (c) 15 13 (d) 17 23 (e) 24 29 (f) 31 60 (g) 75 85 (h) 123 150 10. (a) 4 m2 (b) 625 m2 (c) 3136 students (d) 10201 soldiers 11. (a) 25 cauliflowers (b) 132 students (c) 204 m (d) 798 tiles Project Work 2.1 What is your birth date ? Convert your age into days. Test the numbers of days as square number or not. If it is not square number, who many days should be added in it to make square number? Find. Present in your classroom.

Arithmetic 39 2.2 Cube and Cubic Root of Numbers At the end of this topic, the students will be able to: ¾ find the cube and cubic root of numbers. Learning Objectives Study the following table: Number Cube Numbers Notation Read it as 1 1 13 One cubed 2 8 23 Two cubed 3 27 33 Three cubed 4 64 43 Four cubed 5 125 53 Five cubed and so on From the table: 27 = 33 i.e., 27 = 3 × 3 × 3 27 is the cube of 3 and 3 is the cube root of 27. We write as 33 = 27 and 27 3 = 3 There are situations in daily life where we have to multiply three identical factors i.e., we have to find the cube of a number. There are also the situations where; we have to find the cube root of a number. Consider for example; Hence, we have the definition, The cube root of a number ‘a’ is one of its three identical factors a 3 so that; a 3 × a 3 × a 3 = a. Finding Cubic Roots by Prime Factorization Method The cube root of a perfect cube number is determined by grouping the number into triplets by the prime factorizing method. One of the triplets is the cube root of the number. For example; What is the cube root of 512 ? For this, at first to find the prime factors of 512 as shown in the alongside; 3 3 3

40 The Leading Mathematics - 7 Then the cubic root of 512 is, 512 3 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 3 = 23 × 23 × 23 3 = 8 × 8 × 8 3 = 8 CLASSWORK EXAMPLES Example 1 Find the volume of the cube given in the figure. Solution: Here, Length of the side of the cube (a) = 4 cm Volume of the cube (V) = a3 V = (4 cm)3 = 4 cm × 4 cm × 4 cm = 64 cm3 ∴ The volume of the cube is 64 cubic centimeter. Example 2 A cube has volume 216 cm3 . Find the length of its side. Solution: Here, Volume (V) = 216 cm3 or, a3 = 216 cm 3 or, a3 = (6 cm)3 ∴ a = 6 cm Hence length of the side (a) = 6 cm Here we have worked out the cube root of 216. Here, 216 3 = 6 × 6 × 6 3 = 63 3 = 6 EXERCISE 2.2 Your mastery depends on practice. Practice like you play. 1. Find the cubic numbers of the following numbers: (a) 2 (b) 8 (c) 16 (d) 36 (e) 64 (f) 76 (g) 81 (h) 121 2. Find the cubic roots of the following numbers: (a) 1 (b) 8 (c) 27 (d) 64 (e) 125 (f) 216 (g) 343 (h) 512 (i) 729 (j) 1000 (k) 1331 (l) 1728 (m) 2197 (n) 2744 (o) 3375 (p) 4096 (q) 8000 (r) 13824 3. Find the smallest number which multiplies the following numbers so that the number so obtained is a perfect cube. (a) 4 (b) 25 (c) 72 (d) 169 (e) 256 (f) 484 (g) 968 (h) 8788 4 cm 4 cm 4 cm V = 216 cm3 6 216 6 36 6

Arithmetic 41 4. Find the smallest number which divides the following numbers so that the number so obtained is a perfect cube. (a) 16 (b) 250 (c) 5488 (d) 16875 (e) 16384 (f) 17496 (g) 82944 (h) 93312 5. Find the cubic roots of the following numbers: (a) 125 216 (b) 1728 2197 (c) 195112 32768 (d) 21952 46656 6. Find the volume of the following cubes: (a) 3 cm (b) 9 cm (c) 7. (a) The volume of a cubical box is 3.375 m3 . What is the length of the box? (b) If 13824 cricket balls of the diameter 7 cm each are contained in the cubical container tightly, find the length of the container. 8. Find the cubic number of the natural numbers from 1 to 100 and test the following; (a) Is the cubic number of odd number also odd number? (b) Is the cubic number of even number also even number? ANSWERS 1. (a) 8 (b) 512 (c) 4096 (d) 46656 (e) 262144 (f) 438976 (g) 531441 (h) 1771561 2. (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 10 (k) 11 (l) 12 (m) 13 (n) 14 (o) 15 (p) 16 (q) 20 (r) 24 3. (a) 2 (b) 5 (c) 3 (d) 13 (e) 16 (f) 22 (g) 11 (h) 2 4. (a) 2 (b) 2 (c) 2 (d) 5 (e) 4 (f) 3 (g) 6 (h) 2 5. (a) 5 6 (b) 12 13 (c) 29 16 (d) 7 9 6. (a) 27 cm3 (b) 729 cm3 (c) 1061.208 cm3 7. (a) 1.5 m (b) 168 cm Project Work 2.2 Find any five examples in our daily life related to cube and cubic numbers. Prepare a report and present it in your classroom. 102 cm

42 The Leading Mathematics - 7 2.3 Highest Common Factor (HCF) At the end of this topic, the students will be able to: ¾ find the HCF of the given numbers. Learning Objectives We have seen that a composite number can be factorized. In other words, we can resolve a composite number into factors and prime factors. Highest Common Factor (HCF) Suppose we are given two composite numbers. We resolve them into prime factors. Then two cases may arise as shown below: (i) Greatest factor among common factors What are the factors of the two numbers 12 and 18 ? The factors of 12 are 1, 2, 3, 4, 6, 12. i.e., F(12) = {1, 2, 3, 4, 6, 12} The factors of 18 are 1, 2, 3, 6, 9, 18. i.e., F(18) = {1, 2, 3, 6, 9, 12} Of the three common factors 2, 3 and 6, the highest common factor or greatest common divisor is the number 6. In short, such a factor is denoted by HCF or GCD. This method of finding HCF is called definition method. (ii) Common factors among prime factors What are the prime factors of the two numbers 12 and 18 ? The prime factors of 12 = 2 × 2 × 3 The prime factors of 18 = 2 × 3 × 3 The prime numbers 2 and 3 are common factors. Step-1 Find all the factors of the given numbers. Step-2 Select the common factors from all the factors of each given number. Step-3 Write the greatest common factor from all the factors of the given numbers. Step-1 Find the prime factors of the given numbers. Step-2 Select the common factors from the factors of each given number. Step-3 Write the product of the common factors, which HCF.

Arithmetic 43 Besides, 2 × 3 = 6 is also a common factor of the two numbers 12 and 18. This method of finding HCF is called prime factorization method. The highest common factor or HCF is thus the product of all factors common to the given numbers. Finding HCF Division Method Among two composite numbers 12 and 18, which is smaller? Now, 18 is divided by 12. 12) 18 (1 – 12 6 ) 12 (2 – 12 0 Step-1 Divide the greater number by smaller number. Step-2 Divide the smaller by the remainder. Step-3 Continue the process in the step 2 till the remainder is zero. Step-4 The least divisor is the HCF of the given numbers. CLASSWORK EXAMPLES Example 1 Find the HCF of 18, 42 and 54 Solution: At first, dividing 42 by 18, we get 18 ) 42 ( 2 – 36 6 ) 18 (3 – 18 0 In short, 18 = 2 × 3 × 3 42 = 2 × 3 × 7 54 = 2 × 3 × 3 × 3 ∴ HCF = 2 × 3 = 6 Here, HCF of 18 and 42 is 6. Again, find HCF of 6 and 54. Dividing 54 by 6, we get 6 ) 54 ( 9 –54 0 Here, HCF of 6 and 54 is 6. Hence, HCF of 18, 42 and 54 is 6. Example 2 Find the greatest number of students to whom 36 copies, 54 pens and 174 pencils can be divided equally. Also, find the share of each item among them.

44 The Leading Mathematics - 7 Solution: The greatest number of students is HCF of 36, 54 and 174. Now, dividing 54 by 36, we get 36 ) 54 ( 1 – 36 18 ) 36 ( 1 – 36 0 Again, dividing 174 by 18, we get 18 ) 174 ( 9 – 162 12 ) 18 ( 1 – 12 6 ) 12 ( 2 – 12 0 ∴ The greatest number of students is 6. To find the share of each item, 36 ÷ 6 = 6, 54 ÷ 6 = 9 and 174 ÷ 6 = 29 Each student shares 6 copies, 9 pens ad 29 pencils. EXERCISE 2.3 Your mastery depends on practice. Practice like you play. 1. Find the highest common factor of the following numbers by definition method: (a) 8 and 12 (b) 8 and 27 (c) 10 and 25 (d) 27 and 36 (e) 18 and 45 (f) 60 and 75 (g) 72 and 90 (h) 52 and 104 (i) 54 and 126 (j) 125, 150 and 275 (k) 135, 225 and 315 (l) 192, 216 and 240 2. Find the HCF of the following numbers by factorization method: (a) 9 and 21 (b) 12 and 42 (c) 18 and 60 (d) 28 and 70 (e) 30 and 80 (f) 40 and 130 (g) 84 and 108 (h) 126 and 154 (i) 144 and 208 (j) 126, 154 and 216 (k) 216, 168 and 384 (l) 168, 264 and 700 3. Find the GCD of the following numbers by division method: (a) 6 and 9 (b) 9 and 12 (c) 15 and 42 (d) 16 and 48 (e) 24 and 54 (f) 36 and 84 (g) 54, 72 and 99 (h) 96, 132 and 180 (i) 112, 126 and 224 (j) 128, 144 and 240 (k) 160, 260 and 380 (l) 216, 384 and 600

Arithmetic 45 4. Find the greatest number, which exactly divides: (a) 8 and 18 (b) 24 and 132 (c) 42 and 154 (d) 108, 156 and 240 (e) 175, 225 and 350 (f) 238, 306 and 374 5. (a) There are160 apples and 224 mangoes in a bag. i. What is the greatest number of children to distribute these apples and mangoes equally? ii. How many fruits of each kind will each child get? (b) There are 252 copies, 378 pens and 462 pencils in a shop. i. What is the greatest number of students to distribute stationery goods equally? ii. How many of each kind will each student get? 6. Observe the given situations and answer the following questions. (a) There are 392 chocolates in a box and 728 chocolates in another box. What is the highest number of chocolates that can be picked out at a time from each box so that the baskets will be empty at the same time? (b) A vessel has 36 litres of petrol and another has 19.200 litres of petrol. What is the highest capacity in litres of the container that can empty each vessel? 7. Study the given situations and answer the following questions. (a) A rectangular ground is 18 metres long and 12 metres wide. What is the length of the biggest squared marble needed to pave it with the square marbles of the same size? (b) A rectangular court-yard is 12.5 metres long and 7.5 metres wide. What is the length of the biggest squared slate of the same size needed to pave it? ANSWERS 1. (a) 4 (b) 1 (c) 5 (d) 9 (e) 9 (f) 15 (g) 18 (h) 52 (i) 18 (j) 25 (k) 45 (l) 24 2. (a) 3 (b) 6 (c) 6 (d) 14 (e) 10 (f) 10 (g) 12 (h) 14 (i) 16 (j) 2 (k) 24 (l) 4 3. (a) 3 (b) 3 (c) 3 (d) 16 (e) 6 (f) 6 (g) 9 (h) 12 (i) 6 (j) 16 (k) 20 (l) 24 4. (a) 2 (b) 12 (c) 22 (d) 4 (e) 25 (f) 2 5. (a) 32, 5, 7 (b) 42, 6, 9, 11 6. (a) 7, 13 (b) 3, 1.6 7. (a) 6 m (b) 2.5 m

46 The Leading Mathematics - 7 2.4 Least Common Multiple (LCM) At the end of this topic, the students will be able to: ¾ find the LCM of the given numbers. Learning Objectives We have seen numbers can be built by multiplying prime factors. For example, (a) On multiplying 3 by 2, we get 3 × 2 = 6, i.e. 6 is a multiple of 3 and on multiplying 5 by 2, we get 5 × 2 = 10, i.e. 10 is a multiple of 5. Suppose we have two numbers 6 and 10. We say that, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, ... 90, ... are multiples of 6; and 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ... are multiples of 10. Comparing the two sets (sequences) of multiples, we note that the numbers, 30, 60, 90, ... are common multiples of the numbers 6 and 10. Of all such common multiples, there is the least one. It is 30 in our case. Such a number is called the least common multiple (LCM). This method of finding LCM is called definition method. The least common multiple of given composite numbers is the least of all common multiples of the numbers. It is therefore a multiple of all given numbers. It is therefore a multiple of the highest common factor and the non–common factors. For example, (b) The numbers 6 and 10, i.e., 3 × 2 = 6, and 5 × 2 = 10 have the H.C.F. = 2 and their non–common factors are 3 and 5. The product of the HCF and the non–common factors (HCF) × (Non-common Factors) = 2 × 3 × 5 = 30. This is clearly a multiple of 6 and 10. It is actually the least common multiple or LCM of 6 and 10. This method of finding LCM is called prime factorization method.

Arithmetic 47 Finding LCM by Division Method In this method, we divide the given number by the common prime numbers of them till dividing all of them by the common prime. For example, 2 6, 10 3, 5 Step-1 Arrange all the given numbers separate by comma. Step-2 Divide the given numbers by common prime. Step-3 Continue the process in the step 2 till dividing at least two of them by the common prime. Step-4 Multiply all the divisors and the product is LCM. LCM = 2 × 3 × 5 = 30, where HCF = 2. It is clear that, 6 × 10 = 2 × 30 i.e., First Number × Second Number = HCF × LCM or, Product of Two Numbers = HCF × LCM This is the relation between HCF, LCM and the given two numbers. It is only satisfied when two numbers are given. CLASSWORK EXAMPLES Example 1 Find the LCM of 18, 42 and 54 Solution: Here, the given numbers are 18, 42 and 54. 2 18, 42, 54 3 9, 21, 27 3 3, 7, 9 1, 7, 3 ∴ LCM = 2 × 3 × 3 × 1 ×7 × 3 = 378

48 The Leading Mathematics - 7 “Alternatively” 18 = 2 × 3 × 3 42 = 2 × 3 × 7 54 = 2 × 3 × 3 × 3 LCM = Common factor × Remaining factors = 2 × 3 × 3 × 7 × 3 = 378 Example 2 LCM and HCF of two numbers are 168 and 14 respectively. If one of the numbers is 56, find the next number. Solution: Here, LCM = 168, HCF = 14, One Number = 56, Second Number = ? Now, we know that, First Number × Second Number = HCF × LCM or, 56 × Second Number = 14 × 168 or, Second Number = 14 × 112 56 = 42. Example 3 Three bells ring at the intervals of 12 minutes, 16 minutes and 24 minutes respectively. If they begin together, after what interval of time will they again ring together? Solution: Finding the LCM of 12, 16 and 24 2 12, 16, 24 2 6, 8, 12 2 3, 4, 6 3 3, 2, 3 1, 2, 1 ∴ LCM = 2 × 2 × 2 ×2 × 3 × 1 × 2 × 1 = 96 Thus, all the bells will ring again after 96 minutes i.e., 1 hour 36 minutes. EXERCISE 2.4 Your mastery depends on practice. Practice like you play. 1. Find the lowest common multiple of the following numbers by definition method: (a) 6 and 9 (b) 8 and 18 (c) 10 and 35 (d) 24 and 36 (e) 36 and 45 (f) 60 and 75 (g) 72 and 90 (h) 26 and 156 (i) 108 and 126 (j) 75, 150 and 375 (k) 135, 225 and 315 (l) 192, 216 and 240

Arithmetic 49 2. Find the LCM of the following numbers by factorization method: (a) 15 and 21 (b) 16 and 42 (c) 18 and 60 (d) 28 and 70 (e) 54 and 90 (f) 40 and 130 (g) 42 and 108 (h) 126 and 154 (i) 144 and 208 (j) 126, 154 and 216 (k) 108, 168 and 384 (l) 84, 132 and 350 3. Find the LCM of the following numbers by division method: (a) 6 and 9 (b) 9 and 12 (c) 15 and 42 (d) 16 and 48 (e) 24 and 54 (f) 36 and 84 (g) 54, 72 and 99 (h) 96, 132 and 180 (i) 112, 126 and 224 (j) 128, 144 and 240 (k) 160, 260 and 380 (l) 216, 384 and 600 4. Find the least number, which is exactly divided by the following numbers: (a) 10 and 18 (b) 34 and 119 (c) 88 and 154 (d) 216, 156 and 240 (e) 175, 225 and 350 (f) 119, 153 and 187 5. (a) LCM and HCF of two numbers are 392 and 56 respectively. If one of the numbers is 112, find the next number. (b) HCF and LCM of two numbers are 68 and 408 respectively. If the second number is 102, what is the first number? 6. (a) The product of two numbers is 1536. If the HCF of the two numbers is 16, find the LCM of these two numbers. (b) The product of two numbers is 5488. If the LCM of the two numbers is 196, what is the HCF of these two numbers? 7. (a) Three bells ring at the intervals of 12 minutes, 16 minutes and 24 minutes respectively. If they begin together, after what interval of time will they again ring together? (b) Four bells ring at the intervals of 8 minutes, 12 minutes, 20 minutes and 24 minutes respectively. If they ring together at first, after what interval of time will they again ring together? (c) Three bells ring at the intervals of 12 minutes, 16 minutes and 24 minutes respectively. If they all ring at 10 am, when will they again ring together?

50 The Leading Mathematics - 7 (d) From a bus park, the buses depart to west in every 8 minutes, to east in every 10 minutes and to in south every 12 minutes. If the buses depart from the bus park at 8 am in these three directions, at what time will the buses from the same bus park leave next time at the same time? (e) A scooter is to be refueled after covering every 600 km, engine oil is to be changed after 800 km and service is to be done after every 1800 km. After performing all these three works at the same time, after traveling what distance should all these works be performed at the same time again? 8. (a) Find the least number which leaves a remainder 6 when it is divided by any one of the numbers 8, 10, 16, 20. (b) What is the least number which divided by 24, 36 and 48 will leave in each case a remainder 13? (c) What is the least number to which 15 may be added so that the sum is exactly divisible by 40, 60 and 100? (d) What is the least number from which 12 is subtracted so that the result is exactly divisible by 42, 80 and 192? (e) What number can be subtracted from 1900 so that the difference may be exactly divisible by 10, 15, 50 and 120? ANSWERS 1. (a) 18 (b) 72 (c) 70 (d) 216 (e) 180 (f) 300 (g) 360 (h) 156 (i) 756 (j) 750 (k) 4725 (l) 8640 2. (a) 105 (b) 336 (c) 180 (d) 140 (e) 270 (f) 520 (g) 756 (h) 1386 (i) 1872 (j) 16632 (k) 39520 (l) 23100 3. (a) 18 (b) 36 (c) 210 (d) 48 (e) 216 (f) 252 (g) 2376 (h) 15840 (i) 14112 (j) 5760 (k) 39520 (l) 86400 4. (a) 90 (b) 238 (c) 616 (d) 28080 (e) 3150 (f) 11781 5. (a) 196 (b) 272 6. (a) 96 (b) 28 7. (a) 48 min. (b) 120 min (c) 10:48 am (d) 2 hrs. (e) 7200 km 8. (a) 86 (b) 157 (c) 585 (d) 6732 (e) 100 Project Work 2.4 Ask the ages of your three friends in months. Find the HCF and LCM of their ages in months. Prepare a report and present it in your classroom.