Vedanta Excel in Mathematics Book 9 Final (2078)

Approved by the Government of Nepal, Ministry of Education, Science and Technology,
Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material

vedanta

Excel in

MATHEMATICS

9Book

Author
Hukum Pd. Dahal

Editor
Tara Bahadur Magar

vedanta
Vedanta Publication (P) Ltd.
j]bfGt klAns;] g k|f= ln=

Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np

vedanta

MExAceTl iHnEMATICS

9Book

All rights reserved. No part of this publication may
be reproduced, copied or transmitted in any way,
without the prior written permission of the publisher.

Second and Updated Edition: B. S. 2078 (2021 A. D.)

Published by:

Vedanta Publication (P) Ltd.
j]bfGt klAns;] g k|f= ln=

Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np

Preface

The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical
teaching learning activities and methodologies extracted from Teachers' training, workshops,
seminars, and symposia. It is an innovative and unique series in the sense that the contents of
each textbooks of the series are written and designed to fulfill the need of integrated teaching
learning approaches.

Excel in Mathematics is an absolutely modified and revised edition of my three previous series:
'Elementary mathematics' (B.S. 2053), 'Maths in Action (B. S. 2059)', and 'Speedy Maths' (B. S.
2066).

Excel in Mathematics has incorporated applied constructivism. Every lesson of the whole series
is written and designed in such a manner, that makes the classes automatically constructive
and the learners actively participate in the learning process to construct knowledge themselves,
rather than just receiving ready made information from their instructors. Even the teachers will
be able to get enough opportunities to play the role of facilitators and guides shifting themselves
from the traditional methods of imposing instructions.

Each unit of Excel in Mathematics series is provided with many more worked out examples.
Worked out examples are arranged in the order of the learning objectives and they are reflective
to the corresponding exercises. Therefore, each textbook of the series itself plays the role of a
‘Text Tutor’. There is a proper balance between the verities of problems and their numbers in
each exercise of the textbooks in the series.

Clear and effective visualization of diagrammatic illustrations in the contents of each and
every unit in grades 1 to 5, and most of the units in the higher grades as per need, will be
able to integrate mathematics lab and activities with the regular processes of teaching learning
mathematics connecting to real life situations.

The learner friendly instructions given in each and every learning content and activity during
regular learning processes will promote collaborative learning and help to develop learner-
centred classroom atmosphere.

In grades 6 to 10, the provision of ‘General section’, ‘Creative section - A’, and ‘Creative section -B’
fulfill the coverage of overall learning objectives. For example, the problems in ‘General section’
are based on the knowledge, understanding, and skill (as per the need of the respective unit)
whereas the ‘Creative sections’ include the Higher ability problems.

The provision of ‘Classwork’ from grades 1 to 5 promotes learners in constructing knowledge,
understanding and skill themselves with the help of the effective roles of teacher as a facilitator
and a guide. Besides, the teacher will have enough opportunities to judge the learning progress
and learning difficulties of the learners immediately inside the classroom. These classworks
prepare learners to achieve higher abilities in problem solving. Of course, the commencement
of every unit with 'Classwork-Exercise' plays a significant role as a 'Textual-Instructor'.

The 'project works' given at the end of each unit in grades 1 to 5 and most of the units in higher
grades provide some ideas to connect the learning of mathematics to the real life situations.

The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities
to integrate mental maths and manual maths simultaneously. Moreover, the problems in
‘Section A’ judge the level of achievement of knowledge and understanding, and diagnose the
learning difficulties of the learners.

The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an
important role of ‘placement evaluation’ which is in fact used by a teacher to judge the
level of prior knowledge and understanding of every learner to select their teaching learning
strategies.

The socially communicative approach by language and literature in every textbook,
especially in primary level of the series, plays a vital role as a ‘textual-parents’ to the young
learners and helps them overcome maths anxiety.

The Excel in Mathematics series is completely based on the latest curriculum of mathematics,
designed and developed by the Curriculum Development Centre (CDC), the Government of
Nepal.

I do hope the students, teachers, and even the parents will be highly benefited from the
‘Excel in Mathematics’ series.

Constructive comments and suggestions for the further improvements of the series from the
concerned are highly appreciated.

Acknowledgments

In making effective modification and revision in the Excel in Mathematics series from my
previous series, I’m highly grateful to the Principals, HODs, Mathematics teachers and experts,
PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN, and independent clusters of many
other Schools of Nepal, for providing me with opportunities to participate in workshops,
Seminars, Teachers’ training, Interaction programme, and symposia as the resource person.
Such programmes helped me a lot to investigate the teaching-learning problems and to research
the possible remedies and reflect to the series.

I’m proud of my wife Rita Rai Dahal who always encourages me to write the texts in a more
effective way so that the texts stand as useful and unique in all respects. I’m equally grateful to
my son Bishwant Dahal and my daughter Sunayana Dahal for their necessary supports during
the preparation of the series.

I’m extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, England
who provided me with very valuable suggestions about the effective methods of teaching-
learning mathematics and many reference materials.

Thanks are due to Mr. Tara Bahadur Magar for his painstakingly editing of the series. I am
thankful to Dr. Komal Phuyal for editing the language of the series.

Moreover, I gratefully acknowledge all Mathematics Teachers throughout the country who
encouraged me and provided me with the necessary feedback during the workshops/interactions
and teachers’ training programmes in order to prepare the series in this shape.

I’m profoundly grateful to the Vedanta Publication (P) Ltd. for publishing this series. I would
like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha, and
Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during
the preparation of the series.

Also I’m heartily thankful to Mr. Pradeep Kandel, the Computer and Designing Senior Officer of
the publication house for his skill in designing the series in such an attractive form.

Hukum Pd. Dahal

Contents

S.N Chapter Page No.

1. Set 7 - 26

1.1 Set - Review, 1.2 Notation and specification of sets, 1.3 Types
of sets, 1.4 Cardinal number of set, 1.5 Subset, proper and improper
subsets, 1.6 Universal set, 1.7 Venn-diagrams, 1.8 Set operations by
using Venn diagrams, 1.9 Cardinality relationships of two sets

2. Profit and Loss 27 - 43

2.1 Profit and Loss - Review, 2.2 Profit percent and loss percent -
Review, 2.3 Marked Price (M.P.) and discount, 2.4 Value Added Tax
(VAT)

3. Commission and Taxation 44 - 54

3.1 Commission – Introduction, 3.2 Bonus, 3.3 Taxation, 3.4 Income
Tax, 3.5 Dividend

4. Household Arithmetic 55 - 69

4.1 Introduction, 4.2 Electricity bill, 4.3 Telephone bill, 4.4 Water
bill, 4.5 Calculation of taxi fare in a taximeter

5. Mensuration 70 - 95

5.1 Mensuration – review, 5.2 Area of pathways, 5.3 Area, cost and
quantities, 5.4 Area of 4 walls, floor and ceiling, 5.5 Area and volume
of solids, 5.6 Prisms and their cross sections, 5.7 Estimation of number
of bricks and cost required for building wall

6. Algebraic Expressions 96 - 110

6.1 Factors and Factorisation - Review, 6.2 Highest Common Factor
(H.C.F.) of algebraic expressions, 6.3 Lowest Common Multiple
(L.C.M.) of algebraic expressions, 6.4 Simplification of rational
expressions

7. Indices 111 - 122

7.1 Indices – review, 7.2 Laws of Indices, 7.3 Exponential equation

8. Simultaneous Linear Equations 123 - 132

88.1 Simultaneous equations - review, 8.2 Method of solving
simultaneous equations

9. Quadratic Equations 133 - 143

9.1 Quadratic equation – review, 9.2 Solution of quadratic equations

10. Ratio and Proportion 144 - 159

10.1 Ratio – review, 10.2 Different types of ratios, 10.3 Proportion,
10.4 Properties of proportion

Contents Page No.

S.N Chapter

11. Geometry - Triangle 160 - 194

11.1 Various types of angles, 11.2 Axioms and postulates,
11.3 Types of triangle - review, 11.4 Median and altitude of a triangle,
11.5 Properties of triangles, 11.6 Congruent triangles, 11.7 Conditions
of congruency of triangles, 11.8 Mid-point theorems

12. Geometry - Similarity 195 - 208

12.1 Similar triangles - review, 12.2 Similar polygons,
12.3 Pythagoras Theorem, 12.4 Pythagorean Triple

13. Geometry - Parallelogram 209 - 224
13.1 Special types of Quadrilateral

14. Geometry - Circle 225 - 242

14.1 Circle and it’s various parts, 14.2 Theorems related to chords of
a circle

15. Geometry - Construction 243 - 251
15.1 Construction of quadrilaterals

16. Trigonometry 252 - 264

16.1 Trigonometry - Introduction, 16.2 Trigonometric ratios,
16.3 Relation between trigonometric ratios, 16.4 Values of
trigonometric ratios of some standard angles,

17. Statistics 265 - 294

17.1 Statistics - Review, 17.2 Types of data, 17.3 Frequency table,
17.4 Grouped and continuous data, 17.5 Cumulative frequency
table, 17.6 Graphical representation of data: Histogram, Line
graph, pie-chart, 17.7 Ogive (Cumulative frequency curve),
17.8 Construction of less than ogive and more than ogive,
17.9 Measures of central tendency, 17.10 Arithmetic mean,
17. 11 Median, 17.12 Quartiles, 17.13 Mode

18. Probability 295 - 305

18.1 Probability - Introduction, 18.2 Probability scale,
18.3 Probability of an event, 18.4 Empirical probability
(or Experimental probability)

Revision and Practice Time 306 - 336

Answers 337 - 353

Syllabus 354 - 356

Specification Grid 357

Model Question Set 358 - 360

Unit Set

1

1.1 Set - Review

Let's take a collection of odd numbers less than 10. The members of this collection
are definitely 1, 3, 5, 7, and 9. These members are distinct objects when considered
separately. However, when they are considered collectively, they form a single set of
size five, written {1, 3, 5, 7, 9}. Thus, it is a set of odd numbers less than 10. Here,
any odd number less than 10 is definitely the member of the set. Therefore, a set is a
collection of 'well-defined object'.

1.2 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
E = {2, 4, 6, 8} enclosed inside braces.

2 ∈ N, 5 ∈ N, 4 ∈ N The symbol ‘∈ ‘ denotes the
∈ Membership membership of an element
of the given set.
6 ∈ E, 8 ∈ E

∉ Non- 6 ∉ N, 8 ∉ N, The symbol ‘∉ ‘ denotes
the non-membership of an
membership 3 ∉ E, 5 ∉ E element to the given set.

⊂ Proper {1, 2} ⊂ N, {4, 6, 8} ⊂ E A set that is contained in
subset another set.

⊆ Improper {2, 4, 6, 8} ⊆ E, it A set which is contained in
subset means {2, 4, 6, 8} ⊂ E and equal to another set.
and {2, 4, 6, 8} = E

⊃ Super set N ⊃ {1, 2}, E ⊃ {4, 6, 8} Set N includes {1, 2,} and
Set E includes {4, 6, 8}.

‰ Union N‰E ={1, 2, 3, 4, 5, 6, 8} N ‰ E belongs to set N or
set E.

ˆ Intersection N ˆ E = {2, 4} N ˆ E belongs to both set
N and E.

We usually describe a set by three methods: description, listing (or roster), and
set-builder (or rule) methods.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 7 Vedanta Excel in Mathematics - Book 9

Set

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
roster) set are listed inside curly
brackets { }.

Set-builder A = {x : x ∈ prime numbers, x < 10} A variable is used to describe
(or rule) the common properties of the
elements of a set by using
symbols.

1.3 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 It does not contain any
numbers less than 0. element. It is denoted by empty
W = { } or I . braces { } or by I (Phi)

Unit or singleton set 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.

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
They have exactly the same
Equal sets A = {s, v, u, 3, ª} elements.
Equivalent sets
B = {3, v, ª, s, u} They have equal number of
elements.
?A =B
They have at least one
P = {2, 3, 5, 7} common element. 2 and 10
Q = {1, 4, 9, 16} are the common elements of
?P~Q sets M and N.
They do not have any
Overlapping sets M = {2, 4, 6, 8, 10} common element.
N = {1, 2, 5, 10}
? M and N are overlapping
sets.

Disjoint sets X = {c, cf, O, O}{

Y = {r, 5, h, em, `}

? X and Y are disjoint sets.

Vedanta Excel in Mathematics - Book 9 8 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Set

1.4 Cardinal number of set

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).

Thus, if A = {1, 2, 3, 4, 6, 12}, then n(A) = 6.

1.5 Subset, proper, and improper subsets

Let's consider any three sets, namely, A, B, and C, where A = {1, 2, 3, …, 10},
B = {2, 4, 6, 8}, and C = {1, 2, 3, …, 10}.
Here, sets B and C are the subsets of the sets of A because all elements of B are also
the elements of A and all elements of C are also the elements of A. However, B is the
proper subset of A and C is the improper subset of A.
We write it as B  A and C Ž A.
Thus, '' is the symbol of proper subset and Ž is the symbol of improper subset.
Similarly, the set A is said to be the superset of the subsets B and C.
It is written as A Š B and A ‹ C.
In this way, if B is a proper subset of A, every elements of B is contained in A but
n(B) z n(A).
If C is an improper subset of A, every element of C is contained in A and
n(C) = n(A).

1.6 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.

1.7 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

UU

A AB AB

A ⊂ U (A is a subset of U). A and B are disjoint sets. A and B are overlapping sets.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 9 Vedanta Excel in Mathematics - Book 9

Set U A U A U
B B B
A

CC 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,
but B and C are disjoint sets. but A and C are disjoint sets.

1.8 Set operations by using Venn diagrams

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 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 union of two sets A and B denoted by A ‰ B is the set of all elements that belong
to either to A, or B, or to both A and B.

In set-builder form, union of sets A and B is defined as:

A ‰ B = {x : x  A or x  B}

The shaded regions in the following Venn-diagram represent the union of the given
sets.
UU U

A BA BA BC

A‰B A‰B A‰B‰C

A and B are disjoint sets. A and B are overlapping sets. A, B and C are disjoint sets.
U U
U
AB AB AB

C C
C
A‰B‰C
A‰B‰C A‰B‰C
A, B and C are overlapping sets.
A and B are overlapping, A and B are overlapping,
Vedanta Excel in Mathematics - Book 9 A and C are overlapping, B and C are overlapping,
but B and C are disjoint sets. but A and C are disjoint sets.

10 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Set

2. Intersection of sets

Let's take any two sets P and Q, where P = {2, 4, 6, 8} and Q = {4, 8, 12, 16}. Now,

the intersection of sets P and Q denoted as P ˆ Q = {4, 8}.

Thus, the intersection of two or more sets is made just by listing their common

element/s in a separate set.

The intersection of two sets P and Q denoted by P ˆ Q is the set of the elements

which are common to both the sets.

In set-builder form, intersection of sets P and Q is defined as:

P ˆ Q = {x : x ∈ P and x ∈ Q}

The shaded regions in the following Venn-diagrams represent the intersection of

sets.

UUU

P PQ P Q
Q

R

PˆQ PˆQ P ˆ Q ˆ R

Q is the subset of P. P and Q are overlapping sets. P, Q and R are overlapping sets.

3. Difference of sets

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 ˆ B) Removing the common elements form A
B – A = B – (A ˆ B) Removing the common elements form B

The shaded regions in the following Venn-diagrams represent the difference of the
given sets.

UUUU

A BA BA BA B

A–B A–B B–A B–A
A and B are disjoint A and B are A and B are disjoint A and B are
sets. overlapping sets. sets. overlapping sets.

11Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Set

(iv) Complement of a set

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.
In this way, if A be a subset of a universal set U, the complement of A denoted by A ,
A', Ac is the set of the elements of U which do not belong to the set A.
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 x  B} or, {x : x  U, but x  A ‰ B}
A ˆ B = {x : x  U, but x  A and x  B} or, {x : x  U, but x  A ˆ B}
The shaded regions in the following Venn-diagrams represent the complement of the
given sets.

UUUU

A A B A BA B

A A‰B AˆB A–B

Worked-out examples

Example 1: Let A, B, and C are the subsets of a universal set U. Write the set
operations represented by the shaded regions. Define each operation
by set-builder form.
UU U

a) b) c) A B

A BA B

C

Solution:
a) The non-shaded region is A – B. So, the shaded region is A – B = {x : x  U, but x  A – B}
b) The shaded region is B – A = {x : x  B, but x  A}
c) The shaded region is A ˆ B ˆ C = {x : x  A, x  B and x  C}

Example 2: 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.

Vedanta Excel in Mathematics - Book 9 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Set

(i) A ‰ B and A ‰ B (ii) A ˆ B and A ˆ 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}

(i) A ‰ B = {2, 3, 4, 5, 6, 7, 8, 10} U U

A ‰ B = U – (A ‰ B) A B A B
= {1, 9} 4 3 4 3

62 5 62 5
8 10 7 8 10 7
1 9
19

A‰B A‰B

U U

(ii) A ˆ B = {2} A B A B
A ˆ B = U – (A ˆ B) 4 3 4 3
= {1, 3, 4, 5, 6, 7, 8, 9, 10}
62 5 62 5
8 10 7 8 10 7
9
19 1

AˆB U AˆB

(iii) A – B = A – (A ˆ B) U
A–B = {4, 6, 8, 10}
= U – (A – B) A B A B
= {1, 2, 3, 5, 7, 9} 4 3 4 3

62 5 62 5
8 10 7 8 10 7
9
19 1

A–B A–B

(iv) B – A = B – (A ˆ B) U U
B–A = {3, 5, 7}
= 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
9
19 1

B–A B–A

Example 3: 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 ˆ Q ˆ R c) (P ˆ Q) ‰ R d) (P ‰ Q) ˆ R

Solution:

Here, U = {1, 2, 3, ... 12}

P = {2, 3, 5, 7, 11}, P U
Q = {1, 3, 5, 7, 9} and 11 7 9 Q
R = {1, 2, 3, 4, 5} 2 35 1
a) P ‰ Q = {1, 2, 3, 5, 7, 9, 11}
P ‰ Q ‰ R = {1, 2, 3, 4, 5, 7, 9, 11} 4 R
68 10 12

P ‰ Q‰R

13Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Set

b) P ˆ Q = {3, 5, 7} P U
P ˆ Q ˆ R = {3, 5} 11 7 9 Q
2 35 1

4R
10 12
68
P ˆ Q ˆ R

c) P ˆ Q = {3, 5, 7} d) P ‰ Q ={1, 2, 3, 5, 7, 9, 11}
(P ˆ Q) ‰ R = {1, 2, 3, 4, 5, 7} (P ‰ Q) ˆ R = {1, 2, 3, 5}
(P ‰ Q) ˆ R = {4, 6, 7, 8, 9, 10, 11, 12}
U
P Q P U
11 7 9 11 7 9 Q
2 35 1 2 35 1

4R R
10 12 10 12
68 4
68
(P ˆ Q) ‰ R

(P ‰ Q) ˆ R

EXERCISE 1.1
General section

1. Let A and B are the subsets of a universal set U. Define the following set operations in
set-builder forms.

a) A – B b) A ˆ B c) B – A d) A ‰ B e) A – B

f) A ‰ B g) B – A h) A i) B j) A ˆ B

2. 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  Q, but x  P} b) {x : x  U, but x  Q} c) {x : x  P or x  Q}

d) {x : x  U, but x  P or x  Q} e) {x : x  P, but x  Q} f) {x : x  P and x  Q}

g) {x : x  U, but x  P – Q} h) {x : x  U, but x  P and x  Q}

i) {x : x  U, but x  P}

3. Write the set operations represented by shaded regions shown in the following

Venn-diagrams. b) U c) U d) U
a) U B
A BA BA
AB

e) U f) X U g) U h) U
Q
A B YA BP

Vedanta Excel in Mathematics - Book 9 CR

14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Set

4. 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 ˆ B and A ˆ 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 2 1 U
Q
(i) P ‰ Q ‰ R (ii) P ˆ Q ˆ R (iii) P ‰ Q ‰ R
(vi) (P ˆ Q) ‰ R 4
10 6
(iv) P ˆ Q ˆ R (v) (P ‰ Q) ˆ R 12 3 5

9 15 R

7 11 13 14

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 – P)

and P ‰ Q and verify P ‰ (Q – P) = P ‰ Q by shading.

Creative section - A

6. P and Q are the subsets of the universal set U. 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.

a) P ‰ Q and P ‰ Q b) P ˆ Q and P ˆ Q c) P – Q and P – Q

d) Q – P and Q – P e) P ‰ Q f) P ˆ Q

7. 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.

a) A ‰ B ‰ C and A ‰ B ‰ C b) A ˆ B ˆ C and A ˆ B ˆ C

c) (A ‰ B) ˆ C and (A ‰ B) ˆ C d) A ˆ B ‰ C) and A ˆ B ‰ C)

e) (A – B) ‰ C and (A – B) ‰ C f) A ‰ B – C) and A ‰ B – C)

Creative section - B

8. If U = {1, 2, 3, ... 12}, P = {1, 2, 3, 4, 5, 6}, Q = {2, 4, 6, 8}, and R = {3, 6, 9, 12},

verify the following operations.

a) P ‰ Q ˆ R) = (P ‰ Q) ˆ (P ‰ R) b) P ˆ (Q ‰ R) = (P ˆ Q) ‰ (P ˆ R)

c) P ‰ Q = P ˆ Q d) P ˆ Q ˆ R = P ‰ Q ‰ R

9. 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 ˆ B).

15Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

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1.9 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} 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 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} U

Now, A ‰ B = {a, b, c, d, e, f, h, i} n (A) Aa c f B n (B)
n (only A) be i n (only B)
A ˆ B ={c, e} dh
A ‰ B = {g, j} gj

only A = n0(A) = {a, b, d} n (A ˆ B) n (A ‰ B)
only B = n0(B) = {f, h, i}

Here, n (U) = 10, n (A) = 5, n (B) = 5 and n (A ˆ B) = 2

Now, n (A ‰ B) = 8 = 5 + 5 – 2 = n (A) + n (B) – n (A ˆ 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 ˆ B)
n (only B) = nO(B) = 3 = 5 – 2 = n(B) – n(A ˆ 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 ˆ B)

n (A ‰ B) = n (U) – n (A ‰ B)

n (A ˆ B) = n (A) + n (B) – n (A ‰ B)

n (only A) = no(A) = n(A) – n(A ˆ B)
n (only B) =no(B) = n(B) – n(A ˆ 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)

Vedanta Excel in Mathematics - Book 9 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

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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 ˆ 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 ˆ 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 ˆ Q) = 5 – 2 = 3
i) no (Q) = n(Q) – n(P ˆ 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 ˆ B) = 30, find

a) n (A ‰ B) b) n (A ‰ B) c) no (A) d) no (B)

Solution:

Here, n (U) = 95, n (A) = 64, n (B) = 56 and n (A ˆ B) = 30

a) Now, n (A ‰ B) = n (A) + n (B) – n (A ˆ B)

= 64 + 56 – 30 = 90

b) n (A ‰ B) = n (U) – n (A ‰ B) = 95 – 90 = 5

c) no (A) = n (A) – n (A ˆ B) = 64 – 30 = 34
d) no (B) = n (B) – n (A ˆ B) = 56 – 30 = 26

Example 3: If n(A – B) = 24, n(A ‰ B) = 80, and n(A ˆ 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 ˆ B) = 20

Now, n(A ‰ B) = n0(A) + n(A ˆ B) + n0(B) P Q
36
or, 80 = 24 + 20 + n0(B) 24 20
5
or, n0(B) = 36

Again, n(B) = n0(B) + n(A ˆ 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 ˆ B), no (A) and no (B).
(ii) Illustrate the information in a Venn-diagram.

17Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

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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 ˆ B) (ii) Illustration in Venn-diagram

or, 99 = 63 + 54 – n (A ˆ B) U
AB
or, n (A ˆ B) = 117 – 99 = 18
45 18 36
Also, no (A) = n (A) – n (A ˆ B)
= 63 – 18 = 45

And no (B) = n (B) – n (A ˆ B) 21
= 54 – 18 = 36

Example 5: In a survey of a few number of farmers, it is found that 27 farmers use

only chemical fertilizers, 100 use chemical or bio-fertilizers, and 35 use

both chemical and bio-fertilizers. Find the number of farmers who use

bio-fertilizers.

Solution:

Let C and B be the sets of the farmers who use chemical and bio-fertilizers respectively.

Here, n0(C) = 27, n(C ‰ B) = 100 and n(C ˆ B) = 35
Now, n(C ‰ B) = n0(C) + n(C ˆ B) + n0(B)
or, 100 = 27 + 35 + n0(B)
or, n0(B) = 100 – 62 = 38
Then, n(B) = n0(B) + n(C ˆ B) = 38 + 35 = 73
Hence, 73 farmers use 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.

From the Venn-diagram, Venn-diagram
(i) n (E ‰ D) = 80 – x + x + 95 – x
or, 150 = 175 – x U
ED

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
Hence, 55 students like only educator.

Vedanta Excel in Mathematics - Book 9 18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

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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 ˆ A)
or, 95 = 72 + 53 – n (C ˆ A) U
or, n (C ˆ A) = 125 – 95 CA

= 30 42 30 23
Hence, 30 people like both comedy and action movies.
45

(ii) Also, the number of people who like comedy movies only = no (C)
= n (C) – n (C ˆ A)

= 72 – 30 = 42

And, the number of people who like action moves only = no (A)
= n (A) – n (C ˆ A)

= 53 – 30 = 23

? The number of people who like only one type of movie = 42 + 23 = 65

Example 8: In a survey conducted among some farmers in Ilam district, it was
found that 48% of them have black cardamon farming, 60% have ginger
farming, and 15% of them have none of these farming.
(i) Represent the above information in Venn-diagram.

(ii) If there are 92 farmers who have both types of farming, find the number

of farmers participated in the survey.

Solution:

Let B and G be the sets of the farmers who have black cardamom and ginger farming

respectively.

Here, n (U) = 100 %, n (B) = 48 %, n (G) =60 % and n (B ‰ G ) = 15 %

(i) Now, n (B ‰ G) = n (U) – n (B ‰ G ) = 100 % – 15 % = 85 %

19Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

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Again, n (B ‰ G) = n (B) + n (G) – n (B ˆ G) B 23% U
or, 85 % = 48 % + 60 % – n (B ˆ G) 25% G
or, n (B ˆ G) = 108 % – 85 % = 23 % 37%

(ii) Let the number of farmers participated in the survey be x. 15%

Now, 23 % of x = 92

or, 23 × x = 92
or, 100 x = 400

Hence, 400 farmers participated in the survey.

Example 9: In a survey of 500 tourists who visited Nepal during 'Visit Nepal 2020',
it was found that 168 tourists visited Rara but not Khaptad, 192 visited
Khaptad but not Rara and 45 of them did not visit both places.
(i) Find the number of tourists who visited Rara or Khaptad.
(ii) Find the number of tourists who visited Rara and Khaptad
(iii) How many of them visited Rara?
(iv) Illustrate the result in a Venn-diagram.

Solution:
Let R and K be the sets of the people who visited Rara and Khaptad respectively.

Here, n (U) = 500, no (R) = 168, no (K) = 192 and n (R ‰ K ) = 45
(i) Now, the number of tourists who visited Rara or Khaptad is

n (R ‰ K) = n (U) – n (R ‰ K )

= 500 – 45

= 455

Hence, 455 tourists visited Rara or Khaptad.

(ii) Let the number of tourists who visited Rara and Khaptad be x.

Then, n (R ‰ K) = no (R) + no (K) + n(R ª K) (iv) Venn-diagram
or, 455 = 168 + 192 + x
U
or, x = 95 RK

Hence, 95 tourists visited Rara and Khaptad both places. 168 95 192

(iii) The number of tourist who visited Rara = no (R) + 95 45
= 168 + 95 = 263

Example 10: In an election of a municipality, A and B were two candidates for the post of
the Mayor. There were 36,000 people in the voter list. Voters were supposed
to cast the vote for a single candidate. 21,500 people cast vote for A, 10,400
people cast for B, and 630 people cast vote even for both candidates.

(i) Illustrate these information in a Venn-diagram

(ii) How many people did not cast vote?

(iii) How many votes were valid?

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Solution:

Let A and B be the sets of people who cast vote to the candidates A and B respectively.

Here, n (U) = 36,000 , no (A) = 21,500, no (B) = 10,400 and n(A ˆ B) = 630
(i) U (ii) Number of people who caste vote = n (A ‰ B)

A 630 B n (A ‰ B) = no(A) + no(B) + n(A ˆ B)
21,500 10,400 = 21,500 + 10,400 + 630 = 32,530

Now, number of people who did not caste vote = n(A ‰ B)

3,470 Then, n(A ‰ B) = n(U) – n(A ‰ B)= 36,000 – 32,530 = 3,470

(iii) Again, the number of valid votes = 21,500 + 10,400 = 31,900

Example 11: 45 students of class 9 are taking part in Maths exhibition or in Science
exhibition or in both exhibitions. Out of them, 11 students are taking
part in both exhibitions. The ratio of the number of students who are
taking part in Maths to those who are taking part in Science exhibitions
is 4 : 3.

(i) How many students are taking part in Maths exhibition?

(ii) How many students are taking part in Science exhibition only?

(iii) Show the above information in a Venn-diagram.

Solution:

Let M and S be the sets of students who are taking part in Maths and Science exhibitions

respectively.

Here, n(M ‰ S) = 45, n(M) = 4x, n(S) = 3x, and n(M ª S) = 11

Now, n(M ‰ S) = n(M) + n(S) – n(M ˆ S)

or, 45 = 4x + 3x – 11

or, x = 8

(i) Then, n(M) = 4x = 4 × 8 = 32

Hence, 32 students are taking part in Maths exhibition. (iii) U
S
(ii) Also, n(S) = 3x = 3 × 8 = 24 M

And, n0(S) = n(S) – n(M ˆ S) = 24 – 11 = 13 21 11 13
Hence, 13 students are taking part in Science exhibition.

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 ˆ B) = r, show this information in a Venn-diagram and

show that n (A ‰ B) = n(A) + n(B) – n(A ˆ 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 ).

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d) If M and N are two overlapping subsets of a universal set U, write the relation between

n(M), n(M ˆ N), and no(M).

e) If A and B are two overlapping subsets of a universal set U, write the relation between

n(B), n(A ˆ B) and no(B).

2. From the adjoining Venn-diagram, find the cardinal numbers A U
of the following sets: B

41 3

a) n (U) b) n (A) c) n (B) 29

d) n (A ‰ B) e) n (A ˆ B) f) n (A ‰ B ) 5 7 68
i) no (A) j) no (B)
g) n (A) h) n (B )

3. a) If n (U) = 65, n (A) = 28, n (B) = 45, and n (A ˆ 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 ˆ 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 ˆ Y) = 10, find:

(i) n (X) (ii) n (Y) (iii) n (X ‰ Y) (iv) n (X ‰ Y )

d) A and B are the subsets of a universal set U. If n (A ‰ B ) = 18 %, no (A) = 38 %, and
no (B) = 15 % find,

(i) n (A ‰ B) (ii) n (A ˆ B) (iii) n (A) (iv) n (B)

4. a) A and B are the subsets of a universal set U. If n(U) = 100, no(A) = 30, no(B) = 35, and
n(A ˆ B) = 25, illustrate this information in a Venn-diagram.

b) M and N are the subsets of a universal set U. If n (U) = 65, n (M) = 36, n (N) = 27,
n (M ˆ N) = 12, illustrate this information in a Venn-diagram.

Creative section - A

5. a) A and B are the subsets of a universal set U. If n(U) = 75, n(A) = 40, n(B) = 60, and
A  B, illustrate these information in a Venn-diagram and find:

(i) n(A ‰ B) (ii) n(A ˆ B) (iii) no(B) (iv) n(A ‰ B)
b) M and N are subsets of a universal set U. If n(M) = 70%, n(N) = 57%, and N  M,

illustrate these information in a Venn-diagram and find:

(i) n(M ‰ 1) (ii) n(M ˆ N) (iii) no(M) (iv) n(M ‰ N)

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6. a) A and B are the subsets of a universal set U in which there are n (U) = 54,
n (A) = 32, n (B) = 22, n (A ˆ B) = 9.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the value of n (A ‰ B ).

b) X and Y are the subsets of U. If n (X) = 55 %, n (Y) = 65 %, and n (X ‰ Y ) = 0 %,
(i) illustrate this information in a Venn-diagram.

(ii) Find n (X ˆ Y) , no (X) and no (Y).

7. 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 cannot speak any of two languages?

(iii) How many people can speak Tamang language only?

(iv) How many people can speak the Nepali language only?

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 one song?

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 one of the farming?

Creative section - B

8. a) In a group of 500 students, 280 like bananas, 310 like apples, and 55 dislike 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 do not like tea only.

23Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

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c) In a survey of some people, it was found that 45 % of them were using internet,
60 % were using cellular data, and 15 % of them were not using any type of Wi-Fi
connection.
(i) Represent the above information in a Venn-diagram.
(ii) If there are 60 people who were using both types of Wi-Fi connections, find the
number of people who participated in the survey.

9. 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 200 families, 80 were found using 'A' brand tea only, 75 were found
using 'B' brand tea only, and each family is using at least one of the two brands.

(i) Draw a Venn diagram to illustrate the above information.
(ii) How many families are using both brands?
(iii) How many families are using 'A' brand?
(iv) How many families are using 'B' brand?

10. a) In an election of a municipality X and Y were two candidates for the post of the
Mayor. There were 40,000 people in the voter list. Voters were supposed to cast the
vote for a single candidate. 15,000 people cast vote for X, 23,650 people cast for Y and
350 people cast vote even for both the candidates.

(i) Illustrate the above information in a Venn-diagram

(ii) How many people did not cast vote?
(iii) Find the number of valid votes.

b) There are 900 students in a school. They are allowed to cast vote either only for A or
for B as their school prefect. 36 of them cast vote for both A and B, 483 cast vote for
A and 367 cast vote for B.
(i) How many students did not cast the vote?
(ii) Find the number of valid votes
(iii) Show the information in a Venn-diagram.

11. a) 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.

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b) 54 students of class IX are taking part in sports or in music or in both activities. Out of
them 9 students are taking part in both activities. The ratio of the number of students
who are taking part in sports to those who are taking part in music is 5 : 4.

(i) How many students are taking part in sports?

(ii) How many students are taking part in music only?

(iii) Illustrate this information in a Venn-diagram.

Project work

12. a) Write a universal set of your own choice and its any three overlapping subsets. Show
the elements of these sets in a Venn-diagram.

b) Write a universal set of your own choice and its any three overlapping subsets A, B

and C. Then, verify the following operations.

(i) A ‰ (B ˆ C) = (A ‰ B) ˆ (A ‰ C) (ii) A ˆ (B ‰ C) = (A ˆ B) ‰ (A ˆ C)

(iii) A ˆ B ˆ C = A ‰ B ‰ C
13. 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
14. 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 ˆ B) = 40, n(A ‰ B) = ?

b) n(U) = 190, n(A) = 105, n(B) = 115, n(A ‰ B) = 15, n(A ˆ B) = ?

c) no(A) = 72, n(B) = 100, n(A ˆ B) = 36, n(A ‰ B) = 28, n(U) = ?
Now, solve your verbal problems and illustrate the results in Venn-diagrams.

OBJECTIVE QUESTIONS
1. How can we define (A«B) mathematically?

(A) {x: x±A or x±B} (B) {x: x±A and x±B}

(C) {x: x±A, but x²B} (D) {x: x±U, but x²A}

2. What is the set operation for the mathematical expression {x: x±P and x±Q}?

(A) P ª Q (B) P « Q (C) P – Q (D) (P ª Q)'

3. Which one of the following is NOT correct?

(A) A – B = A – (A ª B) (B) B – A = B – (A ª B)

(C) A – B = (A«B) – B (D) A… = U – B

4. If P = {h, a, v, e}, Q = {h, a, s} and R = {h, a, d}, (P«Q) ªR is

(A) {a, h} (B) {d, s} (C) {d, e, s, v} (D) {a, h, s, v}

25Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Set

5. If U = {x: x †10, x±N}, A = {y: y is a prime number}, B = {z: z is an odd number},
(A « B)… is

(A) {2, 3, 5, 7, 9} (B) {1, 3, 4, 6, 8, 10} (C) {1, 4, 6, 8, 10} (D) {2, 4, 6, 8, 10}

6. 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

7. Which one of the following relation is NOT true on the operations of sets?

(A) A « (B«C) = (AªB)« C (B) A ª (BªC) = (AªB)ª C

(C) A « (BªC) = (A«B)ª (A«C) (D) A ª (B«C) = (AªB) « (AªC)

8. If A ° B, n (A) = 25, and n (A«B) = 40, what is the value of n (B)?

(A) 25 (B) 40 (C) 15 (D) 65

9. If A is a subset of B, n (A) = 20 and n (A‰B) = 50, what is the value of n0 (B)?

(A) 20 (B) 50 (C) 30 (D) 70

10. If A & B are any two sets, in which case is the value of n (A‰B) maximum?

(A) When A is subset of B (B) When B is subset of A

(C) When A and B are overlapping (D) When A and B are disjoint

11. Out of 40 students of a class, 30 liked music, 20 liked dance and 16 students liked
music as well as dance then how many students liked neither music nor dance?

(A) 34 (B) 14 (C) 4 (D) 6

12. In a group of 20 people, 8 go for morning walk only, 6 go for evening walk only and
4 do not go for morning and evening walk. How many people go for both morning and
evening walk?

(A) 6 (B) 4 (C) 2 (D) 10

13. In a group of 200 game lovers, 120 of them like cricket and 100 like football. How many
students like only cricket?

(A) 20 (B) 100 (C) 80 (D) 180

14. In a survey of community, it was found that 45% of the people like agriculture, 25% like
civil service, and 40 % do not like both agriculture and civil service. What percent of
people like only one?

(A) 35% (B) 15% (C) 50% (D) 60%

15. In a class of 40 students, 15 passed in Mathematics and Science both and 20 passed in
only one subject. How many students failed in both the subjects?

(A) 5 (B) 25 (C) 20 (D) 45

Vedanta Excel in Mathematics - Book 9 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Unit Profit and Loss

2

2.1 Profit and Loss - Review
Let a trader buys an electric fan for Rs 1,200 and she sells it for Rs 1,500.
Here, Rs 1,200 is the price paid to buy the fan called cost price (C. P.) and Rs 1,500
is the price at which the fan is sold called selling price (S. P.).
In this case, the trader makes a profit of Rs 300.
Hence, when S. P. is higher than C. P. there is a profit. However, if S. P. is lower
than C. P., there is a loss.
? Profit = S. P. – C. P. and Loss = C. P. – S. P.

2.2 Profit percent and loss percent - Review

In the above illustration, profit = S. P. – C. P. = Rs 1,500 – Rs 1,200 = Rs 300.

When C. P. is Rs 1,200, profit is Rs 300.

When C. P. is Re 1, profit is Rs 300 .
1200
300
When C. P. is Rs 100, profit is Rs 1200 × 100 = Rs 25

Here, Rs 25 is the profit of C. P. of Rs 100.

Since, 25 out of 100 is 25%, the profit percent is 25%.

Thus, profit percent = Profit × 100% o S. P. – C. P. × 100%
C.P. C. P.
Loss C. P. – S. P.
Similarly, loss percent = C.P. × 100% o C. P. × 100%

Worked-out examples

Example 1: A grocer bought 10 crates of eggs for Rs 2,400 and he sold all eggs at
Rs 120 per dozen. Find his profit or loss percent.

Solution:

Here, C.P. of 10 creates i.e. 300 eggs = Rs 2,400
2,400
C.P. of 1 egg = Rs 300 = Rs 8

S.P. of 1 dozen i.e. 12 eggs = Rs 120
120
S.P. of 1 egg = Rs 12 = Rs 10

Since, S.P. > C.P., he made profit.

profit = S.P. – C.P. = Rs 10 – Rs 8 = Rs 2
Profit Rs 2
Now, profit percent = C.P. × 100% = Rs 8 × 100% = 25%.

Hence, his profit percent is Rs 25%.

27Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

Example 2: A fruitseller purchased 1 quintal of organic apples at Mustang for
Rs 9,000 and paid Rs 10 per kg in transportation from Mustang to Pokhara.
5 kg of apples were found damage and he sold the remaining quantity at
Rs 120 per kg. Calculate his profit or loss percent in the total transaction.

Solution:

Here,C.P. of 1 quintal, i.e., 100 kg of apples = Rs 9,000

C.P. of apples with transportation charge = Rs 9,000 + 100 × Rs 10 = Rs 10,000

Salable quantity of apples = 100 kg – 5 kg = 95 kg

S.P. of 95 kg of apples = 95 × Rs 120 = Rs 11,400

Since, S.P. > C.P., he made profit.

Actual profit = S.P. – C.P. = Rs 11,400 – Rs 10,000 = Rs 1,400

Now, profit percent = Actual profit × 100% = Rs 1,400 × 100% = 14%
C.P. Rs 10,000
Hence, his profit percent is 14%.

Example 3: A shopkeeper sells 5 kg of fruits for the cost price of 6 kg of fruits. Find
her profit percent.

Solution: = Rs x
Let, C.P. of 1 kg of fruits = Rs 6x
= Rs 5x
C.P. of 6 kg of fruits
C.P. of 5 kg of fruits

According to the question,

S.P. of 5 kg of fruits = C.P. of 6 kg of fruits

S.P. of 5 kg of fruits = Rs 6x

But, C. P. of 5 kg of fruits = Rs 5x

? Actual profit in 5 kg of fruits = S.P. – C.P. = Rs 6x – 5x = Rs x
Profit x
Now, profit percent = C.P. × 100% = 5x × 100% = 20%

Hence, her profit percent is 20%.
Example 4: Mrs. Nepali bought a second hand scooter and she spent Rs 5,000 for

its repairment. If she sold it for Rs 1,40,400 and gained 8%, at what
price did she buy the scooter?

Solution: Direct process
Here, S. P. of the scooter = Rs 1,40,400 S.P.
C.P. = (100 + profit)%
Profit percent = 8%
Now, C. P. + 8% of C. P. = S. P. = 140400 = 140400
or, 108% of C. P. = Rs 1,40,400 108% 108/100

= Rs 1,30,000

or, C. P. = Rs 1,30,000

Again,C. P. of the scooter without repairment = Rs 1,30,000 – Rs 5,000 = Rs 1,25,000

Hence, she bought the scooter for Rs 1,25,000.

Vedanta Excel in Mathematics - Book 9 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

Example 5: By selling 50 lemons for Rs 300, a man loses 10%. How many lemons

should he sell for Rs 200 to gain 20% in the transaction?

Solution: Direct process
Here, S. P. of 50 lemons = Rs 300 S.P.
C.P. = (100 – loss)%
Loss percent = 10%
Then, C. P. – 10% of C. P. = S. P. of 50 lemons = Rs 300 = Rs 300
(100 – 10)% 90%

or, 90% of C. P. = Rs 300 = Rs 300 = Rs 300 × 100
or, C. P. 90/100 90
? C. P. of 50 lemons 1000
= Rs 10300 = Rs 1000
C. P. of 1 lemon = Rs 13000 3
Again, new S. P. of 1 lemon = Rs 3 × 50
= Rs 20
3
120 20
= C. P. + 20% of C. P. = 100 × Rs 3 = Rs 8

Now,Rs 8 is the S. P. of 1 lemon

Re 1 is the S. P. of 1 lemon
8
1
Rs 200 is the S. P. of 8 × 200 lemons = 25 lemons

Hence, he should sell 25 lemons for Rs 200.

Example 6: Hari bought 10 dozen of glass tumbler. 12 glass tumblers were broken

during transportation. He sold the remaining tumblers at the rate of
Rs 60 and gained 20%. At what rate of cost did he buy the tumblers?
Solution:
Here, the total number of glass tumblers = 10 dozen = 120

The remaining number of tumblers = 120 – 12 = 108

S.P. of 1 glass tumbler = Rs 60 Direct process
S.P. of 108 glass tumblers = 108 × Rs 60 (100 + P)% of C.P. = S.P.

Now, S.P. = C.P. + P% of C.P. or, 120% of C.P. = 108 × 60
120
or, 108 × 60 = C.P. + 20% of C.P. or, 100 × C.P. = 108 × 60

or, 108 × 60 = 6 C.P. C.P. = Rs 5,400
5

or, C.P. = Rs 5,400

Again, C.P. of 120 glass tumblers is Rs 5,400

C.P. of 1 glass tumbler is Rs 5,400 = Rs 45
120

So, he bought the glass tumbler at the rate of Rs 45 each.

Example 7: Mr. Shrestha bought a mobile for Rs 5,500 and sold it at 5% profit. If
Solution: he had bought it for Rs 5,250, what would be his profit or loss percent?

29Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

Here, the first cost price of mobile (C.P.1) = Rs 5,500 Direct process
Profit percent (P%) = 5% S.P. = (100 + P)% of C.P.

Selling price (S.P.) = ? = (100 + 5)% of Rs 5,500

The second cost price (C.P.2) = Rs 5,250 = 105 × Rs 5,500
Profit or loss percent = ? 100

Now, S.P. of mobile = C.P.1 + 5% o1f 50C0.P×.1 = Rs 5,775
= Rs 5,500 +
Rs 5,500

= Rs 5,775

Again, C.P.2 = Rs 5,250 and S.P. = Rs 5,775. So, he made profit.

Profit = S.P. – C.P.= Rs 5,775 – Rs 5,250 = Rs 525

Also, profit percent = Actual profit × 100% = Rs 525 × 100% = 10%
C.P. Rs 5,250

Hence, his profit percent would be 10%.

Example 8: Mrs. Magar sold a cosmetic item for Rs 777 and made a profit of 11%.

If she had sold it for Rs 693, what would be her profit or loss percent?

Solution:
Here, the first selling price of cosmetic item (S.P.1) = Rs 777

Profit percent (P%) = 11%

Cost price of the cosmetic item (C.P.) = ? Direct process

The second selling price of the cosmetic item (S.P.2)= Rs 693 C.P. = S.P.1
Profit or loss percent = ? (100 + 11)%

Now, C.P. + P% of C.P. = S.P.1 = Rs 777
or, C.P. + 11% of C.P. = Rs 777 111%

or, 111 C.P. = Rs 777 = Rs 777 × 100
100 111

or, C.P. = Rs 700 = Rs 700

Again, S.P.2 = Rs 693 and C.P. = Rs 700. So, there is a loss.

Now, actual loss = C.P. – S.P.2 = Rs 700 – Rs 693 = Rs 7

Also, loss percent = Actual loss × 100% = Rs 7 × 100% = 1%
C.P. Rs 700

Hence, her loss percent would be 1%.

Example 9: A trader sold a refrigerator for Rs 28,500 at 5% loss. At what price

should he sell it to gain 5%?

Solution:
Here,the first selling price of the refrigerator set (S.P.1) = Rs 28,500

Loss percent (L%) = 5%
Cost price of the refrigerator (C.P.) = ?

Profit percent (P%) = 5%
The second selling price of the refrigerator (S.P.2) = ?

Vedanta Excel in Mathematics - Book 9 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

Now,

C.P. – L% of C.P. = S.P.1 Direct process
or, C.P. – 5% of C.P. = Rs 28,500

or, 19 C.P. = Rs 28,500 (100 – L)% of C.P. = S.P.1
20
or, 95% of C.P. = Rs 28,500
or, C.P. = Rs 30,000 Rs 28,500
or, C.P. = 95%
Again, S.P.2 = C.P. + P% of C.P.
= Rs 30,000 + 5% of Rs 30,000 C.P. = Rs 30,000

= Rs 32,500

Hence, he should sell the refrigerator for Rs 31,500.

Example 10: Suntali sold a pendrive at a loss of 4%. If she had sold it at Rs 45 more,

she would have gained 5%. Find the cost price of the pendrive.

Solution:

Let, the cost price of the pendrive be Rs x. Direct process
S.P. = (100 – 4)% of C.P.
Now, S.P. of the pendrive = C.P. – L% of C.P.
= 96% of x
= Rs (x – 4% of x) 96x 24x
= 100 = 25
x 24x
= Rs (x – 25 ) = Rs 25

According to question, new S.P. of the pendrive = Rs 24x + Rs 45
25
24x + 1125
= Rs 25

S.P. = C.P. + P% of C.P. Direct process
New S.P. = (100 + 5)% of C.P.
24x + 1125
or, 25 = x + 5% of x or, 24x + 1125 = 105 × x
25 100
or, 24x + 1125 = x + 5 × x
25 100 or, x = Rs 500

or, 24x + 1125 = 21x
25 20

or, 105x = 96x + 4500

? x = 500

Hence, the cost price of the pendrive is Rs 500.

Example 11: Dhurmus bought two digital watches for Rs 2,000. He sold one of them
at 10% profit and the other at 10% loss. Find his gain or loss percent
in this transaction if the selling price of both watches are the same.

Solution:
Let, the C.P.1 of one calculator be Rs x.
∴ The C.P.2 of the other calculator is Rs (2,000 – x).

31Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

In 10% profit, In 10% of loss,

S.P.1 = C.P.1 + 10% of C.P.1 S.P.2 = C.P.2 – 10% of C.P.2

= Rs x + 10% of Rs x = Rs (2,000 – x) – 10% of Rs (2,000 – x)

= Rs 11x = Rs 18,000 – 9x
10 10

According to the question,

S.P.1 = S.P2

or, 11x = 18,000 – 9x
10 10

or, x = Rs 900

Now, S.P.1 = 11x = 11 × 900 = Rs 990
10 10

Since, S.P.1 = S.P.2 , the total S.P. = 2 × Rs 990 = Rs 1,980

Actual loss = C.P. – S.P. = Rs 2,000 – Rs 1,980 = Rs 20

Again, loss percent = Actual loss × 100% = Rs 20 × 100% = 1%
C.P. Rs 2,000

Hence, his loss percent in this transaction is 1%.

EXERCISE 2.1

General section
1. a) If profit is P% and cost price is C.P., what is the selling price?

b) If loss is L% and cost price is C.P. , what is the selling price?
c) If gain is G% and selling price is S.P., what is the cost price?
d) If loss is L% and selling price is S.P., what is the cost of price?
2. From the table given below, calculate the unknown variables:

S.N. C.P. S.P. Actual profit or loss profit or loss percent

a) Rs 200 Rs 220 ? ?

b) Rs 550 Rs 495 ? ?

c) Rs 750 ? ? 8% (profit)

d) Rs 555 ? ? 20% (loss)

e) ? Rs 1,056 ? 10% (profit)

f) ? Rs 720 ? 10% (loss)

g) ? Rs 560 Rs 60 (profit) ?

h) ? Rs 731 Rs 129 (loss) ?

3. a) A dealer purchased a mobile for Rs 4,545 and sold it at 20% profit. Find his
actual profit and selling price of the mobile.

Vedanta Excel in Mathematics - Book 9 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

b) A retailer bought a bicycle for Rs 9,600. If he sold it at 2% loss, how much was
his actual loss? Find the selling price of the bicycle.

4. a) Mr. Rai sells a watch for Rs 2,035 at a profit of 10%. How much money has he
paid to buy the watch? Also find his profit amount.

b) Mrs. Bajracharya sold a 'Palpali Dhaka Topi' for Rs 870 and made a profit of
16%. Find her buying price of the 'Dhaka Topi'. How much is her actual profit?

Creative section - A

5. a) A grocer bought 20 kg of sugar at Rs 70 per kg. He sold 15 kg of sugar at Rs 75
per kg and the remaining quantity at Rs 69 per kg. Find his profit or loss percent
in the transaction.

b) A retailer bought 1000 glass tumblers at Rs 80 each. 50 glass tumblers were
broken and he sold the rest at Rs 96 each. Find his profit or loss percent.

c) Mrs. Rokaya bought 2 quintals of apples in Jumla for Rs 17,000. She paid Rs 15
per kg for the transportation from Jumla to Nepalgunj. 10 kg of apples were
damage and she sold the remaining quantity at Rs 120 per kg. Calculate her
profit or loss percent in the total transaction.

d) A stationer sells 8 pencils for the cost price of 10 pencils, find his gain percent.

e) A fruitseller bought 50 kg of fruits. He sold 30 kg of fruits for the cost price of
35 kg of fruits and he sold the remaining quantity for the cost price of 18 kg of
fruits. Calculate his profit or loss percent in the total transaction.

6. a) Mr. Ghale bought a secondhand scooter for Rs 1,40,000. He spent Rs 7,500 to
repair it. At what price should he sell it to gain 10% on his total investment?

b) A book seller purchased 100 story books at Rs 135 each. She donated 10 books
to a school library. If she sold remaining books at 2% loss, find the selling price
of each book.

7. a) A trader sold 5 mobiles for Rs 35,840 and had a profit of 12%. Find the cost
price of each mobile.

b) Mrs. Limbu bought 180 kg of oranges in Dhankuta. 15 kg of oranges were rotten
and not fit for selling. If she sold the remaining quantity of oranges in Dharan
at the rate of Rs 78 per kg and gained 10%, find the rate of cost price of oranges.

c) A stationer bought 2,000 exercise books. He distributed 200 exercise books to
the students from poor economical background as donation. He sold each of the
remaining exercise books at Rs 10 more than the cost price of each and gained
8%, find the cost price of each exercise book.

d) Bikash purchased 10 pens. He sold 5 pens at 25% profit and the remaining 5
2
pens at 16 3 % loss. If he received Rs 625 in total, find the cost price of each pen.

8. a) Sunayana bought a pen for Rs 19 and sold it at 20% profit to Debasis. If she had
bought it for Rs 20, what would be her profit or loss percent?

b) Pratik bought a sweater for Rs 1,750. He sold it at a profit of 4% to Bishwant.
What would be his profit or loss percent if he had bought it for Rs 2,000?

9. a) Mrs. Yadav sold a bag for Rs 1,368 at 5% loss. Calculate her profit or loss percent
if she had sold it for Rs 1,512.

33Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

b) A grocer sold 5 kg of wheat flour at Rs 55 per kg and gained 10%. If he had sold
it at Rs 52 per kg, what would be his gain or loss percent?

10. a) A shopkeeper sold a mobile set for Rs 9,600 and made a loss of 5%. For what
price should he sell it to gain 10%?

b) Mr. Gurung sold a bicycle for Rs 18,816 and made a profit of 12%. If he wants
to increase his profit by 3%, at what price should he sell the bicycle?

11. a) A stationer buys a gel pen for Rs 20 and sells it at 20% profit. For what price
should he buy it so that he can make 25% profit by selling for the same selling
price.

b) A dealer bought a pendrive for Rs 500. He sold it at 10% loss. If he wants
to make a profit of 12.5% without increasing the selling price, by how much
should the cost price of the pendrive be reduced?

Creative section - B

12. a) A trader sold an article at 10% profit. If he had sold it at 10% loss, it would yield
Rs 140 less than the previous selling price. Find the cost price of the article.?

b) A retailer sold a laptop at a loss of 5%. If he had sold it at Rs 6,000 more, he
would have gained 10%. Find the cost price of the laptop.

c) A shopkeeper sold a computer at a profit of 15%. If he had sold it at Rs 10,000
less, he would have a loss of 1%. At what price did he purchase the computer?

13. a) Mr. Sharma bought two fans for Rs 2,500 each. He sold one of them at 10%
profit and the other at 4% loss. What was his gain or loss percent on the whole?

b) Rajesh Das bought two calculators for Rs 1,000. He sold one of them at 20%
profit and the other at 20% loss. If the selling price of both calculators are same,
find the cost price of each calculator. Also, calculate his gain or loss percent in
the total transaction.

c) The cost price of a box and a pen is Rs 120. The box is sold at 10% profit and
the pen is sold at 10% loss. If the selling price of the box is Rs 52 more than that
of the pen, calculate the profit or loss percent in the whole.

d) Ajaya bought a fan and a heater for Rs 4,000. He sold the fan at a profit of 25%
and the heater at a loss of 5%. If he gained 7% on his total outlay, at what price
did he buy each item?

14. a) A trader purchased a mini sewing machine for Rs1,600 and sold to retailer at
10% profit. The retailer sold it at 5% profit to a consumer. How much did the
consumer pay for it?

b) Anamol sold a watch to Anjolina at 20% profit. Anjolina sold the watch to
Bishal at 10% loss. If Bishal paid Rs 756 to Anjolina, how much did Anamol
pay to buy the watch?

15. a) A shopkeeper decided to make equal rate of profit in each fancy item. If he sold
a jacket costing Rs 4,400 for Rs 5,060, at what price did he sell the shoes which
was purchased for Rs 3,420?

b) A stationer fixes an equal rate of profit in his each item. If he sells a calculator
costing Rs 700 for Rs 756, how much should a customer pay for a novel book
costing Rs 450?

Vedanta Excel in Mathematics - Book 9 34 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

Project work
16. a) Make a group of your friends. Visit a nearby stationery and ask about the cost

price of book, pen, pencil, geometry box, dictionary, calculator, etc and fill in
the given table. Now, find the required selling price of each item to make the
given profit percents.

S.N. Particular C.P. Profit percent S.P.
1. Math Book
2. Exercise book 10 %
3. Geometry box
4. Dictionary 20 %
5. Calculator
6. Pen 5%

25 %

12 %

20 %

b) Write the cost price (C.P.) and selling price (S.P.) of your own to make the given
profit or loss.

C. P. = ............ C. P. = ............ C. P. = ............ C. P. = ............

S. P. = ............ S. P. = ............ S. P. = ............ S. P. = ............

Profit = Rs 50 Loss = Rs 120 Profit% = 10% loss% = 5%

2.3 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.

2.4 Value Added Tax (VAT)

Value added tax (VAT) is an indirect tax which is charged at the time of
consumption of goods and services. Therefore, it is a consumption tax paid by a
consumer while purchasing goods or services.

The VAT rate is given in percent and it is decided by the concerned authority of
the government of a country. The VAT rate may vary from country to country and

35Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

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.

Worked-out examples

Example 1: A group of four friends went to a restaurant and had 4 plates of Mo:Mo,
Solution: 1 plate of french fries, and 4 glasses of fresh juice. If the actual cost
of these items was Rs 1,000, how much should they pay including 10%
service charge and 13% VAT to clear the bill?

Here, the actual cost of the items = Rs 1,000 Direct process
Now, the cost of items with 10 % service charge Cost with 10% service charge
= 110% of Rs 1,000
= Rs 1,000 + 10 % of Rs 1000 = Rs 1,100

= Rs 1,100

Again, the total cost with 13 % VAT Cost with 13% VAT

= Rs 1,100 + 13 % of Rs 1,100 = Rs 1,243 = 113% of Rs 1,100

Hence, Rs 1,243 should be paid to clear the bill. = Rs 1,243

Example 2: The marked price of an electric iron is Rs 1,450. If it is sold for Rs 1,276 by
giving a discount, calculate the discount percent.

Solution:

Here, M.P. of the article = Rs 1,450

S.P. of the article = Rs 1,276

Now, discount amount = M.P. – S.P.

= Rs 1,450 – Rs 1,276

= Rs 174
Discount amount
Again, discount percent = M.P. × 100%

= 174 × 100% = 12%
1,450

Hence, the required discount percent is 12%.

Vedanta Excel in Mathematics - Book 9 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

Example 3: A tablet was sold for Rs 6,545 after a discount of 15%. What was the
marked price of the tablet?
Solution:
Direct process
Let,the marked price of the tablet be Rs x.
S. P.
Now, S.P. of the tablet = M.P. – 15% of M.P. M.P. = (100 – D)%

or, Rs 6,545 = x – 15 x = 6545 × 100
100 (100 – 15)%

or, Rs 6,545 = 17x = 6545 = 7,700
20 85%

or, x = Rs 7,700

Hence, the marked price of the tablet is Rs 7,700.

Example 4: 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, 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. + VAT% of S.P. S.P. with VAT
13
= Rs 4,400 + 100 × Rs 4,400 = (100 + 13)% of S. P.

= Rs 4,972 = 113% of Rs 4,400

Again, Rs 5000 – Rs 4972 = Rs 28 = 113 × Rs 4,400 = Rs 4,972
100

Hence, the shopkeeper will return me Rs 28.

Example 5: The marked price of a mobile set is Rs 18,000 and 15 % discount is allowed.
Solution: How much should a customer pay for it with 13 % VAT?

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
S.P. of the mobile set = M.P. – 15 % of M.P. = Rs 15,300

= Rs 18,000 – 15 × Rs 18,000
100

= Rs 15,300 Direct process

Now, S.P. with VAT = S.P. + VAT % of S.P. S.P. with VAT
= Rs 15,300 + 13 % of Rs 15,300 = (100 + 13)% of S. P.
= Rs 17,289 = 113% of Rs 15,300
= Rs 17,289
Hence, the customer should pay Rs 17,289 for the mobile set.

Example 6: Mr. Dorje paid Rs 31,188 for a washing machine with a discount of 20 %
Solution: including 13 % VAT. What is the marked price of the washing machine?

Let the marked price of the washing machine be Rs x.

37Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

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
5 5 125

Also, the given S.P. = Rs 31,188

or, 113x = Rs 31,188
125

or, x = Rs 34,500
Hence, the marked price of the washing machine is Rs 34,500.

Alternative process Direct process
Let,S.P. without VAT be Rs x.
Then, S.P. without VAT = S.P. with VAT – 13 % of x 113 % of S.P. = Rs 31,188

or, x = Rs 31,188 – 13x or, S.P. = Rs 31,188
100 113%
13x
or, x+ 100 = Rs 31,188 S.P. = Rs 27,600

or, x = Rs 27,600 Again, (100 – 20)% of M.P. = S.P.

Again, let M.P. be Rs y. or, 80 × M.P. = Rs 27,600
100
Then, M.P. – discount % of M.P. = S.P. (without VAT) or, M.P. = Rs 34,500

or, y – 20 % of y = Rs 27,600

or, y = Rs 34,500

Hence, the required marked price is Rs 34,500.

Example 7: The marked price of a radio is Rs 4,000. After allowing certain percent
Solution: of discount with 13% VAT levied the radio is sold at Rs 3,616, find the
discount percent.

Here, M.P. of the radio = Rs 4,000 , VAT percent = 13% , S.P. with VAT = Rs 3,616

Discount percent = ?

Let S.P. without VAT be Rs x.

Then, S.P. with VAT = S.P. + VAT% of S.P. Direct process

or, Rs 3,616 = x + 13% of x 113 % of S.P. = Rs S.P. with VAT

or, Rs 3,616 = x + 13 x or, 113 × S.P. = Rs 3,616
100 100
113x
or, Rs 3,616 = 100 S.P. = Rs 3,200

? x = Rs 3,200

? S.P. without VAT is Rs 3,200.

Also, discount percent = M.P. – S.P. without VAT × 100%
M.P.
Rs 4,000 – Rs 3,200
= 4,000 × 100%

= Rs 800 × 100% = 20%
Rs 4,000

Hence, the required discount percentage is 20%.

Vedanta Excel in Mathematics - Book 9 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

Example 8: The price of a mobile set is fixed 20% above its cost price. When it is sold
allowing 10% discount, there is a gain of Rs 440. Find the marked price of
the mobile set.
Solution:

Let, C.P. of the mobile set be Rs x. Alternative process

Now, M.P. of the mobile set = x + 20% of x M.P. = 120% of x = 6x
5
x
= x + 5 S.P. = (100 – 10)% of M.P.

= 6x S.P. = 90% of M.P.
5
90 6x 27x
Also,S.P. of the mobile set = M.P. – 10% of M.P. = 100 × 5 = 25

= 6x – 10 × 6x Profit = S.P. – C.P.
5 100 5
27x
= 27x Rs 440 = 25 – x
25
or, x = Rs 5,500
Again, profit = S.P. – C.P.
or, 6x 6 × Rs 5,500
or, Rs 440 = 27x – x And, 5 = 5
25
2x = Rs 6,600
Rs 440 = 25

or, x = Rs 5,500

∴ M.P. of the mobile = 6x = 6 × Rs 5,500 = Rs 6,600
5 5

Hence, the marked price of the mobile set is Rs 6,600.

Example 9: Mr. Gupta sold a jacket at a discount of 10%, and made a profit of Rs 200.
Solution: If he had sold it at 15% discount, there would be a loss of Rs 90. Find the
marked price of the jacket.

Let, the marked price (M.P.) of the jacket be Rs x.

In 10% discount, S.P. of the jacket = M.P. – 10% of M.P.

= x – 10% of x = 9x
10
9x
? C.P. of the jacket = S.P. – profit = 10 – Rs 200

Again, in 15% discount, S.P. = x – 15% of x = 17x
20

According to the question,

C.P. – S.P. = Loss

or, 9x – Rs 200 – 17x = Rs 90
10 20
x
or, 20 = Rs 290

or, x = Rs 5,800

Hence, the marked price of the jacket is Rs 5,800.

39Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

EXERCISE 2.2 c) Value Added Tax (VAT)
General section

1. Define: a) Marked price (M.P.) b) Discount

2. a) What will be the selling price of an article when marked price is Rs x and discount
amount is Rs y?

b) If M.P. = a, discount percentage = b% and selling price (S.P.) = c, write the relation
of a, b, and c.

c) Write down the formula to find discount percent when discount amount and marked
price are given.

d) Write the formula for finding the selling price with VAT when selling price excluding
VAT and VAT amount are given.

e) If selling price (S.P.) = p, selling price with VAT = q, what is the rate of VAT?

3. a) The marked price of an article is Rs 280 and 10% discount is allowed. How much is
the discount amount?

b) The labelled price of an umbrella is Rs 420. If 5% discount is allowed, calculate the
selling price of the umbrella.

c) If 15% discount on a book amounts to Rs 45, find the marked price of the book.

d) A shopkeeper sold a sweater for Rs 1,584 allowing 12% discount. Find the marked
price of the sweater.

4. a) The selling price of an article is Rs 1,600 and 13 % VAT is levied on it. How much
should a customer pay for it?

b) The price of a fan is Rs 1,700 and a customer pays Rs 1,921 with VAT. Find the rate
of VAT.

c) When the VAT rate is 15 %, a customer pays Rs 1,449 to buy a watch. Find the cost
of the watch without VAT.

d) If the amount of VAT paid by a customer while buying a bag at 13% VAT rate is
Rs 143, find the cost of the bag without VAT.

Creative section - A

5. 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, find:

(i) the amount of service charge.

(ii) Calculate the VAT amount?

(iii) How much did she pay to clear the bill?

Vedanta Excel in Mathematics - Book 9 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

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?

6. a) The marked price of a bike helmet is Rs 3,000 and 10 % discount is allowed on it. Find
its cost with 13 % VAT.

b) 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?

c) 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?

7. 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.

8. 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?

9. 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?

10. a) The marked price of a cycle is Rs 5,500. After allowing a certain percent of discount
with 10% VAT levied, the cycle is sold at Rs 5,445. Find the discount percent.

b) The marked price of a 'Nepali Thanka' is Rs 2,200. After allowing a certain percent of
discount with 13% VAT levied, the Thanka was sold at Rs 1,988.80. Find the discount
percent.

11. 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.

41Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Profit and Loss

Creative section - B

12. a) After allowing 15% discount on the marked price of a camera, 15% VAT was levied
and sold it. If the difference between the selling price with VAT and selling price after
discount is Rs 1,122, find the marked price of the camera.

b) After allowing 10% discount on the marked price of a projector, 13% VAT was levied
and sold it. If the difference between the selling price with VAT and selling price after
discount is Rs 5,850, find the marked price of the projector.

13. a) The marked price of an article is 25% above its cost price. When it is sold at a discount
of 15%, there is a gain of Rs 200. Find,

(i) the cost price of the article

(ii) the marked price of the article

b) The price of an electric fan is fixed 20% above its cost price. When it is sold allowing
18% discount, there is a loss of Rs 20. Calculate the marked price and the selling price
of the fan.

14. a) The marked price of a cupboard is 25% above its cost price, When it is sold at a gain
of 10%, the profit amounts to Rs 840. Find the marked price of the cupboard.

b) The price of a laptop is fixed 20% above its cost price and sold it at 13% discount to
gain Rs 1,980. How much should a customer pay for it?

15. a) When a pen is sold at a discount of 15%, there is a gain of Rs 10. But if it is sold at
25% discount, there is a loss of Rs 2. Find the marked price of the pen.

b) A trader allowed 12% discount and sold a fan at a loss of Rs 16. If he had sold it at
10% discount he would have gained Rs 20. For what price did he purchase the fan?

c) A retailer made a loss of Rs 20 when he sold a bag at 20% discount. If he had sold it
at 10% discount, he would have gained 8%. Find the marked price of the bag.

16. a) A shopkeeper marked the price of an article a certain percent above the cost price and
he allowed 16% discount to make 5% profit. If a customer paid Rs 9,492 with 13%
VAT to buy the article, by what percent is the marked price above the cost price of the
article?

b) After getting 10% discount a customer paid Rs 2,034 with 13% VAT to buy a bag from
a retailer. If the retailer made a profit of 20%, by how many percent did he mark the
price of the bag above the cost price?

17. a) The marked price of an item is Rs 1,500 and 10% discount is given to make 20%,
profit. By what percent is the discount to be increased to get only 12% profit?

b) The price of a watch is marked Rs 11,250. When it is sold allowing 20% discount,
20% profit is made. By what percent is the discount to be reduced to increase the
profit by 3%?

Vedanta Excel in Mathematics - Book 9 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Profit and Loss

18. a) The marked price of a digital watch is Rs 6,000. Allowing 10% discount and including
same percentage of value added tax, the watch is sold. By how much percent is the
VAT amount less than discount amount?

b) The marked price of a guitar is Rs 5,500. After allowing 10% discount and levying
same percentage of VAT, the guitar is sold. By how much percent is the VAT amount
less than discount amount?

Project work

19. a) Make groups of your friends. Collect different types of goods purchasing bill. Study
about the marked price, rate of discount, rate of VAT or other rates of taxes mentioned
in the bills. Prepare the reports and present in your class.

b) Visit the nearby department store or your local shops. Search and collect the marked
price, rate of discount and rate of VAT on different daily using goods. Prepare a report
and compare your report with that of your friends.

c) Let's become a problem maker and problem solver yourself. Write the values of the
variable of your own and find the unknown variables.

(i) (ii) (iii)
Given: Given: Given:

M. P. = .................. M. P. = .................. M. P. = ..................

Discount= ............. % Discount= ............. % C.P. = ..................

C. P. = .................. Profit % = .................. Profit = ..................%

Profit or loss % = ? C. P. = ? Discount % = ?

(iv) (v) (vi)
Given:
S. P. = .................. Given: Given:
VAT = ..................
S.P. with VAT = ? S. P. = .................. M. P. = ..................

VAT = .................. % Discount = .............%

S.P. with VAT = ? VAT = ..................%

S.P. with VAT = ?

d) Using the above variables make a word problem of each of (i) to (vi) of your own,
related to the real life situations. Then, solve your problems and get the unknown
variables.

43Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Unit Commission and Taxation

3

3.1 Commission – Introduction

Commission is the amount of money paid to an agent for performing a business
service such as buying and selling goods, property, or collection of money.

The commission is usually a percent of the selling price. The percent is called the

commission rate.

For example,

A Publication House pays 5% commission of the total sales of books to its dealer

in a year.

A real estate company pays 2.5% commission to its agent for selling lands and

property.

In this way, we calculate the amount of commission as the given percent of the selling

price.

Amount of commission = percent of commission × total selling price.

Commission percent = Commission Amount × 100%
S.P.

Worked-out examples

Example 1: A real estate company gives 1.5 % commission to its agent on selling a

piece of land for Rs 20,00,000 and 2 % commission for additional amount

of selling price above the fixed price. If the agent sold the land for

Rs 28,50,000, how much commission did the agent receive?

Solution:

Here, the fixed selling price of the land = Rs 20,00,000

The selling price of the land = Rs 28,50,000

Now, the commission received by the agent

= 1.5 % of Rs 20,00,000 + 2 % of (Rs 28,50,000 – Rs 20,00,000)

= 1.5 × Rs 20,00,000 + 2 × Rs 8,50,000
100 100

= Rs 30,000 + Rs 17,000

= Rs 47,000

Hence, the agent received the commission of Rs 47,000.

Vedanta Excel in Mathematics - Book 9 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Commission and Taxation

Example 2: A plywood factory provides commission to its dealers on the basis of the
Solution: following monthly transactions.
Sales up to Rs 5,00,000, 4 % commission.
Sales from Rs 5,00,000 to Rs 7,50,000, 5 % commission.
Sales more than Rs 7,50,000, 6 % commission.
Calculate the commissions of the following monthly transaction of
different dealers.
(i) Rs 4,80,000 (ii) Rs 7,20,000 (iii) Rs 8,60,000

(i) Commission of the sale of Rs 4,80,000 = 4 % of Rs 4,80,000 = Rs 19,200

(ii) Commission of the sale of Rs 7,20,000

= 4 % of Rs 5,00,000 + 5 % of (Rs 7,20,000 – Rs 5,00,000)

= Rs 20,000 + Rs 11,000 = Rs 31,000

(iii) Commission of the sale of Rs 8,60,000

= 4% of Rs 5,00,000 + 5% of (Rs 7,50,000 – Rs 5,00,000) + 6% of (Rs 8,60,000 – Rs 7,50,000)

= Rs 20,000 + Rs 12,500 + Rs 6,600 = Rs 39,100

Example 3: The monthly salary of an employee in a publication house is Rs 30,000
and 2 % commission is provided when the monthly sales is more than
Rs 5,00,000. If the sales of the publication house in a month is Rs 6,75,000,
find the income of the employee in the month.

Solution:
Here, the monthly salary of the employee = Rs 30,000

Commission percent = 2%

Sales of the month = Rs 6,75,000

Now, sales for the commission = Rs 6,75,000 – 5,00,000 = Rs 1,75,000

∴ Amount of commission = 2 % of Rs 1,75,000

= 2 × Rs 1,75,000 = Rs 3,500
100

Again, the income of the employee of the month = Rs 30,000 + Rs 3,500 = Rs 33,500
Hence, the income of the employee in the month is Rs 33,500.

Example 4: The monthly salary of an employee in a hardware shop is
Solution: Rs 27,000 commission and a certain commission is given as per the
monthly sales. If the sales of a month is Rs 5,00,000 and the total income
of the employee of the month including commission is Rs 32,500, calculate
the rate of commission.

Here, the monthly salary of the employee = Rs 27,000

Sales of the month = Rs 5,00,000

Total income of the employee of the month = Rs 32,500

Now, the amount of commission of the month = Rs 32,500 – Rs 27,000 = Rs 5,500

Again, the rate of commission = Amount of commission × 100%
Sales of the month
5,500
Hence, the required rate of commission is 1.1%. = 5,00,000 × 100% = 1.1%

45Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Commission and Taxation

3.2 Bonus

Bonus is an extra amount of money that is given to an employee as a reward
for his/her good performance. It is an incentive given to an employee besides
his/her fixed salary. Bonus is calculated as a certain percent of profit and
it is decided by the board of management of any business organisation.

Example 5: The annual salary of an employee in a business company is Rs 3,02,400.
Besides, the company provides 15% bonus from its net profit at the
end of each fiscal year. If the company made a profit of Rs 60,50,000
in a fiscal year and decided to distribute bonus to its 30 employees
equally, how much income did the employee make in the year?

Solution:

Here, the annual salary of the employee = Rs 3,02,400

Profit of the company = Rs 60,50,000

The total amount of bonus = 15 % of Rs 60,50,000

= Rs 9,07,500

Now, the amount of bonus received by each employee = Rs 907500
30

= Rs 30,250

Again the annual income of the employee = Rs 3,02,400 + Rs 30,250 = Rs 3,32,650
Hence, the employee made an income of Rs 3,32,650 in the year.

EXERCISE 3.1
General section

1. a) If a real estate company gives 3% commission to it's agents. If an agent sold a piece of
land for Rs 12,50,000, how much commission did the agent get?

b) An automobile company gives 10% commission to it's agents for selling second-hand
motorbike. If an agent received Rs 16,290 by selling a scooter, at what price did the
agent sell the scooter?

c) A building owner fixed the cost of his building as Rs 27,50,400 and the price
above the fixed cost goes to an agent as his commission. If the agent sold it for
Rs 29,01,672, find his commission percent.

2. a) A publication house announced to distribute 10 % bonus equally to its 20 employees
from the net profit of Rs 18,36,000 at the end of a fiscal year, find the bonus received
by each employee.

b) A garment factory announced 20 % bonus to its 25 workers from the net profit at the
end of last fiscal year. If every worker received Rs 18,500, how much was the profit of
the factory?

c) A business company distributed bonus to its 24 employees from the net profit of
Rs 16,48,000. If every employee received Rs 8,240, what was the bonus percent?

Vedanta Excel in Mathematics - Book 9 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Commission and Taxation

Creative section
3. a) A real estate company gives 5 % commission on selling a piece of land for

Rs 10,00,000 and 7 % commission for the additional amount of selling price above
the fixed price. If the agent sold the land for Rs 12,99,000, how much commission did
he/she receive from the company?

b) An insurance company offered 2 % commission for the first 10 lakh and 2.5 % for
the rest sum of money collected from new clients by its agents. If an agent is able to
collect a sum of Rs 15,30,000 from his new clients, find his total commission.

c) A noodle factory provides commission to its sales agents on the basis of the following
weekly transactions.

Sales Rate of commissions

Upto Rs 50,000 2.5 %

From 50,000 - Rs 1,00,000 5%

Above Rs 1,00,000 6%

Now, calculate the commission of the following weekly sales.

(i) Rs 48,600 (ii) Rs 72,500 (iii) Rs 1,10,700

4. a) The monthly salary of a salesperson of a subway restaurant is Rs 21,600, and an
additional incentive of 1.5 % on the total monthly sale is provided as commission.

(i) Calculate his/her total income in a month if he/she makes a total sale of
Rs 5,80,000 in that month.

(ii) What should be his/her total sale in the next month so that he/she can receive a
total income of Rs 31,350 in the month?

b) Mr. Bibek is an online salesperson in an online shopping store. His monthly salary is
c) Rs 18,700 and 2% commission is given to him when the monthly sales is more than
5. a) 5 lakh rupees. If the sales of the store in a month is Rs 7,20,000, calculate his total
income of the month.
Mrs. Nepali draws Rs 19,800 as her monthly salary in a wholesale cosmetic shop
and a certain commission is given as per the monthly sales. If the sales of a month is
Rs 12,00,000 and her total income of the month including commission is Rs 31,800,
find the rate of commission.

A cold drink factory provides a bonus of 5% to its employees every month from their
monthly salary. If an employee gets a total income of Rs 23,625 in a month, find
his/her actual monthly salary.

b) A garment factory made a net profit of Rs 48,00,000 in the last year. The management
of the factory decided to distribute 18 % bonus from the profit to its 25 employees.

(i) Find the bonus amount received by each employee.

(ii) By what percent should the bonus be increased so that each employee can receive
Rs 38,400?

(iii) What should be the profit of the company so that it can provide Rs 40,000 to each
employee at 15 % bonus?

47Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Commission and Taxation

c) When a publication house increased its profit from 20 % to 25 % the amount of profit
increased to Rs 52,08,000. If the company decided to distribute 60 % bonus to its 30
employees equally from the increased amount of profit, how much bonus does each
employee receive?

3.3 Taxation

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.
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. Here, among these taxes, we shall discuss about the income tax.

3.4 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 limits 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 income and the tax rate in percent for
the unmarried individuals and married couples based on Nepal Income Tax Rates
2076/77 (Source: www.nbsm.com.np).

Assessed as Individuals Assessed as Couples

Particulars Taxable Income (Rs) Tax Rate Taxable Income (Rs) Tax Rate
1%
First Tax slab 4,00,000 10% 4,50,000 1%
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)
Balance Exceeding
2,00,000 36% 2,00,000 20%
(5,50,000 to 7,50,000)
(5,00,000 to 7,00,000)
12,50,000 30%
13,00,000 (7,50,000 to 20,00,000)

(7,00,000 to 20,00,000) 20,00,000 36%

20,00,000

Vedanta Excel in Mathematics - Book 9 48 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Commission and Taxation

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

(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.

Worked-out examples

Example 1: The monthly income of an unmarried individual is Rs 48,500. If 1 % social
security tax is charged upto 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

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

Now, the taxable income for the next 20% tax = Rs 5,82,000 – (Rs 4,00,000 + Rs 1,00,000)

= Rs 82,000

Then, 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 2: 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

49Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Vedanta Excel in Mathematics - Book 9

Commission and Taxation

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.

EXERCISE 3.2
General section

1. 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 32,300, how much tax should the
couple pay in a year?

Creative section - A

2. Study the income tax table given below and compute the following problems.

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%
Next 1,00,000 10% 1,00,000 10%
Next 20% (4,50,000 to 5,50,000) 20%
Next (4,00,000 to 5,00,000) 30% 2,00,000
Balance Exceeding 36% (5,50,000 to 7,50,000) 30%
2,00,000 12,50,000 36%
(7,50,000 to 20,00,000)
(5,00,000 to 7,00,000) 20,00,000

13,00,000

(7,00,000 to 20,00,000)

20,00,000

a) The monthly income of an individual is Rs 27,900 and one month's salary is provided
as Dahsain bonus. 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 Dahsain bonus 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. She draws Rs 75,000 every
month and a the Dashain bonus of one 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.

Creative section - B

3. a) Mr. Sunil Jha is a Secondary level Mathematic Teacher in a community school. His
monthly salary is Rs 38,700 and one month's salary as Dahsain bonus. 10% of his
salary is deducted to deposit in his provident fund. If his marital status is single,
calculate the annual income tax paid by him.

Vedanta Excel in Mathematics - Book 9 50 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur