Approved by the Government of Nepal, Ministry of Education, Science and Technology,
Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material
vedanta
Excel in
MATHEMATICS
7Book
Author
Hukum Pd. Dahal
Editor
Tara Bahadur Magar
vedanta
Vedanta Publication (P) Ltd.
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Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np
vedanta
Excel in
MATHEMATICS
7Book
All rights reserved. No part of this publication may be
reproduced, copied or transmitted in any way, without
the prior written permission of the publisher.
First Edition: B.S. 2077 (2020 A. D.)
Second Edition: B.S. 2078 (2021 A. D.)
Third Revised and Updated Edition : B.S. 2079 (2022 A. D.)
Price: Rs 461.00
Published by:
Vedanta Publication (P) Ltd.
j]bfGt klAns];g k|f= ln=
Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np
Preface
The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical teaching
learning activities and methodologies extracted from Teachers' training, workshops, seminars and
symposia. It is an innovative and unique series in the sense that the contents of each textbooks of
the series are written and designed to fulfill the need of integrated teaching learning approaches.
Excel in Mathematics is an absolutely modified and revised edition of my three previous series:
'Elementary mathematics' (B.S. 2053), 'Maths In Action (B. S. 2059)' and 'Speedy Maths' (B. S. 2066).
Excel in Mathematics has incorporated applied constructivism. Every lesson of the whole series
is written and designed in such a manner, that makes the classes automatically constructive and
the learners actively participate in the learning process to construct knowledge themselves, rather
than just receiving ready made information from their instructors. Even the teachers will be able
to get enough opportunities to play the role of facilitators and guides shifting themselves from the
traditional methods of imposing instructions.
Each unit of Excel in Mathematics series is provided with many more worked out examples.
Worked out examples are arranged in the hierarchy of the learning objectives and they are reflective
to the corresponding exercises. Therefore, each textbook of the series itself is playing a role of a
‘Text Tutor’. There is a well balance between the verities of problems and their numbers in each
exercise of the textbooks in the series.
Clear and effective visualization of diagrammatic illustrations in the contents of each and every
unit in grades 1 to 5, and most of the units in the higher grades as per need, will be able to integrate
mathematics lab and activities with the regular processes of teaching learning mathematics
connecting to real life situations.
The learner friendly instructions given in each and every learning contents and activities during
regular learning processes will promote collaborative learning and help to develop learner-
centred classroom atmosphere.
In grades 6 to 10, the provision of ‘General section’, ‘Creative section - A’ and ‘Creative
section - B’ fulfills the coverage of overall learning objectives. For example, the problems in
‘General section’ are based on the Knowledge, understanding and skill (as per the need of the
respective unit) whereas the ‘Creative sections’ include the Higher ability problems.
The provision of ‘Classwork’ from grades 1 to 5 promotes learners in constructing knowledge,
understanding and skill themselves with the help of the effective roles of teacher as a facilitator
and a guide. Besides, teacher will have enough opportunities to judge the learning progress and
learning difficulties of the learners immediately inside the classroom. These classworks prepare
learners to achieve higher abilities in problem solving. Of course, the commencement of every
unit with 'Classwork-Exercise' may play a significant role as a 'Textual-Instructor'.
The 'project works' given at the end of each unit in grades 1 to 5 and most of the units in higher
grades provide some ideas to connect the learning of mathematics to the real life situations.
The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities
to integrate mental maths and manual maths simultaneously. Moreover, the problems in ‘Section
A’ judge the level of achievement of knowledge and understanding and diagnose the learning
difficulties of the learners.
The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an important
role of ‘placement evaluation’ which is in fact used by a teacher to judge the level of prior
knowledge and understanding of every learner to make his/her teaching learning strategies.
The socially communicative approach by language and literature in every textbook especially in
primary level of the series will play a vital role as a ‘textual-parents’ to the young learners and
help them in overcoming maths anxiety.
The Excel in Mathematics series is completely based on the latest curriculum of mathematics,
designed and developed by the Curriculum Development Centre (CDC), the Government of Nepal
in 2078 B.S.
I do hope the students, teachers and even the parents will be highly benefited from the ‘Excel in
Mathematics’ series.
Constructive comments and suggestions for the further improvements of the series from the
concerned will be highly appreciated.
Acknowledgments
In making effective modification and revision in the Excel in Mathematics series from my
previous series, I’m highly grateful to the Principals, HODs, Mathematics teachers and experts,
PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN, and independent clusters of many
other Schools of Nepal, for providing me with opportunities to participate in workshops,
Seminars, Teachers’ training, Interaction programme, and symposia as the resource person.
Such programmes helped me a lot to investigate the teaching-learning problems and to research
the possible remedies and reflect to the series.
I’m proud of my wife Rita Rai Dahal who always encourages me to write the texts in a more
effective way so that the texts stand as useful and unique in all respects. I’m equally grateful to
my son Bishwant Dahal and my daughter Sunayana Dahal for their necessary supports during
the preparation of the series.
I’m extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, England who
provided me with very valuable suggestions about the effective methods of teaching-learning
mathematics and many reference materials.
Thanks are due to Mr. Tara Bahadur Magar for his painstakingly editing of the series. I am
thankful to Dr. Komal Phuyal for editing the language of the series.
Moreover, I gratefully acknowledge all Mathematics Teachers throughout the country who
encouraged me and provided me with the necessary feedback during the workshops/interactions
and teachers’ training programmes in order to prepare the series in this shape.
I’m profoundly grateful to the Vedanta Publication (P) Ltd. for publishing this series. I would
like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha, and
Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during
the preparation of the series.
Also I’m heartily thankful to Mr. Pradeep Kandel, the Computer and Designing Senior Officer of
the publication house for his skill in designing the series in such an attractive form.
Hukum Pd. Dahal
S.N Unit Contents Page
7-22
1. Set 23-52
1.1 Set – Looking back, 1.2 Well-defined collection, 1.3 Membership of a set 53-71
and set notation, 1.4 Methods of describing sets, 1.5 Cardinal number of a set,
1.6 Types of sets, 1.7 Relationships between sets, 1.8 Universal set and subset, 72-79
80-100
2. Operations on Whole Numbers
101-113
2.1 Whole numbers - Looking back, 2.2 Decimal or Denary number system, 115-120
2.3 Periods and place, 2.4 Factors and multiples - Looking back, 2.5 Highest 121-128
common factor (H. C. F.), 2.6 Finding H. C. F. by Factorization method, 129-135
2.7 Finding H.C.F. by Division method, 2.8 Lowest common multiple ( L.C.M), 136-164
2.9 Finding L.C.M. by factorisation method, 2.10 Finding L.C.M. by division
method, 2.11 Square and square root, 2.12 Process of finding square root,
2.13 Cube and cube root
3. Integers
3.1, Natural Number, Whole Numbers and Integers - Looking back, 3.2 Integers
3.3 Absolute value of integers, 3.4 Operations on integers, 3.5 Sign rules of
addition and subtraction of integers, 3.6 Properties of addition of integers,
3.7 Multiplication and division of integers, 3.8 Sign rules of multiplication
and division of integers, 3.9 Properties of multiplication of integers, 3.10 Order
of operations,
4. Rational Numbers
4.1 Rational numbers – review, 4.2 Properties of Rational numbers,
4.3 Terminating and non-terminating rational numbers, 4.4 Irrational numbers
5. Fraction and Decimal
5.1 Fraction – Looking back, 5.2 Addition and subtraction of fraction - revision,
5.3 Multiplication of fractions, 5.4 Division of fractions, 5.5 Decimal - revision,
5.6 Terminating and non-terminating recurring decimal, 5.7 Four fundamental
operations on decimals
6. Ratio and Proportion
6.1 Ratio, 6.2 Proportion, 6.3 Types of proportions
7. Unitary Method
7.1 Unitary method - Review
8. Profit and Loss
8.1 Profit and Loss – Looking back, 8.2 Profit and loss per cent,
8.3 Calculation of S.P. when C.P. and profit or loss per cent are given,
8.4 Calculation of C.P. when S.P. and profit or loss per cent are given
9. Laws of Indices
9.1 Laws of indices (or exponents)
10. Algebraic Expressions
10.1 Algebraic terms and expressions – Looking back, 10.2 Types of
algebraic expressions, 10.3 Polynomial, 10.4 Degree of polynomials,
10.5 Evaluation of algebraic expressions, 10.6 Addition and subtraction of
algebraic expressions, 10.7 Multiplication of algebraic expressions, 10.8
Some special products and formulae, 10.9 Division of algebraic expressions,
10.10, Algebraic product and factors, 10.11 Simplification of rational
expressions
S.N Unit Page
165-189
11. Equation, Inequality and Graph
11. 1 Open statement and equation - Looking back, 11.2 Linear equations 190-195
196-209
in one variable, 11.3 Solution to equations, 11.4 Applications of equations, 210-226
11.5 Linear equation with two variables, 11.6 Trichotomy – Review,
11.7 Inequalities, 11.8 Replacement set and solution set, 11.9 Graphical 227-248
representation of solution sets, 11.10 Graph of linear equation 249-252
12. Coordinates 253-272
12.1 Coordinates – Looking back, 12.2 Coordinate axes and quadrants,
12.3 Finding points in all four quadrants, 12.4 Plotting points in all four 273-277
quadrants 278-285
13. Transformation 286-297
13.1 Reflection of geometrical figures, 13.2 Reflection of geometrical 298-310
figures using coordinates, 13.3 Rotation of geometrical figures, 311-312
13.4 Rotation of geometrical figures using coordinates, 13.5 Displacement
14. Geometry: Lines and Angles
14.1 Line and Angels – Looking back, 14.2 Construction of different angles,
14.3 Construction of equal angle using compass, 14.4 Different pairs of
angles, 14.5 Verification of properties of angles, 14.6 Pairs of angles made by a
transversal with parallel lines
15. Plane Figures
15.1 Triangles – Looking back, 15.2 Construction of triangles, 15.3 Properties
of triangles, 15.4 Some special types of quadrilaterals, 15.5 Verification of
properties of special types of quadrilaterals, 15.6 Pythagoras Theorem
16. Congruent Figures
16.1 Congruent figures – Introduction
17. Perimeter, Area and Volume
17.1 Perimeter, Area and Volume – Looking back, 17.2 Perimeter of plane
figures, 17.3 Relation between circumference and diameter of a circle,
17.4 Area of plane figures, 17.5 Nets and skeleton models of regular solids,
17.6 Area of solids, 17.7 Volume of solids
18. Symmetry, Design and Tessellation
18.1 Symmetrical and asymmetrical shapes, 18.2 Line or axis of symmetry,
18.3 Rotational symmetry, 18.4 Order of rotational symmetry, 18.5 Tessellations,
18.6 Types of tessellations
19. Scale Drawing and Bearing
19.1 Scale drawing, 19.2 Scale factor, 19.3 Bearing
20. Statistics
20.1 Statistics – Review, 20.2 Types of data and frequency table, 20.3 Grouped
and continuous data, 20.4 Line graphs, 20.5 Bar graph, 20.6 Average (or Mean)
Answers
Model Question Set
Unit 1 Set
1.1 Set – Looking back
Classwork - Exercise
1. Let's tick(√) the well-defined collections.
a) A collection of High mountains of Nepal.
b) A collection of mountains of Nepal with more than 6000 m altitude.
c) A collection of fruits.
d) A collection of tasty fruits.
2. Let’s write the objects of the following well-defined collections as the
members of sets.
a) A collection of stationery items inside your school bag.
.........................................................................................................................
b) A collection of furniture inside your classroom.
.........................................................................................................................
c) A collection of clours in our national flag.
.........................................................................................................................
d) A collection of clours in a rainbow.
.........................................................................................................................
3. If W = {0, 1, 2, 3, 4, 5} and A = {2, 4, 6}, let's say and write 'true' or 'false'.
a) A ∈ W ................. b) 2 ∈ W .................
c) 6 ∉ W ................. d) {2, 4} ∉ W .................
4. Let's say and list the elements of these sets.
a) {letters of the word ‘SUCCESS’} = ...............................................................
b) {x : x <10, x ∈ odd numbers} = ..............................................................
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1.2 Well-defined collection 0
The diagram shows collection of whole numbers less than 5. It 12
definitely includes the numbers 0, 1, 2, 3 and, 4. Because, the 34
statement ‘whole numbers less than 5’ defines the distinct and
distinguishable objects which are to be included in the collection.
Therefore, it is a well-defined collection. A well-defined collection of objects is
called a set.
Let’s say the answers of the following questions.
(i) Is it possible to write the members of the collection of tall students of your
class?
(ii) Is it possible to write the members of the collection of tasty fruits?
(iii) Discuss, why these collection are not well - defined.
In this way, a collection which is not well-defined is not a set.
1.3 Membership of a set and set notation
A member or an element of a set is any one of the distinct objects that make up that
set.
For example, in a set N = {1, 2, 3, 4, 5} , the members or elements of the set N are
1, 2, 3, 4, and 5.
The membership of a member of a set is denoted by the symbol ‘∈'. For example,
1∈N or 1∈{ 1, 2, 3, 4, 5}. We read it as '1 belongs to set N' or '1 is a member of set N'
or 1 is an element of set N. Similarly, 2∈N, 3∈N, 4∈N and 5∈N.
However, when any element is not a member of a given set, it is denoted by the
symbol ∉. For example: In N = {1, 2, 3, 4, 5}, 6 ∉ N, 7 ∉ N, ... and so on.
Set notation
We denote sets by capital letters, such as, A, B, C, W, N, etc. The members of a set
are enclosed in braces { } and they are separated by commas. For example,
A = {a, e, i, o, u}, W = {0, 1, 2, 3, 4, 5}, and so on.
1.4 Methods of describing sets 2 3
We usually write the members of a set by the following four methods: 5 7
(i) Diagramatic method
In this method, we write the members of a set inside a circular
or oval diagram. A set of prime numbers less than 10 is shown
in the diagram.
(ii) Description method
In this method, we describe the common property (or properties) of the
members of a set inside the braces. For example:
Vedanta Excel in Mathematics - Book 7 8 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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N = {natural numbers less than 10}
P = {prime numbers between 10 and 20}
V = {vowels of English alphabets}, and so on.
(iii) Listing/Roster/Tabular Method
In this method, we list the members of a set inside the braces and the members
are separated by commas. For example:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9}
P = {11, 13, 17, 19}
V = {a, e, i, o, u}, and so on.
(iv) Set-builder/Rule Method
In this method, we use a variable such as x, y, z, p, q, etc. to represent the
members of a set and the common property (or properties) of the members is
described by the variable. For example:
We describe, N ={1, 2, 3, 4, 5, 6, 7, 8, 9}= {x : x is a natural number less than 10}
and read as: “N is the set of all values of x, such that x is a whole number
less than 10.”
Similarly, P = {11, 13, 17, 19} = {y : 10 < y < 20, y ∈ prime number}
V = {a, e, i, o, u} = {z : z is a vowel of English alphabets}
Facts to remember!
(i) A collection of boys in a class is well-defined collection. However, a
collection of tall boys in a class is not well-defined collection.
(ii) Set is a well-defined collection of objects.
(iii) The objects of a set are the members or elements of the set.
(iv) We write the members of a set inside the curly brackets { }.
(v) The membership of objects in a set is denoted by the symbol ∈ and ∉
denotes the non-membership of objects.
(vi) Diagramatic, Description, Listing and Set-builder methods are the four
methods of describing sets.
Worked-out Examples
Example 1: Write the members of the following sets in diagrams.
a) A set of seven colours in a rainbow.
b) A set of prime numbers less than 10. red orange
Solution: yellow green
a) A set of seven colours in rainbow.
blue indigo
violet
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b) A set of prime numbers less than 10. 2
5
37
Example 2: Write the following sets in description method.
a) A = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday}
b) W = {0, 1, 2, 3, 4}
Solution:
a) A = {Seven days of a week}
b) W = {Whole numbers less than 5}
Example 3: Write the following sets in set-builder method.
a) O = {1, 3, 5, 7, 9}
b) S = {1, 4, 9, 16, 25}
Solution:
a) O = {x : x is an odd number less than 10}
b) S = {y : y is a square number less than 26}
Example 4: Write the following sets in listing method.
a) N = {x : x is a natural number between 5 and 10}
b) F = {p : p is a factor of 12}
Solution:
a) N = {6, 7, 8, 9} b) F = {1, 2, 3, 4, 6, 12}
EXERCISE 1.1
General Section -Classwork
1. Let’s tick (√) the well-defined collections.
a) A collection of bigger natural numbers.
b) A collection of natural numbers less than 4.
c) A collection of snacks in a school canteen.
d) A collection of delicious snacks in a school canteen.
2. If A = { 0, 1, 2, 3, 4, 5 } and P = {2, 3, 5, 7}, let’s insert the correct symbol ‘∈’
or ‘∉’ in the blanks..
a) 4 .................. A b) 7 .................. A c) 1 .................. P
d) 5 .................. P e) {2, 3, 5, 7}.............. A f) {2, 3, 5} .................. P
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3. Let’s write the members of the following sets.
a) V = { vowels of English alphabets} V = ........................................
b) Z = { integers between –1 to + 1} Z = ........................................
c) C = { cube numbers between 0 and 50} C = ........................................
d) W = { x : x is a whole number less than 7} W = .......................................
e) M = { y : y is a multiple of 3 less than 20} M = .......................................
Creative Section
4. a) Define set with examples.
b) A collection of domestic animals is a set. However, a collection of big
domestic animals is not a set. How do you justify these statements?
c) Write four methods of describing sets with one example of each method.
5. Let’s write the elements of the following well-defined collections as the
member of sets. Express each set in diagramatic method.
a) A collection of four gases found in the atmosphere.
b) A collection of letters of the word ‘LAPTOP’.
c) A collection of natural numbers less than 10.
d) A collection of composite numbers less than 10.
6. Let’s rewrite these sets in description method.
a) A = {1, 2, 3, 4} b) B = {5, 10, 15, 20, 25}
c) C = {7, 11, 13, 17, 19} d) D = {1, 2, 3, 6}
7. Let’s rewrite these sets in listing method.
a) P = { prime numbers between 10 and 20 }
b) A = { letters of the word ‘FOOTBALL’ }
c) F = {x : x is a factor of 18 }.
d) M = {y : y is a multiple of 3, 5 < y < 10}.
8. Let’s rewrite these sets in set builder method.
a) A= {1, 2, 3, 4, 5} b) B = { 2, 3, 5, 7}
c) C = {1, 4, 9, 16, 25} d) D = {1, 2, 4, 8}
It’s your time - Project work!
9. a) Let’s observe around your classroom and select any three objects as the
members of a set. Then, express the set in diagramatic, description, listing
and set-builder methods.
b) Let’s observe around the kitchen of your house and select any five objects
as the members of a set. Then, express the set in description, roster, and
rule methods.
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c) Let’s write two collections of your own choice which are not well-defined.
Is it possible to make these collections well-defined?
1.5 Cardinal number of a set
Let's write the number of members or element present in the following sets.
a) A = { }, number of members = ........................
b) B = { 5}, number of members = ........................
c) O = { 1, 3, 5, 7 }, number of members = ........................
Thus, the number of members or elements contained by a set is known as its
cardinal number. We denote the cardinal number of set A as n(A), set B as n(B), set
C as n(C), and so on.
1.6 Types of sets
According to the number of elements contained by sets, there are four types of sets.
(i) Empty or null set (ii) Unit or singleton set (iii) Finite set (iv) Infinite set
(i) Empty or null set
Let’s write the number of members of the following sets.
a) W = {whole numbers less than 0}, n(W) = ........................
b) N = {natural numbers between 0 and 1}, n(N) = ........................
Thus, a set which does not have any elements is called an empty set or a null
set. It is represented by the symbol { } or φ (Phi). Similarly,
if A = {even numbers between 2 and 4}, A = { } or φ and n(A) = 0.
if B = {prime numbers less than 2}, B = {} or φ and n(B) = 0, and so on.
(ii) Unit or singleton set
Let’s write the number of members of the following sets.
a) A = {the highest peak of the world}, n(A) = ........................
b) W = {whole number less than 1}, n(W) = ........................
c) P = {prime number between 10 and 12}, n(P) = ........................
In this way, a set that contains exactly one element is called the unit or
singleton set.
Similarly,
if P = {prime and even numbers}, P = {2} and n(P) = 1
if Q = {x : 6 < x < 8, x N}, Q = {7} and n(Q) = 1, and so on.
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(iii) Finite sets
Let’s write the number of members of the following sets.
a) C = {composite numbers less than 10} = {4, 6, 8, 9}, n(C) = .......
b) S = {square numbers less than 50} = {1, 4, 9, 16, 25, 36, 49}, n(S) = .......
c) Z = {integers between –2 to +2} = {–2, –1, 0, 1, 2}, n(Z) = .......
Thus, a set containing finite number of elements or countable number of
elements is called a finite set.
Similarly,
if A = {1, 2, 3, …, 20}, n(A) = 20. It is a finite set.
if B = {x : 2 < x < 31, x prime number}, n(B) = 9. It is a finite set.
(iv) Infinite sets
Let’s discuss the answers of the following questions.
a) W = {whole numbers} = {0, 1, 2, 3, 4, 5, ...}. Can we finish counting the
elements of this set? Does it have finite number of elements?
b) N = {natural numbers} = {1, 2, 3, 4, 5, ...}. Can we finish counting the
elements of this set? Does it have finite number of elements?
c) E = {even numbers} = {2, 4, 6, 8, 10, ...}. Can we finish counting the
elements of this set? Does it have finite number of elements?
Thus, the above mentioned sets are so large that we never finish counting
their members. A set containing infinite number of elements is called an
infinite set. In other words, a set which is not finite is an infinite set.
Facts to remember!
1. The number of elements contained by a set is called its cardinal number.
2. A set having no element is called an empty or null or void set. It is
denoted by { } or φ (phi).
3. A set having exactly one element is called a unit or singleton set.
4. A set having finite number of elements is called a finite set.
5. A set having infinite number of elements is called an infinite set.
EXERCISE 1.2
General Section -Classwork
1. Let's say and write the cardinal numbers of these sets as quickly as possible.
a) A = { 2, 3, 5, 7}, n (A) = .................
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b) B = { even numbers less than 7 }, n (B) = .................
c) P = { letters of the word 'PUPIL' }, n (P) = .................
d) O = { odd numbers between 3 and 5 }, n (O) = .................
2. Let's say and write whether these sets are empty, unit, finite, or infinite as
quickly as possible.
a) A = { odd numbers less than 100 } ...................................................
b) B = { odd numbers more than 100 } ...................................................
c) C = { composite number between 1 and 5 } ...................................................
d) D = { prime number between 7 and 10 } ..................................................
Creative section
3. a) Define cardinal number of a set with an example.
b) Give an example of empty or null set and define it.
c) Give an example of unit or singleton set and define it.
d) Give one example of each of finite and infinite set. Define these two sets.
4. Let’s list the elements and write the cardinal numbers of these sets.
a) A = { composite numbers between 10 and 20}.
b) B = { all possible factors of 12 }.
c) Z = { x : x is an integer, –2 ≤ x ≤ 2}
d) W = { x : x is a whole number, x < 1 }
5. Let’s list the members of the following sets. Mention whether the sets are
empty, unit, finite or infinite.
a) A = {odd numbers between 10 and 20}
b) B = {odd numbers greater than 10}
c) C = {multiples of 5 between 5 and 10}
d) D = {multiples of 5 between 10 and 20}
It's your time - Project work!
6. a) Let's write the whole numbers from 90 to 100. Select the appropriate
numbers to form the following sets. Then, write the types of sets.
(i) A={even numbers} (ii) B={odd numbers}
(iii) C={x : x is a prime number} (iv) D={y : y is a composite number}
(v) E={z : z is a square number} (vi) F={cube numbers}
(vii) G={multiples of 7} (viii) H={x : x is divisible by 11}
b) How are empty set, singleton set, finite set and infinite set existing in our
real life situations? Discuss in the class and list out a few examples of
each of these sets.
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1.7 Relationships between sets
According to the types and number of elements contained by two or more sets, there
are various types of relationships between the sets, such as equal sets, equivalent
sets, disjoint sets, and overlapping sets.
(i) Equal sets
Let’s take any two sets: A = {s, v, u, 3, ª} and B = {ª, u, s, 3, v}.
a) What are the cardinal number of the sets A and B?
b) Do the sets A and B have exactly the same members?
In the given sets A and B, n (A) = 5 and n (B) = 5. Thus, they have the equal
cardinal number and both the sets have exactly the same elements. Therefore,
sets A and B are said to be the equal sets and written as A = B.
Thus, two or more than two sets are said to be equal if they have exactly the
same elements and equal cardinal number.
(ii) Equivalent sets
Let’s take any two sets: A = { c, o, w } and B = { g, o, d}.
a) What are the cardinal numbers of the sets A and B?
b) Do the sets A and B have exactly the same members?
In the given sets A and B, n (A) = 3 and n (B) = 3. They have the equal cardinal
number.
However, the members c, w of set A are not contained by the set B and the
members g, d of the set B are not contained by the set A. So, they are not equal
and they are said to be equivalent sets.
We write it as A ~ B.
Thus, two or more than two sets are said to be equivalent if they have the equal
cardinal number but they do not have exactly the same elements.
(iii) Overlapping sets
Let’s take any two sets: A = {1, 2, 3, 6} and B = { 1, 2, 4, 8}.
a) Do the sets A and B have some common elements?
b) What are the common elements of the sets A and B?
In this way, the elements 1 and 2 are common to the sets A and B. Therefore,
sets A and B are overlapping sets.
AB
Thus, two or more than two sets are said to be overlapping 3 1 4
if they contain at least one element common. The common
elements of overlapping sets are shown in the shaded 62 8
region of the two intersecting diagrams.
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Set
(iv) Disjoint sets
Let’s take any two sets : A = { 3, 6, 9, 12} and B = { 5, 10, 15, 20 }
In these two sets, there is no any common element. Therefore, sets A and B are
disjoint sets. Of course, non overlapping sets are the disjoint sets.
Thus, two or more than two sets are said to be A B
disjoint if they do not have any element common. 3 6 5 10
The elements of disjoint sets are shown in 9 12 15 20
non-intersecting diagrams.
1.8 Universal set and subset
Let’s take a set of natural numbers less than 15.
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
Now, let's select certain elements from this set and make a few other sets.
A = {even numbers less than 15} = {2, 4, 6, 8, 10, 12, 14}
B = {prime numbers less than 15} = {2, 3, 5, 7, 11, 13}
C = {square numbers less than 15} = {1, 4, 9}
D = {possible factors of 12} = {1, 2, 3, 4, 6, 12}
NA N B 14 N C 14 13 N D
2 1 12 5 1 14
12 4 14 1 2 3 12
12 13 4 7 10 34 11 2 3 4 13
3 6 11 5 13 9 5 9 10 7 6
58 10 11 8 12
6 78 9 10 11
79 68
Here, the set of natural numbers less than 15 is known as the universal set.
Furthermore, every element of the sets A, B, C, and D is also an element of the set
N. In such a case, sets A, B, C and D are called the subsets of the set N. We use the
symbol ‘⊂’ to represent a set as a subset of another set. For example:
‘A is a subset of N’ is written as A ⊂ N .
‘B is a subset of N’ is written as B ⊂ N, and so on.
Remember that, every set is a subset of itself and the empty set (f) is a subset of
every set.
On the other hand, if a set is not the subset of a given set, we denote it by the symbol '⊄'.
Thus, a set under the consideration from which many other subsets can be formed is
called a universal set. The set of teachers of a school is a universal set from which the
subsets like set of Maths teachers, set of Science teachers, set of English teachers, etc.
can be formed. We usually denote a universal set by the capital letter 'U'.
Super set
If the set A is a subset of N, N is called the super set of A. It is denoted as N ⊃ A and
read as ‘N is a super set of A’.
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Set
Proper subset
Let's take any two sets, A = {p, e, n, c, i, l} and B = {p, e, n}.
Here, B is a subset of A and B is not equal to A. In such a case, set B is said to be a
proper subset of A.
Thus, the set B is said to be a proper subset of the set A if it contains at least one
element less than A.
We use a symbol ‘⊂’ to represent a set as a proper subset of another set. For example:
B ⊂ A and we read it as ‘B is a proper subset of A’.
Improper subset
Let's take any two set, A = {g,] kf, n} and B = {kf, n, g}]
Here, B is a subset of A and B is equal to A. In such a case, the set B is said to be an
improper subset of A.
Thus, the set B is said to be an improper subset of A, if B is equal to A, i.e., B = A. We
use the symbol ⊆ to represent a set as an improper subset of another set. For example:
B ⊆ A and we read it as 'B is an improper subset of A'.
Number of subsets of a given set
Let's study the following table and draw the conclusion about the number of subsets
of a given set.
Set Subsets No. of elements No. of subsets
A={} {} n(A) = 0 1 ← 2°
B = {a} {a}, f n(B) = 1 2 ← 21
C = {a, b} {a}, {b}, {a, b}, f n(C) = 2 4 ← 22
D = {a, b, c} 8 ← 23
{a}, {b}, {c}, {a, b}, n(D) = 3
{b, c}, {a, c}, {a, b, c}, f
From the above table, we conclude that the number of subsets of a set is given by the
formula 2n, where n is the cardinal number of the given set.
Worked-out Examples
Example 1: List out the elements of the following sets. Write with reasons
whether the sets are equal or equivalent sets using the symbol.
A = {natural numbers less than 5} and
B = {whole numbers less than 4}
Solution:
A = {1, 2, 3, 4} and B = {0, 1, 2, 3}
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Set
Here, n(A) = n(B) = 4
The set A and B have the equal cardinal number. However, they do not have exactly
the same members. Therefore, the sets A and B are equivalent sets and we write it
as A ~ B.
Example 2: List out the elements of the following sets. Write with reasons
whether the sets are overlapping or disjoint.
P = {x : x is a prime number less than 10}
Q = {y : y is a composite number less than 10}
Solution:
P = {2, 3, 5, 7} and Q = {2, 4, 6, 8}
Here, 2 is a member common to the sets P and Q. Therefore, P and Q are the
overlapping sets.
Example 3: Combine the members of the following sets to form a new set. Show
the members in diagrams.
A = {Ram, Hari, Laxmi} and B = {Laxmi, Sita, Lakpa}
Solution: A B
Sita
Combination of the sets A and B = {Ram, Hari, Laxmi, Sita, Lakpa}. Ram Laxmi
Hari Lakpa
Example 4: List out the elements of the following sets. Write the common
elements in a separate set and show in diagrams.
X = {multiplies of 4 less than 20} and
Y = {multiples of 6 less than 20} XY
Solution: 46
X = {4, 8, 12, 16} and Y = {6, 12, 18} 8 12
Set of common multiples of X and Y = {12} 16 18
Facts to remember!
1. Equal sets have exactly the same elements.
2. Equivalent sets have the equal cardinal number.
3. Overlapping sets have at least one common element.
4. Disjoint sets do not have any common element.
5. A set under the consideration from which many other subsets can be
formed is known as a universal set (U).
6. Between a given set and its subset, if the subset contains at least one
element less than the set, it is a proper subset.
7. Between a given set and its subset, if the subset is equal to the given set,
it is an improper subset.
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Set
EXERCISE 1.3
General Section -Classwork
1. Let's say whether these pairs of sets are equal or equivalent sets and write
A = B or A ~ B in the blank spaces.
(a) A = {1, 2, 4, 8 } and B = { 1, 4, 9, 16}, .......................................
(b) A = { 4, 6, 8, 9, 10} and B = { 6, 9, 4, 8, 10}, .......................................
2. Let's say and write whether these pairs of sets are overlapping or disjoint sets.
(a) A = { 1, 3, 5, 15 } and B = {5, 10, 15, 20}, A and B are .........................
(b) P = { 1, 3, 5, 7, 9 } and Q = { 4, 8, 12 }, P and Q are .........................
3. Let's say and write how many subsets of the following sets are possible?
(a) In {1, 2}, number of subsets = ..............
(b) In {g, o, d }, number of subsets = ..............
4. Let's say and write how many proper subsets are possible from these sets?
(a) In {k} number of proper subsets = ..................
(b) In { 1, 4, 9 } number of proper subsets = ..................
5. Let's say and write the possible subsets of these sets.
(a) {p} ...............................................................................................................
(b) { 2, 7} ..............................................................................................................
6. Let's say and write which one is the universal set and its subset in the given
pairs of sets below.
a) A = { students of your class } and B = { boys of your class }
Universal set is .................. and subset is ..................
b) X = { even numbers less than 20 } and W = {whole numbers less than 20}.
Universal set is ....................... and subset is.......................
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Set
Creative section
7. a) Define the following types of sets with examples.
(i) Equal and equivalent sets (ii) Overlapping and disjoints sets
b) Define universal set and subset with examples.
c) Define proper and improper subsets with examples.
8. Let's list out the elements of the following sets and write with reasons whether
they are equal or equivalent sets:
a) A = {days of the week starting with the letter S} and
B = {days of the week starting with the letter T}
b) M = {letters of the word 'FOLLOW'} and N = {letters of the word 'WOLF'}
9. Let's list the elements of the following sets and write with reasons whether
they are overlapping or disjoint sets:
a) O = {odd numbers less than 10} and
S = {perfect square numbers less than 10}
b) X = {x : x is a prime number which exactly divides 30}
Y = {y : y is a prime number which exactly divides 77}
10. Let's write the possible subsets of the following sets and list the proper subsets
separately:
(a) F = {apple} (b) A = {1, 2} (c) E = {a, b, c} (d) T = {1, 2, 3, 4}
11. At first, let's list the elements of each of these sets from the diagrams. Then,
select the common elements and list them in separate sets.
a) A B b) P Q
3 14 m t i
a h
62 8 n
k
12. Let's list the elements and common elements of these pairs of sets and show
them in diagrams.
a) A = { x : x is a whole number less than 10 }and B = { factors of 24 }
b) P = { x : x is a multiple of 4, x ≤ 20 } and
Q = { x : x is a multiple of 5, x ≤ 20 }
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Set
13. Let’s combine the members of the following sets to form new sets. Show the
members in diagrams.
a) A = {Mina, Shiva, Hari} and B = {Ram, Sita, Dolma}
b) M = {3, 6, 9, 12} and N = {5, 10, 15, 20}
c) A = {Maria, Rahim, Bishnu} and B = {Bishnu, Kishan, Dorje}
d) C = {4, 8, 12, 16, 20} and D = {8, 16, 24, 32}
e) E = {1, 2, 3, 6} and F = {1, 2, 4, 8}
14. Let’s list out the elements of the following sets. Write the common elements
in separate sets and show in diagrams.
a) A = {multiples of 2 less than 10} and B = {multiples of 3 less than 10}
b) C = {multiples of 6 less than 50} and D = {multiples of 8 less than 50}
c) P = {factors of 4} and Q = {factors of 6}
d) X = {factors of 12} and Y = {factors of 18}
It's your time - Project work!
15. a) Let's take a universal set under your consideration. Then, write as many
subsets as possible from your universal set.
b) Let's conduct a survey inside your classroom among your friends. Then,
list the name of your friends and make separate sets in the following cases:
(i) Sets of friends who like tea, coffee or milk.
(ii) Sets of friends who like Mo:Mo, chowmein or Thukpa.
(iii) How many overlapping sets are formed? Show them in diagrams.
(iv) How many disjoint sets are formed? Show them in diagrams.
(v) How many equal sets are formed?
(vi) How many equivalent sets are formed?
c) Is there any possibility to form overlapping sets of the sets of teachers
who are teaching different subjects in your school? If so, make these sets
and show in diagrams.
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Set
OBJECTIVE QUESTIONS
Let’s tick (√) the correct alternative.
1. If A = {a, e, i, o, u}, then which of the following is correct?
(A) a ∈A (B) u ∉ A (C) b ∈ A (D) o ∉ A
2. The set P = {y : 5 < y < 15, y ∈ prime number} in roster form is
(A) { 7, 9, 11, 13} (B) {5, 7, 11, 13}
(C) {7, 11, 13} (D) {6, 8, 9, 10, 12, 14}
3. Which of the following set is an empty set?
(A) {odd numbers less than 10} (B) {composite numbers less than 4}
(C) {letters of the word ‘SUCCESS’} (D) {factors of 9}
4. Which one of the following sets is the empty set?
(A) ∅ (B) {∅} (C) {0} (D) {0, ∅}
5. The cardinal number of the set Z = { x : x is an integer, –3 ≤ x ≤ 3} is
(A) 4 (B) 5 (C) 6 (D) 7
6. An example of finite set is
(A) {x: x∈N, x> 5} (B) {x: x∈W}
(C) {x: x∈Z, x<3} (D) {x: x∈N, x < 100}
7. Which one of the following statements is not true?
(A) Every empty set is a finite set (B) Every singleton set is a finite set
(C) Every empty set is an infinite set (D) A finite set may have no element
8. The possible subsets of the set {a, b} are
(A) {a}, {b} (B) {a}, {b}, {a, b}
(C) {a}, {b}, {} (D) {a}, {b}, {a, b}, {}
9. If A = {1, 10, 100}, which of the following sets is the improper subset of
set A?
(A) {1, 10} (B) {10, 100} (C) {100, 10, 1} (D) {}
10. Select the false statements.
(A) The empty set is a subset of any set.
(B) Every set is a subset of itself.
(C) Every set has at least one subset.
(D) Every set has at most one subset.
Vedanta Excel in Mathematics - Book 7 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Unit 2 Operations on Whole Numbers
2.1 Whole numbers - Looking back
Classwork - Exercise
1. Let's tick (√) the correct numeral of the numbers in words.
a) Seven lakh nine thousand: (i) 79000 (ii) 700900 (iii) 709000
b) Three crore six hundred: (i) 30006000 (ii) 30000600 (iii) 30060000
c) Eight million ninety-nine thousand: (i) 8099000 (ii) 899000 (iii) 8990000
2. Let’s tick (√) the correct alternative.
a) If x is the digit at tens and y is at ones place, the number is:
(i) xy (ii) x + y (iii) 10x + y
b) The place name of five in 950038120 is:
(i) Ten-lakhs (ii) Crores (iii) Ten-crores
c) The place value of 9 in 20900470850 is:
(i) 90000000 (ii) 9000000 (iii) 900000000
3. Let's say and write the answers as quickly as possible.
a) The place name of 6 in 2460178953 in Nepali system is ................................
and in International system is ...........................................
b) Rewrite 5639240000 using commas and express in words in Nepali system.
............................................................................................................................
c) Rewrite 8719480000 using commas and express in words in International
system. ..............................................................................................................
Facts to remember!
(i) W = {0, 1, 2, 3, 4, 5, ...} is a set of whole numbers.
(ii) Set of whole numbers is an infinite set.
(iii) Set of whole numbers is the universal set of set of natural numbers.
(iv) Therefore, set of natural numbers is a proper subset of the set of whole
numbers.
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Operations on Whole Numbers
2.2 Decimal or Denary number system
Hindu-Arabic number system is based on decimal or denary number system. In this
system, we use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any whole number can be
written using a combination of these ten digits.
Let’s take 17 blocks of cubes and regroup them into the group of 10 blocks.
17 = 10 + 7 = 1 × 101 + 7 × 10°
Let’s take 39 pencils and regroup them into the group of 10 blocks.
39 = 30 + 9 = 3 × 101 + 9 × 10°
Similarly, 594 = 500 + 90 + 4 = 5 × 102 + 9 × 101 + 4 × 100.
In this way, whole numbers can be regrouped into the base of 10 with some power
of 10. It is called the decimal numeration system or denary system.
Zero (0) is the least whole number, whereas 1 is the least natural number. The
greatest whole number or natural number is infinite.
Place and place value
Each digit of a numeral has its own place. Its place value is obtained multiplying the
digit by its place. For example, let's take a numeral 7425.
7425
It is at ones place and place value is 5 × 1 =5
It is at tens place and place value is 2 × 10 = 20
It is at hundreds place and place value is 4 × 100 = 400
It is at thousands place and place value is 7 × 1000 = 7000
Now, we can write the numeral 7425 in the expanded form in the following way.
7425 = 7 × 1000 + 4 × 100 + 2 × 10 + 5 × 1
= 7 × 103 + 4 × 102 + 2 × 101 + 5 × 10°
In this way, if x, y and z are the digits at hundreds, tens and ones place respectively
in a number, then the number can be expressed as 100x + 10y + z.
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Operations on Whole Numbers
2.3 Periods and place
The tables given below show the periods and places in Nepali system and
International system of numerations.
9 16 7 24 3 8 5 6 30
9 16 7 24 38 5 6 30
From the table, the number name of 916724385630 in Nepali system is:
Nine kharab sixteen arab seventy-two crore forty-three lakh eighty-five thousand
six hundred thirty.
In International system the number name is:
Nine hundred sixteen billion seven hundred twenty-four million three hundred
eighty-five thousand six hundred thirty.
Comparison between Nepali and International numeration system.
100 thousand = 1 lakh , 1 million = 10 lakhs, 10 million = 1 crore
100 million = 10 crore 1 billion = 1 arab, 100 billions = 1 kharab
Placement of commas
We can read and write the larger numbers more easily and comfortably when the
periods of the digits are separated by using commas. Let's take a numeral 507490680
and rewrite it using commas in both Nepali and International system.
Nepali system International system
50,34,90,780 503,490,780
Separating unit period Separating unit period
Separating thousands period
Separating lakhs period Separating thousands period
The number name is fifty crore The number name is five hundred
thirty-four lakh ninety thousand three million four hundred ninety
seven hundred eighty thousand seven hundred eighty.
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Operations on Whole Numbers
EXERCISE 2.1
General Section - Classwork
1. Let's say and write the number names in Nepali system and in International
system.
Numerals Number name in Nepali Number name in
system
a) 100000 International system
b) 4000000
c) 70000000
d) 200000000
e) 5000000000
f) 90000000000
g) 800000000000
2. Let's write the numerals which have the following expanded forms.
a) 2 × 103 + 4 × 102 + 7 × 101 + 5 × 100 = ..........................................
b) 7 × 104 + 5 × 102 + 9 × 101 + 8 × 100 = ..........................................
c) 4 × 105 + 6 × 104 + 1 × 101 + 7 × 100 = ..........................................
d) 9 × 106 +3 × 102 + 5 × 100 = ..........................................
3. a) If x is the digit at tens place and y is the digit at ones place, the two-digit
number formed by these digits is ..............................................
b) If y is the digit at tens place and x is the digit at ones place, the two-digit
number formed by these digits is ..............................................
c) If a is digit at hundreds place, b is at tens place and c is at ones place, the
three-digit number formed by these digit is ..............................................
4. Rewrite these numerals using commas in Nepali system and in International
system.
Numerals Nepali system International system
a) 18576390
b) 420198675
c) 99999999999
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Operations on Whole Numbers
Creative Section A
5. a) The estimated cost of construction of a hydro-power project is
Rs 2562880000. Rewrite this cost in words in Nepali as well as International
numeration system.
b) The world population estimated by US Census Bureau in June 2021 was
about 7905130400. Rewrite this population in words according to Nepali
numeration system.
c) The annual budget of Government of Nepal was about Rs 1647000000000
in the fiscal year 2078/079. Rewrite it in words according to Nepali and
International system.
6. Rewrite these number names in numerals using commas both in Nepali system
and International system.
a) Six kharab eighty-four arab ninety-one crore forty-seven lakh five thousand
three hundred forty.
b) Thirty-seven billion eight hundred fifteen million six hundred sixty-eight
thousand five hundred twenty-one.
7. Let's write the following numbers in the expanded forms:
a) 52063709 b) 400801530 c) 7502600048 d) 23900068407
8. 2.5 arab = 2500000000 = Two arab fifty crore = Two billion five hundred million.
7.38 billion = 7380000000 = Seven billion three hundred eighty million
= Seven arab thirty-eight crore
Now, let's write the values of the following numbers and rewrite the number
names in Nepali or in International systems.
a) 1.6 crore b) 4.75 crore c) 3.4 arab d) 5.13 arab
e) 2.7 million f) 6.99 million g) 7.2 billion h) 8.36 billion
Creative Section B
9. a) Find the difference between one million and two hundred fifty thousand.
Express the difference in words in International and Nepali Systems.
b) Find the difference between two billion and eighty-five million. Express the
difference in words in International and Nepali System.
c) Find the difference between seven crore and seventy-five lakh. Express the
difference in words in Nepali and International Systems.
d) Find the difference between three arab and forty-eight crore. Express the
difference in words in Nepali and International Systems.
10. a) By how much is Rs 3.6 million more than Rs 3.6 lakh? Express in Nepali
System of numeration.
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Operations on Whole Numbers
b) By how much is Rs 7.5 crore more than Rs 7.5 million? Express in
International System of numeration.
It's your time - Project work!
11. U sing the digits from 0 to 9, let's write the greatest and the least ten-digit
numerals.
a) Express these numerals in words in Nepali and International Systems.
b) Express the difference between these numerals in Nepali and International
Systems.
12. Let's visit the available and reliable website. Then, search and find the
following facts and figures:
a) Today's live population of the three most populated countries in the world.
Express the population in Nepali and International Systems of numeration.
b) Find the total population of these three countries and express in words in
Nepali and International System of numeration.
c) The distance between the Sun and its four nearest planets. Express the
distance in words in Nepali and International System of numeration.
2.4 Factors and Multiples - Looking back
Classwork - Exercise
1. Let’s say and write the factors of the following numbers. Then circle the prime
factors.
a) Factors of 4 are ................................ b) Factors of 6 are ................................
c) Factors of 12 are ................................ d) Factors of 18 are ................................
2. Let’s say and write the first five multiples of the following numbers.
a) 2 → .............................................. b) 3 → ..............................................
c) 4 → .............................................. d) 5 → ..............................................
Factors
Let’s take a whole number 24.
The possible factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Among these factors, 2 and
3 are the prime numbers and they are called the prime factors. Thus, the factors of
whole number divide it without leaving a remainder.
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Operations on Whole Numbers
Again, let’s take a whole number 9.
A few multiples of 9 are: 1 × 9 = 9, 2 × 9 = 18, 3 × 9 = 27, 4 × 9 = 36, 5 × 9 = 45, ... Thus,
multiple of a whole number is the number obtained multiplying the whole number
by any natural number.
2.5 Highest Common Factor (H. C. F.)
Let’s take F12 and F18 are the sets of all possible factors of 12 and 18 respectively.
Here, F12 = { 1, 2, 3, 4, 6, 12 } and F18 = { 1, 2, 3, 6, 9, 18}
Now, let’s make another set of the common factors of F12 and F18.
Set of common factors of F12 and F18 = {1, 2, 3, 6}
Among these common factors, 6 is the highest one. So, 6 is called the Highest
Common Factor (H.C.F) of 12 and 18.
Let’s remember the following steps to find H. C. F. from all possible factors of the
given numbers.
Step 1: Find all possible factors of given numbers.
Step 2: List the factors common to the given number.
Step 3: Write the highest/greatest one as H.C.F.
2.6 Finding H. C. F. by Factorization Method
In this method, we should find the prime factors of the given numbers. Then,
the product of the common prime factors is the H.C.F. of the given numbers. For
example:
Find the H.C.F. of 24 and 36.
Finding the prime factors of 24 and 36.
2 24 2 36 Here, 24 = 2 × 2 × 2 × 3 H.C.F. = Product of common
2 12 2 18 36 = 2 × 2 × 3 × 3 prime factors
26 39 ∴ HCF = 2 × 2 × 3 = 12
3 3
2.7 Finding H.C.F. by Division Method
In this method, we divide the larger number by the smaller one. Again, the first
remainder so obtained divides the first divisor. The process is continued till the
remainder becomes zero. The last divisor for which the remainder becomes zero is
the H.C.F. of the given numbers. For example:
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 29 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
Find the H.C.F. of 28 and 48.
28) 48 (1 Dividing the greater number
by the smaller one.
–28
Since the remainder is not zero, dividing
20) 28 (1 the 1st divisor 28 by the remainder 20.
Since, the remainder is not zero, dividing
–20 divisor 20 by the remainder 8.
Since, the remainder is not zero, dividing
8) 20 (2 the divisor 8 by the remainder 4.
Since, the remainder is zero for the divisor 4,
–16 the H.C.F. is 4.
4) 8 (2
–8
∴ H.C.F. = 4 0
Facts to remember!
(i) The Highest Common Factor (H.C.F.) is also called the Greatest Common
Divisor (GCD).
(ii) The H.C.F. of 1 and any number is 1.
(iii) If x is a factor of y then their H.C.F. is x.
(iv) Two numbers are said to be co-primes or relatively prime if their H.C.F.
is 1.
Worked-out examples
Example 1: Write the sets of all possible factors of 18, 30, and 36. Then make a
set of their common factors and find their H.C.F.
Solution: I understood!
1, 2, 3, 6 are the common
Here, F(18) = {1, 2, 3, 6, 9, 18} factors of 18, 30 and 36. Among
F(30) = {1, 2, 3, 5, 6, 10, 15, 30} these common factors, 6 is the
F(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36} Highest Common Factor!!
Now, common factors of F18, F30 and F36 = {1, 2, 3, 6}
∴ H.C.F. of 18, 30 and 36 is 6.
Example 2: Find the H.C.F. of 56, 84, and 140 by prime factorisation method.
Solution:
2 56 2 84 2 140
2 28 2 42 2 70
2 14 3 21 5 35
7 7 7
Vedanta Excel in Mathematics - Book 7 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Now, 56 = 2 × 2 × 2 × 7 Operations on Whole Numbers
84 = 2 × 2 × 3 × 7 Now, it’s clear to us!
140 = 2 × 2 × 5 × 7 H.C.F. is the product of
common factors of the
∴ H.C.F. = 2 × 2 × 7 = 28 given numbers!!
Example 3: Find the greatest number that divides 40 and 56 without leaving a
remainder.
Solution:
Here, the required greatest number is the H.C.F. of 40 and 56.
40 56 1
-40
16 40 2
- 32 Possible factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40
Possible factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56
8 16 2 Common factors of 40 and 56 are 1, 2, 4, and 8
40 ÷ 8 = 5 and 56 ÷ 8 = 7
-16 So, 8 is the required greatest number.
0
∴ H.C.F. = 8
Hence, the required greatest number is 8.
Example 4: Find the greatest number to which, when 9 is added the sum divides
135, 162 and 189 without leaving a remainder.
Solution:
Here, the H.C.F. of 135, 162 and 189 is the sum of the required number and 9.
135 162 1 Again, 27 189 7
-135 - 189
27 135 5 0
-135
0 Interesting!
18 + 9 = 27 and
∴ H.C.F. = 27 135 ÷ 27 = 5, 162 ÷ 27 = 6
Let the required number be x. 189 ÷ 27 = 7
x + 9 = 27
or, x = 27 – 9 = 18
Hence, the required number is 18.
Example 5: Three jars of milk contain 45 l, 60 l and 120 l of milk respectively.
Find the greatest capacity of a vessel that can empty out each jar
with the exact number of fillings.
Solution:
Here, the greatest capacity of the vessel is the H.C.F. of 45 l, 60 l and 120 l.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 31 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
45 60 1 Again, 15 120 8
-45 - 120
15 45 3 0
-45
0
∴ H.C.F. = 15
Hence, the required vessel with the greatest capacity is 15 l.
Example 6: What is the greatest number of students to whom 84 books, 140
exercise books, and 252 pens can be distributed equally? Also, find
the shares of each item among them.
Solution:
Here, the required greatest number of students is the H.C.F. of 84, 140 and 252.
2 84 2 140 2 252 We got it!
2 42 2 70 2 126 84 books, 140 exercise books and 252
3 21 5 35 3 63 pens can be shared equally among
3 21 7 or 14 or 28 students. And, 28 is the
7 7 greatest number of students!!
7
Now, 184 = 2 × 2 × 3 × 7
140 = 2 × 2 × 5 × 7
252 = 2 × 2 × 3 × 3 × 7
∴H.C.F. = 2 × 2 × 7 = 28
Again, to find the shares of each item,
84 ÷ 28 = 3, 140 ÷ 28 = 5, and 252 ÷ 28 = 9
Hence, the required greatest number of student is 28 and each student shares
3 books, 5 exercise books, and 9 pens.
Example 7: A bookseller has 270 English books, 315 Science books and 585
Mathematics books. He wants to sell the books in a box, subject-
wise in equal numbers. What will be the greatest number of the
boxes required so that 5 boxes remain empty?
Solution:
Here, the required greatest number of boxes is 5 more than the H.C.F. of 270, 315
and 585.
270 315 1 Again, 45 585 13
-270 -45
45 270 6 135
-270 –135
0 0
∴ H.C.F. = 45
Hence, the required greatest number of boxes is H.C.F. + 5 = 45 + 5 = 50
Vedanta Excel in Mathematics - Book 7 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
EXERCISE 2.2
General Section - Classwork
1. Between 2 and 4, 2 is a factor of 4. So, 2 is the H.C.F. of 2 and 4.
Between 6 and 18, 6 is a factor of 18. So, 6 is the H.C.F of 6 and 18.
Let’s investigate the idea from the above facts. Then say and write the H.C. F.
of the following pairs of numbers.
(a) 3 and 6 is ............. (b) 7 and 28 is ............. (c) 6 and 30 is .............
(d) 9 and 63 is ............. (e) 12 and 48 is ............. (f) 15 and 90 is .............
2. Let’s take any two numbers 12 and 18.
The smaller number 12 = 2 × 6
6 can also divide 18 exactly. So, the H.C.F of 12 and 18 is 6.
Let’s take other two numbers 36 and 45.
The smaller number 36 = 2 × 18 18 cannot divide 45 exactly.
= 3 × 12 12 cannot divide 45 exactly.
= 4 × 9 9 can also divide 45 exactly.
So, the H.C.F. of 36 and 45 is 9.
Let’s investigate the idea from the above facts. Say and write the H.C.F. of the
following pairs of numbers.
a) 6 and 9 is .............. b) 10 and 15 is .............. c) 12 and 30 is ..............
d) 18 and 24 is .............. e) 24 and 32 is .............. f) 21 and 28 is ..............
Creative Section - A
3. Let’s write the sets of all possible factors and the sets of common factors of
these pairs of numbers, then find their H.C.F.
a) 9, 12 b) 15, 18 c) 12, 16 d) 18, 24 e) 36, 48
4. Let’s find the H.C.F. of the following numbers by factorisation method.
a) 18, 24 b) 40, 50 c) 54, 72 d) 24, 36, 42
e) 35, 56, 70 f) 30, 75, 90 g) 32, 64, 80 h) 96, 144, 216
5. Let’s find the H.C.F. of the following numbers by division method.
a) 12, 18 b) 28, 32 c) 36, 42 d) 52, 65
e) 12, 15, 18 f) 30, 45, 60 g) 156, 221, 390 h) 360, 456, 696
Creative Section - B
6. a) Find the greatest number that divides 45, 60 and 75 without leaving a
remainder.
b) Find the greatest number by which 60, 90 and 120 are exactly divisible.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 33 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
c) Find the greatest number to which, when 7 is added the sum divides 90, 105
and 120 without leaving a remainder.
d) Find the greatest number from which, when 10 is subtracted the difference
divides 160, 192 and 224 without leaving a remainder.
e) Find the greatest number from which when 12 is subtracted, the resulting
number exactly divides the numbers 336, 432 and 528.
7. a) Three jars of milk contain 40 l, 50 l and 60 l of milk. Find the greatest
capacity of a vessel which can empty out each jar with the exact number of
fillings.
b) What is the maximum number of students to whom 48 apples, 60 bananas
and 96 guavas can be distributed equally? Also find the shares of each
fruit.
8. a) A rectangular floor is 20 feet long and 16 feet broad. If it is to be paved
with squared marbles of same size, find the greatest length of each squared
marble.
b) The length, breadth and height of a room are 9 m 80 cm, 8 m 40 cm and
4 m 20 cm respectively. Find the longest tape which can measure the
all three dimensions of the room exactly.
9. a) Shaswat distributed 54 snickers, 72 kit kats and 90 cadburies equally to his
classmates on the occasion of his birthday. What was the greatest number
of his classmates? How many chocolates of each type did every one get?
b) Sunayna is making identical balloon arrangements for a party. She has 56
orange balloons, 40 maroon balloons, and 24 white balloon. She wants each
arrangement to have the same number of each colour. What is the greatest
number of arrangements that she can make if no balloon is left over? Also
find the shares of each coloured balloon.
c) Find the greatest number of old people of a geriatric care centre to whom
50 sweaters, 75 kambals, and 100 warm jackets can be equally distributed.
Also, find the share of each item among them.
d) There are 21 apples, 28 pears, and 49 oranges. These are to be arranged in
heaps containing the same number of fruits. Find the greatest number of
fruits possible to keep in each heap. How many heaps are formed by this
arrangement?
10. a) On the occasion of Teej, the principal of a school organized a Teej program
for her female staffs. She distributes 90 bangles and 108 sweets equally to
the staffs including herself. If there are 20 male staffs in the school, find the
maximum number of staffs of her school.
b) An organisation distributed 240 bottles of sanitizer, 480 pieces of face shield
and 600 pieces of masks, in the bags of item-wise equal number, to a certain
people of a village to prevent from Covid-19. Find the greatest number of
bags required to pack the items so that 10 bags remained empty.
Vedanta Excel in Mathematics - Book 7 34 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
2.8 Lowest common multiple ( L.C.M)
Let’s take, M6 and M9 as the sets of a few multiples of 6 and 9 respectively.
Here, M6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...}
M9 = { 9, 18, 27, 36, 45, 54, 63, 72, 81, 90,...}
Now, let’s make another set A of the common multiples of M6 and M9,
The common multiples of M6 and M9 = { 18, 36, 54, ...}
Among these common multiples, 18 is the lowest one. So, 18 is called the Lowest
Common Multiple (L.C.M.) of 6 and 9.
Thus, the lowest common Multiples (LCM) of two or more natural numbers is the
least natural number which is exactly divided by the given numbers.
2.9 Finding L.C.M. by Factorisation Method
In this method, we should find the prime factors of the given numbers. Then the
product of the common prime factors and the remaining prime factors (which are
not common) is the L.C.M. of the given numbers. For example:
Find the L.C.M. of 36 and 60.
2 36 2 60 Here, 36 = 2 × 2 × 3 × 3
2 18 2 30 60 = 2 × 2 × 3 × 5
39 3 15
∴ L.C.M. = 2 × 2 × 3 × 3 × 5 = 180
3 5
2.10 Finding L.C.M. by Division Method
In this method, we arrange the given numbers in a row and they are successively
divided by the least common factors till all the quotients are 1 or prime factors.
Then, the product of these prime factors is the L.C.M. of the given numbers.
For example:
Find the L.C.M of 28, 30, and 35
2 28, 30, 35 2 is the least common prime factor of 28 and 30.
5 is the least common prime factor of 15 and 35.
5 14, 15, 35 7 is the least prime factor of 14 and 7.
7 14, 3, 7
2, 3, 1
∴ L.C.M. = 2 × 5 × 7 × 2 × 3 = 420
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Operations on Whole Numbers
Facts to remember!
(i) L.C.M. is always greater than or equal to the largest of the given numbers.
(ii) L.C.M. of the given numbers is always a multiple of their H.C.F.
(iii) If x is a factor of y, or y is the multiple of x, then, their L.C.M. is y.
(iv) If x and y are any prime numbers, then, their L.C.M. is x × y
(v) If x and y are composite number and they do not have any common
prime factor, then, their L.C.M. is x × y.
Worked-out examples
Example 1: Write the sets of a few multiples of 6, 9, and 12. Make a set of their
Solution: common multiples and find their L.C.M.
Here, M(6) = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, …}
M(9) = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …}
M(12) = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, …}
Now, The common multiples of M(6), M(9), M(12) = {36, 72, 108, …}
∴ L.C.M. of 6, 9, and 12 is 36.
Example 2: Three measuring rods are 30 cm, 45 cm, and 60 cm long. Find the
shortest length of cloth which can be measured exactly by any one
Solution: of these rods.
Here, the shortest length of the cloth is the L.C.M. of 30 cm, 45 cm, and 60 cm.
2 30, 45, 60 30 cm 45 cm 60 cm
3 15, 45, 30 30 cm 60 cm 90 cm 120 cm 150 cm 180 cm
5 5, 15, 10 45 cm 90 cm 135 cm 180 cm
1, 3, 2 60 cm 120 cm 180 cm
∴ L.C.M. = 2 × 3 × 5 × 3 × 2 = 180
Hence, the required shortest length of the cloth is 180 cm.
Example 3: Find the least number which leaves a remainder 7 when it is divided
Solution: by any one of the numbers 20, 30, and 40.
Here, the least number divisible by 20, 30, and 40 is their L.C.M. Then, the required
least number = (L.C.M. of 20, 30, 40) + 7.
2 20, 30, 40 ∴ L.C.M. = 2 × 2 × 5 × 3 × 2 = 120
Hence, the required least number = 120 + 7 = 127.
2 10, 15, 20
5 5, 15, 10
1, 3, 2
Vedanta Excel in Mathematics - Book 7 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
Example 4: In a school Auditorium, students are to be kept in the rows for
watching a program. Find the least number of students required for
either the rows of 16 students in each row or the rows of 24 students
in each row or the rows of 30 students in each row so that non of the
Solution: students are left out of these three types of rows arrangements.
Here, the required least number of students is the L.C.M. of 16, 24 and 30.
2 16, 24, 30 We got it!
2 8, 12, 15 240 ÷ 16 = 15 rows of 16 students
2 4, 6, 15 240 ÷ 24 = 10 rows of 24 students
3 2, 3, 15 240 ÷ 30 = 8 rows of 30 students
2, 1, 5
∴ L.C.M. = 2 × 2 × 2 × 3 × 2 × 5 = 240
Hence, the required least number of students = 240
EXERCISE 2.3
General Section -Classwork
1. i) 6 is a multiple of 3. So, L.C.M. of 3 and 6 is 6.
ii) 2 and 3 are prime numbers. So, L.C.M. of 2 and 3 = 2 × 3 = 6.
iii) 4 is a composite number and 5 is a prime number. They are not exactly
divisible to each other. So, L.C.M. of 4 and 5 = 4 × 5 = 20.
iv) 4 and 9 are composite numbers and they do not have any common prime
factors So, L.C.M. of 4 and 9 = 4 × 9 = 36.
Let’s investigate the ideas from the above facts. Say and write the L.C.M. of
a) 2 and 6 is ............. b) 4 and 8 is ............. c) 6 and 18 is .............
d) 2 and 5 is ............. e) 5 and 7 is ............. f) 7 and 11 is .............
g) 5 and 6 is ............. h) 3 and 8 is ............. i) 9 and 10 is .............
j) 4 and 9 is ............. k) 9 and 8 is ............. l) 10 and 21 is .............
2. Let’s take any two composite numbers 10 and 15. They have a common prime
factor. Take the bigger number 15 and think of its multiples.
15 × 1 = 15 is not the multiple of 10.
15 × 2 = 30 is the multiple of 10.
So, L.C.M. of 10 and 15 is 30.
Let’s follow the similar process and say and find the L.C.M. of
a) 6 and 9 is ....................... b) 8 and 12 is .......................
c) 10 and 15 is ....................... d) 12 and 18 is .......................
e) 8 and 20 is ....................... f) 15 and 25 is .......................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 37 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
Creative Section - A
3. Let’s write the sets of the first ten multiples and the sets of common multiples
of these pairs of numbers. Then, find their L.C.M.
a) 2, 3 b) 3, 4 c) 4, 6 d) 6, 9 e) 8, 12
4. Let’s find the L.C.M. of the following numbers by factorization method.
a) 6, 9 b) 9, 12 c) 12, 18 d) 14, 21
e) 4, 6, 8 f) 8, 12, 18 g) 10, 15, 25 h) 14, 21, 35
5. Let’s fnd the L.C.M. of the following numbers by division method.
a) 8, 12 b) 14, 21 c) 30, 45 d) 12, 15, 20
e) 12, 16, 24 f) 20, 30, 40 g) 45, 60, 90 h) 72, 90, 120
Creative Section - B
6. a) Find the least number which is exactly divisible by 12, 16, and 24.
b) Find the smallest number which is when divided by 25, 30, and 75 leaves
no remainder.
7. a) If three buckets of capacities 10l, 12l, and 15l can fill a drum in exact number
of fillings, find the least capacity of the drum.
b) Three measuring rods are 40 cm, 50 cm and 80 cm long. Find the shortest
c) length of cloth which can be measured exactly by any one of these rods.
8. a) After travelling every 400 km, a motorcycle needs to fill petrol. After every
b) 1600 km, it needs to change mobil. After every 2400 km, it needs servicing.
If these works are done together on a day. After how many kilometres of
9. a) travelling are all theses works repeated again ?
b) Three bells ring at an interval of 10, 15, and 20 minutes respectively. If they
all ring together at 6:00 a.m., at what time do they ring again together?
From Kathmandu bus park, buses moved to Pokhara in an interval of every
30 minutes, Birgunj in every 45 minutes, and Mahendranagar in every
1 hour. If the buses move to these places at the same time at 3:00 p.m. at
what time do the buses again move from the bus park at a time?
In a school Auditorium, there are three types of rows arrangements possible
for watching programs: rows of 20 students per row, or rows of 25 students
per row, or rows of 40 students per row. Find the least number of students
required for any one of these arrangements so that all the students can be
accommodated in each of these arrangements.
A mathematics teacher wants to make three possible groups of students for
a project work: Either the groups of 4 students in each group or 5 students
in each group or 8 students in each group. What is the least number of
students required to make any one of these three groups so that non of the
students are left to be in the groups?
Vedanta Excel in Mathematics - Book 7 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
10. a) Find the least number which leaves a remainder 5 when it is divided by
any one of the numbers 10, 12 , and 15.
b) Find the least number which leaves a remainder 4 when it is divided by any
one of the number 12, 16, and 24.
11. a) Find the least number with which when 5 is added, the sum is exactly
divisible by 12, 18, and 20.
b) Find the least number with which when 20 is added, the sum is exactly
divisible by 30, 40, and 60.
It’s your time : Project work!
12. Let’s do the following activities and answer the given questions.
a) Let’s take 4 green cards and 6 yellow cards.
(i) Divide these cards equally between your four friends. Is it possible?
(ii) Divide these cards equally between your three friends. Is it possible?
(iii) Divide these cards equally between your two friends. Is it possible?
(iv) What is the H.C.F. of 4 and 6?
b) Let’s take the number of the following pairs of coloured card and repeat the
similar activities to find the H.C.F. of the each pair of numbers.
(i) 2 green and 4 yellow cards (ii) 6 green and 9 yellow cards
(iii) 8 green and 12 yellow cards (iv) 10 yellow and 15 green cards.
13. a) Let’s take two call bells and mark them as A and B. Now, let’s do the
following activities and answer the given questions.
(i) Let’s ring both the bells at a time and see at what time you rang the bell.
(ii) Then, ring bell A after each interval of 2 minutes and bell B after each
interval of 3 minutes. In this way, after how many minutes do you ring
both the bells at a time?
(iii) Again, ring bell A in every 4 minutes intervals and bell B in every 6
minutes intervals.
In this way, after how many minutes do you ring both the bells at a
time? Now, did you understand one of the uses of L.C.M. in our real life
situations?
b) Let’s take any two pairs of composite numbers between 5 and 15.
(i) Let’s find the H.C.F. and L.C.M. of each pair of your composite numbers.
(ii) Find the product of H.C.F. and L.C.M. of each pair of numbers. Also,
find the product of each pair of numbers.
Could you investigate a fact about the H.C.F. and L.C.M. of any pair of
numbers? Write a report about it and discuss in the class.
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Operations on Whole Numbers
2.11 Square and square root
Prefect square numbers
Let’s study the following examples and investigate the idea of perfect square
numbers.
← 1 squa red = 1 2 = 1 × 1 = 1 (1 is a perfect square number)
← 2 squa red = 2 2 = 2 × 2 = 4 (4 is a perfect square number)
← 3 sq uared = 32 = 3 × 3 = 9 (9 is a perfect square number)
Thus, a perfect square number is the product of two identical numbers. 1, 4, 9, 16,
25, 36, 49, …, etc. are the examples of perfect square numbers.
Square root
Let’s study the following examples and investigate the idea of square root.
1 = 12 = 1 × 1 → So, 1 is the square root of 1. → 1 = 1
4 = 22 = 2 × 2 → So, 2 is the square root of 4. → 4 = 2
9 = 32 = 3 × 3 → So, 3 is the square root of 9. → 9 = 3
16 = 42 = 4 × 4 → So, 4 is the square root of 16. → 16 = 4
Thus, a perfect square number is the product of two identical numbers and one of
the identical numbers is the square root of the square number.
The radical symbol ( ) is used to denote the square root of a number.
2.12 Process of finding square root
We usually use fractorisation method and division method to find the square root of
the given perfect square number.
Factorisation method to find square root
We usually use this method to find the square root of smaller perfect square numbers.
Let’s learn the process from the example given below.
Find the square root of 225 by factorisation method.
3 225 Tofindthesquareroot,weshouldmake thepairsof
3 75 two identical factors and the product of one of
5 25 the factors taken from each pair is the square
root of the number.
5
Now, 225 = 3 × 3 × 5 × 5
∴ 225 = 3 × 5 = 15
Vedanta Excel in Mathematics - Book 7 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
Division method to find square root
We use division method to find the square roots of the larger numbers. It is also
useful to find the square root of decimals. Let’s learn the process from the example
given below.
(i) Starting from the unit place make the number pairs by using
bar mark just over each pair.
Find the square root of (ii) Take the first pair 68 and think of the largest perfect square
(i) 6889 (ii) 64516 which is equal to 68 or less than 68. It is 64.
8 68 89 83 (iii) Think of the square root of 64, it is 8.
+ 8 – 64
(iv) Write 8 as the divisor and quotient. Multiply 8 × 8 = 64 and
write just below 68, then subtract. The remainder is 4.
163 489 (v) Add 8 + 8 = 16 in the divisor side. It is the trial divisor.
+ 3 – 489 (vi) Bring down the other pair 89. Now, 489 is the new dividend.
166 0
6889 = 83 (vii) 48 is 3 times divisible by 16. So, write 3 in the quotient and
also in the divisor.
(viii) Find the product of 163 × 3 which is 489, and write it just
below 489 and subtract. The remainder is 0. Thus, 83 is the
square root of 6889.
2 645 16 254 The greatest perfect square less than 6 is 4
+2 –4 and its square root is 2.
45 245 24 ÷ 4 = 6, 46 × 6 = 276 > 245
+5 –225 so, 45 × 5 = 225
504 2016 20 ÷ 5 = 4 , 504 × 4 = 2016.
+ 4 –2016
508 0
Worked-out examples
Example 1: Find the square of a) 24 b) 85 c) 120
Solution:
a) Square of 24 = 242 = 24 × 24 = 576
b) Square of 85 = 852 = 85 × 85 = 7225
c) Square of 120 = 1202 = 120 × 120 = 14400 12×12 = 144, then 120×120=14400
Example 2: Find the square root of 1764 by prime factorisation method.
Solution: Now, 1764 = 2 × 2 × 3 × 3 × 7 × 7 I got it!
2 1764 The product of one of the
2 882 = 22 × 32 × 72 identical factors taken from
3 441 each pair is the square root.
3 147 ∴ 1764 = 2 × 3 × 7 = 42
7 49
7
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 41 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
Example 3: Find the square root of 11449 by division method.
Solution:
1 114 49 107
+ 1 –1
20 14 While making the number pairs in
+0 –0 11449 from the unit place, 1 is left
unpaired. So, it is divided at first.
207 1449
+7 –1449
214 0
∴ 11449 = 107 × 2 18 × 3 75 (b) 144
441
Example 4: Simplify a) 8
Solution:
a) 8 × 2 18 × 3 75
= 4 × 2 × 2 9 × 2 × 3 25 × 3 Think of the factors of 8, 18 and 75 such
that one of the factors is a perfect square.
=2 2 ×2×3 2 ×3×5 3 Find the square root of each perfect square factor.
= 180 12 2 × 6 × 15 = 180 and 2 × 2 × 3 = 12
= 180 × 4 × 3 Think of the factors of 12 such that one of
the factors is a perfect square.
= 180 × 2 3 = 360 3
b) 414441 = 22 × 22 × 32 144 = 2 × 2 × 2 × 2 × 3 × 3
32 × 72 = 22 × 22 × 32
441 = 3 × 3 × 7 × 7
= 2× 2 = 4 = 32 × 72
7 7
Example 5: Find the square root of 3.24 2 324 2 100
Solution: 2 162 2 50
Here, 3.24 = 324 3 81 5 25
100
3 27 5
∴ 3.24 = 324 3 9
100
3
= 22 × 32 × 32 = 2×3×3 324 = 2 × 2 × 3 × 3 × 3 × 3
22 × 52 2×5 = 22 × 32 × 32
100 = 2 × 2 × 5 × 5
= 18 = 1.8 = 22 × 52
10
Vedanta Excel in Mathematics - Book 7 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
Example 6 : There are 40 pupils in class 7. If every pupil collects as much as
money as their numbers for a charity show, how much money do
they collect altogether?
Solution:
Here, Number of pupils = 40
sum of money collected from each pupil = Rs 40
Thus, the required sum of money = Rs (402) = Rs ( 40 × 40) = Rs 1600.
Hence, they collect Rs 1,600 altogether.
Example 7. The commander arranged his 784 soldiers in the square form for
march pass. How many soldiers were there in each row.
2
Solution: 784
2 392
Let the number of soldier in each row be x.
2 196
Then, total number of soldiers = x × x 2 98
or, x2 = 784 7 49
7
or, x2 = 22 × 22 × 72
or, x2 = (2 × 2 × 7)2 ∴ 784 = 2 × 2 × 2 × 2 × 7 × 7
∴ x = 2 × 2 × 7 = 28 = 22 × 22 × 72
Hence, the number of soldiers in each row was 28. 784 = 2 × 2 × 7 = 28
Example 8. Find the smallest number by which when 180 is multiplied it becomes
a perfect square.
Solution:
2 180
2 90
3 45
3 15 In 180 = 22 × 33 × 5, two numbers 22 and 32
are perfect square but 5 is not. So, to make 5 a
5 perfect square it is multiplied by 5.
∴ 180 = 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5
Hence, the required smallest number is 5.
Example 9. Find the least number which is to be subtracted from 6090 to make
Solution: it a perfect square.
7 60 90 78
– 49
+7 I understood!
148 1190 6090 is not a perfect square because of the remainder
+8 –1184 6. So, the remainder 9 should be subtracted to make
156 it a perfect square.
6
So, the required least number to be subtracted is 6.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 43 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
Example 10. Find the least number which is to be added to 3993 to make it a
Solution: perfect square.
Here, 6 39 93 63 3993 is greater than 632. So, by
+ 6 – 36 adding the least number it should be
123 393 equal to 642 = 4096.
Now, what should be added to 3993
+ 3 –369
126 24 to make it 4096?
It is 4096 – 3993.
Here, 642 = 4096
∴ The required least number to be added = 4096 – 3993 = 103
EXERCISE 2.4
General Section - Classwork
1. Let’s investigate the idea from the given examples and apply it to find the
squares of the given numbers.
Example: 32 = 9 , 302 = 900, 3002 = 90000, 30002 = 9000000
30 has one zero. So its square has two zeros. 300 has two zeros . So, its square
has four zeros !!
a) 42 = ........ , 402 = ........ , 4002 = ........, 40002 = ........
b) 52 = ........ , 502 = ........, 5002 = ........, 50002 = ........
c) 72 = ........ , 702 = ........, 7002 = ........, 70002 = ........
d) 82 = ........ , 802 = ........, 8002 = ........, 80002 = ........
2. Let’s investigate the idea from examples and apply it to find the square
roots of the given numbers.
Example : = 9 = 3, √900 = 30, √90000 = 300 , √9000000 = 3000
900 has two zeros and its square root has one zero. 90000 has four zeros and
its square root has two zeros.
a) 4 = ........ , 400 = ........ , 40000 = ........, 4000000 = ........
b) 16 = ........ , 1600 = ........ , 160000 = ........, 16000000 = ........
c) 36 = ........ , 3600 = ........ , 360000 = ........, 36000000 = ........
d) 81 = ........ , 8100 = ........ , 810000 = ........, 81000000 = ........
Creative Section - A b) 88 c) 124
3. Lets find the squares of a) 45
4. Let’s observe the example given below and learn to identify whether the given
numbers are perfect square or not.
196 By factorization of 196 = 2 × 2 × 7 × 7 = 22 × 72
As the pairing of identical factor is possible, 196 is a perfect square.
Vedanta Excel in Mathematics - Book 7 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
216 By factorization on 216 = 2 × 2 × 2 × 3 × 3 × 3 = 22 × 2 × 32 × 3
As 2 and 3 are left unpaired, 216 is not a perfect square number.
Let’s apply the above process to test whether the following numbers are perfect
squares.
a) 225 b) 392 c) 324 d) 432 e) 576
5. Let’s find the square roots of these numbers by factorization method.
a) 64 b) 100 c) 144 d) 324 e) 784 f) 1225
6. Let’s find the square roots of these numbers by division method.
a) 196 b) 256 c) 441 d) 1225 e) 4096 f) 24964
7. Let’s simplify.
a) 3256 b) 49 c) 225256 d) 256
e) 2 × 6 64 625
f) 2 3 × 15 g) 7 × 2 14 h) 2 × 3 × 12
i) 3 × 5 × 12 j) 6 × 5 × 10 k) 2 8 × 3 18 × 50
8. Let’s simplify and find the values of the following.
a) 22 × 32 × 62 b) 42 × 92 × 32 c) 82 × 92 × 102
42 22 × 62 42 × 52 × 62
9. Let’s find the square roots of the following fractions.
49 18010 c) 169 d) 12151 e) 3116
a) 64 b) 196
10. Let’s find the square roots of the following decimal numbers.
(a) 0.16 (b) 0.81 (c) 1.44 (d) 1.96 (e) 3.24
11. a) In a morning assembly students are arranged in the square form. If there
are 25 students in each row, find the number of students assembled in the
ground.
b) If 45 rose plants are planted along the length and the breadth of a square
floriculture garden, how many plants are there in the garden?
c) The length of a square pond is 30 m. Find its area.
d) When a certain number of children are arranged in a square ground, there
are 14 children along the length and 14 children along the breadth. How
many more children are needed to arrange 15 children along the length and
the breadth?
12. a) 400 students are assembled in the square form. How many students are there
in each row ?
b) The area of a square garden is 625 m2. Find its length.
c) In an afforestation program on the ‘World Environment Day’ every student
from different schools planted as many plants as their number. If they planted
4225 plants altogether, how many students took part in the program?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 45 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
d) Every student of a school donated as much money as their number to make
a fund for coronavirus victims. If they collected Rs 13,225 altogether, how
many students donated money in the fund?
13. a) Find the smallest number by which each of the following numbers is
multiplied to make them perfect squares.
(i) 32 (ii) 192 (iii) 245 (iv) 448 (v) 720
b) Find the smallest number by which each of the following numbers is
divided to make them perfect squares.
(i) 98 (ii) 125 (iii) 243 (iv) 384 (v) 756
Creative Section - B
14. a) Find the least number which is a perfect square and exactly divisible by
10, 12 and 15.
b) Find the least number which is a perfect square and exactly divisible by
18, 24 and 36.
15. a) Find the least number that must be subtracted from 2120 so that the
result is a perfect square.
b) What is the smallest number to be subtracted from 4400 so that the
result is a perfect square?
c) Find the least number that must be added to 2477 so that the sum will be a
perfect square.
d) What is the smallest number to be added to 13431 so that the sum will be a
perfect square
It’s your time - Project work!
16. a) Let’s find the square of the numbers from 1 to 10. Then observe the digits at
ones place of each square number. Now, write a short report about the fact
that you have discovered and present in your class. Can we apply this fact
in square of any other bigger numbers? Discuss in your class.
b) Let’s find the square of any five even numbers and odd numbers separately.
What types of square numbers did you find in these two separate cases?
Write a short report and discuss in the class.
c) Let’s write any three 2-digit any three 3-digit square numbers. Divide each
square number by 4 and observe the remainders. Write a short report about
your observations and present in the class. Can we use this fact to identify
whether a given number is perfect square or not? Discuss in the class.
d) Let’s find the sum of the following consecutive odd numbers.
(i) 1 + 3 (ii) 1 + 3 + 5 (iii) 1 + 3 + 5 + 7 (iv) 1 + 3 + 5 + 7 + 9
(v) 1 + 3 + 5 + 7 + 9 + 11 (vi) 1 + 3 + 5 + 7 + 9 + 11 + 13
Can you discover any fact from these, sums of consecutive odd numbers? Write
a short report and discuss in the class.
Vedanta Excel in Mathematics - Book 7 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
17. a) Let’s study the tricky process of finding square numbers of the numbers
ending with 5.
152 → 1 × (1 + 1) = 2 and 52 = 25, So, 152 = 225
952 → 9 × (1 + 9) = 90 and 52 = 25, So, 952 = 9025
Now, let’s find the square of remaining two-digit numbers ending with 5.
b) A tricky way of finding square root of any bigger number!
Let’s take a square number 576.
Let’s group the last pair of digits (76) and the rest digit (5).
5 76 → Unit digit of 576 is 6. So, the unit digit of 576 will be either 4 or 6
[42 = 16 or 62 = 36]
Now, take the rest digit 5 which is in the between the square of 2 and 3.
So, the ten’s digit of 576 must be 2.
5 76 4
6
5 is in between Again, 2 × 3 = 6 and 5 < 6, so, the unit digit of 576 must
the square of 2 be the smaller option which is 4.
and 3.
∴ 576 = 24
Now, let’s apply the above tricks and find the square root of following numbers
mentally as soon as possible.
(i) 225 (ii) 256 (iii) 324 (iv) 441 (v) 729 (vi) 1296
2.13 Cube and cube root
Let’s study the following illustrations and investigate the idea of cube numbers and
their cube roots.
12 3
1 22 3
1 3
It is a cube of 8 unit cubes.
It is a unit cube. 2 × 2 × 2 = 23 It is a cube of 27 unit cubes.
1 × 1 × 1 = 13 = 8 is a cube number.
= 1 is a cube number. 3 × 3 × 3 = 33
= 27 is a cube number.
Again,
13 = 1 × 1 × 1 = 1 (or 13) is the cube number and 1 is its cube root.
23 = 2 × 2 × 2 = 8 (or 23) is the cube number and 2 is its cube root.
33 = 3 × 3 × 3 = 27 (or 33) is the cube number and 3 is its cube root.
Thus, the product of three identical numbers is the cube of the number (or cubic
number) and one of the identical numbers is the cube root of the cubic number.
The cube root of a number is denoted by the symbol 3 . For example,
3 1 = 1, 3 8 = 2, 3 27= 3, 3 64 = 4, 3 125= 5, and so on.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 47 Vedanta Excel in Mathematics - Book 7
Operations on Whole Numbers
Worked-out examples
Example 1 : Find the cube of a) 9 b) 12 c) 400
Solution:
Cube of 9 = 93 = 9 × 9 × 9 = 729
Cube of 12 = 123 = 12 × 12 × 12 = 1728
Cube of 400 = 4003 = 400 × 400 × 400 = 64000000
Example 2 : Find the cube root of 5832.
Solution
Finding the prime factors of 5832,
2 5832 To find the cube root, we should
2 2916 make the group of three identical
2 1458 factors and the product of one of
3 729 the factors taken from each group
3 243 is the cube root.
3 81
3 27
39
3
Now, 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
= 23 × 33 × 33
∴ 3 5832 = 2 × 3 × 3 = 18
Example 3 : Test whether 1080 is a perfect cube number or not.
Solution:
To find the prime factors of 1080,
2 1080 1082 = 2 × 2 × 2× 3 × 3 × 3 × 5
2 540
2 270
3 135 = 23 × 33 × 5
3 45 Here, 5 is left to make group of three identical factors.
3 15 So, 1080 is not a perfect cube number.
5 We got it!
To find the cube root of a decimal
Example 4 : Find the cube root of 0.125. number, we should convert it into
Solution: the fraction.
Here, 0.125 = 125
1000
∴ 3 0.125 = 3 125
1000
= 3 53 = 5 = 0.5
103 10
Vedanta Excel in Mathematics - Book 7 48 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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