Approved by the Government of Nepal, Ministry of Education, Science and Technology,
Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material
vEexdcaenltain
MATHEMATICS
9Book
vedanta
Vedanta Publication (P) Ltd.
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vEexdcaenltain
MATHEMATICS
9Book
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Second and Updated Edition: B. S. 2078 (2021 A. D.)
Third Revised and Updated Edition: B. S. 2079 (2022 A. D.)
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Publisher's words
Vedanta, the emergent publication house, is committed to cater quality
school-level textbooks and reference materials that are well-researched and
thoughtfully prepared. We make the utmost endeavours to ensure the readers
that our products are error-free. We believe, the processes of updating,
upgrading and modification of textbooks are the most significant aspects of
quality education. Therefore, our textbooks which are completely based on the
latest curriculum developed by Curriculum Development Centre (CDC), Nepal
are also updated, upgraded and modified as per time.
The textbooks of Vedanta Excel in Mathematics series are originally written
by Mr. Hukum Pd. Dahal, who has been writing mathematics text books since
B.S. 2052 (1995 A.D.) to till the date. Besides the Excel in Mathematics series,
Mr. Dahal has written three more series: Elementary Mathematics, Maths in
Action and Speedy Maths.
Recently, according to the new curriculum developed by CDC, Nepal,
Vedanta Excel in Mathematics Book 9 is updated, upgraded and modified by
Mr. Tara Bdr. Magar, who is also involving as the editor of all the textbooks of
the series.
-Vedanta Publication (P) Ltd.
Forewords
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 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.
Furthermore, as per the need of ICT in teaching and learning mathematics, we have included
'Vedanta ICT Corner' to fulfil this requirement in classes 4, 7 and 9 this year and it will be
included in the remaining classes in the next year.
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.
We 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 the
previous series, We are highly grateful to the Principals, HODs, Mathematics teachers and
experts, PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN, UNIPS and independent
clusters of many other Schools of Nepal, for providing us with opportunities to participate in
workshops, Seminars, Teachers’ training, Interaction programme, and symposia as the resource
person. Such programmes helped us a lot to investigate the teaching-learning problems and to
research the possible remedies and reflect to the series.
We are extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, England
who provided us with very valuable suggestions about the effective methods of teaching-
learning mathematics and many reference materials.
We are thankful to Dr. Komal Phuyal for editing the language of the series.
Moreover, we gratefully acknowledge all Mathematics Teachers throughout the country who
encouraged us and provided us with the necessary feedback during the workshops/interactions
and teachers’ training programmes in order to prepare the series in this shape.
We are profoundly grateful to the Vedanta Publication (P) Ltd. for publishing this series.
We 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, we are 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.
-Authors
Contents
S.N Chapter Page No.
1. Set 7 - 27
1.1 Set – Looking back, 1.2 Set- Review, 1.3 Notation and specification
2. of sets, 1.4 Methods of describing a set, 1.5 Cardinal number of sets
1.6 Types of sets, 1.7 Relationship between sets, 1.8 Universal set,
3. 1.9 Venn-diagrams, 1.10 Set operations, 1.11 Cardinality relationships
of two sets
4.
Taxation 28 - 48
5.
2.1 Tax - Introduction, 2.2 Income Tax, 2.3 Tax withholding,
2.4 Marked price (M.P.) and Discount, 2.5 Value Added Tax (VAT)
6.
Bonus and Dividend 49 - 60
7.
3.1 Commission – Introduction, 3.2 Bonus, 3.3 Dividend
8. Household Arithmetic 61 - 74
4.1 Introduction, 4.2 Electricity bill, 4.3 Telephone bill, 4.4 Water
9. bill, 4.5 Calculation of taxi fare in a taximeter
Mensuration (I): Area 75 - 94
5.1 Area – Looking back, 5.2 Mensuration, 5.3 Perimeter and area of
plane figures-review, 5.4 Area of triangle, 5.5 Area of 4 walls, floor
and ceiling
Mensuration (II): Prism 95 - 107
6.1 Prism – Looking back, 6.2 Area and volume of solids, 6.3 Prisms,
6.4 Surface area and volume of triangular prism
Mensuration (III): Cylinder and Sphere 108 - 128
7.1 Cylinder – Looking back, 7.2 Cylinder, 7.3 Area of cylinder,
7.4 Volume of cylinder, 7.5 Curved surface area, total surface area
and volume of a hollow cylinder, 7.6 Half cylinder or semi-cylinder,
7.7 Area and Volume of a Sphere, 7.8 Hemisphere and great circle,
7.9 Surface area and volume of a hollow hemispherical object,
7.10 Area and Volume of cylinder having hemispherical ends,
7.11 Cost estimation
Sequence and Series 129 - 147
8.1 Sequence - Introduction, 8.2 General term of sequence, 8.3 Series, 148 - 161
8.4 Sigma Notation, 8.5 Partial Sum, 8.6 Arithmetic Sequence,
8.7 Terms and Common Difference of an A.P., 8.8 Geometric
Sequence, 8.9 Terms and Common Ratios of a G.P.
Factorisation of Algebraic Expressions
9.1 Factorisation-Looking back, 9.2 Factors and Factorisation -
Review
10. Highest Common Factor (H.C.F.) and 162 - 172
Lowest Common Multiples (L.C.M.)
10.1 H.C.F. and L.C.M.-looking back, 10.2 H.C.F. of algebraic
expressions, 10.3 L.C.M. of algebraic expressions
11. Indices 173 - 181
11.1 Indices – review, 11.2 Laws of Indices, 11.3 Exponential
equation
12. Simultaneous Linear Equations 182 - 196
12.1 Simultaneous equations - review, 12.2 Method of solving
simultaneous equations, 12.3 Application of simultaneous linear
equations
13. Geometry - Triangle 197 - 230
13.1 Triangle-Looking back, 13.2 Types of triangles - review,
13.3 Median and altitude of a triangle, 13.4 Properties of triangles,
13.5 Triangle inequality property, 13.6 Congruent triangles,
13.7 Conditions of congruency of triangles, 13.8 Similar triangles-
review
14. Geometry - Parallelogram 231 - 243
14.1 Special types of Quadrilateral
15. Geometry - Construction 244 - 250
15.1 Quadrilaterals, 15.2, Construction of Rhombus, 15.3 Construction of
scalene quadrilaterals, 15.4 Construction of trapezium
16. Geometry - Circle 251 - 267
16.1 Circle and it’s various parts, 16.2 Theorems related to chords of
a circle
17. Statistics (I): Classification and Graphical Representation of data 268 - 289
17.1 Statistics – Looking back, 17.2 Statistics, 17.3 Types of
data, 17.4 Frequency table, 17.5 Grouped and continuous data,
17.6 Cumulative frequency table, 17.7 Graphical representation of
data: Histogram, Frequency polygon, Ogive (Cumulative frequency
curve)
18. Statistics (II): Measure of Central Tendency 390 - 303
18.1 Central tendency – Looking back, 18.2 Arithmetic mean,
18. 3 Median, 18.4 Quartiles, 18.5 Mode, 18.6 Range
19. Probability 304 - 315
19.1 Probability – Looking back, 19.2 Probability-Introduction,
19.3 Probability scale, 19.4 Probability of an event, 19.5 Empirical
probability (or Experimental probability)
20. Trigonometry 316 - 328
20.1 Trigonometry - Introduction, 20.2 Trigonometric ratios,
20.3 Relation between trigonometric ratios, 20.4 Values of
trigonometric ratios of some standard angles
Revision and Practice Time 329 - 347
Answers 348 - 364
Model Questions 366 - 368
Unit 1 Set
Classwork - Exercise
1.1 Set – Looking back
1. Let A = {2, 3, 5, 7}. Insert the appropriate symbol or in the blank space.
a) 2 …. A b) 4 …. A c) 10 …… A d) 7 … A
2. Re-write the following sets in roster form.
a) A = {the letters in the word ‘PARALLELOGRAM’},
A = {...............................................................................}
b) B = {the prime numbers less than 10}, B = {...............................................}
c) C = {x: x = 4n, n ∈ N and n<5}, C = {..............................................}
3. Identify and write whether the following sets are empty (null), unit
(singleton), finite or infinite.
a) A = {the prime number between 90 and 100} ...................................
b) B = {x: x is a perfect square number} ...................................
c) C = {all leap years between 2076 and 2080} ...................................
d) D = {y: y is a common factor of 36 and 48} ...................................
1.2 Set - Review
Let’s take a collection of multiples of 5 between 20 and 50. The members of this
collection are definitely 25, 30, 35, 40, 45. These members are distinct objects when
considered separately. However, when they are considered collectively, they form a
single set of size five, written {25, 30, 35, 40, 45}. It is a set of multiples of 5 between
20 and 50. Here, any multiple of 5 between 20 and 50 is definitely the member of
the set. Therefore, a set is a collection of ‘well-defined and distinct objects’.
The German mathematician and logician Georg Cantor (1845 –1918)
created modern set theory between the years 1874 to 1897. Today, it
is used in almost every branch of Mathematics.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 7 Vedanta Excel in Mathematics - Book 9
Set
1.3 Notation and specification of sets
We usually denote sets by capital letters and the members or elements of the sets
are enclosed inside the braces { } separated with commas. The table given below
shows a summary of the use of symbols in sets. Here, we are taking any two sets, i.e.,
N = {1, 2, 3, 4, 5} and E = {2, 4, 6, 8}, to discuss about the use of symbols.
Symbol Name Example Explanation
{} Set
∈ N = {1, 2, 3, 4, 5} The members of the sets are
Membership E = {2, 4, 6, 8} enclosed inside braces.
∉
2 ∈ N, 5 ∈ N, 4 ∈ N, The symbol ‘∈‘denotes the
⊂ 6 ∈ E, 8 ∈ E membership of an element of
⊆ the given set.
Non-membership 6 ∉ N, 8 ∉ N, 3 ∉ E,
⊃ 5∉E The symbol ‘∉‘denotes
the non-membership of an
Proper subset {1, 2} ⊂ N, element to the given set.
Improper subset {4, 6, 8} ⊂ E
A set that is contained in
Super set {2, 4, 6, 8} ⊆ E, it another set.
means {2, 4, 6, 8} ⊂ E
and {2, 4, 6, 8} = E A set which is contained in
and equal to another set.
N ⊃ {1, 2},
E ⊃ {4, 6, 8} Set N includes {1, 2,} and Set
E includes {4, 6, 8}.
1.4 Methods of describing a set
We usually describe a set by three methods: description, listing (or roster), and set-builder
(or rule) methods.
Method Example Explanation
Description A is a set of prime numbers less than Words description of common
10. properties of elements of a set.
Listing (or A = {2, 3, 5, 7} The distinct elements of a set are
roster) listed inside curly brackets { }.
Set-builder A = {x : x ∈ prime numbers, x < 10} A variable is used to describe the
(or rule) common properties of the elements
of a set by using symbols.
Vedanta Excel in Mathematics - Book 9 8 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
1.5 Cardinal number of set
Let’s take a set A = {2, 3, 5, 7, 11, 13, 17, 19}. Since, it has 8 elements. The cardinal number
of the set A is 8. The number of elements contained by a set is called its cardinal number.
If A be a given set, it’s cardinal number is represented as n(A). For example: If A = {1, 2, 3,
4, 6, 12}, then n(A) = 6.
1.6 Types of sets
On the basis of the number of elements contained in sets, we classify sets into four types:
empty (or null) set, singleton (or unit) set, finite set, and infinite set.
Type of sets Examples Explanation
Empty or null set The set of whole numbers It does not contain any element.
Unit or singleton set less than 0. It is denoted by empty braces { }
W = { } or φ or by φ(Phi)
The set of even numbers It contains only one element.
between 3 and 5, P = {4}
Finite set W = {0, 1, 2, 3, ..., 100} It contains finite number of
elements.
Infinite set W = {0, 1, 2, 3, ...} It contains infinite number of
elements.
Facts to remember
1. {0} and {∅} are not null sets rather these are singleton sets.
2. The empty set ∅, has no element. So, it’s a finite set.
3. The sets of natural numbers (N), whole numbers (W), integers (Z), rational numbers
(Q), irrational numbers (P) and real numbers (R) are infinite.
1.7 Relationship between sets
On the basis of the types of elements contained in two or more sets, the types of their
relationship can be defined in the following ways.
Type of relationship Examples Explanation
Equal sets
Equivalent sets A = {क, ख, ग, घ, ङ} They have exactly the same
Overlapping sets B = {घ, ख, ङ, क, ग} ∴ A = B elements.
P = {2, 3, 5, 7}
Disjoint sets Q = {1, 4, 9, 16} ∴ P ~ Q They have equal number of
elements.
M = {2, 4, 6, 8, 10}
N = {1, 2, 5, 10} They have at least one common
∴ M and N are overlapping element. 2 and 10 are the
sets. common elements of sets M
and N.
X = {अ, आ, इ, ई}
Y = {च, छ, ज, झ} They do not have any common
∴ X and Y are disjoint sets. element.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 9 Vedanta Excel in Mathematics - Book 9
Set
1.8 Universal set
Let’s consider a set of natural numbers less than 20.
From this set, we can make many other subsets such as set of even numbers less than 20,
set of odd numbers less than 20, set of prime numbers less than 20, and so on. In this case,
the set of natural numbers less than 20 is considered as a universal set and denoted by U.
Thus, a set under the consideration from which many other subsets can be formed is
known as a universal set. A universal set is the set of all the elements of any group under
consideration.
1.9 Venn-diagrams
We can represent sets and set operations by using diagrams like rectangle,
circle, or oval shape. The idea of representation of sets in diagrams was
first introduced by Swiss Mathematician Euler. It was further developed
by the British Mathematics John Venn. So, such diagrams are famous
as Venn Euler diagrams or simply Venn-diagrams. In Venn-diagram, the
universal set is represented by rectangle and its subsets are represented by
circles or ovals inside the rectangle.
Let’s study the following set relationship by using Venn-diagrams.
U U U
AA BA B
A is subset of U U A and B are disjoint sets A and B are overlapping sets
AB U U
AB AB
C C C
A, B and C are overlapping sets. A and B are overlapping, A and B are overlapping,
A and C are overlapping, B and C are overlapping,
1.10 Set operations but B and C are disjoint sets. but A and C are disjoint sets.
Like the operations of addition, subtraction, multiplication, division on numbers in
Arithmetic, there are four types of operations on sets:
1. Union of sets 2. Intersection of sets
3. Difference of sets 4. Complement of a set
These operations are well known as set operations. Now, let’s learn about these sets
operations by using Venn-diagrams.
1. Union of sets
Let’s study the following examples and investigate the idea about union of sets.
Vedanta Excel in Mathematics - Book 9 10 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
a) Suppose that the sport and cultural committees are recently formed with a few
number of students of class 9. The set of students who are involving in the sport
committee is S = {Raju, Pinky, Ganesh, Lakpa, Diya, Biswant, Rahul} and the set
of students who are involving in cultural committee is C = {Salina, Diya, Suntali,
Ojaswi, Rahul, Smrity}.
If a joint meeting of these committees is held and a new committee is formed, who
will be the members of the joint committee?
Obviously, the list of the students of the joint committee is {Raju, Pinky, Ganesh,
Lakpa, Diya, Biswant, Binaya, Rahul, Salina, Suntali, Ojaswi, Smrity}. Of course, it
is the union of two committees S and C.
b) Let’s take any two sets A and B, where A = {1, 2, 3, 4, 5}, B = {1, 3, 5, 7}.
Now, the union of sets A and B denoted as A ∪ B = {1, 2, 3, 4, 5, 7}.
Thus, the union of two or more sets is made by grouping their elements together in
a single set. In the case of overlapping sets, the common elements are listed only
once while making the union.
The shaded regions in the following Venn-diagrams represent the union of the given sets.
A U A U BU
B B A
A∪B A∪B A∪B
A and B are disjoint sets A and B are overlapping sets A is subset of B
U U U
A BC AB
C A∪B∪C C
A∪B∪C A, B and C are disjoint sets. A∪B∪C
A, B and C are overlapping sets. A and B are overlapping,
A and C are overlapping,
but B and C are disjoint sets.
Facts to remember
1. The union of two sets A and B denoted by A ∪ B is the set of all elements that
belong to either to set A, or set B, or to both sets A and B.
In set-builder form, union of sets A and B is defined as:
A ∪ B = {x : x ∈ A or x ∈ B}
2. A ∪ A = A 3. A ∪ ∅ = A 4. A ∪ U = U
5. A ⊆ (A ∪ B) and B ⊆ (A ∪ B) 6. A ∪ B = A if B ⊆ A and A ∪ B = B if A ⊆ B
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 11 Vedanta Excel in Mathematics - Book 9
Set
Worked-out Examples
Example 1: If A = {x: x is a prime number less than 20} and B = {y: y is a factor of 30}
are the subsets of a universal set U, find A∪B and show it in a Venn-
diagram.
Solution:
Here, A = {x: x is a prime number less than 20}
= {2, 3, 5, 7, 11, 13, 17, 19}
B = {y: y is a factor of 30} = {1, 2, 3, 5, 6, 10, 15, 30} 16 B U
Now,
A ∪ B = {2, 3, 5, 7, 11, 13, 17, 19} ∪ {1, 2, 3, 5, 6, 10, 15, 30} A7 2 10
= {1, 2, 3, 5, 6, 7, 10, 11, 13, 15, 17, 19, 30} 11 3 15
The shaded region represents A ∪ B. 13 17 5
19 30
Example 2: If F = {x: x is a multiple of 5, 20<x<50} and T = {y: y is a multiple of 10,
20<y<50}, find F ∪ T and show it in a Venn-diagram.
Solution:
Here, F = {x: x is a multiple of 5, 20<x<50} = {25, 30, 35, 40, 45}
T = {y: y is a multiple of 10, 20<y<50} = {30, 40} FU
Now, T 35
F ∪ T = {25, 30, 35, 40, 45} ∪ {30, 40}
= {25, 30, 35, 40, 45} 25
The shaded region represents F ∪ T. 30
40
45
2. Intersection of sets
Let’s study the following examples and investigate the idea about intersection of sets.
a) The lists of a few animals that live on land (L) and in water (W) are shown in the
table given below.
Land (L) Water (W)
cat, duck, horse, frog, rabbit, cow, fish, frog, crocodile, duck, turtle,
crocodile, elephant, tiger octopus, sea-horse
Now, the list of animals that live both on land and in water is {duck, frog, crocodile}
The, new set so formed is the intersection of sets of animals that live in land (L) and
the set of animals that live in water (W).
b) Let’s take any two sets P and Q, where P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 7, 9}.
Now, the intersection of sets P and Q denoted as P ∩ Q = {3, 5, 7}.
Thus, the intersection of two or more sets is formed just by listing their common
element/s in a separate set.
Vedanta Excel in Mathematics - Book 9 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
The shaded regions in the following Venn-diagram represent the intersection of the
given sets.
PQ U P UP U
Q Q
R
P∩Q P∩Q P∩Q∩R
Q is subset of P P and Q are overlapping sets
P, Q and R are overlapping sets.
Facts to remember
1. The intersection of two sets A and B denoted by A ∩ B is the set of all elements that
belong to both the sets A and B.
In set-builder form, intersection of sets A and B is defined as:
A ∩ B = {x : x ∈ A and x ∈ B}
2. A ∩ B = A 3. A ∩ f = f 4. A ∩ U = A
5. (A ∩ B) ⊆ A and (A ∩ B) ⊆ B 6. A ∩ B = A if A ⊆ B and A ∩ B = B if B ⊆ A
Example 3: If A = {odd numbers less than 10} and B = {prime numbers less than 15}are
the subsets of a universal set U, find A∩B and show it in a Venn-diagram.
Solution: 2B U
Here, A = {odd numbers less than 10} = {1, 3, 5, 7, 9}
A1 3 11
B = {prime numbers less than 15} = {2, 3, 5, 7, 11, 13} 9 5
Now,
A ∩ B = {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7, 11, 13} 7 13
= {3, 5, 7}
The shaded region represents A ∩ B.
Example 4: If M= {x: x = 100n, n∈N and n<5} and N = {y: y= 200n, n∈N and n<3},
find M ∩ N and show it in a Venn-diagram.
Solution: U
Here, M = {x: x = 100n, n∈N and n<5} = {100, 200, 300, 400} M 300
N= {y: y = 200n, n∈N and n<3} = {200, 400} N
Now, 100 200
M ∩ N = {100, 200, 300, 400} ∩ {200, 400} 400
= {200, 400}
The shaded region represents M ∩ N.
3. Difference of sets B Januka Bina Surendra U
Bishnu Ramesh S
a) The sets of teachers teaching in basic Sunita
level (B) and secondary level (S) of a Kamala Hari Nischal
school are shown in the Venn-diagram.
Ambika
Let's make the following sets.
(i) The set of teachers who teach in
both the levels.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 13 Vedanta Excel in Mathematics - Book 9
Set
(ii) The set of teachers who teach in either basic or secondary level.
(iii) The set of teachers who teach in basic level only.
(iv) The set of teachers who teach in secondary level only.
b) Let A = {1, 2, 3, 4, 5} and B = {2, 3, 5, 7, 11} be any two sets.
Then, the difference between the sets A and B denoted as A – B = {1, 4}.
Also, the difference between the sets B and A denoted as B – A = {7, 11}.
Thus, the difference of two sets A and B denoted by ‘A – B’ is the set of the elements
of only A which do not belong to B.
Similarly, the difference of two sets B and A denoted by B – A is the set of the
elements of only B which do not belong to A.
In set-builder form, the difference of sets A and B is defined as:
A – B = {x: x∈A, but x∉B}
Also, the difference of sets B and A is defined as:
B – A = {x: x ∈B, but x ∉A}
In this way,
A – B = A – (A∩B) ←Removing the common elements form A
B – A = B – (A∩B) ←Removing the common elements form B
The shaded regions in the following Venn-diagrams represent the difference of the
given sets.
UU U
A B AB BA
B–A A–B U B–A U
A and B are disjoint sets B is subset of A B A is subset of B B
A A
U
AB
A–B A–B B–A
A and B are overlapping sets A and B are disjoint sets A and B are overlapping sets
The union of A – B and B – A is called symmetric difference between the sets
A and B. It is denoted by A∆B. Thus, A∆B = (A – B) ∪ (B – A)
The shaded regions in the following Venn-diagram represent the symmetric
difference between the given sets.
UU U
A BB AA B
∪=
A – B B–A B∆A
Vedanta Excel in Mathematics - Book 9
14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Facts to remember
1. A – A = ∅ 2. A – ∅ = A
3. A – B = A and B – A = B if A and B are disjoint sets.
4. In general, A – B ≠ B – A but A – B = B – A only when A = B
5. A D B = B D A
Example 5: Let P = {x: x is a positive even number and x≤12}, Q = {y: y is a factor of 18}
and R = {z: z = 3n, n∈W and n ≤ 2} are three sets. Find (i) P – Q
(ii) Q – P (iii) Q – R (iv) R – P
Also, represent the operations by shading in Venn-diagrams.
Solution:
Here, P ={x: x is a positive even number and x≤12} = {2, 4, 6, 8, 10, 12},
Q = {y: y is a factor of 18} = {1, 2, 3, 6, 9, 18} and
R = {z: z = 3n, n∈W and n ≤ 2} = {1, 3, 9} P4 U
Now, 8 10 1Q
(i) P – Q = {2, 4, 6, 8, 10, 12} – {1, 2, 3, 6, 9, 18} 12 29 3
= {4, 8, 10, 12} 6
The shaded region represents P – Q. 18
(ii) Q – P ={1, 2, 3, 6, 9, 18}–{2, 4, 6, 8, 10, 12} P4 U
= {1, 3, 9, 18} 8 10 1Q
12 29 3
The shaded region represents Q – P. 6
18
(iii) Q – R ={1, 2, 3, 6, 9, 18}–{1, 3, 9} Q R6 U
= {2, 6, 18} 2
1 3 18
The shaded region represents Q – R. 9
(iv) R – P ={1, 3, 9}–{2, 4, 6, 8, 10, 12} R U
= {1, 3, 9} 1 P
The shaded region represents R – P. 24
3 68
4. Complement of a set 9
10 12
a) If the universal set, U is the set of all students of class 9 and A is the set of students
who are present in class on Sunday of a certain week. Then, we can list out the
students of class 9 who are absent on that day and we can form a new set of
absentees.
b) Let A = {1, 3, 5, 7, 9} be a subset of a universal set U = {1, 2, 3, …, 10}.
Then, the complement of the set A denoted as A = {2, 4, 6, 8, 10}.
Thus, the complement of set A is the difference of U and A. The complement of
set A can also be denoted by A’ or Ac.
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Set
In set-builder form, the complement of a set A is defined as:
A = {x : x∈U, but x∉A} which is U – A.
Similarly, the set builder forms of the complements of union and intersection of set A
and set B are:
A ∪ B = {x: x∈U, but x ∉ (A or B)} or, {x: x∈ U, but x∉A∪B}
A ∩ B = {x: x∈U, but x ∈(A and B)} or, {x: x∈ U, but x∉A∩B}
The shaded regions in the following Venn-diagrams represent the complement
of the given sets.
U U U U
AA BA BA B
A A∪B A∩B A–B
Facts to remember
1. If A be a subset of a universal set U, the complement of A denoted by A or A’ or Ac
is the set of the elements of U which do not belong to the set A.
1. A = U – A = A 2. U = U – U = ∅ 3. ∅ = U – ∅ = U
4. A ∪ A = U 5. A ∩ A = ∅
6. A – B = A ∩ B and B – A = A ∩ B 7. A ∪ B = A ∩ B and A ∩ B = A ∪ B
Example 6: Let A = {2, 4, 6, 8, 10} and B = {2, 3, 5, 7} are the subsets of the universal
set of U = {1, 2, 3, 4, ... 10} . Compute the following set operations with
Venn-diagrams.
(i) A ∪ B and A ∪ B (ii) A ∩ B and A ∩ B
(iii) A – B and A – B (iv) B – A and B – A
Solution:
Here, A = {2, 4, 6, 8, 10}, B = {2, 3, 5, 7} and U = {1, 2, 3, 4, … 10}
U
(i) A ∪ B = {2, 3, 4, 5, 6, 7, 8, 10} B U
A 3 A B
A ∪ B = U – (A ∪ B) 4 4 3
62
= {1, 9} 8 10 5 62 5
7 8 10 7
1 9
Vedanta ICT Corner 19
Please! Scan this QR code or A∪B A∪B
browse the link given below:
https://www.geogebra.org/m/dbgxdh3p A U A U
(ii) A ∩ B = {2} 4 B 4 B
A ∩ B = U – (A ∩ B) 3 3
62 62
8 10 5 8 10 5
7 7
1
19 9
= {1, 3, 4, 5, 6, 7, 8, 9, 10} A∩B A∩B
Vedanta Excel in Mathematics - Book 9 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
U U
(iii) A – B = A – (A ∩ B) A B A B
= {4, 6, 8, 10} 4 3 4 3
A – B = U – (A – B)
= {1, 2, 3, 5, 7, 9} 62 5 62 5
8 10 7 8 10 7
19 19
A–B A–B
(iv) B – A = B – (A ∩ B) U U
= {3, 5, 7}
B – A = U – (B – A) A B A B
= {1, 2, 4, 6, 8, 9, 10} 4 3 4 3
62 5 62 5
8 10 7 8 10 7
19 19
B–A B–A
Example 7: S is a subset of a universal set U. If U = {x: x ∈N, x≤10} and S = {y: y is a
perfect square number}, find: (i) S (ii) S ∪ S (iii) S ∩ S (iv) S
Solution:
Here, U = {x: x ∈N, x≤10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and S = {y: y is a perfect square number} = {1, 4, 9}
Now, (i) S = U – S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 4, 9} = {2, 3, 5, 6, 7, 8, 10}
(ii) S ∪ S = {1, 4, 9}∪{2, 3, 4, 5, 6, 7, 8, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}=U
(iii) S ∩ S = {1, 4, 9}∩{2, 3, 4, 5, 6, 7, 8, 10} = ∅
(iv) S = U – S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {2, 3, 5, 6, 7, 8, 10} = {1, 4, 9} = S
Example 8: P and Q are the subsets of a universal set U.
If U = {–4, –3, –2, –1, 0, 1, 2, 3, 4}, P = {–4, –2, 0, 2, 4} and
Solution: Q = {–2, –1, 0, 1, 2}, verify that P ∪ Q = P ∩ Q .
Here, U = {–4, –3, –2, –1, 0, 1, 2, 3, 4}, P = {–4, –2, 0, 2, 4}, and Q = {–2, –1, 0, 1, 2}
Now, P∪Q = {–4, –2, 0, 2, 4}∪{–2, –1, 0, 1, 2}={–4, –2, –1, 0, 1, 2, 4}
Also, P ∪ Q = U – (P∪Q)
= {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–4, –2, –1, 0, 1, 2, 4}
= {–3, 3} … (i)
Again, P = U – P = {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–4, –2, 0, 2, 4} = {–3, –1, 3}
Q = U – Q = {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–2, –1, 0, 1, 2} = {–4, –3, 3, 4}
∴P ∩ Q = {–3, –1, 3} ∩ {–4, –3, 3, 4} = {–3, 3} … (ii)
From (i) and (ii), P ∪ Q = P ∩ Q Verified
Example 9: P, Q, and R are the subsets of the universal set U. If U = {1, 2, 3, ... 12},
P = {2, 3, 5, 7, 11}, Q = {1, 3, 5, 7, 9}, and R = {1, 2, 3, 4, 5}, find
a) P ∪ Q ∪ R b) P ∩ Q ∩ R c) (P ∩ Q) ∪ R d) (P ∪ Q) ∩ R
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 17 Vedanta Excel in Mathematics - Book 9
Set
Solution: P U
Here, U = {1, 2, 3, ... 12} P = {2, 3, 5, 7, 11}, 11 7 9 Q
Q = {1, 3, 5, 7, 9} and R = {1, 2, 3, 4, 5} 2 35 1
a) P ∪ Q = {1, 2, 3, 5, 7, 9, 11}
4 R
P ∪ Q ∪ R = {1, 2, 3, 4, 5, 7, 9, 11} 68 10 12
U P ∪ Q∪R
b) P ∩ Q = {3, 5, 7} P Q Vedanta ICT Corner
P ∩ Q ∩ R = {3, 5} 11 7 9 Please! Scan this QR code or
2 35 1 browse the link given below:
4R https://www.geogebra.org/m/zbtapnzy
10 12
6
8
P∩Q∩R
c) P ∩ Q = {3, 5, 7} d) P ∪ Q ={1, 2, 3, 5, 7, 9, 11}
(P ∩ Q) ∪ R = {1, 2, 3, 4, 5, 7} (P ∪ Q) ∩ R = {1, 2, 3, 5}
U (P ∪ Q) ∩ R = {4, 6, 7, 8, 9, 10, 11, 12}
Q
P U
11 7 9
2 35 1 P Q
11 7 9
4R 2 35 1
68 10 12 4 R
68 10 12
(P ∩ Q) ∪ R
(P ∪ Q) ∩ R
EXERCISE 1.1
General section
1. Let P and Q are the subsets of a universal set U. Write the set operations defined by
the following set-builder forms.
a) {x: x ∈ P or x∈Q} b) {x: x∈P and x ∈ Q} c) {x: x∈P but x ∉Q}
d) {x: x ∈ Q but x∉P} e) {x: x∈U but x ∉ P} f) {x: x∈U but x ∉Q}
g) {x: x ∈ U but x ∉P or Q} h) {x: x∈U but x ∉P and Q} i) {x: x∈U but x ∉P – Q}
2. Write the set operations represented by shaded regions shown in the following
Venn-diagrams.
a) U b) U c) U d) U
B
A BA BA BA
18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Vedanta Excel in Mathematics - Book 9
e) U f) X U g) U h) Set
A B YA BP U
Q
CR
3. a) A and B are the subsets of the universal set U. From the given diagrams, list the
elements of the following set operations. U
(i) A ∪ B and A ∪ B (ii) A ∩ B and A ∩ B A 1 B
(iii) A – B and A – B (iv) B – A and B – A 5 3 2
6
7
9 10
48
b) P, Q and R are the subsets of the universal set U. List the elements of the following set
operations from the given diagram. P8 U
Q
(i) P ∪ Q ∪ R (ii) P ∩ Q ∩ R (iii) P ∪ Q ∪ R 2 1
(vi) (P ∩ Q) ∪ R 4
10 6
12 3 5
(iv) P ∩ Q ∩ R (v) (P ∪ Q) ∩ R 9 15 R
7 11 13 14
4. a) If A = {n, e, p, a, l} and B = {b, h, u, t, a, n}, find
(i) A∪B (ii) A∩B (iii) A – B (iv) B – A.
Also, represent them in Venn-diagrams.
b) Let P = {x: x ∈N and x<10} and Q= {y: y is a factor of 8}, find
(i) P ∪ Q (ii) P ∩ Q (iii) P – Q (iv) Q – P.
Also, show these operations in Venn-diagrams.
c) If M = {x: x is an odd numbers between 10 and 20} and Q= {y: y is a prime number
between 15 and 25}, find and show the following operations in Venn-diagrams.
(i) M ∪ N (ii) M ∩ N (iii) M – N (iv) N – M
5. a) Assuming that A and B are two overlapping sets, draw two separate Venn-diagrams
to verify A ∪ B = B ∪ (A – B) by shading.
b) Let P and Q are two overlapping sets. Draw two separate Venn-diagrams of P – Q
and P – (P ∩ Q) and verify P – Q = P – (P ∩ Q) by shading.
6. a) If A = {1, 2, 4, 8} and B = {4, 6, 8, 10}, find (A – B) ∪ (B – A).
b) Find the symmetric difference between the following sets.
(i) A = {m, a, t, h} and B = {m, i, n, d, e, r}
(ii) P = {2, 3, 5, 7, 11} and Q={1, 3, 5, 11}
7. a) If U = {1, 2, 3, …, 10}, A ={1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, find the following
sets.
(i) A (ii) B (iii) A ∪ B (iv) A ∩ B
(v) A ∪ B (vi) A ∩ B (vii) A (viii) B
b) If U = {1, 2, 3, …, 15} and A ={2, 4, 6, 8, 10, 12, 14}, find:
(i) A (ii) A ∪ A (iii) A ∩ A (iv) U
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Set
Creative section
8. a) If U = {1, 2, 3, ... 10}, P = {1, 2, 3, 4, 5}and Q = {2, 4, 6, 8}, list the elements of the
following set operations and represent them by shading in Venn-diagrams.
(i) P ∪ Q and P ∪ Q (ii) P ∩ Q and P ∩ Q (iii) P – Q and P – Q
(iv) Q – P and Q – P (v) P ∪ Q (vi) P ∩ Q
b) A = {1, 3, 5, 7, 9, 11}, B = {1, 2, 3, 4, 5, 6, 7} and C = {3, 6, 9, 12, 15} are the
subsets of the universal set U = {1, 2, 3, ... 15}. List the elements of the following set
operations and illustrate them in Venn-diagrams by shading.
(i) A ∪ B ∪ C and A ∪ B ∪ C (ii) A ∩ B ∩ C and A ∩ B ∩ C
(iii) (A ∪ B) ∩ C and (A ∪ B) ∩ C (iv) A ∩ (B ∪ C) and A ∩ (B ∪ C)
(v) (A – B) ∪ C and (A – B) ∪ C (vi) A ∪ (B – C) and A ∪ (B – C)
9. a) If U = {0, 1, 2, …., 10}, A = {2, 3, 5, 7}and B = {1, 3, 5, 7, 9}, verify the following
operations.
(i) A ∪ B = A ∩ B (ii) A ∩ B = A ∪ B
b) If a universal set U = {x : x ∈ N, x ≤ 10}, A = {y : y = 2n, n ∈ N, n < 5} and
B = {z : z = 3n, n ∈ N, n<4 }, prove that:
(i) A – B = B – A (ii) A D B = A D B
10. a) If P = {1, 2, 3, 4, 5, 6}, Q = {2, 4, 6, 8}, and R = {3, 6, 9, 12}, explore the relationship
between the following operations.
(i) P ∪ (Q ∩ R) and (P ∪ Q) ∩ (P ∪ R) (ii) P ∩ (Q ∪ R) and (P ∩ Q) ∪ (P ∩ R)
(iii) P – (Q ∪ R) and (P – Q) ∩ (P – R) (iv) P – (Q ∩ R) and (P – Q) ∪ (P – R)
b) A, B and C are the subsets of a universal set U. If U = {x: x∈N, x≤12},
A = {odd numbers less than 10}, B = {prime numbers less than 12}, and
R = {prime numbers less than 6}, verify the following operations.
(i) A ∩ B ∩ C = A ∪ B ∪ C (ii) A ∪ B ∪ C = A ∩ B ∩ C
11. a) If A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} are two disjoint sets verify that
n(A∪B) = n(A) + n(B).
b) If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 6, 12} are two overlapping sets, show that
n(A∪B) = n(A) + n(B) – n(A∩B)
Project work and activity section
12. Conduct a survey among your at least 20 friends whether they like to play volleyball or
basketball or both. Represent the set of students who like to play volleyball by V and the
set of students who like to play basketball by B.
Vedanta Excel in Mathematics - Book 9 20 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Draw Venn-diagrams and show each of the following set operations by shading.
(i) Set of students who like to play volleyball or basketball or both, V ∪ B.
(ii) Set of students who like to play volleyball as well as basketball, V ∩ B.
(iii) Set of students who like to play volleyball but not basketball, V – B.
(iv) Set of students who like to play basketball but not volleyball, B – V
(v) Set of students who like to play none of these games, V ∪ B .
(vi) Set of students who like to play only one of these games, V ∆ B.
13. Ask to any two of your friends about the brands of mobile sets that their family members
use. Assume that the sets of brands of mobiles used by family members of one friend by
A and another by B.
Find (i) A ∪ B (ii) B ∪ A (iii) A ∩ B (iv) B ∩ A
14. Suppose, A and B are overlapping sets and the subsets a universal set U.
Draw the separate Venn-diagrams in the chart paper and shade to verify the following
operations.
Vedanta ICT Corner
(i) (A – B) ∪ (B – A) = (A∪B) – (A∩B) Please! Scan this QR code or
(ii) A ∪ B = A ∩ B browse the link given below:
(iii) A – B = A ∪ B ∪ B https://www.geogebra.org/m/mxm83pss
1.11 Cardinality relationships of two sets
The cardinality of a set is a measure of the number of members of the set. For example,
the set A = {1, 3, 5, 7, 9} contains 5 members, and therefore A has a cardinality of 5.
The number of members of a set is called its cardinality. Certain relationships can be
generalised by taking the cardinalities of different sets.
(i) Cardinality relations of union of two disjoint sets
Let U = {a, b, c, d, e, f, g, h, i, j} is a universal set. A = {a, b, c, d} and
B = {e, f, g, h, i} are the subsets of U. U
Now, A ∪ B = {a, b, c, d, e, f, g, h, i} A d B
Here, n (U) = 10, n (A) = 4 and n (B) = 5 a e
Now, n (A ∪ B) = 9 = 4 + 5 = n (A) + n (B) fi
Also, n (A ∪ B) = 1 = 10 – 9 = n (U) – n (A ∪ B) b gh
c
j
Thus, if A and B are any two disjoint subsets of a universal set U, then
n (A ∪ B) = n (A) + n (B)
n (A ∪ B) = n (U) – n (A ∪ B)
(ii) Cardinality relations of union of two overlapping sets
Let A = {a, b, c, d, e} and B = {c, h, i, e, f} are two subsets of a universal set
U = {a, b, c, d, e, f, g, h, i, j}
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 21 Vedanta Excel in Mathematics - Book 9
Set
Now, A ∪ B = {a, b, c, d, e, f, h, i} U
A ∩ B ={c, e} n (A) Aa c f B n (B)
A ∪ B = {g, j} n (only A) be i n (only B)
only A = n0(A) = {a, b, d} dh
only B = n0(B) = {f, h, i} gj
n (A ∩ B) n (A ∪ B)
Here, n (U) = 10, n (A) = 5, n (B) = 5 and n (A ∩ B) = 2
Now, n (A ∪ B) = 8 = 5 + 5 – 2 = n (A) + n (B) – n (A ∩ B )
Also, n (A ∪ B) = 2 = 10 – 8 = n (U) – n (A ∪ B)
Again, n (only A) = nO(A) = 3 = 5 – 2 = n(A) – n(A ∩ B)
n (only B) = nO(B) = 3 = 5 – 2 = n(B) – n(A ∩ B)
Thus, if A and B are any two overlapping subsets of a universal set U,
n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
n (A ∪ B) = n (U) – n (A ∪ B)
n (A ∩ B) = n (A) + n (B) – n (A ∪ B)
n (only A) = no(A) = n(A) – n(A ∩ B)
n (only B) =no(B) = n(B) – n(A ∩ B)
Furthermore, if the universal set U contains only the members of A and B,
n (A ∪ B) = 0 and n (U) = n (A ∪ B)
Worked-out Examples
Example 1: From the adjoining Venn-diagram, find the U
cardinal numbers of the following sets. Pa df Q
e g
a) n (P) b) n (Q) c) n (P ∪ Q) b
c h i
d) n (P ∩ Q) e) n (P ∪ Q) f) n (P )
j
g) n (Q) h) no (P) i) no (Q) kl
Solution:
a) n (P) = 5 b) n (Q) = 6 c) n (P ∪ Q) = 9
d) n (P ∩ Q) = 2 e) n (P ∪ Q) = 3 f) n (P ) = n (U) – n (P) = 12 – 5 = 7
g) n (Q ) = n (U) – n (Q) = 12 – 6 = 6 h) no (P) = n(P) – n(P ∩ Q) = 5 – 2 = 3
i) no (Q) = n(Q) – n(P ∩ Q) = 6 – 2 = 4
Example 2: A and B are the subsets of a universal set U. If n (U) = 95, n (A) = 64,
n (B) = 56, n (A ∩ B) = 30, find
a) n (A ∪ B) b) n (A ∪ B) c) no (A) d) no (B)
Vedanta Excel in Mathematics - Book 9 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Solution:
Here, n (U) = 95, n (A) = 64, n (B) = 56 and n (A ∩ B) = 30
a) Now, n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
= 64 + 56 – 30 = 90
b) n (A ∪ B) = n (U) – n (A ∪ B) = 95 – 90 = 5
c) no (A) = n (A) – n (A ∩ B) = 64 – 30 = 34
d) no (B) = n (B) – n (A ∩ B) = 56 – 30 = 26
Example 3: If n(A – B) = 24, n(A ∪ B) = 80, and n(A ∩ B) = 20, then find n(B). Illustrate
this information in a Venn-diagram.
Solution:
Here, n(A – B) = n0(A) = 24, n(A ∪ B) = 80 and n(A ∩ B) = 20
Now, n(A ∪ B) = n0(A) + n(A ∩ B) + n0(B) A B
36
or, 80 = 24 + 20 + n0(B) 24 20
or, n0(B) = 36
Again, n(B) = n0(B) + n(A ∩ B)
= 36 + 20 = 56
Example 4: If n (U) = 120, n (A) = 63, n (B) = 54 and n (A ∪ B) = 99, complete the
following: (i) Find n (A ∪ B), n (A ∩ B), no (A) and no (B).
(ii) Illustrate the information in a Venn-diagram.
Solution:
Here, n (U) = 120, n (A) = 63, n (B) = 54 and n (A ∪ B) = 99
(i) Now, n (A ∪ B) = n (U) – n ( A ∪ B)
= 120 – 99 = 21
Again, n (A ∪ B) = n (A) + n (B) – n (A ∩ B) (ii) Illustration in Venn-diagram
or, 99 = 63 + 54 – n (A ∩ B)
or, n (A ∩ B) = 117 – 99 = 18 U
Also, no (A) = n (A) – n (A ∩ B) AB
= 63 – 18 = 45
45 18 36
And no (B) = n (B) – n (A ∩ B) 21
= 54 – 18 = 36
Example 5: In a survey of a few number of farmers, it is found that 50 farmers use
chemical fertilizers, 65 use bio-fertilizers, and 35 use both chemical
and bio-fertilizers. Find the number of farmers who use chemical or
bio-fertilizers.
Solution:
Let C and B be the sets of the farmers who use chemical and bio-fertilizers respectively.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 23 Vedanta Excel in Mathematics - Book 9
Set
Here, n(C) = 50, n(B) = 65 and n(C ∩ B) = 35
Now, n (C ∪ B) = n(C) + n(B) – n(C ∩ B)
= 50 + 65 – 35
= 115 – 35
= 80
Hence, 80 farmers use chemical or bio-fertilizers.
Example 6: In a group of 150 students who like at least one profession: educator
or doctor. 80 of them like educator and 95 like doctor. By drawing
Venn-diagram, find
(i) How many students like both the professions?
(ii) How many students like only educator?
Solution:
Let E and D be the sets of the students who like educator and doctor respectively.
Let n(E ∩ D) = x,
From the Venn-diagram, Venn-diagram U
(i) n (E ∪ D) = 80 – x + x + 95 – x D
or, 150 = 175 – x E
or, x = 175 – 150 = 25
80-x x 95-x
Hence, 25 of them like both the profession.
(ii) no (E) = 80 – x = 80 – 25 = 55
Example 7: In a survey, 140 cinema lovers were asked what they would like comedy
or action movies. 72 of them said comedy movies and 53 said action
movies. However, 45 people said they did not like any type of movies.
(i) Find the number of people who like both comedy and action movies.
(ii) Find the number of people who like only one type of movie.
(iii) Illustrate the result in Venn-diagram.
Solution:
Let C and A be the sets of the people who like comedy and action movies respectively.
Here, n (U) = 140, n (C) = 72, n (A) = 53 and n (C ∪ A ) = 45
(i) Now, n (C ∪ A) = n (U) – n (C ∪ A )
= 140 – 45 = 95 (iii) Venn-diagram
Again, n (C ∪ A) =n (C) + n (A) – n (C ∩ A)
or, 95 = 72 + 53 – n (C ∩ A) U
or, n (C ∩ A) = 125 – 95 = 30 CA
42 30 23
Hence, 30 people like both comedy and action movies. 45
Vedanta Excel in Mathematics - Book 9 24 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
(ii) Also, the number of people who like comedy movies only = no (C)
= n (C) – n (C ∩ A)
= 72 – 30 = 42
And, the number of people who like action moves only = no (A)
= n (A) – n (C ∩ A)
= 53 – 30 = 23
∴ The number of people who like only one type of movie = 42 + 23 = 65
EXERCISE 1.2
General section
1. a) A and B are any two disjoint sets. If n(A) = x and n(B) = y, find n(A ∪ B).
b) If n(A) = p, n(B) = q, and n (A ∩ B) = r, show this information in a Venn-diagram and
show that n (A ∪ B) = n(A) + n(B) – n(A ∩ B).
c) If X and Y are two overlapping subsets of a universal set U, write the relation between
n(U), n (X ∪ Y), and n (X ∪ Y ).
d) If A and B are two overlapping subsets of a universal set U, write the relation between
n(A), n(A ∩ B) and no(A). A 1 U
4 2 B
2. From the adjoining Venn-diagram, find the cardinal 3
numbers of the following sets:
69
a) n (U) b) n (A) c) n (B)
8
d) n (A ∪ B) e) n (A ∩ B) f) n (A ∪ B ) 57
g) n (A) h) n (B ) i) no (A) j) no (B)
3. a) If n (U) = 65, n (A) = 28, n (B) = 45, and n (A ∩ B) = 20, find
(i) n (A ∪ B) (ii) n (A ∪ B ) (iii) no (A) (iv) no (B)
b) P and Q are the subsets of a universal set U. If n (P) = 55 %, n (Q) = 50 %, and
n (P ∪ Q ) = 15 %, find: (i) n (P ∪ Q) (ii) n (P ∩ Q) (iii) n (only P) (iv) n (only Q)
c) X and Y are the subsets of a universal set U. If n (U) = 88, no (X) = 35,
no (Y) = 30, and n (X ∩ Y) = 10, find:
(i) n (X) (ii) n (Y) (iii) n (X ∪ Y) (iv) n (X ∪ Y )
Creative section
4. a) In a survey of 600 people in a village of Dhading district, 400 people said they can
speak Tamang language, 350 said Nepali language and 200 of them said they can
speak both the languages.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) How many people can speak Tamang language only?
(iii) How many people can speak Nepali language only?
(iv) How many people cannot speak any of two languages?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 25 Vedanta Excel in Mathematics - Book 9
Set
b) In a survey of 1500 people, 775 of them like Nepal Idol, 975 liked Comedy Champion,
and 450 people liked both the shows.
(i) Show the above information in a Venn-diagram.
(ii) How many people did not like both the shows?
c) In a group of 250 music lovers, 135 of them like folk songs, and 150 like modern
songs. By drawing a Venn-diagram, find:
(i) how many people like both the songs?
(ii) How many people like only folk songs?
d) In a survey of a group of farmers, it was found that 80 % farmers have crops farming,
30% farmers have animals farming, and every farmer has at least one farming.
(i) Represent the information in a Venn-diagram.
(ii) What percent of farmers had both farming?
(iii) What percent of farmers had only animals farming?
5. a) In a group of 500 students, 280 like bananas, 310 like apples, and 55 do not like both
the fruits.
(i) Find the number of students who like both the fruits.
(ii) Find the number of students who like only one fruit.
(iii) Show the result in a Venn-diagram.
b) In a survey of 900 students in a school, it was found that 600 students liked tea, 500
liked coffee, and 125 did not like both drinks.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of students who like both drinks.
(iii) Find the number of students who like one of these drinks only.
6. a) In a group of 75 students, 20 liked football only, 30 liked cricket only and 18 did not
like any of two games?
(i) How many of them liked at least one game?
(ii) Find the number of students who liked both the games.
(iii) How many of them liked football?
(iv) How many of them liked cricket?
(v) Represent the result in a Venn diagram.
b) In a survey of 750 tourist who visited Nepal during 'Visit Nepal 2020', it was found
that 260 tourists visited Pokhara but not Sauraha, 240 visited Sauraha but not Pokhara
and 125 of them did not visit both places.
(i) Find the number of tourists who visited Pokhara or Sauraha.
(ii) Find the number of tourist who visited Pokhara and Sauraha.
(iii) How many tourists visited Pokhara.
(iv) Show the result in a Venn-diagram.
Vedanta Excel in Mathematics - Book 9 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Project work and activity section
7. Conduct a survey inside your classroom and collect the data about how many of your
friends like football, cricket, and both football and cricket. Then, tabulate the data and
find the following numbers by using cardinality relation of two sets:
a) Number of friends who like football and cricket.
b) Number of friends who do not like any of these two games.
c) Number of friends who like only (i) football (ii) cricket
d) Show your data in a Venn-diagram
8. Make one verbal problem reflecting to the real life situations by using each of the following
sets of known and unknown variables:
a) n(U) = 310, n(A) = 120, n(B) = 180, n(A ∩ B) = 40, n(A ∪ B) = ?
b) n(U) = 190, n(A) = 105, n(B) = 115, n(A ∪ B) = 15, n(A ∩ B) = ?
c) no(A) = 72, n(B) = 100, n(A ∩ B) = 36, n(A ∪ B) = 28, n(U) = ?
Now, solve your verbal problems and illustrate the results in Venn-diagrams.
OBJECTIVE QUESTIONS
Let’s tick (√) the correct alternative.
1. In set builder form, A∪B is written as
(A) {x: x ∈ A or x∈B} (B) {x: x ∈ A and x ∈B}
(C) {x: x∈A, but x∉B} (D){x: x ∈ B, but x ∉A}
2. The set operation defined by {x: x∈P and x∈Q} is
(A) P∩Q (B) P ∪ Q (C) P – Q (D) (P ∩ Q)′
3. If A = {x : x ∈N, x < 5} and B = {y: y ∈ N, 3 < y < 7} then A ∪ B is
(A) {4} (B) {1, 2, 3} (C) {5, 6} (D) {1, 2, 3, 4, 5, 6}
4. Which of the following statements is true?
(A) A ∪ B = A ∩ B (B) A ∩ B =A – B (C) A ∪ B = A – B (D) A ∩ B ⊆ A
5. If A ⊂ B then A ∩ B is equal to
(A) A (B) B (C) A D B (D) A ∪ B
6. If U = {1, 2, 3, 4, 5} and A = {2, 3} then A'' is
(A) {} (B) {1, 4, 5} (C) {2, 3} (D) {1, 3, 5}
7. Which of the following relations is not correct?
(A) A ∪ B = B ∪ A (B) A ∩ B = B ∩ A (C) A – B = B – A (D) A ∪ ∅ = A
8. Which of the following relations is correct?
(A) A ∩ ∅ = ∅ (B) A ∪ ∅ = ∅ (C) A ∪ U = A (D) A ∩ U = U
9. Which of the following relations is correct?
(A) ∅' = ∅ (B) A ∪ A' = ∅ (C) A ∩ A = U (D) A ∪ A' = U
10. Which one of the following relation is always true on the operations of sets?
(A) A ∪ B = A ∪ B (B) A ∩ B = A ∩ B
(C) A ∩ B = A ∪ B (D) A ∩ B = A – B
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 27 Vedanta Excel in Mathematics - Book 9
Unit 2 Taxation
2.1 Tax – Introduction
Let's study a few examples of taxes.
a) Bimal paid Rs 5,000 for bluebook renewal tax of his bike.
b) Nikita paid 1% of her yearly income as social security tax.
c) Anil paid 25% tax for his normal business.
d) Mr. Chaudhary sold a television with 13% VAT.
e) Discuss with your family members and the various types of taxes paid by
your family.
f) Discuss with your friends about the various types of taxes paid by their
families.
The government does not have its own money. Its receipts come from individual
income taxes, corporate income taxes, estate and gift taxes, social insurance taxes,
excise taxes, etc. The taxes we pay are used by government for transport, education,
health, law and order, culture, media and sport, trade and industry, environment, etc.
The Inland Revenue Department (IRD) under the Ministry of Finance of the
Government of Nepal is responsible for the administration of Value Added Tax,
Income Tax, and Excise Duty.
Facts to remember
1. A tax is a compulsory financial charge or some other type of levy imposed
upon taxpayer by a government authority in order to fund various public
expenditures.
2. Tax is a compulsory contribution to state revenue, levied by the government
on citizen’s income and business profits, or added to the cost of some goods,
services, and transactions.
Here, among these taxes, we shall discuss about the income tax and Value Added
Tax (VAT).
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Taxation
2.2 Income Tax
An income tax is a tax imposed on individuals or taxpayers as per their annual income.
The rates of income tax vary with the specified limits of taxable incomes. The specified
limit of income to levy different rates of tax is decided by the government on the basis
of minimum needs of individuals. The minimum needs of an individual are justified
according to his/her personal details, e.g. marital status, number of dependents, etc.
1% Social Security Tax is levied on the annual income up to the certain limit.
In this way, to calculate the income tax, at first the social security taxable income is
deducted from the total income to find the taxable income above the minimum limit.
Then, the tax is levied as per the rate of tax in percent.
Taxable income = Total income – Social security taxable income
Income tax = Rate of tax (in %) × Taxable income
The table given below shows the taxable Vedanta ICT Corner
income and the tax rate in percent for Please! Scan this QR code or
the unmarried individuals and married browse the link given below:
couples based on Nepal Income Tax Rates https://www.geogebra.org/m/bzvyteuh
2078/79 (Source: www.nbsm.com.np).
Tax Rates for Natural person for the Fiscal Year 2078/79 (2021/22)
(Only for Employment Income)
Assessed as Individuals Assessed as Couples
Particulars Taxable Income (Rs) Tax Rate Taxable Income (Rs) Tax Rate
First Tax slab 4,00,000 1% 4,50,000 1%
10%
Next 1,00,000 10% 1,00,000 20%
(4,50,000 to 5,50,000)
(4,00,000 to 5,00,000) 30%
36%
Next 2,00,000 20% 2,00,000
(5,50,000 to 7,50,000)
(5,00,000 to 7,00,000)
Next 13,00,000 30% 12,50,000
(7,50,000 to 20,00,000)
(7,00,000 to 20,00,000)
Balance Exceeding 20,00,000 36% 20,00,000
(Only for Proprietorship firm)
Assessed as Individuals Assessed as Couples
Particulars Taxable Income (Rs) Tax Rate Taxable Income (Rs) Tax Rate
0%
First Tax slab 4,00,000 10% 4,50,000 0%
Next 1,00,000
Next 20% 1,00,000 10%
Next (4,00,000 to 5,00,000) 30% (4,50,000 to 5,50,000) 20%
Balance Exceeding 2,00,000
2,00,000 36% (5,50,000 to 7,50,000)
(5,00,000 to 7,00,000) 12,50,000 30%
(7,50,000 to 20,00,000)
13,00,000
20,00,000 36%
(7,00,000 to 20,00,000)
20,00,000
The following incomes are entitled for the Tax rebate
(i) The premium paid to the insurance company (ii) Provident fund
(iii) Citizen investment trust (iv) Donation
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Taxation
(v) Religious functions (vi) Remote area allowance
(vii) 75% of the foreign allowance (viii) Medical expenditure
Note: We can find the detail about taxation in the website http://www.ird.gov.np.
2.3 Tax Withholding
Tax withholding, is an income tax paid to the government by the payer of the income
rather than by the recipient of the income. The tax is withheld or deducted from the
income due to the recipient. The withholding taxes are charged on payments of
service fee, interest, dividends, royalties, rent or even the sale of real estate.
The interest paid by a bank or financial institution to any natural person for deposit
which is sourced in Nepal and is not related to operation of a business, then 5% final
withholding tax applies.
Simple interest
When we borrow money from a bank, we should pay interest to the bank. When
we deposit money in a bank, the bank pays interest to us. The interest which is
calculated from the original borrowed (or deposited) sum is called simple interest.
Let’s review the following terms which are needed to calculate simple interest.
(i) Principal (P) – It is the deposited or borrowed sum of money.
(ii) Rate of interest (R) – It is the interest of Rs 100 for 1 year. It is expressed as the
percent per year (or per annum, p.a.)
(iii) Time (T) – It is the duration for which principal is deposited or borrowed.
(iv) Interest (I) – It is the simple interest on Rs P at R% p.a. in T years.
(v) Amount (A) – It is the total sum of principal and interest. So, A = P + I
Facts to remember
(i) If I is the simple interest on the principal Rs. P for the duration of T year at the
rate of R% per annum, then I = PTR
100
(ii) The account holder is charged 5% tax on interest paid by the bank or any financial
institutions as withholding tax.
Worked-out Examples
Example 1: Mr. Thakur has a barber shop. His annual income is Rs 6,40,000. If the
tax up to Rs 4,50,000 is exempted, 10% tax is charged for the income of
Rs 4,50,001 to Rs 5,50,000 and 20% tax is charged for the income of
Rs 5,50,001 to Rs 7,00,000. Calculate the annual income tax paid by him.
Solution:
Here, the annual income of the individual = Rs 6,40,000
Vedanta Excel in Mathematics - Book 9 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Taxation
Taxable income Tax rate
Up to Rs 4,50,000 0%
Rs 5,50,00 – Rs 4,50,000 = Rs 1,00,000 10%
Rs 6,40,000 – Rs 5,50,000 = Rs 90,000 20%
Now, the income tax for the income of Rs 1,00,000 = 10% of Rs 1,00,000
= Rs 10,000
Again, the income tax for the next income of Rs 90,000 = 20% of Rs 90,000
= Rs 18,000
The total income tax paid by him = Rs 10,000 + Rs 18,000 = Rs 28,000
Hence, he pays the income tax of Rs 28,000 in a year.
Example 2: The monthly income of an unmarried individual is Rs 48,500. If 1 % social
security tax is charged up to the annual income of Rs 4,00,000. Then 10% and
20% taxes are charged for the next incomes of Rs 1,00,000 and Rs 2,00,000
respectively. Calculate the annual income tax paid by the individual.
Solution:
Here, the annual income of the individual = Rs 12 × Rs 48,500 = Rs 5,82,000
= Rs 4,00,000 + (Rs 5,00,000 – Rs 4,00,000) + (Rs 5,82,000 – Rs 5,00,000)
= Rs 4,00,000 + Rs 1,00,000 +Rs 82,000
Now, the social security tax for the first Rs 4,00,000 = 1% of Rs 4,00,000 = Rs 4,000
Also, the income tax for the next income of Rs 1,00,000 = 10% of Rs 1,00,000
= Rs 10,000
Again, the income tax for the next income of Rs 82,000 = 20% of Rs 82,000
= Rs 16,400
The total income tax paid by the individual = Rs 4000 + Rs 10,000 + Rs 16,400
= Rs 30,400
Hence, the individual should pay the income tax of Rs 30,400 in a year.
Example 3: Mr. Limbu is an unmarried secondary level Science teacher. His monthly
salary is Rs 39,990 with allowance of Rs 2,000. He gets a festival expense
which is equivalent to his basic salary of one month and 10% of his salary
excluding allowance and festival expense is deducted as provident fund.
If 1% social security tax is levied upon the income of Rs 4,00,000, 10%
and 20% taxes are levied on the next incomes of Rs 1,00,000 and up to
Rs 2,00,000 respectively, how much income tax should he pay in a year?
Solution:
Here, his monthly basic salary = Rs 39,990 – Rs 2,000 = Rs 37,990
Festival expense = Basic salary of one month = Rs 37,990
Monthly provident fund = 10% of Rs 37,990 = Rs 3,799
After deducting provident fund, his monthly income = Rs 39,990 – Rs 3,799 = Rs 36,191
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Taxation
Taxable income of the year with festival expense = 12 × Rs 36,191 + Rs 37,990
= Rs 4,72,282
= Rs 4,00,000 + Rs 72,282
Now, the social security tax for the first Rs 4,00,000 = 1% of Rs 4,00,000 = Rs 4,000
Again, the income tax for remaining taxable income of Rs 72,282 = 10% of Rs 72,282
= Rs 7,228.20
The total income tax paid by Mr. Limbu = Rs 4000 + Rs 7,228.20
= Rs 11,228.20
Hence, Mr. Limbu should pay the income tax of Rs 11,228.20 in a year.
Example 4: Mrs. Gautam is the Branch Manager of a commercial bank. Her monthly
salary is Rs 95,400 and 10% of her salary is deducted as provident fund. She
pays Rs 24,520 as the premium of her life insurance. If 1% social security
tax is levied upon the income of 4,50,000, 10%, 20% and 30% taxes are
levied on the next incomes of Rs 1,00,000, Rs 2,00,000 and upto Rs 12,50,000
respectively, how much income tax should she pay in a year?
Solution:
Here, her monthly income after deducting the provident fund = Rs 95,400 – 10% of Rs 95,400
= Rs 85,860
Her annual income after deducting provident fund = 12 × Rs 85,860
= Rs 10,30,320
Her taxable income after deducting premium of insurance = Rs 10,30,320 – Rs 24,520
= Rs 10,05,800
The social security tax for the first Rs 4,50,000 = 1% of Rs 4,50,000
= Rs 4,500
The income tax for the next income of Rs 1,00,000 = 10% of Rs 1,00,000 = Rs 10,000
The income tax for the next income of Rs 2,00,000 = Rs 20% of Rs 2,00,000 = Rs 40,000
Now, the taxable income for the next 30% tax
= Rs 10,05,800 – (Rs 4,50,000 + Rs 1,00,000 + Rs 2,00,000)
= Rs 2,55,800
Then, the income tax for the next income of Rs 2,55,800 = 30% of Rs 2,55,800 = Rs 76,740
The total income tax paid by her = Rs 4,500 + Rs 10,000 + Rs 40,000 + Rs 76,740
= Rs 1,31,240
Hence, she should pay the income tax of Rs 1,31,240 in a year.
Example 5: The present rate of income tax fixed by Inland Revenue Department (IRD)
is given below.
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Taxation
For an individual For couple
Income slab Tax rate Income slab Tax rate
Up to Rs 4,00,000 1% Up to Rs 4,50,000 1%
Rs 4,00,001 to Rs Rs 5,00,000 10% Rs 4,50,001 to Rs Rs 5,50,000 10%
Rs 5,00,001 to Rs Rs 7,00,000 20% Rs 5,50,001 to Rs Rs 7,00,000 20%
The monthly salary of Suresh, an unmarried servant in a bank, is Rs 35,000 and his
annual income is equivalent to his salary of 15 months. Similarly, the monthly salary
of Bina, a married civil servant is Rs 42,000 and her annual income is equivalent to
her salary of 13 months including festival expense. According to the above rates of
tax, who pays more income tax and by how much?
Solution:
For Suresh, annual income = 15×Rs 35,000 = Rs 5,25,000
= Rs 4,00,000 + (Rs 5,00,000 – Rs 4,00,000) + (Rs 5,25,000 – Rs 5,00,000)
= Rs 4,00,000 + Rs 1,00,000 + Rs 25,000
The social security tax for the first Rs 4,00,000 = 1% of Rs 4,00,000 = Rs 4,000
The income tax for the next income of Rs 1,00,000 = 10% of Rs 1,00,000 = Rs 10,000
The income tax for the next income of Rs 25,000 = Rs 20% of Rs 25,000 = Rs 5,000
∴ The total income tax paid by him = Rs 4,000 + Rs 10,000 + Rs 5,000 = Rs 19,000
Also,
For Bina, annual income = 13×Rs 42,000 = Rs 5,46,000
= Rs 4,50,000 + (Rs 5,46,000 – Rs 4,50,000)
= Rs 4,50,000 + Rs 96,000
The social security tax for the first Rs 4,50,000 = 1% of Rs 4,50,000 = Rs 4,500
The income tax for the next income of Rs 96,000 = 10% of Rs 96,000 = Rs 9,600
∴ The total income tax paid by her = Rs 4,500 + Rs 9,600 = Rs 14,100
Difference in income taxes = Rs 19,000 – Rs 14,100 = Rs 4,900
Hence, Suresh pays Rs 4,900 more income tax than Bina.
Example 6: Mr. Gurung deposited Rs. 7,00,000 for 4 years in his fixed account at a
commercial bank. The bank pays him the simple interest at the rate of
10% p.a. How much net interest would he get if 5% of interest is charged
as income tax?
Solution:
Here, principal (P) = Rs 7,00,0000, time (T) = 4 years and rate (R)=10% p.a.
Now, simple interest (I) = PTR
100
Rs 700000 × 4 × 10
= 100 = Rs 2,80,000
Also, rate of tax = 5%
∴Tax amount = 5% of Rs 2,80,000 = Rs 14,000
Net interest = Rs 2,80,000 – Rs 14,000 = Rs 2,66,000
Hence, he would receive the net interest of Rs 2,66,000.
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Taxation
EXERCISE 2.1
General section
1. a) The annual income of a sole proprietor of a grocery shop is Rs 10,00,000. If the
tax is exempted up to Rs 4,50,000, what is his/her taxable income?
b) The yearly income of an individual is Rs 4,44,000 with Rs 24,000 allowance.
What is his/her taxable income?
c) The yearly income of an officer is Rs 4,55,880. If he accumulates Rs 45,588 in
provident fund and he pays Rs 25,000 as premium of his life insurance in the
year, what is his taxable income?
d) The monthly income of a government servant is Rs 77,280 and he gets the festival
expense of one month’s salary, what is his taxable income?
2. a) The monthly salary of an individual is Rs 25,450. If 1% social security tax is
charged upto the annual income of Rs 4,00,000, calculate the income tax paid by
the individual.
b) 1 % social security tax is charged upto the yearly income of Rs 4,50,000 to a
married couple. If the monthly income of a couple is Rs 33,500, how much tax
should the couple pay in a year?
Creative section - A
3. Inland Revenue Department (IRD) has fixed the following rates of income tax for
Proprietorship firm. Use it to calculate the income taxes.
For an individual For couple
Income slab Tax rate Income slab Tax rate
0%
Up to Rs 4,00,000 0% Up to Rs 4,50,000 10%
Rs 4,00,001 to Rs 5,00,000 10% Rs 4,50,001 to Rs 5,50,000
Rs 5,00,001 to Rs 7,00,000 20% Rs 5,50,001 to Rs 7,50,000 20%
Rs 7,00,001 to Rs 20,00,000 30% Rs 7,50,001 to Rs 20,00,000 30%
a) Mr. Baral has a stationery shop. His annual income is Rs 6,40,000. If he is
unmarried, how much income tax should he pay? Find it.
b) Mr. Yadav is still unmarried but he is the proprietor of a furniture factory. He
earned Rs 15,00,000 last year, how much income tax did he pay last year?
c) Mrs. Adhikari is the proprietor of boutique training centre. If her annual income
is Rs 6,75,000, how much income tax does she pay?
d) Mr. Manandhar is a married person. He has a registered computer repair service
centre. He earned Rs 9,25,000 in this year. How much tax is charged on his
income?
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Taxation
4. Study the given income tax rates fixed by IRD and workout the following problems.
Assessed as individual Assessed as couple
Particulars Taxable income (Rs) Tax rate Taxable income (Rs) Tax rate
First tax slab 4,00,000 1% 4,50,000 1%
Next 1,00,000 10% 1,00,000 10%
(4,00,001 to 5,00,000) (4,50,001 to 5,50,000)
Next 2,00,000 20% 2,00,000 20%
(5,00,001 to 7,00,000) (5,50,001 to 7,50,000)
Next 13,00,000 30% 12,50,000 30%
(7,00,001 to 20,00,000) (7,50,001 to 20,00,000)
Balance 20,00,000 36% 20,00,000 36%
exceeding
a) The monthly income of an unmarried civil officer is Rs 37,990 and one month’s
salary is provided as Dashain expense. How much income tax should he/she pay
in a year?
b) The monthly salary of a married couple is Rs 40,500 plus a festival expense of
Rs 30,000. Calculate the income tax paid by the couple in a year.
c) Mrs. Gurung is a bank Manager in a development bank. Her monthly is Rs 50,000.
If her annual income is equivalent to her 15 month’s salary, find her income tax
in a year.
d) The monthly salary of an individual employee of an INGO is Rs 1,80,000.
Calculate the income tax paid by the individual in a year.
5. a) Mrs. Thakuri deposited Rs. 2,00,000 in her fixed account at a bank for 3 years.
The bank pays her the simple interest at the rate of 10% p.a. How much net
interest would she get if 5% of interest is charged as income tax?
b) Mr. Thapa deposits Rs 50,000 in a bank at the rate of 8% p.a. How much net
interest will he get after 4 years if he has to pay 5% of his interest as income tax?
c) In the beginning of BS 2076, Dolma deposited Rs 1,20,000 in her account at the
rate of 9% p.a. If she paid 5% of her interest as income tax, how much total
amount did she receive in the beginning of BS 2079?
d) On the occasion of daughter’s 14th birthday, Dharmendra deposits Rs 25,000 in
his daughter's bank account at the rate of 6% p.a. If 5% of the interest is charged
as income tax, how much amount will she withdraw on her 16th birthday?
6. a) Mrs. Majhi deposited a certain amount in her bank account at the rate of
6.5% p.a. If she paid 5% of her interest as income tax and received Rs 4940 net
interest after 4 years, how much money was deposited by her?
b) Madan Bahadur deposited a sum of money at his bank account at the rate of
10% p.a.. After 5 years, he received Rs 1900, the net interest when 5% of the total
interest was charged as income tax. Find, how much sum was deposited by him?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 35 Vedanta Excel in Mathematics - Book 9
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Creative section-B
7. a) Mr. Khatiwada is an unmarried secondary level mathematics teacher in a
community school. His monthly salary is Rs 39,990 with Rs 2,000 allowance
and gets one month’s basic salary as festival expense. If 10% and next 13% of his
basic salary is deposited in his provident fund and civil investment trust (CIT)
respectively, how much income tax should he pay in this year?
b) Mrs. Anjali Subba is a medical doctor in a government hospital. Her monthly
salary is Rs 50,000 including Rs 2,000 allowance and she receives festival
expense equivalent to her one month’s basic salary. 10% of her basic salary is
deducted as provident fund and she pays Rs 48,500 annually as the premium of
her insurance. How much income tax should she pay in a year?
c) After deducting 10% provident fund, a married person draws Rs 40,500 salary
per month and one month’s salary as festival expense, the person pays Rs 14,500
annually as the premium of his/her insurance. Calculate the annual income tax
paid by the person.
d) Mr. Sayad Sharma an unmarried employee of a UN Project draws monthly salary
of Rs 51,000 after deducting 10% salary in his provident fund and 5% in citizen
investment trust. He also receives the festival expense of one month’s salary. He
pays Rs 22,000 annually as the premium of his life insurance. How much income
tax does he pay in a year?
8. a) Mr. and Mrs. Pandey are a married couple. Mr. Pandey is the mayor of a
municipality and his monthly salary is Rs 48,000 with Rs 2,000 allowance.
Mrs. Pandey is the sole proprietor of a beauty-parlor and her annual income is
Rs 6,20,000. Who pays more income tax and by how much?
b) The monthly salary of Ms. Chhiring, an unmarried servant in a bank, is Rs 30,000
and her annual income is equivalent to her salary of 15 months. Similarly, the
monthly salary of Sumesh, a married civil servant is Rs 40,000 and his annual
income is equivalent to his 13 month’s salary including festival expense. Who
pays more income tax and by how much?
Project work and activity section
9. a) Let's ask the monthly salary of your Mathematics, Science, and English teachers.
Then, complete the table given below.
Name of Marital Monthly Festival Provident Insurance
teachers status Salary expense fund
Now, calculate the annual income tax paid by each subject teacher.
b) If your parents are involving in any government or private service, let's ask their
monthly salary, bonus, provident fund, insurance, etc. Then, calculate the income tax
paid by them in a year.
Vedanta Excel in Mathematics - Book 9 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Taxation
2.5 Marked Price (M.P.) and Discount
Have you ever seen the price tagged on fancy items, the price printed on books, packets of
noodles, the price on catalogue of television, refrigeration, etc? The price on the label of an
article or product is called the marked price (M. P.). It is also called list price. This is the
price at which the article is intended to be sold. However, there can be some reduction given
on this price and the actual selling price of the article may be less than the marked price.
The amount of reduction in the marked price is called discount. Discount is usually given as
a certain percent of marked price.
The formulae given below are useful to workout the problems of marked price and discount.
Discount amount = discount percent of M.P. (D% of M.P.)
Selling price = M.P. – D% of M.P.
Discount percent (D%) = Discount amount × 100%
M.P.
2.6 Value Added Tax (VAT) Deurali Electronics
Tahachal-13, Kathmandu
Let's study the adjoining bill given by a Ph. No. 9841240225
shopkeeper to a customer.
(i) How much is the selling price of the
mobile without VAT?
(ii) What is the VAT rate and amount of VAT Mobile 6,000 6000
in the bill?
(iii) How much price should the customer pay
with VAT? Rs. In words: ........................................... Total 6000
...S..i.x...t.h..o..u..s..a..n..d...o..n..l..y...................................
............%
Value added tax (VAT) is an indirect tax which Taxable Amount
................................................................
is charged at the time of consumption of goods Ownership Tax
and services. Therefore, it is a consumption tax Sub Total 6000
paid by a consumer while purchasing goods or 13 % VAT 780
services. The VAT rate is given in percent and Grand Total 6780
it is decided by the concerned authority of the
government of a country. The VAT rate may vary from country to country and even in a
country, it may be changed from time to time. For example, when VAT was introduced for
the first time in Nepal on 16 November, 1997, the rate of VAT was 10%, but it is 13% now.
VAT is one of the sources of government revenue of a country for its administrative expenses,
welfare, development expenses, and so on. VAT is levied on the actual selling price of goods.
VAT amount = Rate of VAT × selling price (S.P.)
S.P. with VAT = S.P. + VAT% of S.P.
If goods are sold by giving a certain discount, at first the amount of discount is to be deducted
from the given marked price to find actual selling price. Then, VAT is levied on the actual
selling price. i.e.,
Actual S.P. = M.P. – D% of M.P. S.P. with VAT = S.P. + VAT% of S.P
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 37 Vedanta Excel in Mathematics - Book 9
Taxation
The process of calculating actual selling price is shown in the following diagram.
Cost Price (C.P.) + Profit + Transportation cost
– Loss + Local tax
+ Service charge
+ Commission Actual S.P.
+ Insurance
Marked Price (M.P.) – Discount
The purchaser should not pay VAT on tax-exempt goods and services. The following goods
and services are VAT exempted in Nepal.
(a) Goods and services of basic needs which include rice, pulses flour, fresh fish,
meat, eggs, fruits, flowers, edible oil, piped water, wood fuel etc.
(b) Basic agricultural products are also tax-exempt, for example, paddy, wheat, maize,
millet, cereals and vegetables.
(c) The expense of buying goods and services required to grow basic agricultural
products are tax-exempt. This includes live animals, agricultural inputs including
machinery, manure, fertilizer, seeds, and pesticides.
(d) Social welfare services including medicine, medical services, veterinary services
and educational services.
(e) Goods made for the use of disabled persons.
(f) Air Transport.
(g) Educational and cultural goods and services such as books and other printed
materials, radio and television transmissions, artistic goods, cultural programmes,
non-professional sporting events and admissions to educational and cultural
facilities.
(h) Personal services are also tax-exempt. These are services provided, for example,
by actors and other entertainers, sportsmen, writers, translators and manpower
supplies agents.
(i) Exemption from VAT is also extended to the purchase and renting of land and
buildings
(j) Financial and insurance services.
(k) Postage and revenue stamps, bank notes, cheque books.
Worked-out Examples
Example 1: Calculate the VAT amount on the selling price of Rs 3,600 at the rate of 13%.
Solution:
Here, the amount of VAT = VAT% of S.P.
= 13% VAT of Rs 3,600 = Rs 468
Hence, the required VAT amount is Rs 468.
Example 2: The selling price of a mobile is Rs 7,500. How much should a customer pay
for it with 13% VAT?
Solution:
Here, the selling price of mobile = Rs 7,500 and VAT rate = 13%
Vedanta Excel in Mathematics - Book 9 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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Now, the cost of the mobile with VAT = S.P. + VAT% of S.P. Direct process
= Rs 7,500 + 13% of Rs 7,500 S.P. with VAT=113% of S.P.
= Rs 7,500 + 13 × Rs 7,500 = 113 × Rs 7,500
100 100
= Rs 8,475
= Rs 8,475
Hence, the customer should pay Rs 8,475 for the mobile.
Example 3: The marked price of a bicycle is Rs 4,400 and the shopkeepers levies 13%
Solution: VAT on it. If you give Rs 5,000 to the shopkeeper while buying it, what
change will the shopkeeper return to you?
Here, M.P. of a bicycle = S.P. of bicycle = Rs 4,400 [ No discount is allowed]
Rate of VAT = 13%, and S.P. with VAT = ? Direct process
Now, S.P. with VAT
S.P. with VAT = S.P. + VAT% of S.P. = (100 + 13)% of S. P.
13 = 113% of Rs 4,400
= Rs 4,400 + 100 × Rs 4,400
= Rs 4,972
Again, Rs 5000 – Rs 4972 = Rs 28 = 113 × Rs 4,400 = Rs 4,972
100
Hence, the shopkeeper will return Rs 28 to me.
Example 4: A family had dinner in a restaurant. If the cost of the dinner was Rs 1,500,
how much did the family pay with 10% service charge and 13% VAT?
Solution: Vedanta ICT Corner
Here, cost of the dinner = Rs 1,500 Please! Scan this QR code or
browse the link given below:
Service charge = 10%
VAT rate = 13% https://www.geogebra.org/m/xjgvxst3
Now, the cost of the dinner with service charge = S.P. + 10% of S.P.
= Rs 1,500 + 10% of Rs 1,500 = Rs 1650
Again, the cost of the dinner with service charge and VAT = Rs 1650 + 13% of Rs 1650
= Rs 1,864.50
Therefore, the family should paid Rs 1,864.50.
Example 5: Mrs. Shrestha purchased a watch for Rs 7,360 with 15% VAT. Find the cost of
the watch without VAT?
Solution: Direct process
Here, the cost of the watch with 15% VAT = Rs 7,360 Cost of watch without VAT
Let the cost of the watch without VAT = Rs x cost of watch with VAT
(100 + VAT)%
Now, x + 15% of x = Rs 7,360 =
or, 115x = Rs 7,360 = Rs 7,360 = Rs 6,400
100 115%
or, x = Rs 6,400 = Rs 6,400
Hence, the cost of the watch without VAT is Rs 6,400.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 39 Vedanta Excel in Mathematics - Book 9
Taxation
Example 6: If the cost of an article with VAT is Rs 4,746 and without VAT is Rs 4,200, find
the VAT rate.
Solution:
Here, the cost of the article with VAT = Rs 4,746
And, the cost of the article without VAT = Rs 4,200
Now, the amount of VAT = Rs 4,746 – Rs 4,200 = Rs 546
∴ Rate of VAT = VAT amount × 100%
cost without VAT
= Rs 546 × 100% = Rs 13%
Rs 4200
Hence, the required VAT rate is 13%.
Example 7: A trader bought an electric oven for Rs 16,000 and sold at a profit of 20% to
a customer with 13% VAT. How much did the customer pay for the oven?
Solution: Direct process
Here, C.P. of the oven = Rs 16,000. S.P. of the oven = 120% of C.P.
Profit percent = 20% = 120 × Rs 16,000
100
S.P. of the oven = Rs 16,000 + 20 × Rs 16,000
= Rs 19,200 100 = Rs 19,200
Now, S.P. of the oven with 13% VAT= S.P. + 13% of S.P. Direct process
13
= Rs 19,200 + 100 × Rs 19,200 S.P. with VAT = 113% of S.P.
= Rs 21,696
= 113 × Rs 19,200
100
Hence, the customer paid Rs 21,696 for the oven.
= Rs 21,696
Example 8: A shopkeeper bought a television set for Rs 24,000 and fixed its price to
make 15% profit. If the television was sold for Rs 31,188 with VAT, calculate
the rate of VAT.
Solution:
Here, S.P. for the shopkeeper = Rs 24,000 + 15% of Rs 24,000
= Rs 24,000 + Rs 3,600 Direct process
= Rs 27,600 S.P. of television
Also, S.P. with VAT = Rs 31,188 = 115% of C.P.
∴ VAT amount = Rs 31,188 – Rs 27,600 = Rs 3,588
= 115 × Rs 24,000
100
Again, rate of VAT = VAT amount × 100% = Rs 27,600
S.P.
= Rs 3,588 × 100%
Rs 27,600
= 13%
Hence, the required rate of VAT is 13%.
Example 9: The marked price of a mobile set is Rs 18,000 and 15 % discount is allowed.
How much should a customer pay for it with 13 % VAT?
Vedanta Excel in Mathematics - Book 9 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Taxation
Solution:
Here, M.P. of the mobile set = Rs 18,000 Direct process
S.P. = (100 – 15)% of M. P.
Discount percent = 15 % = 85% of Rs 1,800
= Rs 15,300
S.P. of the mobile set = M.P. – 15 % of M.P.
Direct process
= Rs 18,000 – 15 × Rs 18,000 S.P. with VAT
100 = (100 + 13)% of S. P.
= 113% of Rs 15,300
= Rs 15,300 = Rs 17,289
Now, S.P. with VAT = S.P. + VAT % of S.P.
= Rs 15,300 + 13 % of Rs 15,300
= Rs 17,289
Hence, the customer should pay Rs 17,289 for the mobile set.
Example 10: Mr. Dorje paid Rs 31,188 for a washing machine with a discount of 20 %
including 13 % VAT. What is the marked price of the washing machine?
Solution:
Let the marked price of the washing machine be Rs x.
Now, S.P. of the washing machine = M.P. – discount % of M.P.
= x – 20 % of x = 4x
5
Again, S.P. with VAT = 4x + 13 % of 4x = 113x Vedanta ICT Corner
5 5 125
Also, the given S.P. = Rs 31,188 Please! Scan this QR code or
or, browse the link given below:
or, 113x = Rs 31,188 https://www.geogebra.org/m/k7kkap33
125
x = Rs 34,500
Hence, the marked price of the washing machine is Rs 34,500.
Alternative process
Let, S.P. without VAT be Rs x.
Then, S.P. without VAT = S.P. with VAT – 13 % of x
or, x = Rs 31,188 – 13x Direct process
100
13x 113 % of S.P. = Rs 31,188
or, x+ 100 = Rs 31,188
Rs 31,188
or, x = Rs 27,600 or, S.P. = 113%
Again, let M.P. be Rs y. S.P. = Rs 27,600
Then, M.P. – discount % of M.P. = S.P. (without VAT) Again, (100 – 20)% of M.P. = S.P.
or, 18000 × M.P. = Rs 27,600
or, y – 20 % of y = Rs 27,600 or, M.P. = Rs 34,500
or, y = Rs 34,500
Hence, the required marked price is Rs 34,500.
Example 11: Sunayana purchased a fancy item for Rs 4,800 and sold it for Rs 6,780 with
13% VAT. Find her profit or loss percent.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 41 Vedanta Excel in Mathematics - Book 9
Taxation
Solution:
Here, C.P. of the fancy item = Rs 4,800
S.P of the item with VAT = Rs 6,780
Rate of VAT = 13%
Let the S.P. of the item without VAT be Rs x. Direct process
∴ x + 13% of x = Rs 6,780 S.P. without VAT = S.P. with VAT
113%
13x
or, x+ 100 = Rs 6,780 = Rs 6,780
113%
or, 113x = Rs 6,780
100 = Rs 6,000
or, x = Rs 6,000
Thus, S.P. without VAT = Rs 6,000
Now, profit = S.P. – C.P. Direct process
= Rs 6,000 – 4,800 Profit percent = S. P. – C. P. × 100%
C. P.
= Rs 1,200 = Rs 6,000 – Rs 4,800 × 100%
Rs 4,800
Also, profit percent = Actual profit × 100% = Rs 1,200 × 100%
C.P. Rs 4,800
= Rs 1,200 × 100% = 25% = Rs 25%
Rs 4,800
Hence, the required profit percent is 25%.
Example 12: A wholesaler sold a Honda generator for Rs 3,00,000 to a retailer. The
retailer spent Rs 6,000 for transportation and Rs 4,000 for the local tax. If
the retailer sold it at a profit of Rs 20,000 to a customer, how much did the
customer pay for it with 13% VAT?
Solution:
Here,
For the retailer, C.P. of the generator = Rs 3,00,000
Transportation cost = Rs 6,000, local tax = Rs 4,000 and profit amount = Rs 20,000
∴ VAT eligible price = C.P. + transportation cost + local tax + profit
= Rs 3,00,000 + Rs 6,000 + Rs 4,000 + Rs 20,000= Rs 3,30,000
Now, S.P. with VAT = Rs 3,30,000 + 13% of 3,30,000 = Rs 3,72,900
Hence, the customer paid Rs 3,72,900 for the generator.
Example 13: Pashupati supplier purchased some building materials for Rs 5,55,000 and
sold them at 5% profit to Lumbini supplier. The Lumbini supplier spent
Rs 7,000 for transportation and Rs 4,250 for the local tax and sold at a profit
of 10% to a customer. How much did the customer pay for the materials
with 13% VAT?
Solution:
Here, for Pashupati supplier,
C.P. of the building materials = Rs 5,55,000
Vedanta Excel in Mathematics - Book 9 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Taxation
∴ S.P. at 5% profit = C.P. + 5% of C.P.
= Rs 5,55,000 + 5% of Rs 5,55,000 = Rs 5,82,750
For Lumbini supplier, C.P. of the materials =S.P. of Pashupati supplier = Rs 5,82,750
C.P. with transportation cost and local tax = Rs 5,82,750 + Rs 7,000 +Rs 4,250
= Rs 5,94,000
∴Actual S.P. = Rs 5,94,000 + 10% of Rs 5,94,000 = Rs 6,53,400
Now, C.P. for customer with VAT = S.P. for Lumbini supplier with 13% VAT
= Rs 6,53,400+13% of 6,53,400= Rs 7,38,342
Hence, the customer paid Rs 7,38,342 for the materials with VAT.
Example 14: A shopkeeper allowed 10% discount and sold a rice cooker for Rs 3,051
with 13% VAT and made a profit of 20%. By what percent is the discount to
be reduced to increase the profit by 4%?
Solution: Direct process
Let, S.P. without VAT be Rs x.
S.P. without VAT = S.P. with VAT
Here, discount percent = 10% 113%
3051
S.P. with VAT = Rs 3,051 = 113% = Rs 2,700
VAT = 13% M. P. = S.P. without VAT
90%
2700
Now, ( 100 + 13)% of x = Rs 3,051 = 90% = Rs 3,000
or, 113 × x = Rs 3,051 C.P. = S.P. without VAT
100 120%
2700
or, x = Rs 2,700 = 120% = Rs 2,250
∴ S.P. without VAT = Rs 2,700 New S.P. = 124% of C.P.
Again, (100 – 10)% of M.P. = S.P. without VAT = 124 × Rs 2250
100
90
or, 100 × M.P. = Rs 2,700 = Rs 2790
or, M.P. = Rs 3,000 New discount = Rs 3000 – Rs 2790
Also, profit = 20% = Rs 210
∴ (100 + 20)% of C. P. = S.P. without VAT N ew discount percent = 210 × 100%
3000
120
or, 100 × C.P. = Rs 2,700 = 7%
or, C.P. = Rs 2,250 ∴ Reduction in discount percent
Now, profit = 20% + 4% = 24% = 10% – 7% = 3%
∴ New S.P. = (100 + 24)% of C.P. = 124 × Rs 2,250 = Rs 2,790
100
And, new discount = M.P. – new S.P. = Rs 3,000 – Rs 2,790 = Rs 210
Then, new discount percent = New discount × 100% = 210 × 100% = 7%
M.P. 3000
∴ Reduction in discount percent = 10% – 7% = 3%
Hence, the discount is to be reduced by 3%.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 43 Vedanta Excel in Mathematics - Book 9
Taxation
Example 12: A business person hired a room in a shopping mall at Rs 40,000 rent
per month and started a business of electronic items. She invested
Rs 25,00,000 to purchase different electronic items in the first phase
and labelled the price of each item 40% above the cost price. She then
allowed 10% discount on each item and sold to customers. Her monthly
miscellaneous expenditure was Rs 18,000 and the items of worth 10% of
the investment remained as stocks after three months. Find her net profit
or loss percent.
Solution:
Here, the amount of investment = Rs 25,00,000
Stocks after three months = 10% of Rs 25,00,000 = Rs 2,50,000
∴ The investment excluding stocks = Rs 25,00,000 – Rs 2,50,000 = Rs 22,50,000
Now, M. P. of the items = 140% of Rs 22,50,000 = Rs 31,50,000
Discount percent = 10%
∴ S.P. of the items = 90% of M.P. = 90 × Rs 31,50,000 = Rs 28,35,000
100
∴ Gross profit = Rs 28,35,000 – Rs 22,50,000 = Rs 5,85,000
Again, the rent of room in 3 months = 3 × Rs 40,000 = Rs 1,20,000
Miscellaneous expenditure in 3 months = 3 × Rs 18,000 = Rs 54,000
∴ Total expenditure = Rs 1,20,000 + Rs 54,000 = Rs 1,74,000
Now, n et profit = Gross profit – total expenditure = Rs 5,85,000 – Rs 1,74,000 = Rs 4,11,000
Then, net profit percent = net profit × 100% = 4,11,000 × 100% = 18.27%
investment 22,50,000
Hence, her net profit percent is 18.27%.
EXERCISE 2.2
General section
1. a) If R% be the rate of VAT and Rs x be the selling price, write the formula to find
amount of VAT.
b) If Rs x be the selling price and Rs y be the amount of VAT, write the formula to find
VAT percent.
c) If Rs P be the selling price and R% be the VAT rate, write the formula to find selling
price with VAT.
d) If marked price (M.P.) = Rs x, discount = Rs y and VAT = Rs z, what is the selling
price including VAT?
2. a) Find the VAT amount from the table given below.
S.N. Particulars S.P. without VAT VAT rate VAT amount
(i) Mobile set Rs 22,000 13% …………
(ii) Camera Rs 35,000 13% …………
(iii) Television Rs 40,000 10% …………
Vedanta Excel in Mathematics - Book 9 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Taxation
b) Find the selling price without VAT from the table given below.
S.N. Particulars VAT amount VAT rate S.P. without VAT
(i) Radio Rs 585 13% …………
(ii) Bicycle Rs 975 13% …………
(iii) Laptop Rs 9,900 15% …………
3. a) The selling price of a watch is Rs 3,000. What will be the VAT amount on it at the
rate of 13%?
b) Calculate the VAT amount on a tablet costing Rs 15,000 at the rate of 13%.
c) The catalogue price of a refrigerator is Rs 28,500. How much amount of VAT is levied
on it at the rate of 13%?
4. a) The cost of a fan is Rs 1,600. If Mrs. Khadka purchased it with 13% VAT, how much
did she pay for it?
b) The selling price of a radio is Rs 4,000. How much should a customer pay for it with
13% value added tax?
c) The marked price of a pen-drive is Rs 700 and the shopkeeper levies 13% VAT on it.
If you give a 1,000 rupee note, what change will the shopkeeper return to you?
5. a) The cost of a rice cooker with 13% VAT is Rs 4,068. Find its cost without VAT.
b) Mr. Magar purchased a mobile set for Rs 11,155 with 15% VAT exclusive. Find the
cost of the mobile without VAT and also calculate the VAT amount.
c) Mrs. Maharjan bought a refrigerator for Rs 26,442 with 13% VAT. How much did she
pay for the VAT?
6. a) If the cost of a watch with VAT is Rs 5,130 and without VAT is Rs 4,500, find the VAT
rate.
b) Malvika purchased a fancy bag for Rs 7,119 with VAT. If its cost without VAT is
Rs 6,300, calculate the rate of VAT.
c) If the cost of a computer with VAT is Rs 67,800 and without VAT is Rs 60,000, find
the VAT rate.
Creative section-A
7. a) Find the selling price of the following articles with VAT.
(i) (ii) (iii) iv)
M.P. =Rs 25,000 M.P. =Rs 40,000 M.P. =Rs 85,000 M.P. =Rs 1,20,000
Discount rate = 10% Discount rate = 15% Discount rate = 14% Discount rate = 14%
VAT rate = 13% VAT rate = 13% VAT rate = 13% VAT rate = 13%
b) The marked price of a bike helmet is Rs 3,000 and 10 % discount is allowed on it.
Find its cost with 13 % VAT.
c) The price of a blanket is marked as Rs 5,500. If the shopkeeper allows 20 % discount
and adds 13 % VAT, how much does a customer pay for the blanket?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 45 Vedanta Excel in Mathematics - Book 9
Taxation
d) A trader bought a motorbike for Rs 2,40,000 and fixed its price 20 % above the cost
price. Then, he allowed 10 % discount and sold to a customer. How much did the
customer paid for it with 13 % VAT?
8. a) A shopkeeper bought a television for Rs 16,000 and sold at a profit of 20% to a
customer with 13% VAT. How much did the customer pay for the television?
b) Mrs. Lama marked the price of a cosmetic item 25% above its cost price. If the cost
price of the cosmetic item was Rs 4,400, at what price did she sell it with 13% VAT?
c) Mr. Sharma bought a computer for Rs 50,000 and fixed its price 15% above the cost
price. How much did the customer pay for the computer including 13% value added
tax?
9. a) Mrs. Kandel went to a restaurant with her family. They had three plates of Mo.Mo
at Rs 120 per plate, one plate chicken chilly at Rs 220 per plate, and three bottles
of cold drink at Rs 40 per bottle. If 13% VAT is levied on the bill after adding 10%
service charge on the bill, how much did she pay to clear the bill?
b) A group of three friends had two plates of chicken chilly, two plates of french fry,
two plates of Mo:Mo and a few glasses of fresh juice in a restaurant. If the cost of
these items amounts to Rs 900, how much should they pay with 10 % service charge
and 13 % VAT to clear the bill?
10. a) A retailer allows 15 % discount on the marked price of an electric fan. If a customer
pays Rs 2,244 with 10 % VAT, find the marked price of the fan.
b) Allowing 16 % discount on the marked price of a television and levying 13 % VAT, a
buyer has to pay Rs 18,984 to buy it. Find the marked price of the television.
c) Allowing 15% discount and including same percentage of VAT, the laptop was sold
at Rs 64,515. Find the marked price of the laptop.
11. a) After allowing 5 % discount on the marked price of a gift item, 10 % VAT is charged
on it. Now, its price became Rs 1,672. How much amount was given in the discount?
b) Mrs. Gurung sold her goods for Rs 16,950 allowing 25 % discount and then levied on
13 % VAT, what was the amount of discount?
12. a) After allowing 20 % discount on the marked price of a computer, 15 % VAT was
levied on it. If its price becomes Rs 26,496, what amount was levied in the VAT?
b) After allowing 10 % discount on the marked price of an iPod and levying 13 % value
added tax, the price of the iPod becomes Rs 7,119. Find the value added tax.
c) A tourist paid Rs 5,610 for a carved window made up of wood with a discount
of 15% including 10% value added tax (VAT). How much does he get back while
leaving Nepal?
13. a) A mobile price is tagged Rs 5,000. If a customer gets 12% discount and adding certain
percent VAT reaches as Rs 4,972, find out the VAT percentage.
b) The marked price of a bag is Rs 2,000. The price of the bag becomes Rs 1,921 after
15% discount and adding VAT amount. Find the rate of VAT.
14. a) Mrs Karki purchased a sari for Rs 8,000 and sold it for Rs 11,300 with 13% VAT. Find
her profit or loss percent.
b) A supplier bought a scanner machine for Rs 35,000 and sold it for Rs 47,460 with
13% VAT. Find the profit or loss percent of the supplier.
Vedanta Excel in Mathematics - Book 9 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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Vedanta Excel in Mathematics Book 9 Final Re_CTP
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