Approved by the Government of Nepal, Ministry of Education, Curriculum
Development Centre, Sanothimi, Bhaktapur as an Additional Material
vedanta
EXCEL in
MATHEMATICS
6Book
Author
Hukum Pd. Dahal
Editor
Tara Bahadur Magar
vedanta
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
vedanta
EXCEL in
MATHEMATICS
6Book
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 Revised (According to New Curriculum) Edition: B.S. 2078 (2021 A.D.)
Price: Rs 423.00
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 order of the learning objectives and they are reflective
to the corresponding exercises. Therefore, each textbook of the series itself plays the role of a
‘Text Tutor’. There is a proper 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 content and activity 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’
fulfill 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, the 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' plays 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 select their teaching learning strategies.
The socially communicative approach by language and literature in every textbook, especially in
primary level of the series, plays a vital role as a ‘textual-parents’ to the young learners and helps
them overcome 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.
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 are 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
Contents
Page No.
Unit Set 9
1 1.1 Set - Looking back, 1.2 Membership of a set, 1.3 Set notation, 1.4 methods of 19
writing members of set, 1.5 cardinal number of sets, 1.6 Types of sets, 1.7 Relationships
Unit between sets
2 Operations on whole numbers
Unit 2.1 Number system - Looking back, 2.2 Hindu- Arabic numeration system, 2.3 Natural
numbers and whole numbers, 2.4 Place, place value and face value, 2.5 Use of
3 commas, 2.6 Expansion of numbers in terms of power of 10, 2.7 The greatest and the
least numbers, 2.8 Fundamental operations - Looking back, 2.9 Order of operations,
2.10 Use of brackets in simpli ication
Properties of Whole Numbers 34
3.1 Various types of numbers - Looking back, 3.2 Prime and Composite numbers,
3.3 Factors and Multiples, 3.4 Prime factors, 3.5 Test of divisibility, 3.6 Prime factorisation,
3.7 Highest Common Factor (H.C.F), 3.8 Lowest Common Multiple (L.C.M.), 3.9 Perfect
square and square root, 3.10 Cube numbers and cube roots
Unit Integers 57
4 4.1 Integers - Introduction, 4.2 Operations on integers
Unit Fraction and Decimal 65
5 5.1 Fraction - Looking back, 5.2 Equivalent fractions, 5.3 To ind the fractions equivalent 114
to the given fraction, 5.4 Like and unlike fractions, 5.5 To convert unlike fractions into 120
like fractions, 5.6 Comparison of fractions, 5.7 Proper, improper fractions and mixed
numbers, 5.8 Reducing fractions to their lowest terms, 5.9 Addition and subtraction
of fractions, 5.10 Multiplication and division of fractions, 5.11 Tenths, hundredths,
and thousandths, 5.12 Decimals, 5.13 Conversion of Fractions into decimals,
5.14 Conversion of decimals into fractions, 5.15 Place and place values of decimals,
5.16 Addition and subtraction of decimals, 5.17 Multiplication and division of decimals,
5.18 Rounding off decimal numbers (Approximation)
Unit Unitary Method
6 6.1 Unitary Method - Looking back, 6.2 Unit number of quantity and unit value
6.3 More number of quantity and more value
Unit Percent
7 7.1 Percent - Looking back , 7.2 Percent and percentage, 7.3 Conversion of fractions or
decimals to percent, 7.4 Conversion of percent to fractions or decimals, 7.5 To express
a given quantity as the percent of whole quantity, 7.6 To ind the value of the given
percent of a quantity, 7.7 To ind a quantity whose value of certain percent is given
Unit Profit and Loss 131
8 8.1 Cost price (C. P.) and selling price (S. P.) - Looking back , 8.2 Pro it and loss,
8.3 Discount
Unit Algebra 135
9 9.1 Constant and variables - Looking back, 9.2 Algebraic terms and expressions, 157
9.3 Coef icient, base and exponent of algebraic terms, 9.4 Evaluation of algebraic
Unit expressions , 9.5 Like and unlike terms, 9.6 Addition and subtraction of algebraic 177
terms, 9.7 Addition and subtraction of algebraic expressions, 9.8 Multiplication of 182
10 algebraic terms, 9.9 Multiplication of polynomials by monomials, 9.10 Multiplication 188
of polynomials, 9.11 Division of algebraic terms, 9.12 Division of polynomials by
monomials, 9.13 Division of polynomials by binomials 203
Equation, Inequality and Graph 220
229
10.1 Equation - Looking back, 10.2 Mathematical statements, 10.3 Open
mathematical statements,10.4 Equation and solution of an equation, 10.5 The facts 254
for solving equations, 10.6 Process of solving equations, 10.7 Verbal problems - Use 260
of equations, 10.8 Trichotomy, 10.9 Negation of trichotomy, 10.10 Trichotomy rules, 268
10.11 Inequalities, 10.12 Solution of inequalities 281
283
Unit Coordinates
11 11.1 Ordered pairs - Looking back, 11.2 Coordinates
Unit Geometry: Point and Line
12 12.1 Point, line, ray, line segment and plane - review, 12.2 Intersecting line segments,
12.3 Parallel line segments, 12.4 Perpendicular line segments
Unit Geometry: Angle
13 13.1 Angles (review), 13.2 Angles formed by a revolving line, 13.3 Measurement of
angles, 13.4 Types of angles, 13.5 Different pairs of angles, 13.6 Angles made by a
transversal with straight line segments, 13.7 Pairs of angles made by a transversal with
parallel line segments
Unit Geometry: Triangles and Polygons
14 14.1 Triangle – review, 14.2 Types of triangles by sides, 14.3 Types of triangles
by angles, 14.4 Sum of the angles of a triangle, 14.5 Exterior angle of a triangle,
14.6 Quadrilaterals, 14.7 Some special types of quadrilaterals, 14. 8 Tangram, 14.9 Sum
of the angles of a quadrilateral, 14.10 Polygons, 14.11 Circle
Unit Geometry: Construction
15 15.1 Construction of angles, 15.2 Bisecting a given angle, 15.3 Construction of
perpendiculars, 15.4 Construction of a line parallel to given line, 15.5 Construction of
regular polygons
Unit Perimeter, Area and Volume
16 16.1 Perimeter, area and volume - Looking back, 16.2 Length and Distance,
16.3 Perimeter of plane igures, 16.4 Area of plane igures, 16.5 Introduction of solid
Unit igures - review, 16.6 Faces, edges and vertices of solid igures, 16.7 Construction of
some models of solids, 16.8 Area of solids, 16.9 Volume of solids
17
Symmetrical Figures, Design of Polygons and Tessellations
17.1 Symmetrical igures, 17.2 Design of polygons
Unit Statistics Presentation of data, 18.3 Frequency table,
18 18.1 Collection of data, 18.2
18.4 Bar graph, 18.5 Average
Answers
Syllabus
Evaluation Model Questions
Unit Set
1
1.1 Set - Looking back
Classwork - Exercise
1. Let's tick the well-defined collections.
a) A collection of delicious fruits.
b) A collection of fruits.
c) A collection of high mountains which are more than 7,000 m.
d) A collection of high mountains.
2. Let's tell and write the members of these sets inside curly brackets and
name the sets
a) The set of the first five letters of 'Nepali Barnamala'.
............................................................................................................
b) The set of four planets closer to the Sun in solar system.
............................................................................................................
c) The set of prime numbers less than 10.
...........................................................................................................
d) A = { x : x is an even number less than 10 }.
...........................................................................................................
3. Let's rewrite the following sets in set - builder form.
a) W = {0, 1, 2, 3, 4} ...........................................................,,,,,,...........
b) A = {1, 3, 5, 7, 9} ..............................................................,,,,,,,........
Let's study the following illustrations and investigate the idea about sets.
It is a collection of stationery items. Any type of stationery can be
included in this collection. So, it is a well-defined collection. A
well-defined collection of objects is called a set.
It is a collection of even numbers less than 10. It definitely includes 2 4
the members like 2, 4, 6 and 8. It is also a collection of well defined 6 8
members because we are able to say whether any member can be
included in this collection or not. Therefore, it is also a set.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 9 Vedanta Excel in Mathematics - Book 6
Set
On the other hand, ‘a collection of tall students in class 6 is not a set. In this case,
we are not able to say that whether a student 4 feet tall, 4.5 feet tall, etc. can be
included or not in the collection because the term ‘tall’ is not well-defined. So,
it does not clearly distinguish the members of the collection.
1.2 Membership of a set
Let's take a set of prime numbers less than 10.
P = {2, 3, 5, 7}
Here, 2, 3, 5 and 7 are the members (or elements) of the set P.
Each member of this set belongs to the set P.
We denote the membership of an element of a set by the symbol '∈'. For example:
2 belongs to the set P is written as 2 ∈ P.
3 belongs to the set P is written as 3 ∈ P, and so on.
However, in P = {2, 3, 5, 7}, 4 does not belong to the set P.
We write it as 4 ∉ P. Similarly, 6 ∉ P, 8 ∉ P, and so on.
1. 3 Set notation
(i) We denote sets by capital letters like A, B, C, N, W, etc. For example, a set
of whole numbers can be denoted by W, a set of natural numbers by N,
and so on.
(ii) The members or elements of a set are enclosed in braces { } and they are
separated by commas (,). For examples,
W = {0, 1, 2, 3, 4, 5}, M = {3, 6, 9, 12}. , V = {a, e, i, o, u}, and so on.
1.4 Methods of writing members of set
We usually write the members of a set by the following four methods.
(i) Diagrammatic method
In this method, we write the members of a set inside a circular, 1
or oval, or rectangular diagram. A set of square numbers less 9
than 10 are shown in the given diagram.
4
(ii) Description method
In this method, we describe the common property (or properties) of the
members of the set inside the braces. For example,
A = {square numbers less than 10} , B = {factors of 18}, and so on.
(iii) Listing method
In this method, we list the members of a set inside the braces and the
members are separated by commas. For example
A = {1, 4, 9}, B = {1, 2, 3, 6, 9, 18}, etc.
Vedanta Excel in Mathematics - Book 6 10 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
(iv) Set - builder method
In this method, we represent the members of a set by a variable (x, y, z,
p, q, …) .Then the common property (or properties) of the members are
described by the variable inside the braces. For example:
A = {1, 4, 9} o A = {x : x is a square number less than 10}.
We read it as 'A is the set of all values of x such that (:) x is a square
number less than 10'.
B = {1, 2, 3, 6, 9, 18} o B = {y : y is a factor of 18}
We read it as 'B is the set of all values of y such that (:) y is a factor of 18'.
Now, let’s recall to describe sets by these four methods at a glance.
Methods Examples
Diagrammatic method
419
Descriptive method A={square numbers less than 10}
Listing method A = {1, 4, 9}
Set-builder method A = {x : x is a square number less than 10}
Worked-out examples
Example 1 : Let's express the following sets in descriptive method.
a) A = {4, 6, 8, 9} b) B = {a, e, i, o, u}
Solution :
a) A = {4, 6, 8, 9} o A = {composite numbers less than 10}
b) B = {a, e, i, o, u} o A = {vowels of English alphabets}
Example 2 : Let's describe the following sets in listing method.
a) P = {letters of the word ‘teacher’}
b) Q = {x : x is a cube number, x < 30}
Solution :
a) P = {letters of the word 'teacher'} o P = {t, e, a, c, h, r} In the word ‘teacher’
the letter e is repeated.
Such repeated members
b) Q = {x : x is cube number, x < 30} o Q = {1, 8, 27} are listed only one time.
Example 3 : Describe the following sets in set-builder method.
a) P = {prime numbers less than 10} b) F = {1, 2, 3, 4, 6, 12}
c) M = {3, 6, 9, 12, 15}
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 11 Vedanta Excel in Mathematics - Book 6
Set
Solution :
a) P = {prime number less than 10} o P = {x : x is a prime number, x < 10}
b) F = {1, 2, 3, 4, 6, 12} o F = {x : x is a factor of 12}
c) M = {3, 6, 9, 12, 15} o M = {x : x is the first five multiples of 3} or
M = {x : x is a multiple of 3, x < 16}
EXERCISE 1.1
General Section - Classwork
1. Let's tick (√) to the well-defined collections.
a) A collection of tasty fruits. b) A collection of fruits.
c) A collection of long rivers of Nepal.
d) A collection of rivers of Nepal.
2. If N = {1, 2, 3, 4, 5} and O = {1, 3, 5, 7, 9}, let's tell and write 'True' or 'False'
in the blank spaces.
a) 4 N ................... b) 4 O .................. c) 5 N ...................
d) 7 O ................... e) 6 N ................... f) 1 O ...................
3. If W = {0, 1, 2, 3, 4} and E = {2, 4, 6, 8}, let's insert the appropriate symbol
' ' or '' in the blank.
a) 6 ................... E b) 3 ................... E c) 5 ................... W
d) 1 ................... W e) 4 ................... W f) 7 ................... E
4. Let's tell and write the members of these sets in listing method.
a) A = {prime numbers between 10 and 20} A = ........................................
b) B = {letters of the word 'elephant'} B = ........................................
c) C = {x : x is an odd number less than 5} C = ........................................
d) D = {y : y is a factor of 12} D = ........................................
Creative Section
5. Let's answer the following questions.
a) What is a set? Give an example of a set.
b) Is the collection of nice Nepali songs a set? Why?
c) Write three methods of writing sets. Give one example of each method.
d) How do we write a set in set-builder method?
6. Let's write the following membership or non-membership by using set
notation symbols.
Example: (i) 4 belongs to the set W. 4 ∈ W
(ii) b is not a member of {a, e, i, o, u}. b ∉ {a, e, i, o, u}
Vedanta Excel in Mathematics - Book 6 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
a) 9 belongs to the set N. b) u is a member of set V.
c) 3 is not an element of Z. d) 7 does not belong to {2, 4, 6, 8}
e) 'e' belongs to {n, e, p, a, l} f) 8 is not a member of {1, 3, 5, 7, 9}
7. Let's describe the following sets in listing method.
a) W = {the whole numbers less than 6}
b) M = {the first five multiples of 4}
c) A = {the letters of the word ‘MATHEMATICS’}
d) P = {x : x is a prime number x < 20}
e) B = {y : y is a composite number, y < 10}
8. Let's express the following sets in descriptive method.
a) A = {s, v, u, 3, ª} b) N = {1, 2, 3, 4, 5, 6, 7, 8, 9}
c) P = {2, 3, 5, 7} d) V = {a, e, i, o, u}
e) F = {1, 2, 7, 14} f) D = {4, 8, 12, 16}
9. Let's express the following sets in set-builder method.
a) W = {0, 1, 2, 3, 4} b) S = {1, 4, 9, 16, 25}
c) O = {1, 3, 5, 7, 9} d) A = {a, b, c, d, e}
e) F = {1, 3, 5, 15} f) M = {3, 6, 9, 12, 15, 18}
It's your time - Project work!
10. a) Let's write the whole numbers upto 20. Select the appropriate numbers
to form the following sets in listing and set-builder forms.
(i) Set of composite numbers (ii) Set of prime numbers
(iii) Set of even numbers (iv) Set of odd numbers
(v) Set of square numbers (vi) set of cube numbers
(vii) Set of multiples of 5 (viii) Set of factors of 20
b) Let's observe around your classroom and select any four objects as
the members of a set. Then express the set in descriptive, listing and
set-builder methods.
1.5 Cardinal number of sets
Let's study the following illustrations and investigate the idea about cardinal
number of sets.
In V = {a, e, i, o, u}, cardinal number of set V = n(V) = 5
In P = {2, 3, 5, 7}, the cardinal number of set P = n(P) = 4
In A = {1, 4, 9}, the cardinal number of set A = n(A) = 3
Thus, the number of elements contained by a set is known as its cardinal
number. The cardinal number of a set A is denoted by n(A).
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 13 Vedanta Excel in Mathematics - Book 6
Set
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
An empty or null set does not contain any element. For example:
A = {oceans in Nepal}, B = {natural number less than 1}, and so on.
An empty set is denoted by { } or I (phi, a greek alphabet).
If A = {oceans in Nepal}, then A = { } or I and n(A) = 0.
If B = {natural numbers less than 1}, then B = { } or I and n(B) = 0
(ii) Unit set (or singleton set)
A unit set or singleton set contains exactly one element. For example:
A = {the highest peak of the world}, i.e. A = {Sagarmatha} and n(A) = 1
B = {prime number between 6 and 10}, i.e. B = {7} and n(B ) = 1, and so on.
(iii) Finite set
A finite set contains a finite number of elements. It means, the process of
counting of it's elements surely comes to an end. For example:
W = {whole numbers less than 10}, i.e. W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and
n(W) = 10
S = {square numbers less than 50}, i.e. S = {1, 4, 9, 16, 25, 36, 49} and n(S) = 7
(iv) Infinite set
A set which is not finite is called an infinite set. It contains infinite number
of elements and the process of counting of it's elements does not come to an
end. For example:
N = {natural numbers}, i.e. N = {1, 2, 3, 4, 5, …}
P = {prime numbers greater than 1}, i.e. P = {2, 3, 5, 7, 11, …}, and so on.
1.7 Relationships between sets
According to the number of elements and types of elements contained by two or
more sets, there are four types of relationships of sets.
(i) Equal sets
In any two sets A and B, where, A = {a, e, i, o, u} and B = {u, i, a, o, e},
the sets A and B have the equal cardinal numbers and exactly the same
elements. Therefore, sets A and B are called equal sets.
It is written as A = B.
Two or more sets are said to be equal if they contain exactly the same
elements.
Vedanta Excel in Mathematics - Book 6 14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
(ii) Equivalent sets
In any two sets P and Q, where, P = {1, 3, 5, 7} and Q = {2, 4, 6, 8},
the elements of the sets P and Q are not exactly the same, however, the
cardinal number of both of them is equal. Therefore, sets P and Q are
called equivalent sets. It is written as P a Q.
Two or more sets are said to be equivalent if they have equal cardinal
number.
All equal sets are equivalent but all equivalent sets may not be equal.
(iii) Overlapping sets
Let’s take any two sets A and B, where,
A = {a, e, i, o, u} and B = {a, b, c, d}.
Here, the element a is common to both sets. Therefore, the sets A and B
are called the overlapping sets.
Two or more sets are said to be overlapping if they have at least one
element common. AB
The given intersecting circles represent overlapping eb
sets. The shaded region contains the common ia
element. c
uo d
(iv) Disjoint sets Overlapping sets
Let’s take any two sets M and N, where M = {5, 10, 15, 20} and
N = {2, 4, 6, 8}.
Here, sets M and N do not have any common MN
element. It means, they are not overlapping.
Therefore, set M and N are called disjoint sets. 5 10 2
15 20 6
The given non-intersecting circles represent the
disjoint sets M and N. 4
8
Disjoint sets
Worked-out examples
Example 1 : State with reasons, whether the following pairs of sets are equal
or equivalent.
a) A = {x : x is an odd number less than 10} and B = {0, 1, 2, 3, 4}
b) P = {x : x is an even number, x < 10} and Q = {2, 4, 6, 8}
Solution :
a) A = {x : x is an odd number less than 10}={1, 3, 5, 7, 9}, So, n (A)= 5
B = {0, 1, 2, 3, 4} ; so, n (B) = 5
Here, sets A and B do not have exactly the same elements. Therefore,
they are not equal sets.
But, n (A) = n (B) = 5. Therefore, they are equivalent sets (A a B)
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 15 Vedanta Excel in Mathematics - Book 6
Set
b) P = {x : x is an even number, x < 10} = {2, 4, 6, 8}
Q = {2, 4, 6, 8}
Sets P and Q have exactly the same elements. So, they are equal sets.
Example 2 : Let’s list the elements of the sets A and B A 2 1B
from the given diagrams. Also find a set C 8 4 3
that contains the common elements of the
10 6 7 5
sets A and B. C
Solution :
Here, A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6, 7}
Also, C = {2, 4, 6} is the set of common elements of sets A and B.
Example 3 : From the given sets P = {2, 3, 5, 7} and Q = {1, 3, 5, 7, 9}, list the
common elements in separate set R. Show the elements of P and
Q in diagrams.
Solution : PQ
Here, P = {2, 3, 5, 7} and Q = {1, 3, 5, 7, 9}
31
5
2 7 9
? R = {3, 5, 7} is the set of common elements of P and Q. R
EXERCISE 1.2
General Section - Classwork
1. Let's tell and write whether the following sets are empty (null), unit
(singleton), finite or infinite.
a) A = {2, 4, 6, 8, 10, ...} ............................................
b) B = {2, 3, 5, 7, 11, 13} ............................................
c) C = {Mt. Everest} ...........................................
d) D = {composite number between 4 and 6} ............................................
2. Let's tell and write the correct answers in the blank spaces.
a) If N = {11, 12 ,13, 14}, n (N) = ....................................................
b) If S = {1, 4, 9, … 49}, n (S) = ....................................................
c) If V = {i}, subsets of V are .......................... and ..........................
d) If A = {teachers} and B = {mathematics teachers}, then universal set is
......... and its subset is ...........
Vedanta Excel in Mathematics - Book 6 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
3. Let's tell and write whether the following sets are equal or equivalent.
a) A = {e, f, g, h}, B = {g, f, h, e}, A and B are ....................................
b) X = {3, 6, 9, 12}, Y = {1, 3, 5, 7}, X and Y are ................................
4. Let's tell and write whether the following sets are overlapping or disjoint.
a) P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8}, P and Q are ........................
b) A = {5, 10, 15, 20}, B = {4, 8, 12, 16, 20}, A and B are ........................
5. If N = {1, 2, 3, 4, 5, … 10} and A = {2, 4, 6, 8, 10}, insert or in the
blank spaces.
a) B = {2, 3, 5, 7}, B ............... N b) C = {4, 8, 12}, C ............... A
c) {3, 6, 9, 12} ................... N d) {4, 6, 8} ................... A
Creative Section
6. Let’s list the elements of these sets and write whether they are empty
(null), singleton, finite or infinite sets.
a) A = {prime number between 5 and 7}
b) B = {multiples of 2 less than 20}
c) C = {multiples of 2 greater than 20}
d) D = {square numbers between 10 and 20}
7. Let’s list the elements and write the cardinal numbers of these sets.
a) A = {Natural numbers less than 10}
b) B = {factors of 18}
c) C = {letters of the word ‘apple’}
d) D = {x : x is a square number, x < 75}
8. Let’s list the elements and write with reasons whether the following pairs
of sets are equal or equivalent.
a) A = {Whole numbers less than 5} and B = {1, 2, 3, 4, 5}
b) P = {x : x is a prime number, x < 10} and Q = {7, 5, 3, 2}
9. Let’s list the elements and write with reasons whether the following pairs
of sets are overlapping or disjoint.
a) A = {factors of 12} and B = {factors of 18}.
b) P = {first five multiples of 4} and Q = {first five multiples of 7}.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 17 Vedanta Excel in Mathematics - Book 6
Set
10. Let’s list the elements of each pair of overlapping sets. Then make a set of
common elements in each case.
AB PQ M N
63
b i 2 17 2 12 9
c ao 3 48
4 59 15
d eu 10
11. From the given sets, list the common elements in separate sets. Show the
elements and the common elements of each pair of sets in diagrams.
a) A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {2, 3, 5, 7}
b) P = {2, 4, 6, 8, 10} and Q = {4, 8, 12, 16}
It’s your time - Project work!
12. a) Let's conduct a survey among your at least 10 classmates and list out
how many of them are like football or basketball. Make separate sets of
the name of your classmates and answer the following questions.
(i) What is the cardinal number of the set of students who like
football?
(ii) What is the cardinal number of the set of students who like
basketball?
(iii) What is the cardinal number of the set of students who like both
games?
(iv) What is the cardinal number of the set of students who do not like
both games?
b) Let’s write a set of natural numbers upto 10. Then, select the appropriate
elements to make the following sets.
(i) Any two pairs of disjoint sets. (ii) Any two pairs of overlapping sets.
c) Let's write the whole numbers upto 20. Select the appropriate elements
to form the following sets.
(i) Set of even numbers less than 10 and set of multiples of 4 less
than 15.
(ii) Set of prime numbers less than 20 and set of multiples of 3 less
than 20
Vedanta Excel in Mathematics - Book 6 18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Unit Operations on Whole Numbers
2
2.1 Number system - Looking back
Classwork - Exercise
1. Let’s count the blocks of thousands, hundreds, tens and ones. Then
write the numerals and number names.
a) .1...0...0...3.....................
.O....n..e......t..h..o...u...s...a...n..d......a...n..d......t..h..r..e...e..............
b) ...............................
.......................................................................
c) ...............................
.......................................................................
d) ...............................
.......................................................................
e)
...............................
.......................................................................
2. Let’s tell and write the answers as quickly as possible.
a) How many hundreds are there in 1 thousand? .....................................
b) How many thousands are there in 1 lakh? .....................................
c) How many lakhs are there in 1 crore? .....................................
d) How many crores are there in 1 arab? .....................................
e) How many lakhs are there in 1 million? .....................................
f) How many millions are there in 1 crore? .....................................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 19 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
2.2 Hindu- Arabic numeration system
There are ten digits in Hindu-Arabic numeration system. The digits are
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This numeration system was developed by
Hindus and it was spread out by the Arabs all over the world. Therefore, the
numeration system is well-known as Hindu-Arabic numeration system. The
system is based on grouping of tens. Therefore, it is also known as Base Ten
or Decimal System.
2.3 Natural numbers and whole numbers
Natural number
We count the number of objects by 1, 2, 3, 4, 5, …, 10, …, 100 and so on.
Therefore, these are the counting numbers. The counting numbers are also
called the natural numbers.
1 is the least natural number and the greatest natural number is infinite.
The set of natural number is denoted by N.
? N = {1, 2, 3, 4, 5, …}
Whole number
Suppose, you have 2 apples. You eat 1 apple and you give 1 apple to your
friend.
Now, how many apples are left with you?
How much is left when 5 is subtracted from 5?
The answer of each of these questions is ‘None’.
In counting, ‘none’ means zero (0). Therefore, zero also counts the number of
objects. However, it counts ‘there is no any number of objects’.
Thus, counting numbers include zero (0) also. The counting numbers
(or natural numbers) including zero are called the whole numbers. The set of
whole numbers is denoted by W.
? W = {0, 1, 2, 3, 4, 5, …}
The least whole number is 0 and the greatest whole number is infinite.
2.4 Place, place value and face value
Let’s take a numeral 3214.
3214
Ones =4×1 =4
Tens = 1 × 10 = 10
Hundreds = 2 × 100 = 200
3 × 1000 2 × 100 1 × 10 4 × 1 Thousands = 3 × 1000 = 3000
Vedanta Excel in Mathematics - Book 6 20 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
Here, ones, tens, hundreds and thousands are the places of digits of the
given numeral. A digit itself at any place is it’s face value. The product of
face value of a digit and its place is the place value.
Let’s study the following example and learn more about place, face value
and place value.
Numeral Places Face values Place Values
342576
Ones 6 6×1=6
Tens 7 7 × 10 = 70
Hundreds 5 5 × 100 = 500
Thousands 2 2 × 1000 = 2000
Ten thousands 4 4 × 10000 = 40000
Lakhs 3 3 × 100000 = 300000
There are two types of place name systems.
(i) Nepali place name system (ii) International place name system
In both the systems, the places higher than hundreds are grouped in the
particular periods.
Periods and places in Nepali System
Periods and places in International System
Now let’s learn to compare the periods and place names of Nepali and
International place name systems.
Nepali system International system Place value Notation in power of 10
Ones 10°
Tens Ones 1 101
Hundreds 102
Thousands Tens 10 103
Ten thousands 104
Lakhs Hundreds 100 105
Ten lakhs 106
Crores Thousands 1000 107
Ten crores 108
Arabs Ten thousands 10000 109
Ten arabs 1010
kharabs Hundred thousands 100000 1011
Ten kharabs 1012
Millions 1000000
Ten millions 10000000
Hundred millions 100000000
Billions 1000000000
Ten billions 10000000000
Hundred billions 100000000000
Trillions 1000000000000
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 21 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
2.5 Use of commas
It is easier to read and write larger numbers when the periods of the digits are
separated by using commas (,). For example :
In Nepali system International system
72,58,36,194 725,836,194
using comma to separate the using comma to separate
digits at units period the digits at units period
using comma to separate the
using comma to separate the digits at thousands period
digits at thousands period
Number name is:
using comma to separate Seven hundred twenty-five million
the digits at lakhs period eight hundred thirty-six thousand one
hundred ninety-four.
Number name is:
Seventy-two crore fifty-eight lakh
thirty-six thousand one hundred
ninety-four.
2.6 Expansion of numbers in terms of power of 10
Let’s study the following illustrations and learn to write the notation of power
of 10.
In 1, there is no zero. It is written as 100
In 10, there is one zero. It is written as 101
In 100, there are two zeros. It is written as 102
Similarly, in 10000000, there are seven zeros. Therefore, it is written as 107.
Now, let's take a number 85,349 and expand it.
85349 8 8 8 8
5 5 5 5
33 3 3
44 4 4
99 9 9
? 85349 = 8 u 104 + 5 u 103 + 3 u 102 4 u 101 + 9 u 100
Let’s study the following examples and learn more about the expansion of
numbers in power of 10.
700 = 7 × 102, 701 = 7 × 102 + 1 × 100, 720 = 7 × 102 + 2 × 101
4000 = 4 × 103, 4005 = 4 × 103 + 5 × 100, 4032 = 4 × 103 + 3×101 + 2×100
90060 = 9 × 104 + 6 × 101, 90537 = 9 × 104 + 5 × 102 + 3 × 101 + 7 × 100
Vedanta Excel in Mathematics - Book 6 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
2.7 The greatest and the least numbers
Among 0 to 9, the greatest digit is 9. Therefore, 9 is the greatest 1-digit number.
99 is the greatest 2-digit number, 999 is the greater 3-digit number and so on.
1 is the least counting number. Therefore, 1 is the least 1-digit number. 10 is the
least 2-digit number, 100 is the least 3-digit number, and so on.
To write the greatest number by using different digits, they are arranged in the
decreasing order.
Similarly, in the case of writing the least number by using different digits, they are
arranged in increasing order. For example,
The greatest five-digit number formed by 4, 1, 0, 9, 7 is 97410.
The least five-digit number formed by 4, 1, 0, 9, 7 is 10479.
Remember !
01479 is not the smallest five-digit number.
Because, 01479 is same as 1479 which is a four-digit number !!
Worked-out examples
Example 1 : How many millions are there in 3 crore ?
Solution
Here, 3 crore = 3,00,00,000 = 30,000,000
= Thirty million
Example 2 : Express 59047000 in the expanded form of power of 10.
Solution
Here 59047000 = 5 × 107 + 9 × 106 + 4 × 104 + 7 × 103
Note : In this expanded form, 0 × 105 + 0 × 102 + 0 × 101 + 0 × 10° is not
necessary to write.
Example 3 : If x, y and z are the digits at hundreds, tens and ones place
respectively, write a 3-digit number formed by these digits.
Solution
Here, the place value of x = x × 100 = 100x
the place value of y = y × 10 = 10y
the place value of z = z × 1 = z
So, the required number is 100x + 10y + z.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 23 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
Example 4: Find the sum of the greatest and the least numbers of 6 digits.
Solution
The greatest number of 6-digits = 999999
The least number of 6-digits = + 100000
Sum = 1099999
Example 5: Find the difference of the greatest and the least numbers of
7 digits formed by 4, 2, 0, 1, 7, 5, 9.
Solution
Here, the greatest number = 9754210
the least number = – 1024579
difference = 8729631
EXERCISE 2.1
General Section - Classwork
1. Let’s tell the answers and write in the blanks as quickly as possible.
a) The digits in Hindu-Arabic number system are .........................................
b) The face value, place and place value of 5 in 3517248 are ........................,
........................................ and ................................... respectively.
c) The place value of 9 in 46904317 is ..........................., whereas, it is
............................. in 49604317.
d) If the expanded form of a numeral is 3 × 104 + 2 × 102, the numeral
is ................................
2. Let's tell and write the answers as quickly as possible.
a) The least natural number is ............................
b) The least whole number is ............................
c) The greatest whole number is ............................
d) Between N = {1, 2, 3, …} and W = {0, 1, 2, 3, …}, ............................
the universal set is
e) The greatest number of 5-digits is ............................
Vedanta Excel in Mathematics - Book 6 24 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
f) The least number of 6-digits is .............................
g) The greatest number of 4-digits formed by 8, 3, 0, 6 is .............................
h) The least number of 5-digits formed by 7, 0, 2, 4, 9 is ............................
i) How many millions are there in 40 lakh? .............................
j) How many crores are there in 3 hundred million ? .............................
Creative Section - A
3. Write the place names of coloured digits in Nepali as well as in
International numeration systems. Also write the face value and place value
of the coloured digits.
a) 743809165 b) 41062598700
4. Re-write the numerals using commas according to Nepali as well as
International system. Also write the number names in both systems.
a) 2705891436 b) 50398702416 c) 400876593120
5. Write the numerals using commas of these numerations.
a) Six arab forty-five crore seventy nine lakh thirty-five thousand.
b) Eight hundred fifty-nine billion one hundred twenty-four million ninety
thousand.
6. a) The cost of construction of a road is Rs 158657000. Express this cost in
words in
(i) Nepali numeration system (ii) International numeration system.
b) The estimated budget of a hydro-electric project in Nepal is twenty-eight
billion five hundred forty-two million rupees.
(i) Express the budget in figure.
(ii) Re-write the value of the budget in Nepali numeration system.
7. a) How many lakhs are there in 4 million ?
b) How many crores are there in 20 million ?
c) How many arabs are there in 30 billion?
d) How many millions are there in 5 crore ?
e) How many billions are there in 7 arab?
8. Express these numerals in the expanded forms of power of 10
Eg. 300 = 3 × 102 5200 = 5 × 103 + 2 × 102
703045 = 7 × 105 + 3 × 103 + 4 × 101 + 5 × 10°
a) 500 b) 2700 c) 305260 d) 90180370 e) 638042957
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 25 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
9. From these expanded form, write the numerals.
Eg. 6 × 103 + 5 × 101 = 6050
4 × 105 + 3 × 103 + 8 × 102 = 403800
a) 3 × 102 + 2 × 101 + 5 × 10°
b) 7 × 104 + 4 × 102 + 6 × 10°
c) 2 × 107 + 5 × 104 + 1 × 103 + 9 × 101
d) 8 × 109 + 9 × 105 + 3 × 102 + 7 × 101
10. a) Find the sum of the greatest and the least numbers of 6 digits.
b) Find the difference between the greatest and the least numbers of
7-digits formed by 3, 0, 1, 8, 5, 9, 4.
11. a) If x is at tens place and y is at ones place, write the number formed by
these digits.
b) If a is at hundreds place, b is at tens place and c is at ones place, write
the number formed by these digits.
Creative section - B
12. a) Find the sum and difference between the place value and face value of
5 in the number 35086941.
b) Find the sum and difference between the place values of two sevens in
the number 6784037525.
c) Find the sum and difference between the place values of nine in the
numbers 49560327 and 1024968.
13. a) Find the sum and difference between the number 98 and the number
obtained by reversing its digits.
b) What are the sum and difference between the numbers 473 and the
number obtained by reversing the digits at ones and hundreds places?
14. a) By how much is 6345987 smaller than 1 crore?
b) By how much is 9966004321 larger than 1 billion?
It’s your time - Project work!
15. Let’s visit to the available website and search the current population of
Nepal, India and China.
a) Write the population in numerals.
b) Express the population in words in Nepali as well as International system
of numeration.
c) Compare the population of these countries.
16. Let’s search the current population of your district and write it in numeral
as well as in words.
Vedanta Excel in Mathematics - Book 6 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
2.8 Fundamental operations - Looking back
Classwork - Exercise
1. Let’s investigate and apply tricky ways, of addition and subtraction Then
complete the sums as quickly as possible. These are some tricks!
a) 26 + 15 = ........... b) 34 + 42 = ........... 27 + 25 = (30 + 25) – 3
= 52
c) 39 + 45 = ........... d) 52 + 43 = ........... 27 + 25 = (30 + 25) – 3
e) 25 – 13 = ........... f) 47 – 24 = ........... = 52
g) 59 – 35 = ........... h) 71 – 44 = ...........
36 – 14 = (36 – 10) – 4
= 26 – 4 = 22
69 – 36 = (69 – 30) – 6
= 39 – 6 = 33
2. Let’s investigate and apply tricky ways of multiplication and division.
9×6 8×7 36÷12 15÷5
9 × 60 = 540 8 × 700 = 5600, 3600 ÷ 12 = 300, 15000 ÷ 500 = 30
a) 50 × 7 = ........... b) 600 × 8 = ........... c) 800 × 70 = ...........
d) 630 ÷ 7 = ........... e) 7200 ÷ 90 = ........... f) 56000 ÷ 800 = ...........
3. Let’s investigate and apply tricky ways, then multiply as quickly as
possible.
a) 6 × 13 = .......... b) 8 × 16 = ........... 8 × 17 = 8 × 10 + 8 × 7
c) 11 × 15 = .......... d) 13 × 12 = ........... = 80 + 56 = 136
e) 15 × 12 = .......... f) 16 × 13 = ............ 14 × 12= 14 × 10 + 14 × 2
= 140 + 28 = 168
4. Quiz time!
a) The sum of two numbers is 18 and the bigger number is 13.
The smaller number is ............................
b) The difference of two number is 6 and the smaller number is 25.
The bigger number is ............................
c) The sum of two numbers is 16 and the difference is 2.
The numbers are ............................ and ............................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 27 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
d) If x + y + z = 27 and x + y = 17, then z = ............................
e) The product of two numbers is 45 and one of them is 9.
The other number is ............................
f) Multiplier is 6 and the product is 54, multiplicand is ............................
g) The quotient of 72 divided by a number is 9, the number is ..................
h) If p is the divisor, q is the quotient and r is the remainder, then the
dividend is .............................................................
5. Puzzle time!
a) Let’s fill in the missing numbers to complete the sums.
i) ii) – = iii) – 7 =
+ 12 = 30
+ + + – – –+ – +
+ =9
17 + = – 10 = 8
= = == = == = =
+ 15 = 42 – = 28 – 6 = 19
iv) × v) ÷ 8 = 12 vi) ÷ = 4
8 = 40
× × × ÷ ÷ ÷ × ××
10 ÷ =
= ×= ÷=
==
= == = ==
× 10 = 240
16 ÷ =4 120 ÷ =8
2.9 Order of operations
Addition, subtraction, multiplication and division are four fundamental
mathematical operations. When a problem contains more than one operation, it
is called a mixed operation. While simplifying a problem with mixed operations,
we need to follow a proper order to get the correct answer. Such mixed operations
are performed in the following order.
Simplify : 27 – 54 ÷ 9 + 5 × 6
Solution :
27 – 54 ÷ 9 + 5 × 6 Division is the first operation (D)
= 27 – 6 + 5 × 6 Multiplication is the second operation (M)
= 27 – 6 + 30 Addition or subtraction the third operation (A or S)
= 21 + 30 or 57 – 6 = 51
Thus, the simplification of a mixed operation is performed under the ‘DMAS’ rule.
Vedanta Excel in Mathematics - Book 6 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
EXERCISE 2.2
General Section - Classwork
1. Let’s tell and write the answers as quickly as possible.
a) 30 × 60 = ........... b) 20 × 40 = ........... c) 40 × 12 = .........
d) 200 × 60 = ........... e) 300 × 70 = ........... f) 50 × 150 = .........
g) 200 ÷ 40 = ........... h) 6000 ÷ 30 = ........... i) 14000 ÷ 700 = .........
j) 18000 ÷ 600 = ........... k) 2400 ÷ 120 = ........... l) 75000 ÷ 150 = .........
2. Let’s tell and write the answers as quickly as possible.
a) Add 7 with the difference of 12 and 9. = ...................
b) Subtract 7 from the sum of 9 and 8 = ...................
c) Add 5 with product of 4 and 6 = ..................
d) Divide 63 by 9, then multiply the quotient by 5. = ..................
3. Let’s simplify mentally, then tell and write the answers as quickly as
possible.
a) 4 + 6 – 3 = .......... b) 11 – 6 + 4 = .......... c) 7 + 5 × 4 = ........
d) 9 × 5 + 5 = ........ e) 8 × 15 ÷ 5 = ........ f) 12 ÷ 4 × 2 = ........
g) 8 + 3 × 7 – 6 = .... h) 5 × 2 + 3 – 6 = ..... i) 10 – 3 × 2 + 1 = ........
4. Let’s insert the appropriate sign in the box to get the given answer.
a) 5 7 – 3 = 9 b) 10 4 × 2 = 2
c) 6 12 ÷ 4 = 18 d) 30 6 × 4 = 20
5. Let's tell and write the correct numerals in the blanks.
a) 8 + ............. – 5 = 10 b) .............× 6 + 2 = 26
c) 20 – 24 ÷ ............. = 14 d) 3 × ............. ÷ 5 = 6
6. Let’s add or multiply and complete these puzzles.
a) + 9 7 b) × 5
20
6 15 42
15 24
12 17 9 72
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 29 Vedanta Excel in Mathematics - Book 6
Operations on whole numbers
Creative Section
7. Let’s make mathematical expressions and simplify.
a) 5 is subtracted from the sum of 8 and 7.
b) 7 is added to the product of 5 and 6.
c) 9 is subtracted from the product of 7 and 3.
d) 8 is added to the quotient of 36 divided by 4.
e) The product of 7 and the quotient of 40 divided by 5.
8. Let’s simplify these mixed operations.
a) 27 ÷ 3 × 4 + 12 – 5 b) 48 ÷ 8 × 7 + 5 – 10
c) 10 × 9 + 63 ÷ 7 – 20 d) 40 + 4 × 3 – 32 ÷ 8
e) 35 – 12 + 6 × 36 ÷ 4 f) 8 × 5 – 15 + 10 × 27 ÷ 9
g) 3 + 8 × 72 ÷ 8 – 10 – 5 h) 15 – 6 + 7 × 63 ÷ 9 – 40
9. Let’s make mathematical expressions. Then simplify and solve these
problems.
a) Sunayana bought 6 pencils at Rs 8 each and she gave a Rs 50 note to
the shopkeeper for the payment. How much change did the shopkeeper
return her?
b) Anil bought 6 exercise books at Rs 40 each and an eraser for Rs 10.
He gave a Rs 500 note to the shopkeeper for the payment. How much
change did the shopkeeper return him?
c) The cost of 1 kg of rice is Rs 90 and 1 kg of sugar is Rs 80. Find the total
cost of 5 kg of rice and 2 kg of sugar.
d) On a day, there were 27 students present in class six. 16 of them were
girls and the rest were boys. If only 3 boys were absent on that day, find
the number of boys in class six.
It’s your time - Project work!
10. a) Let’s make any four your own mixed expressions using all four signs
(+, –, ×, ÷) in each expression. Simplify them and get the correct
answers.
b) Let’s rewrite these simplifications and find the mistakes. Then complete
the simplification in the correct way.
35 – 10 + 5 27 – 6 – 4 9 + 3 × 2 18 – 5 × 3 40 ÷ 5 × 2
= 35 – 15 = 27 – 2 = 12 × 2 = 13 × 3 = 40 ÷ 10
= 20 = 25 = 24 = 39 =4
Vedanta Excel in Mathematics - Book 6 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on whole numbers
2.10 Use of brackets in simplification
Let’s study the following worked-out examples and learn about the use of
brackets in simplification of mixed operations.
Worked-out examples
Example 1: Find the product of 7 and the sum of 4 and 5.
Solution
Here, the mathematical expressions is 7 × (4 + 5) but not 7 × 4 + 5.
Now, 7 × (4 + 5) = 7 × 9 But, 7 × 4 + 5 = 28 + 5 = 33, which is the
= 63 wrong answer for the given problem.
In this problem, at first we need to find the sum of 4 and 5. Then the sum is
multiplied by 7. Therefore, to find the sum at first, we enclose 4 + 5 in the
brackets ( ).
Example 2: Find 6 times the difference between 25 and 20 is divided by 5.
Solution
Here, the mathematical expression is {6 × (25 – 20)} ÷ 5
Now, {6 × (25 – 20)} ÷ 5 = {6 × 5} ÷ 5
= 30 ÷ 5 = 6
In this case, we write the difference between 25 and 20 inside the small bracket
( ). Then we write the 6 times the difference between 25 and 20 inside the middle
brackets or braces { }.
Example 3 : Simplify 24 ÷ 2 [70 ÷ 5 {4 +(12 – 18 ÷ 6 × 3)}]
Solution
24 ÷ 2 [70 ÷ 5 {4 +(12 – 18 ÷ 6 × 3)}]
= 24 ÷ 2 [70 ÷ 5 {4 + (12 – 3 × 3)}] 18 ÷ 6 = 3 inside ( ).
= 24 ÷ 2 [70 ÷ 5{4 + (12 – 9)}] 3 × 3 = 9 inside ( ).
= 24 ÷ 2 [70 ÷ 5{4 + 3}] 12 – 9 = 3 inside ( )
= 24 ÷ 2 [70 ÷ 5{7}] 4 + 3 = 7 inside { }
= 24 ÷ 2 [70 ÷ 35] 5 {7} = 5 of 7 = 35
= 24 ÷ 2 [2] 70 ÷ 35 = 2 inside [ ]
= 24 ÷ 4 2 [2] = 2 of 2 = 4
=6
Example 4 : Simplify by making mathematical expressions.
The sum of 32 and 18 is divided by 5 and the sum of quotient and
11 is divided by 7.
Solution
The sum of 32 and 18 is divided by 5 = (32 + 18) ÷ 5
The sum of the quotient and 11 = {(32 + 18)÷ 5} + 11
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 31 Vedanta Excel in Mathematics - Book 6
Operations on Whole Numbers
The sum is divided by 7 = [{(32 + 18)÷ 5} + 11] ÷ 7
= [{50 ÷ 5} + 11] ÷ 7
= [10 + 11] ÷ 7
= 21 ÷ 7 = 3
Example 5: Bishwant has 200 rupees. He buys 2 pens at Rs 25 each and
4 exercise books at Rs 35 each. How much money is left with
him? Solve it by making mathematical expression.
Solution
Here, the expression = 200 – {(2 × 25) + (4 × 35)}
= 200 – {50 + 140} = 200 – 190 = Rs 10
Therefore, Rs 10 is left with him.
EXERCISE 2.3
General Section - Classwork
1. Let's tell and write the correct answer as quickly as possible.
a) 9 is subtracted from the sum of 8 and 7 is ...................
b) 5 is added to the difference of 15 and 6 is ...................
c) 4 times the sum of 5 and 3 is ...................
d) The quotient of 36 divided by the sum of 4 and 5 is ...................
e) The quotient of 72 divided by the product of 2 and 4 is ...................
2. Let's simplify mentally, tell and write the answers as quickly as possible.
a) 5 + (6 – 2) = .............. b) 12 – (3 + 7) = .................
c) 4 × (4 + 5) = .............. d) 15 ÷ (9 – 4) = .................
e) {3 + (6 – 2)} × 3 = ............ f) 24 ÷ {2 × (3 + 1)} = .................
3. Let's enclose the operation which is to be performed at first by using
brackets to get the given answer.
a) 3 × 4 + 6 Ans: 30 b) 7 × 14 – 8 Ans: 42
c) 5 ×7 + 3 ÷ 10 Ans: 5 d) 45 ÷ 8 – 5 × 3 Ans: 5
Vedanta Excel in Mathematics - Book 6 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Operations on Whole Numbers
Creative Section
4. Let's simplify.
a) 7 + (15 – 6) b) 6 × (5 + 4)
c) 4(2 + 6) + 4 – 3(9 – 7) d) 32 ÷ (14 – 6) × 3
e) 40 ÷ 5 (16 – 48 ÷ 4) f) 35 – {15 – (19 + 5)÷3}
g) 62 + 2 {56 ÷ (4 × 2) – 5} h) 5 {13 + 12 ÷ 3 (2 × 2) – 4}
i) 39 – 4 {16 ÷ (7 – 3)} – 23 j) 25 – [90 – 5{9 –(14 – 12)}] ÷ 5
k) 55 ÷ 11 [120 ÷ 2{4 + (10 + 5 – 7)}]
l) (7 × 6) ÷ 2 [{45 ÷ 3(7 × 2 – 15 + 6)} + 4]
5. Let's make the mathematical expressions, then simplify.
a) 4 times the sum of 7 and 2.
b) 7 times the difference of 12 and 8.
c) 9 is subtracted from 2 times the sum of 7 and 4.
d) 48 is divided by 3 times the difference of 15 and 11.
e) The sum of 60 and 30 is divided by 10 and the difference of quotient
and 3 is divided by 2.
6. Let's make the mathematical expressions. Simplify them and find the
correct answers.
a) A sick person takes 10 ml of medicine three times a day. How much
medicine does she/he take in 15 days?
b) The distance between Dakshes’s house and his school is 7 km. How
many kilometres does he travel in 6 days?
c) Mrs. Chamling earns Rs 6,000 in a week. She spends Rs 250 for food and
Rs 75 for transportation everyday. How much money does she save in a
week?
d) Sunayana had 20 marbles. She lost 6 marbles and she divided the rest of
them among her 7 friends. How many marbles would each get ?
e) Pratik has 300 rupees. He buys 2 boxes at Rs 60 each and 3 pens at
Rs 30 each. How much money is left with him ?
f) There are two baskets each of them containing 18 apples. 10 apples are
rotten and not fit for eating. You add 4 more apples and then divided
among 6 friends. How many apples would each get.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 33 Vedanta Excel in Mathematics - Book 6
Unit Algebra
9
9.1 Constant and variables - Looking back
Classwork - Exercise
Let’s tell and tick the correct answers as quickly as possible.
1. a) x represents the number of provinces of Nepal. x is (constant/variable)
b) x represents the heights of the students in your class. x is (constant/
variable)
2. a) 2ab is a (monomial/binomial) expression.
b) 2a + b, is a (monomial/binomial) expression.
3. a) In x2 , coefficient is .........., base is .........., power is .............
b) In 7y3, coefficient is ............., base is .........., power is ..............
4. a) If x = 2, y = 3, then, x + y = ................. and x y = .................
b) If l = 8, b = 5 then l × b = ................. and 2(l + b) = .................
What does the number 5 represent? Does it represent six, four, or any other
number of things?
The numbers such as 1, 2, 3, 4, 5, ... etc. always represent the fixed number of
things. These numbers are called constants. We may also use letters like x, y, z, a,
b, c,... in the place of numbers. If a letter represents a fixed value (number), it is
considered as constant. However, if the letter represents many values (numbers),
it is called variable. For example,
(i) x represents the sum of 7 and 3. Here x = 10. Here, x represents a fixed
value. So, x is a constant.
(ii) x represents the prime numbers less than 10. Here, the values of x may be 2,
3, 5, or 7. Thus, x represents many numbers. Therefore, it is a variable.
9.2 Algebraic terms and expressions
Let’s study the following illustrations and investigate the idea of algebraic
expressions.
‘5 times x’ is an expression o 5x is an algebraic expression.
‘y is added to 3’ is an expression o y + 3 is an algebraic expression.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 135 Vedanta Excel in Mathematics - Book 6
Algebra
Thus, an algebraic expression is a mathematical statement formed by the four
fundamental operations (+, –, ×, ÷) on constants and variables.
Let’s consider an algebraic expression 2a + 3b – 6.
Here, 2a, 3b and 6 are called the terms of the expression. In this expression,
there are 3 terms.
Depending on the number of terms, an expression may be monomial, binomial,
trinomial or polynomial.
An algebraic term itself is a monomial expression. So,
Monomial expression a monomial expression contains only a single term.
a
For example, 5x, 6ab, b , 8p÷q etc. are monomial
expressions.
An algebraic expression containing two unlike terms is
Binomial expression called a binomial expression. a + b, 3p – 4q, ab + 7xy,
etc. are binomial expressions.
An algebraic expression containing three unlike terms
Trinomial expression is called a trinomial expression. 2x + 3y – 5, a + 4b – 9c,
xy – yz + zx, etc. are trinomial expressions.
An algebraic expression containing two or more than two unlike terms is also
called a multinomial expression.
Binomial and trinomial are also the multinomial. For example:
x + y, xy – 5, abc + a, etc. are multinomial of two terms.
a + b + c, xy + x + y, mn – 5 m + 7, etc are multinomial of three terms.
2a – 3b + c – 8, 2xy – 4yz + 7zx + 9, etc are multinomial of four terms.
9.3 Coefficient, base and exponent of algebraic terms
Let’s consider an algebraic term 3x2.
Her, 3 is called the coefficient, x is the base and 2 is the exponent. The coefficient
3 in 3x2 tells that the term x2 is added 3 times.
i.e. x2 + x2 + x2 + = 3x2
Similarly, the exponent 2 of x in 3x2 tells that the base x is multiplied 2 times.
i.e. 3 u x u x = 3x2 .
A power is a product of repeated multiplication of the same base. The exponent
of a power is the number of times the base is multiplied.
Vedanta Excel in Mathematics - Book 6 136 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
5a3 x×x×x
Exponent
Base x3 exponent
Coefficient base
Let’s take another algebraic term y. power
Here, y = 1.y1. So, in y, its coefficient is 1 and exponent is also 1.
If the coefficient of a base is a number, it is called a numerical coefficient. In 3x,
3 is the numerical coefficient of x. If the coefficient of a base is a letter, it is called
a literal coefficient. In ax, a is the literal coefficient of x.
EXERCISE 9.1
General Section – Classwork
1. Let’s tell and write whether the letters represent ‘constant’ or ‘variable’.
a) x represents the natural numbers between 3 and 5. x is a ........................
b) x represents the natural numbers between 6 and 9. x is a ........................
c) y represents the even prime numbers. y is a ........................
d) y represents the composite numbers less than 10. y is a ........................
2. Let’s tell and write whether the following expressions are monomial,
binomial or trinomial.
a) 7xyz is ......................................... expression.
b) 2a + 5b – c is a ......................................... expression.
c) xy – ab is a ......................................... expression.
pq ......................................... expression.
d) r is a ......................................... expression.
e) 5x ÷ y is a
3. a) In 5a2, coefficient is .............. base is .............. power is ..............
b) In 4x3, coefficient is .............. base is .............. power is ..............
c) In y, coefficient is .............. base is .............. power is ..............
d) In 6ax, numerical coefficient is ............... literal coefficient is ..............
4. Let’s tell and write the algebraic expressions as quickly as possible.
a) Sum of a and b = .........................................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 137 Vedanta Excel in Mathematics - Book 6
Algebra
b) Difference of a and b = .........................................
c) product of a and b = .........................................
d) The quotient of a divided by b = .........................................
e) Two times the sum of x and y = .........................................
f) Five tomes the product of m and n = .........................................
Creative Section
5. Let’s take the terms x, y and z. Make monomial, binomial and trinomial
expressions of your own using these terms.
6. Rewrite the following statements in algebraic expressions.
a) Product of x and y is added to z.
b) Three times the sum of x and y is increased by 5.
c) Two times the difference of p and q is decreased by 4.
d) Product of p and q is subtracted from r.
e) Five times the product of a and b is increased by x
f) The sum of x and y is divided by 2 and decreased by 7.
7. a) The present age of Anamol is x years.
(i) How old was he 2 years before?
(ii) How old will he be after 2 years ?
(iii) If his father is four times older than him, how old is his father ?
b) The breadth of a room is b metre. If its length is 5 metres longer than its
breadth, represent the length of the room by an expression.
c) The marks obtained by A in maths is x. The marks obtained by B is double
than that of A and marks obtained by C is double than that of B. Represent
the marks obtained by B and C by expressions.
8. Rewrite the following formulae in algebraic expressions.
a) The perimeter of a triangle is sum of its three sides a, b and c of the
triangle. What is the formula of perimeter of the triangle ?
b) The area of a triangle is half of the product of base (b) and height (h).
What is the formula of area of the triangle ?
c) The perimeter of a square is four times of its length (l). What is the
perimeter of the square?
d) The area of a square is the square of its length (l). What is the area of
the square?
Vedanta Excel in Mathematics - Book 6 138 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
e) The perimeter of a rectangle is two times the sum of its length (l) and
breadth (b). What is the perimeter of the rectangle ?
f) The area of a rectangle is the product of its length (l) and breadth (b).
What is the area of the rectangle ?
It's your time - Project work!
9. Let's write any two different monomial expressions. Write the coefficient,
base and exponent of each expression.
Expressions Coefficient Base Exponent
9.4 Evaluation of algebraic expressions
We obtain the value of a term or an expression by replacing the variable/s
of the term or expression with numbers. It is called evaluation of a term or
expression. For example:
If x = 2 and y = 3, then 4x = 4 × 2 = 8, 2(x + y) = 2(2 + 3) = 2 × 5 = 10
(xy)2 = (2 × 3)2 = 62 = 36, (x – y)2 = (2 – 3)2 = (– 1)2 = 1 and so on.
Worked-out examples
Example 1: If y = 2x + 1 and x is a variable of the set B = {1, 2, 3}, find the
possible values of y.
Solution :
When x = 1, then y = 2x + 1 = 2 × 1 + 1 = 2 + 1 = 3
When x = 2, then y = 2x + 1 = 2 × 2 + 1 = 4 + 1 = 5
When x = 3, then y = 2x + 1 = 2 × 3 + 1 = 6 + 1 = 7
So, the required values of y are 3, 5 and 7.
Example 2: If a = 3b, express 4a + 5b in terms of b and evaluate the
expression when b = 2.
Solution :
Here, a = 3b
∴ 4a + 5b = 4 × 3b + 5b = 12b + 5b = 17b
Now, when b = 2, then 17b = 17 × 2 = 34
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 139 Vedanta Excel in Mathematics - Book 6
Algebra
Example 3: If x = 2y = 4z, express x + 3y + z in terms of z and evaluate the
expression when z = 3.
Solution :
Here, x = 2y = 4z
∴ x = 4z and 2y = 4z
4z
y = 2 = 2z
Now, x + 3y + z = 4z + 3 × 2z + z = 11z
Again, when z = 3, then 11z = 11 × 3 = 33
EXERCISE 9.2
General Section – Classwork
1. Let's input x = 1, 2, 3, ... Then tell and write outputs.
Input (x) Outputs
1
X+1 X+5 2X 3X
................. .................
................. .................
2 ................. ................. ................. .................
3 ................. ................. ................. .................
4 ................. ................. ................. .................
5 ................. ................. ................. .................
2. Let's tell and write the values as quickly as possible.
a) If x = 2 and y = 3, then 4x = ..........., 5y = ........... , 2xy = ...............
b) If x = – 1 and y = 2, then x + y = ................, x – y = ...............
c) If a = 3 and b = 2, then (ab)2 = .............., (a + b)2 = ...............
d) If l = 5 and b = 4, then l × b = ................, 2 (l + b) = ...............
3. Let’s tell and write the answers as quickly as possible.
a) If x = 2y, then x + y in terms of y = .......................
b) If x = 3y, then 2xy in terms of y = ......................
c) If a = 2b and b = 3, then a + b = ................
d) If p = 3q and q = 2, then p – q = ....................
Vedanta Excel in Mathematics - Book 6 140 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
Creative Section - A
4. If x = 2 and y = 3 and z = 4, evaluate the following expressions.
a) x + y + z b) 2x + 3y – z c) 3(x – y + z) d) 5xy
e) x2 + y2 f) y2 + z2 – x2 g) x+ 2y h) (x–y+z)2
5. If = 3, z y
= 4 and = 5, find the values of following expressions.
a) 2 – b) 3 + 2 – c) ( + ) d) ( + ) ÷
6. a) The area of a square = l2. Find the area of squares in sq. cm.
(i) l = 4cm (ii) l = 7cm (iii) l = 3.5 cm (iv) l = 4.2 cm
b) The perimeter of a square = 4l. Find the erimeter of squares in cm.
(i) l = 6cm (ii) l = 9cm (iii) l = 8.5 cm (iv) l = 4.3 cm
7. a) Area of rectangle = l × b. Find the area of rectangles in sq. cm.
(i) l = 7cm, b = 4cm (ii) l = 8cm, b = 6cm (iii) l = 7.5 cm, b = 4cm
b) perimeter of rectangle = 2 (l + b). Find the perimeter of rectangles in cm.
(i) l = 6cm, b = 4cm (ii) l = 9cm, b = 5cm (iii) l = 6.4cm, b = 3.7 cm
8. The area of four walls of a room = 2h(l + b). Find the area of four walls in sq. m.
(i) l = 10 m, b = 8 m, h = 5 m (ii) l = 12 m, b = 7.5 m, h = 4 m
9. a) The area of a circle = Sr2, where S = 22 . Find the area of circles in sq. cm.
7
(i) r = 7 cm (ii) r = 14 cm (iii) r = 21 cm
b) The perimeter of a circle = 2Sr, where S = 22 . Find the perimeter of circle
in cm. 7
(i) r = 7 cm (ii) r = 14 cm (iii) r = 21 cm
10. a) The volume of a cube = l3. Find the volume of cubes in cubic cm.
(i) l = 2 cm (ii) l = 4 cm (iii) l = 5 cm
b) The area of a cube = 6l2. Find the area of cubes in sq. cm.
(i) l = 3 cm (ii) l = 5 cm (iii) l = 6 cm
11. a) The volume of a cuboid = l u b u h. Find the volume of cuboids in cubic cm.
(i) l = 5 cm, b = 4 cm, h = 3 cm
(ii) l = 10 cm, b = 7 cm, h = 5.5 cm
b) The area of a cuboid = 2(lb + bh + lh). Find the area of cuboids in sq. cm.
(i) l = 8 cm, b = 5 cm, h = 2 cm
(ii) l = 10 cm, b = 6 cm, h = 4 cm
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 141 Vedanta Excel in Mathematics - Book 6
Algebra
Creative Section - B
12. If x = 5 cm, find the length of the following line segments.
13. Write the algebraic expressions to represent the perimeters of the following
figures. If x = 2, y = 3 and z = 5, find the perimeters of the figures.
14. a) x is a variable on the set A = {1, 2, 3}, that is, x can be replaced by 1, 2
and 3. Evaluate the expressions (i) x + 5 (ii) 2x – 1
b) If y = 2x + 1 and x is a variable on the set B = {2, 4, 6}, find the possible
values of y.
15. a) If x = 2y, express 2x + 5y in terms of y and evaluate the expression when
y = 3.
b) If a = b + 3, express a + b in terms of b and evaluate the expression when
b = 3.
c) If x = 2a + 1, show that 3x – 6a + 7 = 10
It's your time - Project work!
16. a) Let's write a monomial expression. Replace the variable of the expression
by the natural numbers less than 4 and evaluate the expression.
b) Let's write a binomial expression of the form ax + b, where a is a numeral
coefficient and b is an integer. Replace the variable by the whole numbers
less than 4 and evaluate your expression.
9.5 Like and unlike terms
1 apple and 2 apples are like (same) type of things. Similarly, x and 2x are like terms.
3 pens and 4 pens are like (same) type of things. Similarly, 3y and 4y are like terms.
The algebraic terms having the same base and equal power are called like terms.
For examples:
3x, 5x, 8x, etc. are like terms because they have same base x and equal power 1.
5a2, 9a2, 12a2, etc. are like terms because they have the same base a and equal
power 2.
2 apples and 2 pens are unlike (different) types of things. Similarly, 2x and 2y are
unlike terms. 2x and 2y have unlike bases.
Vedanta Excel in Mathematics - Book 6 142 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
Again, 2x, 3x2, 5x2 etc. have the same base x but each base does not have equal
power. So, they are also unlike terms. Similarly, 5x2, 3y2, 4z2, etc. are unlike
terms because these terms do not have the same base, even though each base
has the equal power.
Thus, the algebraic terms which have different bases or different powers are
called unlike terms.
9.6 Addition and subtraction of algebraic terms
Let's study the following illustrations and learn to add coefficients while adding
like terms.
2 apples + 3 apples = 5 apples 2x + 3x = 5x
4 oranges + 5 oranges = 9 oranges 4y + 5y = 9y
6 pens – 2 pens = 4 pens 6a – 2a = 4a
8 books – 3 books = 5 books 8p – 3p = 5p
Thus, to add or subtract algebraic terms, we should add or subtract the coeffi-
cients of like terms. For example:
2x + 3x = 5x, 4x2 + 6x2 = 10x2, 3ab + 2ab + ab = 6ab
8y – 5y = 3y, 9y3 – 4y3 = 5y3, 7pqr – 6pqr = pqr
But, we cannot add or subtract the coefficients of unlike terms.
For example:
2x + 3y + x + 4y = 3x + 7y, 4a + 5a + 8 = 9a + 8 and so on.
EXERCISE 9.3
General Section A – Classwork
1. Let’s tell and write whether these terms are like or unlike.
a) 4x, 7x, – 3x ................. terms b) 3x, 2x2, xy ................. terms
c) 3a2, 4a3, 6a ................. terms d) p2, 3p2, 8p2 ................. terms
e) 2xy, 5xy, 3xy ................. terms f) 2x2y, 5xy2, 3 2y2 ................. terms
2. Let’s tell and write the answers as quickly as possible.
a) a + 2a = ........... b) x2 + x2 = ............ c) 3xy + 2xy = ...........
d) 5x – 3x = ........... e) 4a2 – a2 = ............ f) 9pq2 – 4pq2 = ..........
3. a) What should be added to 2x to get 5x? ..................
b) What should be subtracted from 8x to get 3x ? ..................
Creative Section
4. Add or subtract.
a) 4x + 7x b) 5a2 + 9a2 c) 5xy + 3xy d) 3abc + 6abc
e) 10p3 – 3p3 f) 12mn – 7mn g) 9x2y – 4x2y h) 8xy2 – 7xy2
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 143 Vedanta Excel in Mathematics - Book 6
Algebra
5. Simplify b) 2p2 + 3p2 + p2 + 4p2
a) a + 2a + 3a
c) 2x + 3y + x + y d) 6m + 2n + n – 4m
e) 3xy + yz – xy – 2yz – 5 f) 5p2 – 2q2 + 2p2 – 3q2
6. a) What should be added to 9x to get 15x ?
b) What should be added to p to get p + q ?
c) What should be subtracted from 12a to get 5a ?
d) What should be subtracted from 3x to get 3x – 2y ?
Game Time!
7. Complete this obstacle course.
START x Add 4x Subtract 6x Add 5x
Subtract 3x Same answer in Subtract x
Add 7x both ways? If
not, start again.
Add 6x
Subtract 2x Add 9x Subtract 3x FINISH
It's your time - Project work!
8. Let's write any two like terms with the variable x. Find the sum and difference
of your like terms.
Like terms Sum Difference
..............., ............... ............................ ............................
..............., ............... ............................ ............................
9.7 Addition and subtraction of algebraic expressions
While adding or subtracting two or more binomials, trinomials or multinomial,
the like terms should be arranged in the same column, then, their coefficients
should be added or subtracted. Alternatively, we can arrange the expressions
horizontally and the addition or subtraction of like terms can be performed.
Worked-out examples
Example 1: Add 7x2 + 4xy + 5y2 and 2x2 – 3xy – 8y2.
Solution:
Vedanta Excel in Mathematics - Book 6 144 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
Addition by horizontal arrangement Addition by vertical arrangement
7x2 + 4xy + 5y2 + 2x2 – 3xy – 8y2 7x2 + 4xy + 5y2
= 7x2 + 2x2 + 4xy – 3xy + 5y2 – 8y2 2x2 – 3xy – 8y2
= 9x2 + xy – 3y2 9x2 + xy – 3y2
Example 2: Subtract 2ab – 4bc + 7 from 5ab – 4bc – 2.
Solution:
Subtraction by horizontal arrangement Subtraction by vertical arrangement
5ab – 4bc – 2 – (2ab – 4bc + 7) 5ab – 4bc – 2
= 5ab – 4bc – 2 – 2ab + 4bc – 7 ±2ab 4bc ± 7
= 5ab – 2ab – 2 – 7 = 3ab – 9 3ab – 9
Example 3: What should be added to 3x – 4 to get 8x + 5?
Solution: Let’s think, what should be
added to 7 to get 10? It’s 3 and
The required express to be added is it is 10 – 7.
8x + 5 – (3x – 4) It’s my good idea to work out
such problems!
= 8x + 5 – 3x + 4
= 8x – 3x + 5 + 4 = 5x + 9
Example 4: What should be subtracted from 8x – 3y + 2 to get 5x + 2y – 1?
Solution:
The required expression to be subtracted is
8x – 3y + 2 – (5x + 2y – 1) Let’s think, what should be
= 8x – 3y + 2 – 5x – 2y + 1 subtracted from 9 to get 7? It’s 2
= 8x – 5x – 3y – 2y + 2 + 1 and it is 9 – 7.
= 3x – 5y + 3
I can also work out such questions.
Example 5 : If x = m + 2 and y = n – m, show that x + y = n + 2.
Solution :
Here, x = m + 2 I got it !
y=n–m In x + y, x is replaced by m + 2
Now, x + y = m + 2 + n – m and y is replaced by n – m.
= n + 2 proved.
EXERCISE 9.4
General Section – Classwork
Let’s tell and write the correct answers as quickly as possible.
1. a) Sum of x + 1 and x + 2 = ............................................
b) Sum of 3x + 5 and x – 4 = ............................................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 145 Vedanta Excel in Mathematics - Book 6
Algebra
c) Sum of a2 + b2 and a2 – b2 = ............................................
d) Difference of 2x + 3 and x + 1 = ............................................
e) Difference of 4y + 7 and 3y + 2 = ............................................
f) Difference of 8a + 5 and 4a – 3 = ............................................
2. Complete the following table.
+ a+2 a–1 a+b
a+5 2a – b + 2 2a – 4 2a + b + 5
a–3
a–b
Creative Section - A
3. Add.
a) x + 3 and x + 2 b) 2x + 5 and 3x + 1 c) 3a + 8 and 4a – 5
d) 4 + 3a and 5 – 8a e) 9m – 2 and m – 7 f) 11 – 2n and 4 – 5n
g) 3a + 4b and 7a + 2b h) 8a – 5b and a – 3b i) p2 + q2 and 2p2 + 3q2
j) 3a2 – 5b2 and 2a2 + 7b2 k) x2 – xy + y2 and 2x2 + 6xy – y2
l) x2 + x – 2 and 2x2 – 5x + 7 m) 2ab – 3bc + 4 and ab + 4bc – 5
n) 8abc – 5ab + 3a and 2a + 7ab – 4abc
4. Subtract.
a) x + 1 from 5x + 4 b) 2x + 3 from 5x + 1
c) 6a – 7 from 9a + 2 d) 2a + 5 from 6a – 1
e) 3p – 2 from 5p – 7 f) 8 – 3p from 1 – p
g) 5m + 3n from 7m – 2n h) 9mn – 5p from 6mn – p
i) 4x2 – 3y2 from 9x2 + 4y2 j) 3a3 + 5b3 from 8a3 – b3
k) 2x3 – 5x2 + x – 6 from 4x3 + 2x2 – 3x + 7
l) 4ax – 6by + 7 from 6ax + 2by – 3
m) abc + 2ab – 3bc – ca from 3ab – ab + bc + 2ca
n) xyz – 4xy + 5 from 5xyz + xy + 3yz – 4
5. a) What should be added to 2x – 3 to get 5x + 7?
b) What should be added to 3a + 4b to get 7a – 5b?
c) What should be subtracted from 5m – 3n to get 2m + 5n?
Vedanta Excel in Mathematics - Book 6 146 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
d) What should be subtracted from 5x – 3y to get 2x – 5y?
e) To what expression 2a – 3b + 1 must be added to get 4a + 7b – 3?
f) From what expression x2 + 5y2 – 3xy must be subtracted to get 2x2 – y2 + 4xy?
Creative Section - B
6. a) If x = a + 7 and y = b – a, show that x + y = b + 7.
b) If x = 2m – n and y = m + n, show that x – y = m – 2n.
c) If x2 = 2a2 – b2, and y2 = 2b2 – c2 and z2 = 2c2 – a2,
show that: x2 + y2 + z2 = a2 + b2 + c2.
7. Simplify:
a) 4x + 3y – (3x + y) b) 5a – 3b – (a + 6b)
c) 8p + q – (5p – 3q) d) 6a – 5x – (2x – 3a)
e) – 7m – 2n – 5 – (–8m – n +1) f) –a – 2t – (t – 4a) – (–3a – 5t)
g) –(b – z) – (–2b + z) – (–3b – z) h) –(a – b) – (–4b + 3c) – (c – 2a)
8. a) The sides of a triangle are x + 4, 2x – 3 and 3x + 1, find its perimeter.
b) The sides of a triangle are 2x + 3y, x + 2y and 7x – 2y, find its perimeter.
It's your time - Project work!
9. a) Let's write two binomial expressions of the form ax + b, where a is a
natural number and b is an integer. Find the sum and difference of your
expressions.
b) Let's write two trinomial expressions of the form ax + by + c, where a, b,
and c are integers. Find the sum and difference of your expressions.
9.8 Multiplication of algebraic terms
While multiplying the algebraic terms, the coefficients of the terms are multiplied
and the exponents of the same bases are added. For example:
Example 1: Multiply 3x by 2x. 3 × 2 = 6 (Coefficient are
multiplied.)
Solution: x × x = x1+1 = x2 (Exponents of
the same bases are added.)
Here, 3x u 2x = 6x2
x2 x2 x2 x
The multiplication of 3x × 2x can also be 2x
shown diagrammatically.
From diagram, x2 x2 x2 x
3x × 2x = x2 + x2 + x2 + x2 + x2 + x2 = 6x2 x xx
3x
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 147 Vedanta Excel in Mathematics - Book 6
Algebra
Example 2: Multiply 4a2b by – 5ab2 I got it!
Solution:
Here, 4a2b u (– 5ab2) = – 20a3b3 4 × (–5) = –20
a2 × a = a2 + 1 = a3
b × b2 = b1 + 2 = b3
9.9 Multiplication of polynomials by monomials
Of course, monomial, binomial and trinomial with positive exponents of variables
are also called polynomials. To multiply a polynomial by a monomial, each term
of the polynomial is separately multiplied by the monomial. For example:
Example 3: Multiply: (a + b) × c. ac + bc c
Solution:
ac bc
(a + b) × c = a × c + b × c ab
= ac + bc a+b
Example 4: Multiply 4a2 + 3b2 by 2ab
Solution: It’s easier!
Here, 2ab × (4a2 + 3b2) = 2ab × 4a2 + 2ab × 3b2 Each term of 4a2 + 3b2 is
separately multiplied by 2ab
= 8a3b + 6ab3.
Example 5: Multiply 7x2y2 – 4xy + 3 by – 2xy
Solution:
Here, –2xy(7x2y2 – 4xy + 3) = –2xy × 7x2y2 – 2xy(–4xy) – 2xy × 3
= –14x3y3 + 8x2y2 – 6xy
Example 6: If x = p + 3 and y = 2p, show that xy = 2p2 + 6p.
Solution: It’s easier!
Here, x = p + 3 and y = 2p In xy, x is replaced by p + 3 and
∴ xy = (p + 3) × 2p y is replaced by 2p.
= 2p × p + 2p × 3 = 2p2 + 6p proved.
Example 7: If x = 2a, y = 3a and a = 4, find the value of 6xy .
Solution:
Here, x = 2a and y = 3a
∴ 6xy = 6 × 2a × 3a = 36a2 = 6a
When a = 4, then 6a = 6 × 4 = 24 148 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Vedanta Excel in Mathematics - Book 6
Algebra
EXERCISE 9.5
General Section – Classwork
1. Area of rectangle = length (l) × breadth (b). Let’s find the area of the
following rectangles.
a) b)
3y Area (A) = ............ 5q Area (A) = ............
4x
c) d) 2p (2x+1) 2a Area (A) = ............
2x Area (A) = ............ 5a Area (A) = ............
3x
3x
e) f)
x Area (A) = ............
(2x + 1)
2. Volume of cuboid (V) = length (l) × breadth (b) × height (h). Let’s find
the volume of following cuboids.
a) b) 3z
c
b 2y
a 4x
V = ..................... V = .....................
c) d)
x 2p
2x 3p
2x 5p
V = ..................... V = .....................
3. Let's tell and write the products as quickly as possible.
a) x × x = ........... b) 2x × x = .......... c) 3a × 2a = ...........
d) p2 × p = .......... e) 3b2 × 4b2 = ........... f) 2x3 × 5x2 = ...........
g) x (x + 1) = ................... h) 2x(x – 3) = ...................
i) 3x(x2 + 2) = ................... j) 2a(a2 – 5) = ...................
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 149 Vedanta Excel in Mathematics - Book 6
Algebra
4. Look at the targets and investigate the operation. Tell and write the
answers as quickly as possible. a
ay
x xa ay y
2x 4a aa
x2 a4
a
x 2p y 2x
2y 3y
2x
xx p 3x
xx 3x x
Creative Section - A
5. Multiply.
a) x × x b) 2x × 3x c) 3x × 4x
d) x × 2x2 e) 2y × 3y2 f) 3xy × 2xy
g) 4a2b × 3ab2 h) –5xyz × 4xyz i) 3xy × (–5x2y2)
6. Simplify.
a) x u x u x2 b) 2a2 u 3a u 5a c) 3x u 2y u 2x u y
d) p2 u 5q u 2p u 3q2 e) 3b3 u c2 u 2b2 u 4c3 f) xyz u 2xy u yz u 5zx
g) (– 3ab) u (– 2bc) u (– abc) h) (–5qr) × (– 2pqr) u pq
i) 6x2y u (– xy2) u xyz u (– 3yz2)
7. Simplify.
a) 2x × 2x + 2x × 5 b) 3a × 2a – 3a × 4 c) 5x2 × x2 + 5x2 × 8
d) ab × 2a – ab × 3b e) ab × 5a2 – ab × 3b2 f) 10x2y2 × 3x – 10x2y2 × 2y
g) –2x × 4x – (–2x) × 3 h) –pq × 2p – (–pq) × 3q i) –3abc × 2a2 + (–3abc) × 5bc
8. Multiply.
a) 3x (2x + 5) b) 4a (5a – 6) c) 6p (8p2 – 1)
d) 2y2 (y2 + 7) e) 6m2 (2 – 3m2) f) 7b3 (8 + 3b2)
g) xy (x – y) h) 2xy (2x + 3y) i) 2ab (3a2 – 4b2)
j) – 3pq (2p + 5q) k) – 2bc (7b – 3c) l) – 5xy (5x2 – 4y2)
m) – 4x (x2 – 2x + 1) n) – 5a (3a2 + 3a – 5) o) – 3xy (4x2 – 5xy – 3y2)
Creative Section - B
9. a) If x = a + 5 and y = 2a, show that xy = 2a2 + 10a.
b) If p = 5x and q = 3x – 1, show that 2pq = 30x2 – 10x.
c) If a = x – 3, b = 3x and x = 2, find the value of 4ab.
Vedanta Excel in Mathematics - Book 6 150 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Algebra
d) If m = n = 2a and a = 3, find the value of mn.
e) If p = 2x, q = 3x and x = 3, find the value of 6pq .
10. Simplify. b) x(x – 1) + 3(x – 1)
a) x(x + 1) + 2(x + 1) d) a(2a + 1) – a(2a + 1)
c) y(y + 2) + 1(y + 2) f) a(a – b) – b(a – b)
e) x(x + y) + y(x + y)
9.10 Multiplication of polynomials
In the case of multiplication of two binomial expressions, each term of a binomial
is separately multiplied by each term of another binomial. Then the product is
simplified. The multiplication of any two polynomials is also worked out in the
similar process.
Worked-out examples
Example 1: Multiply a) (a + b) by (a + b) b) (2m + 3n) by (3m – 2n)
Solution:
a) By horizontal arrangement By vertical arrangement
(a + b) (a + b) = a (a + b) + b (a + b) a+b
= a2 + ab + ab + b2 ×a+b
= a2 + 2ab + b2
a2 + ab Add
+ ab + b2
a2 + 2ab + b2
b) By horizontal arrangement By vertical arrangement
(3m – 2n) (2m + 3n)= 3m (2m + 3n) – 2n(2m + 3n) 2m + 3n
= 6m2 + 9mn – 4mn – 6n2 × 3m – 2n
= 6m2 + 5mn – 6n2 6m2 + 9mn Add
– 4mn – 6n2
Example 2: Multiply (2a – 3b + 5) by (4a – b). 6m2 + 5mn – 6n2
Solution:
By horizontal arrangement By vertical arrangement
(4a – b) (2a – 3b + 5) 2a – 3b + 5
= 4a (2a – 3b + 5) – b (2a – 3b + 5)
= 8a2 – 12ab + 20a – 2ab + 3b2 – 5b × 4a – b
= 8a2 – 14ab + 3b2 + 20a – 5b 8a2 – 12ab + 20a
– 2ab + 3b2 – 5b
8a2 – 14ab + 3b2 + 20a – 5b
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 151 Vedanta Excel in Mathematics - Book 6
The words you are searching are inside this book. To get more targeted content, please make full-text search by clicking here.
Vedanta Excel in Mathematics Book 6 Final (2078)
Related Publications
Discover the best professional documents and content resources in AnyFlip Document Base.
Search