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.
Vanasthali, Kathmandu, Nepal
+977-01-4382404, 01-4362082
[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.
Published by:
Vedanta Publication (P) Ltd.
Vanasthali, Kathmandu, Nepal
+977-01-4382404, 01-4362082
[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.
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, HOD, 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 programmes 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 very valuable suggestions about the effective methods of teaching-learning
mathematics and many reference materials.
Grateful thanks are due to Mr. Tara Bahadur Magar for his painstakingly editing of the series.
Moreover, I gratefully acknowledge all Mathematics Teachers throughout the country who
encouraged me and provided me 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. to get this series published. I would
like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha,
Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during
the preparation of the series.
Last but not the least, 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 5
1 1.1 Set - Looking back, 1.2 Membership of a set, 1.3 Set notation, 1.4 methods of
writing members of set, 1.5 cardinal number of sets, 1.6 Types of sets, 1.7 Relationships
Unit between sets, 1.8 Universal set and subset
2 Operations on whole numbers 17
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,
Unit 2.10 Use of brackets in simpli ication
4 Properties of Whole Numbers 32
3.1 Various types of numbers - Looking back, 3.2 Prime and Composite numbers, 55
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, 3.11 Sequence and pattern
of numbers
Integers, Rational and Irrational Numbers
4.1 Integers - Introduction, 4.2 Operations on integers, 4.3 Rational
numbers – Introduction, 4.4 Terminating and non-terminating rational numbers,
4.5 Irrational numbers,
Unit Fraction and Decimal 68
5 5.1 Fraction - Looking back, 5.2 Equivalent fractions, 5.3 To ind the fractions equivalent
to the given fraction, 5.4 Like and unlike fractions, 5.5 To convert unlike fractions into
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 Simpli ication of fractions,
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 Ratio and Proportion 108
6 6.1 Ratio - Looking back, 6.2 Meaning of ratio, 6.3 Proportion, 6.4 Types of proportions
Unit Unitary Method 114
7 7.1 Unitary Method - Looking back, 7.2 Unit number of quantity and unit value,
7.3 Time and work
Unit Percent 122
8 8.1 Percent - Looking back , 8.2 Percent and percentage, 8.3 Conversion of fractions or
decimals to percent, 8.4 Conversion of percent to fractions or decimals, 8.5 To express
a given quantity as the percent of whole quantity, 8.6 To ind the value of the given
percent of a quantity, 8.7 To ind a quantity whose value of certain percent is given
Unit Profit and Loss 132
140
9 9.1 Cost price (C. P.) and selling price (S. P.) - Looking back , 9.2 Pro it and loss, 9.3 Pro it
or loss percent, 9.4 Discount, 9.5 Discount percent , 9.6 Value Added Tax (VAT)
Unit
Simple Interest
10
10.1 Simple Interest - introduction, 10.2 Calculation of simple interest
Unit Algebra 144
11 11.1 Constant and variables - Looking back, 11.2 Algebraic terms and expressions,
11.3 Coef icient, base and exponent of algebraic terms, 11.4 Evaluation of algebraic
Unit expressions , 11.5 Like and unlike terms, 11.6 Addition and subtraction of algebraic
terms, 11.7 Addition and subtraction of algebraic expressions, 11.8 Multiplication of
12 algebraic terms, 11.9 Multiplication of polynomials by monomials, 11.10 Multiplication
of polynomials, 11.11 Division of algebraic terms, 11.12 Division of polynomials by
Unit monomials, 12.13 Division of polynomials by binomials,
13 Equation, Inequality and Graph 165
Unit 12.1 Equation - Looking back, 12.2 Mathematical statements, 12.3 Open
mathematical statements,12.4 Equation and solution of an equation, 12.5 The facts
14 for solving equations, 12.6 Process of solving equations, 12.7 Verbal problems - Use
of equations, 12.8 Trichotomy, 12.9 Negation of trichotomy, 12.10 Trichotomy rules,
Unit 12.11 Inequalities, 12.12 Solution of inequalities
15 Coordinates 185
Unit 13.1 Ordered pairs - Looking back, 13.2 Coordinates, 13.3 Re lection of geometrical
igures, 13.4 Rotation of geometrical igures, 13.5 Displacement
16
Geometry: Point and Line 193
Unit
14.1 Point, line, ray, line segment and plane - review, 14.2 Intersecting line segments,
17 14.3 Parallel line segments, 14.4 Perpendicular line segments
Unit Geometry: Angle 199
18 15.1 Angles (review), 15.2 Angles formed by a revolving line, 15.3 Measurement of
angles, 15.4 Types of angles, 15.5 Different pairs of angles, 15.6 Angles made by a
Unit transversal with straight line segments, 15.7 Pairs of angles made by a transversal with
parallel line segments
19
Geometry: Triangles and Polygons 214
Unit
16.1 Triangle – review, 16.2 Types of triangles by sides, 16.3 Types of triangles
20 by angles, 16.4 Sum of the angles of a triangle, 16.5 Exterior angle of a triangle,
16.6 Quadrilaterals, 16.7 Some special types of quadrilaterals, 16.8 Sum of the angles
Unit of a quadrilateral, 16.9 Polygons
21 Geometry: Construction 226
17.1 Construction of angles, 17.2 Bisecting a given angle, 17.3 Construction of
perpendiculars, 17.4 Construction of a line parallel to given line, 17.5 Construction of
regular polygons,
Perimeter, Area and Volume 235
18.1 Perimeter, area and volume - Looking back, 18.2 Perimeter of plane igures,
18.3 Area of plane igures, 18.4 Introduction of solid igures - review, 18.5 Faces, edges
and vertices of solid igures, 18.6 Construction of some models of solids, 18.7 Area of
solids, 18.8 Volume of solids,
Symmetrical Figures, Design of Polygons and Tessellations 255
19.1 Symmetrical igures, 19.2 Design of polygons
Statistics Presentation of data, 261
20.1 Collection of data, 20.2 20.3 Frequency table,
20.4 Bar graph, 21.5 Average
269
Scale, Drawing and Bearing
21.1 Scale drawing - Introduction, 21.2 Scale factor, 21.3 Bearing - Introduction 273
287
Answers
Evaluation Model Questions
Unit Set
1
1.1 Set - Looking back
Classroom - 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 7000 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) A set of the first five letters of 'Nepali Barnamala'.
............................................................................................................
b) A 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.
5 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
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 6
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}
7 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 (¹) 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?
Vedanta Excel in Mathematics - Book 6 8
Set
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}
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 method.
9 Vedanta Excel in Mathematics - Book 6
Set
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).
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.
Vedanta Excel in Mathematics - Book 6 10
Set
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.
(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
Overlapping sets
(iv) Disjoint 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
11 Vedanta Excel in Mathematics - Book 6
Set
1.8 Universal set and Subset
Let's take a set of natural numbers less than 11.
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Now, let's select certain elements from this set and make some other sets.
A = {2, 4, 6, 8} Set of even numbers less than 10.
B = {1, 3, 5, 7, 9} Set of odd number less than 10.
C = {2, 3, 5, 7} Set of prime numbers less than 10.
D = {1, 2, 5, 10} Set of factors of 10 and so on.
N N N N
A B C D
2 4 10 13 2 3 10 1 29
1 68 9 2 9 7 5 10 157 9 3 5 10 8
3 57 4 68 4 68 4 67
Here, the set of natural numbers less than 11 is known as the universal set.
The sets A, B, C and D are the subsets of the universal set. Every element of the
subsets A, B, C and D is also an element of the universal set.
‘A is a subset of N’ is written as A N.
‘B is a subset of N’ is written as B N.
‘C is a subset of N’ is written as C N.
‘D is a subset of N’ is written as D N.
Thus, a set from which elements are selected to form many other subsets is
called universal set. It is usually denoted by the capital letter U.
Similarly, the set of students of a school is a universal set, from which the
subsets like set of girls, set of boys, set of cricket players, set of singers, etc. can
be formed.
We use the symbol to denote ‘is a subset of’ the given set.
If a set is not the subset of a given set, we denote it by the symbol .
Vedanta Excel in Mathematics - Book 6 12
Set
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)
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 write the possible subsets of the set S = {1, 4, 9}
Solution : I must remember!
An empty set is a
Here, the possible subsets of S = {1, 4, 9} are subset of every set!
A = {1}, B = {4} C = {9} D = {1, 4}
E = {1, 9} F = {4, 9} G = {1, 4, 9}, H = { } or I
Example 3 : Let’s list the elements of the sets A and B A 1B
from the given diagrams. Also find a set C 8 3
that contains the common elements of the 2
sets A and B. 4
10 6 7 5
Solution : C
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 4 : 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. PQ
Solution : 31
Here, P = {2, 3, 5, 7} and Q = {1, 3, 5, 7, 9} 5
2 7 9
? R = {3, 5, 7} is the set of common elements of P and Q. R
13 Vedanta Excel in Mathematics - Book 6
Set
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 ...........
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}
Vedanta Excel in Mathematics - Book 6 14
Set
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}.
10. Let’s write which one is the universal set or subset in the following pairs of
sets.
a) W = {whole numbers less than 20} and O = {odd numbers less than 20}.
b) A = {2, 4, 6, 8, …, 20} and E = {even numbers less than 30}.
11. Let's write all possible subsets of the following sets.
a) {u} b) {a, m} c) {g o d}
12. 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
13. 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}
15 Vedanta Excel in Mathematics - Book 6
Set
It’s your time - Project work!
14. a) Let’s survey on the following cases among your at least 10 classmates.
Then list the name of your classmates and make separate sets of these
cases.
(i) Sets of students who like basketball or football or cricket.
(ii) Set of students who like both basketball and football.
(iii) Set of students who like both basketball and cricket.
(iv) Set of students who like both football and cricket.
Now, draw diagram and show the names of your classmates in the
diagrams.
BFC
B FB CF C
You can draw bigger circles in a chart paper to write the names of your
classmates comfortably.
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. Then copy the given diagrams and show the
elements in the diagram.
(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
(i) (ii)
Vedanta Excel in Mathematics - Book 6 16
Unit Operations on Whole Numbers
2
2.1 Number system - Looking back
Classroom - Exercise
1. Let’s count the blocks of thousands, hundreds, tens and ones. Then
write the numerals and number names.
a) ...............................
.......................................................................
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? .....................................
17 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 18
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
19 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 20
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 the 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.
21 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 is .............................
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 22
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
23 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 is 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 24
Operations on whole numbers
2.8 Fundamental operations - Looking back
Classroom - 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
= 52
e) 25 – 13 = ........... f) 47 – 24 = ........... 36 – 14 = (36 – 10) – 4
= 26 – 4 = 22
g) 59 – 35 = ........... h) 71 – 44 = ........... 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 ............................
25 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 Additions 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 26
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
27 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.
10. It’s your time - Project work!
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 28
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
29 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 30
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 6000 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.
31 Vedanta Excel in Mathematics - Book 6
Properties of Whole Numbers
17. 22 = 4 = 1 + 2 + 1 32 = 9 = 1 + 2 + 3 + 2 + 1
Investigate the rules from the above illustrations. Then, complete these as
quickly as possible.
a) 42 = 16 = .....................................................................................................
b) 52 = 25 = .....................................................................................................
c) 62 = 36 = .....................................................................................................
d) 72 = 49 = ....................................................................................................
18. Let’s complete the given chart and investigate the fact.
Number Factor No. of factors Remark
1 1 1 Odd
2 1, 2 2 Even
3 2 Even
1, 3
4 3 Odd
5 1, 2, 4 2 Even
6 1, 5
7 ................ ................
8 ................
9 ................ ................
10 ................
................ ................
................
................ ................
................
................ ................
................
From the above chart, let’s answer the following questions and draw out the
conclusion.
a) List out the numbers which have exactly two factors.
b) List out the numbers which have even number of factors but more than
two factors.
c) List out the numbers which have odd number of factors.
d) List out the square numbers from the chart.
e) Draw out your conclusion about the factors of a square numbers.
3.11 Sequence and pattern of numbers
1. Sequence of numbers
Consider a few numbers : 1, 3, 5, 7, 9, ...
Let’s find the difference between each consecutive pair of numbers.
3 – 1 = 2, 5 – 3 = 2, 7 – 5 = 2, 9 – 7 = 2, ...
Here, the difference between each consecutive pair of number is 2. So, these
numbers are in a fixed pattern. The numbers 1, 3, 5, 7, 9, ... are said to be in a
sequence.
53 Vedanta Excel in Mathematics - Book 6
Properties of Whole Numbers
Let’s take another sequence of numbers : 2, 4, 8, 16, 32, ... In this sequence, the
next number of each consecutive pair is 2 times the earlier one.
For example,
2 × 2 = 4, 2 × 4 =8, 2 × 8 = 16, 2 × 16 = 32, ...
Making and using rules for sequences of numbers which have common
differences
(i) Consider a sequence 1, 3, 5, 7, ...
Here, the common difference of each consecutive pair of numbers is 2.
Now, let’s try to investigate the rule to find any number of term of the
sequence.
Suppose, the number of terms of the sequence be ‘n’.
Since the common difference is 2, the first term of the rule must be
2n + ... or 2n – ...
In the sequence 1, 3, 5, 7, ...
To get the first term 1 n = 1 and 2n – 1 = 2 × 1 – 1 = 1
To get the second term 3 n = 2 and 2n – 1 = 2 × 2 – 1 = 3
To get the third term 5 n = 3 and 2n – 1 = 2 × 3 – 1 = 5 and so on.
Thus, the nth term rule of the sequence 1, 3, 5, 7, ... is 2n – 1
Now, let’s use this rule to find 50th term of the sequence 1, 3, 5, 7, ...
Here, n = 50. So, 2n – 1 = 2 × 50 – 1 = 99
(ii) Consider another sequence of numbers 2, 4, 6, 8, ...
Since the common difference is 2, the first term of the rule must be
2n + ... or 2n – ...
To get the first term 2 n = 1 and 2n = 2 × 1 = 2
To get the second term 4 n = 2 and 2n = 2 × 2 = 4
To get the third term 6 n = 3 and 2n = 2 × 3 = 6 and so on.
Thus, the nth term rule of the sequence 2, 4, 6, 8, ... is 2n.
Using this rule, the 24th term = 2n = 2 × 24 = 48.
(iii) Consider a next sequence 2, 6, 10, 14, ...
Here, the common difference is 4.
So, the nth term is 4n + ... or 4n – ...
The first term is 2 = 4n + ... or 4n – ... = 4 × 1 – 2.
So, the missing number in 4n – ... is 2.
Thus, the nth term rule for the sequence 2, 6, 10, 14, ... is 4n – 2.
Vedanta Excel in Mathematics - Book 6 54
Properties of Whole Numbers
EXERCISE 3.5
General Section - Classwork
1. Tell and write the common difference (c.d.) of the numbers of these
sequences.
a) 5, 7, 9, 11,... c.d. = ........... b) 4, 7, 10, 13, ... c.d. = .................
c) 2, 7, 12, 17, ... c.d. = .......... d) 3, 7, 11, 15, ... c.d. = ................
2. Investigate the common differences. Then, tell and write the next two
numbers of these sequences.
a) 3, 5, 7, 9, ......................... b) 6, 9, 12, 15, ........................
c) 5, 9, 13, 17, ...................... d) 7, 12, 17, 22, ......................
3. Tell and write the first terms of the nth term rules of these sequences.
a) 2, 5, 8, 11, ... ................. b) 5, 7, 9, 11, ... ................
c) 1, 5, 9, 13, ... ................. d) 3, 8, 13, 18, ... ................
4. Match the sequences with the nth term rules.
3, 7, 11, 15, ... n+3
4, 7, 10, 13, ... 7n
3, 5, 7, 9, ... 4n – 1
4, 5, 6, 7, ... 3n + 1
7, 14, 21, 28, ... 2n + 1
Creative Section
1. Use the given nth term rules to find the first five terms of each sequence by
replacing n = 1, 2, 3, 4, 5.
a) n + 1 b) n + 2 c) 2n d) 3n e) 2n + 1
f) 2n – 1 g) 2n + 2 h) 3n + 1 i) 3n – 1 j) 4n + 1
2. Use the given nth term rules to find the terms asked for in each of these
sequences.
a) 2n + 3 (1st, 2nd, 3rd) b) 3n – 2 (4th, 5th, 6th) c) 4n + 1 (5th, 10th, 15th)
d) 4n – 3 (3rd, 6th, 9th) e) 5n – 2 (2nd, 4th, 6th) f) 3n + 5 (1st, 2nd, 3rd)
3. Find the rule for the nth term, and check it for the first two terms of these
sequences. Use the rule to find term asked for in each of the sequences.
a) 1, 3, 5, 7, … (10th term) b) 2, 4, 6, 8, … (15th term)
c) 4, 6, 8, 10, … (20th term) d) 3, 6, 9, 12, … (5th term)
e) 5, 9, 13, 17, … (8th term) f) 4, 10, 16, 22, … (7th term)
g) 2, 7, 12, 17, … (30th term) h) 5, 12, 19, 26, … (50th term)
55 Vedanta Excel in Mathematics - Book 6
Unit Integers, Rational and Irrational Numbers
4
4.1 Integers - Introduction
Let’s use a positive or negative number to represent the following
conditions.
a) a profit of Rs 20. …................
b) a loss of Rs 5. …................
c) 18° C above 0° C. …................
d) 2° C below 0° C. …................
e) Taking 5 steps forward …................
f) Taking 3 steps backwards …................
g) a drop of Rs 25 in the price of L P gas. …................
h) a deposit o f Rs 5000 in the bank account. …................
We have already learned about the following sets of numbers.
N = {1, 2, 3, 4, 5, …} is the set of natural numbers.
W = {0, 1, 2, 3, 4, 5, …} is the set of whole numbers.
Now, let’s consider any two whole numbers 4 and 9.
Here, 4 + 9 = 13 (13 is a member of the set of whole numbers)
9 – 4 = 5 (5 is a member of the set of whole numbers)
9 u 4 = 36 (36 is a member of the set of whole numbers)
But, now let’s try to subtract 9 from 4. Is it possible? Look at the following number line.
From the number line, 4 – 9 = – 5
Here, – 5 is not a member of the set of whole numbers. 9
Then, in which set does – 5 belong? 4
– 5 belongs to the set of Integers. 5
Thus, the set of all numbers both positive and negative including zero (0) is
called the set of integers.
The set of integers is denoted by the letter ‘Z’.
The number line given below represents the integers.
Z = {…, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, …} is the set of integers.
Vedanta Excel in Mathematics - Book 6 56
Integers, Rational and Irrational Numbers
On the number line, the numbers towards right form 0 (zero) are positive
integers. Thus Z+ = {+ 1, +2, +3, +4, +5, ...} is the set of positive integers. The
positive integers are always greater than 0.
On the other hand, the numbers towards left from 0 (zero) are negative integers.
Thus, Z– = {–1, –2, –3, –4, –5, …} is the set of negative integers. The negative
integers are always less than 0. So, –1, –2, –3, etc. are less than 0.
4.2 Operations on integers
It is easier to understand the fundamental operations (addition, subtraction,
multiplication and division) of integers on number lines. Let’s study and learn
from the following illustrations.
1. Addition rule
(i) The positive integers are always added. The sum holds the positive (+)
sign.
For example, I investigated the rule
(a) (+ 2) + (+ 5) = 2 + 5 = 7 (+) + (+) Add = + sum
(+ 3) + (+ 4) = + 9
(+2) + (+5)
+2 +5
(b) (+ 3) + (+ 4) + (+ 2) = 3 + 4 + 2 = 9
(+3) + (4) + (2)
+3 +4 +2
(ii) The negative integers are always added. I also investigated!
(–) + (–) Add = – sum
The sum holds the negative (–) sign. (– 2) + (– 6) = – 8
For example,
(a) (– 4) + (– 5)= – 4 – 5 = – 9
57 Vedanta Excel in Mathematics - Book 6
Integers, Rational and Irrational Numbers
(b) (– 3) + (– 2) + (– 6) = – 3 – 2 – 6 = – 11
(iii) The positive and negative integers are always subtracted.
The difference holds the sign of the bigger integer.
For example, + 8 is bigger. So, the
(a) (+ 8) + (– 3) = 8 – 3 = 5 difference 5 holds
positive (+) sign.
+8
–3
(b) (– 9) + (+ 5) = – 9 + 5 = – 4 –9 is bigger. So, the
difference 4 holds
–9 negative (–) sign.
+5
2. Multiplication and division rules
(i) The product or quotient of two positive integers is always positive.
For example, I can remember now!
(a) (+ 3) u (+ 2) = 3 u 2 = 6 (+) × (+) = + product
(+) ÷ (+) = + quotient
(b) (+ 8) ÷ (+ 4) = 8 ÷ 4 = 2 (+8) ÷ (+4)
1 time
Vedanta Excel in Mathematics - Book 6 58
Integers, Rational and Irrational Numbers
(ii) The product or quotient of a positive and a negative integers is always
negative.
For example, I can remember!
(+ 4) u (– 2) = (– 4) u (+ 2) = – 8 (+) × (–) = (–) × (+) = – product
(+ 15) ÷ (– 3) = (– 15) ÷ (+ 3) = – 5 (+) ÷ (–) = (–) ÷ (+) = – quotient
(iii) The product or quotient of two negative integers is always positive.
For example, It's interesting!
(– 5) u (– 3) = 15 (–) × (–) = + product
(– 12) ÷ (– 4) = 3 (–) ÷ (–) = + quotient
Worked-out examples
Example 1: Simplify a) (+ 5) + (– 2) + (+ 8) + (– 6)
b) (– 3) u (+ 9) u (– 2)
Solution:
a) (+ 5) + (– 2) + (+ 8) + (– 6) b) (– 3) u (+ 9) u (– 2)
= (+ 13) + (– 8) = + 5 = 5 = (– 27) u (– 2) = + 54 = 54
Example 2: Evaluate (a) (– 3)2 (b) (– 2)5
Solution:
a) (– 3)2 = (– 3) u (– 3) = + 9 = 9
b) (– 2)5 = (– 2) u (– 2) u (– 2) u (– 2) u (– 2)
= (+ 4) u (+ 4) u (– 2) = (+ 16) u (– 2) = – 32
59 Vedanta Excel in Mathematics - Book 6
Integers, Rational and Irrational Numbers
EXERCISE 4.1
General Section – Classwork
1. Let’s tell and write the correct answers in the blank spaces.
a) The positive integers lie towards ........................... from 0 (zero) in a
number line.
b) The ........................... integers lie towards left from 0 (zero) in a number
line.
c) Integers between – 3 and + 3 are .............................................................
d) Integers greater than – 2 and less than +1 are ........................................
e) The ascending order of the integers 2, –3, 0, –1, 4 is
....................................................................................................................
f) The descending order of the integers 0, –2, 3, –1, 2, –3, 1 is
....................................................................................................................
2. A loss of Rs 40 is written as – 40. Let’s tell and write the appropriate
integers as quickly as possible.
a) a loss of Rs 15 is .....................
b) a gain of Rs 100 is .....................
c) a drop of Rs 10 in a movie ticket is .....................
d) a rise of Rs 20 in a sack of rice is .....................
e) 16°C above 0° C is .....................
f) 3°C below 0°C is .....................
g) 5 steps forward from a fixed point .....................
h) 2 steps backward from a fixed point .....................
3. Let’s tell and insert appropriate sign > or < between two integers.
a) + 4 +9 b) + 6 +1 c) +2 –3
d) –7 +5 e) –5 –3 f) –2 0
4. Let’s tell and write True of False for the following statements.
a) 0 is less than every positive integer. ........................
b) 0 is a positive integers. ........................
c) 0 is the smallest integer ........................
Vedanta Excel in Mathematics - Book 6 60
Integers, Rational and Irrational Numbers
d) – 1 is the greatest negative integer ........................
e) The opposite of – 4 is 4 ........................
f) The opposite of +6 is +9 ........................
g) – 5 is greater than 0 ........................
h) The sum of an integer and its opposite is always zero .........................
i) The sum of two negative integers is positive ........................
j) The product of two negative integers in negative ........................
5. Let’s tell and write the correct answers as quickly as possible.
a) 3 more than – 4 = .................. b) – 3 more than – 4 = ..................
c) 3 less than – 4 = .................. d) – 3 less than – 4 = ..................
e) 8 – 5 = ..............., 5 – 8 = ............., – 5 – 8 = ..............
f) – 2 × 5 = ..............., 2 × (– 5) = ............., – 2 × (– 5) = ..............
g) – 16 ÷ 8 = ..............., 16÷ (– 8) = ............... , – 16 ÷ (– 8) = ..............
Creative Section
6. Make mathematical expressions from these number lines, then simplify.
Hint: (– 4) + (– 3) = – 7
Hint: 2 × (– 3) = – 6
61 Vedanta Excel in Mathematics - Book 6
Integers, Rational and Irrational Numbers
7. Simplify. b) (+ 7) + (– 9) c) (– 6) + (– 8)
a) (+ 7) + (– 4) e) (– 9) + (+ 4) + (– 3) f) 2 – 8 – 6 – 11 + 12
d) (+8) + (+ 6) + (– 5) h) (– 4) × (– 6) i) (– 2) ×(– 7) × (– 3)
g) (+3) × (– 5) k) (– 72) ÷ (+ 9) l) (– 63) ÷ (– 7)
j) (+ 18) ÷ (– 6)
8. Evaluate. c) (– 1)4 d) (– 2)3 e) (–2)4 f) (–2)5
a) (– 1)2 b) (– 1)3 i) (– 4)2 j) (– 4)3 k) (–5)2 l) (–5)3
g) (– 3)2 h (– 3)3
9. a) A man runs 10 km due East and A 10 km B
B
then 4 km due West . Find. His
position with respect to his 4 km
starting point.
6 km
Hint: 10 km – 4 km = 6 km East from the starting point.
b) Mrs. Yadhav walks 18 km due North and then 12 km due South. Find
her position with respect to his starting point.
c) Mr. Chamling travels 20 km due East and then 25 km due west. Find
his position with respect to his starting point.
d) Sunayana walks 8 km due East, then turns round and goes 4 km west of
the starting point. Then, she again turns back and returns to the starting
point. What is the total distance travelled by her?
10. a) The temperature of a body first rises by 20° C and then falls by 24° C.
Find the final temperature of the body, if its initial temperature is 10° C.
Hint : Final temperature = 10° + (+ 20°) + (– 24°) = 6° C.
b) The temperature of a body first rise by 25° C and then falls by 30° C.
Find the final temperature of the body, if its initial temperature is
(i) 8°C. (ii) 2°C (iii) 0°C. (iv) – 3° C
Vedanta Excel in Mathematics - Book 6 62
Integers, Rational and Irrational Numbers
c) The temperature of a body first falls by 15° C and then rises by 20°C.
Find the final temperature of the body, if its initial temperature was :
(i) 28° C (ii) 12° C (iii) 5° C. (iv) – 2°C
11. It’s game time!
Play with one of your friends.
Let’s fill in the addition and multiplication tables separately. Then, compare
your completed tables with your competitor. Winner will be declared if
anyone completes both the tasks as quickly as possible with correct sums.
Addition Table
+ +5 +4 +3 +2 +1 0 –1 –2 –3 –4 –5
+5
+4
+3
+2
+1
0
–1
–2
–3
–4
–5
Multiplication Table
× +5 +4 +3 +2 +1 0 –1 –2 –3 –4 –5
+5
+4
+3
+2
+1
0
–1
–2
–3
–4
–5
63 Vedanta Excel in Mathematics - Book 6
Integers, Rational and Irrational Numbers
4.3 Rational numbers – Introduction
Let’s consider any two integers 2 and 4.
Now study the following operations on these integers
2+4 =4+2 = 6 (6 is also an integer)
2 – 4 = – 2, 4 – 2 = 2 (– 2 and 2 are also integers)
2×4 =4×2 = 8 (8 is an integer)
4÷ 2 = 4 =2 (2 is an integer)
2 (21 is not an integer)
1
2÷ 4 = 2 = 2
4
Thus, when an integer is divided by another integer the quotient is not always
an integer. The fact indicates another set of numbers that can also include such
quotients which are not integers. The set of such numbers is called the set of
Rational numbers. p
q
Any numbers which can be expressed in the form , where p and q are integers
and q ≠ 0, are called rational numbers. The set of rational numbers is denoted
by the letter ‘Q’.
? Q = {…, – 3, – 5 , – 2, – 3 , – 1, – 1 , 0, 1 , 1, 3 , 2, 5 , 3, …}
2 2 2 2 2 2
The set of rational numbers is the wider set that includes the sets of natural
numbers (N), whole numbers (W) and integers (Z).
4.4 Terminating and non-terminating rational numbers
When a rational number is decimalised, the decimal so obtained may be
terminating or non-terminating decimal. If the decimal is non–terminating, a
digit or a block of digits after the decimal point repeat after certain intervals.
Such decimals are called non-terminating recurring decimals. For example:
5 = 2.5 (Terminating decimal)
2 (Terminating decimal)
1 (Terminating decimal)
4 = 0.25 (Non-terminating recurring decimal)
(Non-terminating recurring decimal)
3 = 0.375 (Non terminating recurring decimal)
8 (Non terminating recurring decimal)
2
3 = 0.6666…
5 = 0.8333…
6
2
7 = 0.285714 285714285…
3 = 0.272727…
11
Thus, the rational numbers are either terminating decimals or non-terminating
recurring decimals.
Vedanta Excel in Mathematics - Book 6 64
Integers, Rational and Irrational Numbers
Example 1: Show the rational numbers 3, 3 , 5 , – 1 , – 9 in a number line.
2 2 2 2
Solution:
Here, 3 = 1.5, 5 = 2.5
2 2
– 1 = –0.5 and – 9 = –4.5
2 2
3 3
2
9 1 5
2 2 2
4.5 Irrational numbers
Let’s study the following illustrations and try to investigate the idea of irrational
numbers
4 =2 (2 is a rational number)
25 = 5 (5 is a rational number)
(83 is a rational number. The decimal is terminating)
3 = 0.375
8
On the other hand,
2 = 1.414213562… (Decimal is non-terminating non-recurring. 2 is not a
rational number. It is an irrational number.)
3 = 1.7320508070… (Decimal is non-terminating non-recurring. 3 is not a
rational number. It is an irrational number.)
Thus, the numbers which are not rational are called irrational numbers.
2 , 3 , 5 , 6 , 7, etc. are a few examples of irrational numbers.
When irrational numbers are decimalised, the decimals apqrefonrmon. -terminating
non-recurring. So, irrational numbers cannot be expressed in
In this way, all the sets of number system, for example, natural numbers, whole
numbers, integers, rational numbers and irrational numbers are defined under
the set of Real number system.
Real numbers
Rational numbers Non-Integers Irrational numbers
Integers 2 , 3 , 8 , 2 , etc.
Negative integers Positive integers 1 , 2 , 3 , 7 , etc.
2 3 5 10
Natural numbers Whole numbers
1, 2, 3, 4, 5,... 0, 1, 2, 3, 4, 5,...
65 Vedanta Excel in Mathematics - Book 6
Integers, Rational and Irrational Numbers Q 0.6
EXERCISE 4.2 1
General Section A – Classwork 2Z
1. If N is the set of natural numbers W is –1
the set of whole numbers, Z is the set
of integers and Q is the set of rational W
numbers. Let’s observe the relations of
N, W, Z and Q in the diagram and write 1 -2 N
‘True’ or ‘False’ as quickly as possible.
3 0 1,2,3, 4, ...
-3
-4 -5 ... 0.33
3 ...
4
a) Every natural number is a rational number. .............................
b) Every rational number is a natural number. .............................
c) Every whole number is a rational number. .............................
d) Every rational number is a whole number. .............................
e) Every integer is a rational number . .............................
f) Every rational number is an integer. .............................
g) Every rational number is a real number. .............................
h) Every real number is a rational number. .............................
i) Set of irrational numbers is a subset of real numbers .............................
j) Set of real numbers is as subset of irrational numbers .............................
2. Let's tell and write whether the decimals of these fractions are terminating or
non – terminating recurring.
a) 1 .................................... b) 1 ......................................
2 5
c) 1 ....................................... d) 1 .........................................
4 3
e) 2 ...................................... f) 4 ...........................................
3 7
Vedanta Excel in Mathematics - Book 6 66
Integers, Rational and Irrational Numbers
3. Let's tell and write whether these numbers are ‘Rational’ or ‘Irrational.’
a) 4 ........................ (b) 3 ........................ c) 7 ........................
(f) – 2.57 ........................
(d) 3 ........................ e) 1 ........................
5 4
Creative Section - A
4. Answer the following questions.
a) What do you mean by rational number? Give an example.
b) Write down a rational number whose numerator is the greatest number of
one-digit and the denominator is the smallest number of two digits.
c) Write down a rational number whose numerator is the greatest negative
integer and the denominator is the smallest prime number.
5. Express these rational numbers in decimal. State whether they are terminating
or non-terminating recurring.
a) 1 (b) 3 (c) 9 (d) 1 (e) 5 (f) 6 (g) 13
2 4 8 3 6 7 11
6. Show the rational numbers 2, 12, 25, – 21, – 7 in a number line.
2
It's your time - Project work!
7. a) Divide the numbers 1 to 8 by 9. Observe the decimals and discuss about
your observation of non- terminating recurring decimals.
b) Divide the numbers 1 to 10 by 11 and discuss about your observation of
non-terminating recurring decimals so obtained.
8. a) Let’s write any four rational numbers having the denominator 2 or power
of 2, then express them in decimals. Are the decimals always terminating?
Write the remarks of your investigation.
b) The decimals of rational numbers with denominators 3 or multiple of 3 and
numerator is not the multiple denominator, are always non – terminating
recurring. Verify it by four examples.
67 Vedanta Excel in Mathematics - Book 6
Unit Algebra
11
11.1 Constant and variables - Looking back
Classroom - 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.
Vedanta Excel in Mathematics - Book 6 144
Algebra
11.2 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.
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.
11.3 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 .
145 Vedanta Excel in Mathematics - Book 6
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