Approved by the Government of Nepal, Ministry of Education, Curriculum
Development Centre, Sanothimi, Bhaktapur as an Additional Material
vedanta
EXCEL in
MATHEMATICS
10Book
Author
Hukum Pd. Dahal
Editor
Tara Bahadur Magar
vedanta
Vedanta Publication (P) Ltd.
jb] fGt klAns];g k|f= ln=
Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np
vedanta
EXCEL in
MATHEMATICS
10Book
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 Enlarged 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
Unit Set Page No.
1 1.1 Set and its types - review, 1.2 Set operations and use of Venn-diagrams, 5
1.3 Cardinality relations of sets
Unit Tax and Money Exchange 25
2 2.1 Taxation- review, 2.2 Value Added Tax (VAT)-Review,
2.3 Marked price, discount and VAT, 2.4 Currency exchange- Introduction,
2.5 Money exchange by using chain rule
Unit Compound Interest 48
3 3.1 Principal and interest - review, 3.2 Compound interest, 3.3. Interest
compounded half-yearly and quarter-yearly
Unit Population Growth and Depreciation 62
4 4.1 Population growth - Introduction, 4.2 Depreciation
Unit Mensuration (I): Area of Plane Surface 75
5 5.1 Area of triangles and quadrilaterals - review, 5.2 Perimeter and area of
plane igures - review, 5.3 Area of triangle
Unit Mensuration (II): Cylinder and Sphere 86
6 6.1 Surface area and volume of solid objects, 6.2 Area and volume of a
cylinder, 6.3 Curved surface area, total surface area and volume of a hollow
cylinder, 6.4 Half cylinder or semicylinder, 6.5 Area and volume of a sphere,
6.6 Hemisphere and great circle, 6.7 Surface area and volume of a hollow
hemispherical object, 6.8 Volume of material contain by a hollow sphere,
6.9 Area and volume of cylinder having hemispherical ends
Unit Mensuration (III): Prism and Pyramid 101
7 7.1 Prism and Pyramid, 7.2 Surface area and volume of triangular prisms,
7.3 Pyramids, 7.4 Surface area and volume of pyramids, 7.5 Cone, 7.6, Surface
area and volume of cone, 7.7 Mensuration in household activities
Unit Highest Common Factor and Lowest Common Multiple 128
8 8.1 H.C.F. of algebraic expressions, 8.2 L.C.M. of algebraic expressions
Unit Simpliϐication of Rational Expressions 133
9 9.1 Rational expressions - review, 9.2 Simpli ication of rational expressions
Unit Indices 141
10 10.1 Indices - review, 10.2 Laws of indices, 10.3 Exponential equation
Unit Surds 152
167
11 11.1 Surds - review, 11.2 Laws of surds, 11.3 Like and unlike surds, 189
11.4 Simpli ication of surds, 11.5 Rationalisation, 11.6 Conjugates, 203
Unit 11.7 Simple surds equations
209
12 Equations 237
Unit 12.1 Simultaneous equation - review, 12.2 Application of 258
simultaneous equations, 12.3 Quadratic equation- review,
13 12.4 Application of quadratic equations 277
Unit Geometry - Area of Triangles and Quadrilaterals 299
352
14 13.1 Relation between the area of triangles and quadrilaterals 377
379
Unit Geometry - Construction 380
15 14.1 Construction of a triangle whose area is equal to the area of a given
quadrilateral, 14.2 Construction of a parallelogram whose area is equal to
Unit the area of a given triangle, 14.3 Construction of a triangle whose area is
equal to the area of a given parallelogram
16
Geometry - Circle
Unit
15.1 De inition of terms related to circle, 15.2 Theorems related to arcs
17 and the angle subtended by them, 15.3 Tangent and secant, 15.4 Angles in
alternate segments
Unit
Trigonometry
18
16.1 Trigonometric ratios of an acute angle of a right angled triangle - review,
16.2 Value of trigonometric ratios of some standard angles, 16.3 Height
and distance, 16.4 Angle of elevation and angle of depression, 16.5 Area of
triangle, 16.6 Area of a triangle when its two sides and angle contained by
them are given
Statistics
17.1 Statistics - review, 17.2 Measures of central tendency,
17.3 Mean of grouped and continuous data 17.4 Median - review,
17.5 Median of grouped and continuous series, 17.6 Quartiles, 17.7 Quartiles
of grouped and continuous series, 17.8 Ogive (cumulative frequency curve),
17.9 Construction of less than ogive and more than ogive, 17.10 Use of less
than and more than ogives to ind median and quartiles
Probability
18.1 De initions of basic terms on probability - review,
18.2 Law of addition of mutually exclusive event, 18.3 Law of addition of
none-mutually exclusive events, 18.4 Multiplication law of probability for
independent events, 18.5 Probability tree diagram, 18.6 Multiplication law
of probability of dependent events
Revision and Practice Time
Answer
Syllabus
Speciϐication grid
Model Question
Unit Set
1
1.1 Set and its types - review
A set is a collection of well defined objects (or members or elements). On the basis of
the number of elements of sets, there are four types of sets:
Type of sets Introduction Examples
Empty or
null sets It does not contain any element. It is The set of natural numbers
Unit or
singleton set denoted by empty braces {} or by I less than 1.
Finite set
(Phi) N = { } or I
Infinite set
It contains only one element. The set of prime numbers
between 4 and 6.
It contains finite number of N = {1, 2, 3, 4, 5, ..., 50}
elements. E = {2, 4, 6, 8, 10, …, 100}
It contains infinite number of W = {0, 1, 2, 3, 4, 5, …}
elements. O = {1, 3, 5, 7, 9, …}
On the basis of types of elements contained by two or more sets, the types of their
relationship can be defined in the following ways:
Type of sets Introduction Examples
Equal sets
Equivalent They have exactly the A = {a, e, i, o, u} and B = {u, i, o, e, a}
sets
Overlapping same elements. ?A=B
sets
They have equal N = (1, 2, 3, 4, 5) and P = {2, 3, 5, 7, 11}
Disjoint sets
number of elements. ? N ~ P
They have at least M = {a, b, c, d, e} and N = {a, e, i, o u}
one element common. The common elements of M and N are a and
e. Therefore, M and N are overlapping sets.
They do not have any O = {1, 3, 5, 7, 9} and E = {2, 4, 6, 8, 10}
element common. Here, sets O and E are disjoint sets.
Subset, proper, and improper subsets
Let's take two sets: A = {1, 2, 3, ..., 10} and B = {1, 3, 5, 7}.
Here, every element of set B is contained by set A. Therefore, B is a subset of A. It is
denoted as B A. However, A is called the superset of B and denoted as A B.
Again, in A = {1, 2, 3, …, 10} and B = {1, 3, 5, 7}, n(A) = 10 and n(B) = 4.
Here, n(B) ≠ n(A), therefore, n(B) is called a proper subset of A.
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Furthermore, if A = {1, 2, 3, …, 10} and B = {1, 2, 3, …, 10}, n(B) = n(A).
In this case B is said to be the improper subset of A. It is denoted as B A.
Thus, a proper subset contains at least one element less than its parent set.
An improper subset contains every element of it's parent set.
Universal set
A set under the consideration from which many other subsets can be formed is known
as a universal set. For example, a set of whole numbers less than 10 can be a universal
set from which we can make many other subsets such as set of even numbers less
than 10, set of square numbers less than 10, and so on. A universal set is usually
denoted by the capital letter 'U' or by [ (Xi, a Greek alphabet).
1.2 Set operations and use of Venn-diagrams
There are four main set operations. They are:
(i) Union of sets (ii) Intersection of sets
(iii) Difference of sets (iv) Complement of sets
Sets and set operations can also be represented by using Venn-diagrams. The idea of
representation of sets and set operations in diagrams was first introduced by Euler
John Venn in the 20th century.
Let's learn the operations of sets by using Venn-diagrams.
Set operations Use of Venn-diagrams
(i) Union of sets
The union of sets A and B is the set of UU
all members that belong either to A or
to B or to both A and B. A A BB
A B = {x : x A or x B} B
(ii) Intersection of sets C
The intersection of sets A and B is the
set that contains the elements common AB ABC
to A and B.
A B = {x : x A and x B} U U
(iii) Difference of two sets A A BB
The difference of sets A and B is the set B
of the elements of A which do not belong
to B. C
A – B = {x : x A, but x B}
B – A = {x : x B, but x A} AB ABC
(iv) Complement of a set U
If A be a subset of a universal set U, U
then the complement of A is the set of
the elements of U which do not belong A BA B
to A.
A = {x : x U, but x A} A–B U B–A U
A A
B
A
A AB
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1.3 Cardinality relations of sets
The cardinal number of a set is called its cardinality. Certain relations between two
or more sets can be generalised by taking the cardinalities of the sets. Such relations
are very much useful in the application of set theory.
1. Cardinality relations of union of two sets
If A and B are two overlapping sets in a universal set U,
n(A B) = n(A) + n(B) – n(A B) U
Proof AB
Let n(A) = a, n(B) = b and n(A B) = c
Then, from the Venn-diagram, a–c c b–c
n(A B) = (a – c) + c + (b – c)
=a+b–c
= n(A) + n(B) – n(A B) Proved
Let's learn a few more relations which are derived from the cardinality relation of
union of two sets.
(i) If A and B are disjoint sets, U
n(A B ) = n(A) + n(B) A B
(ii) n(A B) = n(A) + n(B) – n(A B)
n (A) n (B)
n (only A) n (only B)
(iii) n(A B) = no(A) + no(B) + n(A B) n (A B)
(iv) n(only A) = no(A) = n(A – B) = n(A) – n(A B) n (A B)
(v) n(only B) = no(B) = n(B – A) = n(B) – n(A B)
(vi) n(A B) = n(U) – n(A B )
Worked-out examples
Example 1: If n(U) = 150, n(A) = 60, n(B) = 80 and n(A B) = 110, find
(i) n(A B) (ii) n(A B) (iii) no(A) (iv) no(B).
Draw a Venn-diagram to illustrate the above information.
Solution: n(U) = 150
Here,
n(A) = 60, n(B) = 80, n(A B) = 110 Venn-diagram
(i) Now, n(A B) = n(U) – n(A B)
U
= 150 – 110 = 40
AB
(ii) Also, n(A B) = n(A) + n(B) – n(A B) 30 30 50
= 60 + 80 – 110 = 30 40
(iii) Again, no(A) = n(A) – n(A B) = 60 – 30 = 30
(iv) And, no(B) = n(B) – n(A B) = 80 – 30 = 50
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 7 Vedanta Excel in Mathematics - Book 10
Set
Example 2: In an examination, 80 % students got grade A in Science, 90 % got grade A
in Mathematics, and every student got grade A at least in one subject. Find
the percent of students who got grade A:
(i) in both subjects,
(ii) In Mathematics only.
(iii) Show the information in a Venn-diagram.
Solution:
Let S and M denote the sets of students who got grade A in Science and Mathematics
respectively.
Here, n(S) = 80 %, n(M) = 90 % and n(U) = n(S M) = 100 % (iii) Venn-diagram70%
(i) Now, n(S M) = n(S) + n(M) – n(S M ) U
= 80 % + 90 % – 100 % = 70 % SM
(ii) Again, no(M) = n(M) – n(S M ) 10% 20%
= 90 % – 70 % = 20 %
Hence, 70 % of the students got grade A in both subjects and 20 %
of them got grade A in Mathematics only.
Example 3: In a survey of 75 SEE graduates, 42 are willing to join Management, 36 like
Science and 5 of them are not willing to join any of these faculties. Find the
number of students who are willing to join either Management or Science
faculty, and only Management faculty by using a Venn-diagram.
Solution:
Let M and S denote the sets of students who are willing to join Management and Science
respectively.
Here, n(U) = 75, n(M) = 42, n(S) = 36 and (M S) = 5
Now, n(M S) = n(U) – n(M S) U
= 75 – 5 = 70 MS
Again, n(M S) = n(M) + n(S) – n(M S) 34 8 28
= 42 + 36 – 70 =8 5
Also, no(M) = n(M) – n(M S) = 42 – 8 = 34
Hence, 8 students are willing to join either of the faculties and 34 students are willing to
join only Management faculty.
Example 4: In a group of 112 people, 56 like cold drinks only, 45 like hot drinks only and
each person likes at least one of the two drinks.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) How many people like both the drinks?
(iii) How many people like cold drinks?
(iv) How many people like hot drinks?
Vedanta Excel in Mathematics - Book 10 8 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Solution:
Let, C and H denote the set of students who like cold drinks and hot drinks respectively.
Here, n(U) = n(C H) = 112, no(C) = 56 and no(H) = 45
(i) Now, n(C H) = no(C) + no(H) + n(C H)
or, 112 = 56 + 45 + n(C H) C U
H
or, n(C H) = 11
(ii) From the Venn-diagram, 11 people like both the drinks. 56 11 45
(iii) n(C) = no(C) + n(C H)
= 56 + 11 = 67
? 67 people like cold drink.
(iv) n(H) = no(H) + n(C H)
= 45 + 11 = 56
? 56 people like hot drinks.
Example 5: During 'Visit Nepal 2020', 75 tourists of a Travel Company preferred to visit
Sauraha or Pokhara or both places. Out of them 20 preferred both places.
The ratio of the number of tourists who preferred Sauraha to those who
preferred Pokhara is 2 : 3.
(i) Find the number of tourists who preferred Sauraha.
(ii) Find the number of tourists who preferred Pokhara only.
(iii) Represent the above information in a Venn-diagram.
Solution:
Let S and P denote the sets of tourists who preferred to visit Sauraha and Pokhara respectively.
U
Here, n(U) = n(S P) = 75 and n(S P) = 20
Let, n(S) = 2x and n(P) = 3x SP
Now, n(S P) = n(S) + n(P) – n(S P) 18 20 37
or, 75 = 2x + 3x – 20
or, x = 19 = n(S) = 2 × 19 = 38
(i) The number of tourists who preferred to visit Sauraha
(ii) Also, the number of tourists who preferred to visit Pokhara = n(P) = 3 × 19 = 57
? The number of tourists who preferred to visit Pokhara only = no(P) = n(P) – n(S P)
= 57 – 20 = 37
Example 6: In a survey, it was found that the ratio of the people who liked electric bike
and petrol bike is 5 : 4 out of which 45 people liked both types of bikes,
35 liked petrol bike only, and 60 liked none of the bikes.
(i) Represent the above data in a Venn-diagram.
(ii) Find the number of people who participated in the survey.
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Solution:
Let E and P denote the sets of people who liked electric bike and petrol bike respectively.
Here, n(E P) = 45, no(P) = 35 and n(E P) = 60
Now, n(P) = no(P) + n(E P)
(i) U
= 35 + 45 = 80 P
E
Let n(E) = 5x and n(P) = 4x
Then, 4x = 80 55 45 35
60
or, x = 20
? n(E) = 5x = 5 × 20 = 100
(ii) From the Venn-diagram, n(U) = 55 + 45 + 35 + 60 = 195
Hence, 195 people participated in the survey.
Example 7: In a survey of some farmers, it is found that 48 % of them use organic fertilizers,
64 % use chemical fertilizers and 10 % of them use none of the fertilizers.
(i) Represent the above information in a Venn-diagram.
(ii) If there are 44 farmers who use both types of fertilizers, find the number of
farmers who participated in the survey.
Solution:
Let A and B be the sets of the farmers who use organic and chemical fertilizers respectively.
Here, n(U) = 100 % n(A) = 48 %, n(B) = 64 % and n(A B ) = 10 %
(
(i) Now, n(A B) = n(U) – n(A B ) U
( B
= 100 % – 10 % A
= 90 % 26% 22% 42%
Again, n (A B) = n(A) + n(B) – n(A B) 10%
= 48 % + 64 % – 90 %
= 22 %
(ii) Let the number of farmers who participated in the survey be x.
Then, 22% of x = 44
or, x = 200
Hence, 200 farmers participated in the survey.
Example 8: In a locality of hilly region of Nepal, 60% families do not have electricity
facility, 55% do not have drinking water supply facility, and 36 families do
not have both the facilities. If none of the families have both the facilities,
find:
(i) the number of families who have electricity facility only.
(ii) Represent the above information in a Venn-diagram.
Vedanta Excel in Mathematics - Book 10 10 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Solution:
Let the sets of families who do not have electricity and drinking water supply facilities are
E and W respectively.
Here, n(U) = n (E W) = 100 %, n(E) = 60 % and n(W) = 55 %
(i) Now, n(E W) = n(E) + n(W) – n(E W) (ii) U U
= 60 % + 55 % – 100 % = 15 % E WE W
Let the total number of families be x. 45% 15% 40% 108 36 96
Then, 15 % of x = 36
or, x = 240
Again, percent of families who have electricity facility only = (55 – 15) % or (100 – 60)%
= 40 %
The number of families who have electricity facility only = 40 % of 240
= 96
Example 9: In an examination, 45 % students got grade B in History only and 35 %
got grade B in Geography only. If 12 % students got grade D in both the
subjects, find:
(i) The percent of students who got grade B in both the subjects.
(ii) What percent of students got grade B in History?
(iii) Represent all the results in a Venn-diagram.
Solution:
Let the sets of examinees who got grade B in History and Geography are H and G respectively.
Here, n(U) = 100 %, no(H) = 45 %, no(G) = 35 % and n(H G) = 12 %
(
(i) Now, n(H G) = n(U) – n(H G)
(
= 100 % – 12 %
= 88 %
? Percentage of students who got grade B in both subjects
Also, n(H G) = no(H) + no(G) + n(H G) (iii) 8% U
or, n(H G) = n(H G) – no(H) – no(G) G
H
= 88 % – 45 % – 35 % 35%
45% 12%
=8%
(ii) Again, percentage of students who got grade B in History
= no(H) + n(H G)
= 45 % + 8 %
= 53 %
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Set
Example 10: In a survey of people who recently returned Nepal from foreign
Solution: employments, it was found that 60% of them were planning for dairy
farming, 50 % for poultry farming, 25% for both types of framing and
21 people were not interested in both of these farming.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of people participated in the survey.
(iii) Find the number of people who were interested in only one type of
farming.
Let the sets of people who were planning for diary and poultry farming are D and P respectively.
Here, n(U) = 100 %, n(D) = 60 %, n(P) = 50 % , n(D P) = 25 % and n(D P ) = 21
(
(i) Now, n(D P) = n(D) + n(P) – n(D P) U
= 60 % + 50 % – 25 % = 85 % DP
? n(D P ) = n(U) – n(D P) = 100% – 85% = 15% 35% 25%25%
( 15%
(ii) Let, the number of people participated in the survey be x.
Then, 15 % of x = 21
or, x = 140
Hence, 140 people participated in the survey.
(iii) Again, from the Venn-diagram,
no(D) = 35 % of 140 = 49 and no(P) = 25% of 140 = 35
The number of people who were interested in only one type of farming = 49 + 35 = 84
Example 11: In an election of a municipality, two candidates X and Y stood for the
post of a Mayor. There were 27,900 voters in the voter list and they were
allowed to cast vote for a single candidate. 11,340 people cast vote for
X, 13,500 people cast for Y and 1,260 people cast vote even for both the
candidates.
(i) Illustrate this information in a Venn-diagram.
(ii) Find the number of people who did not cast vote.
(iii) Find the number of valid votes.
Solution:
Let the sets of people who cast votes for X and Y be A and B respectively. A U
11340 B
Here, n(U) = 27,900, no(A) = 11,340, no(B) = 13,500 and 1260
n(A B) = 1,260. 13500
(i) The given Venn-diagram illustrates the above information. 1800
(ii) From the Venn-diagram,
The number of people who cast vote = 11340 + 1260 + 13500 = 26,100
? The number of people who did not cast vote = n(U) – 26,100 = 27,900 – 26100 = 1,800
(iii) Again, the number of valid votes = no(A) + no(B)
= 11,340 + 13,500 = 24,840
Vedanta Excel in Mathematics - Book 10 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Example 12: A marketing company found that, of 500 households surveyed, 140 used
Solution: neither brand A nor B soaps, 110 used only brand A soap and for every
household that used both brands of soap, 4 used only brand B soap.
(i) How many households used both brands of soaps?
(ii) How many households used only one brand of soap?
(iii) Draw a Venn-diagram to show the above information.
Let, A and B are sets of households surveyed.
Here, n(U) = 500, n(A B ) = 140 and no(A) = 110
(
Now, n(A B) = n(U) – n(A B ) = 500 – 140 = 360
(
Let, n(A B) = x, then no(B) = 4x
(i) Now, n(A B) = no(A) + no(B) + n(A B)
or, 360 = 110 + 4x + x
or, 5x = 250
or, x = 50
? 50 households used both brands of soaps. (iii) A U
B
(ii) Again, no(A) + no( B) = 110 + 4x 110 50
= 110 + 4 × 50 = 310 200
140
? 310 households used only one brand of soap.
EXERCISE 1.1
General section
1. a) In the given Venn-diagram, F and B are the sets of students U
who play football and basketball respectively. Let's find the F B
number of students in these sets operations. 12 18 10
8
(i) n(F B) (ii) n(F B) (iii) n(F B )
(
(iv) no(F) (v) n(B – A) (vi) n(U)
b) In the adjoining Venn-diagram, N and H are the sets of people U
who like Nepali and Hindi movies respectively. If n(U) = 135 N H
and n(N H) = 120, find: 85 – x x 55 – x
(i) n(N H) (ii) n(N H) (iii) no(N) (iv) no(H)
(
2. a) If n(U) = 50, n(A) = 25, n(B) = 27 and n(A B) = 10, find:
(i) n(A B) (ii) n(A B ) (iii) n(A – B) (iv) no(B)
(
b) If n(U) = 60, n(A) = 30, n(B) = 40 and n(A B) = 55, find:
(i) n(A B) (ii) n(A B ) (iii) no(A) (iv) n(only B)
(
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 13 Vedanta Excel in Mathematics - Book 10
Set
c) P and Q are the subsets of a universal set U. If n(U) = 90, n(P) = 45, n(Q) = 35
and n(P Q ) = 15, illustrate this information in a Venn-diagram and find:
(
(i) n(P Q) (ii) n(P Q) (iii) no(P) (iv) n(Q – P)
d) If n(A) = 65, n(B) = 80 and A B, find:
(i) n(A B) (ii) n(A B) (iii) n(A – B) (iv) n(B – A)
e) If no(A) = 12, no(B) = 15, n(A B ) = 11 and n(U) = 45, find:
(
(i) n(A B) (ii) n(A B) (iii) n(A) (iv) n(B)
Creative section - A
3. a) In a survey of 125 students of class 10 in a school, 65 preferred to visit Lumbini,
75 preferred to visit Pokhara, and 25 preferred both the places in their annual
excursion.
(i) Show the above information in a Venn-diagram.
(ii) Find the number of students who liked neither of two places.
(iii) Find the number of students who liked only Lumbini.
b) During 'Visit Nepal 2020', among 7,500 Chinese tourists who visited Nepal, 60% of
them had already visited Bhutan, 50 % had visited Srilanka, and 30 % have visited
both the countries.
(i) Illustrate this information in a Venn-diagram.
(ii) How many tourists have not visited any of these two countries?
c) In a group of 240 game lovers, 135 like cricket and 120 like football. By drawing a
Venn-diagram, find:
(i) how many people like both the games?
(ii) How many people like only cricket?
d) In a school, every student has to participate at least one of the activities, athletics
or music. In a class of 45 students, 27 participated in athletics and 24 participated
in music.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) How many students participated in only one activity?
4. a) In a survey of 750 students in a school, it was found that 520 students liked tea,
450 liked coffee, and 90 did not like both the drinks.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of students who liked both the drinks.
(iii) Find the number of students who did not like tea only.
b) In a survey of a community of 2,500 people, 2,100 speak Nepali, 1,800 speak
Newari, and 200 people speak neither of these two languages.
(i) Draw a Venn-diagram to represent the given information.
(ii) How many people speak only one language?
Vedanta Excel in Mathematics - Book 10 14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
c) Among 54 SEE appeared students, 18 got 'A' grade in Maths only. 25 got 'A' grade
in English only and 7 students did not get 'A' grade in these two subjects.
(i) Represent the above information in a Venn-diagram.
(ii) Find the number of students who got 'A' grade in both of these subjects.
(iii) How many students got 'A' grade in Maths?
d) In a survey of a group of people, 50 % are using cellular data and 60% are using
Wi-Fi. If 30 % of them neither use cellular data nor Wi-Fi,
(i) represent the above information in a Venn-diagram.
(ii) find the percent of people who are using cellular data as well as Wi-Fi.
e) In a survey of a community, it was found that 65% of people liked folk songs, 55%
liked modern songs, and 10% of people did not like both types of songs.
(i) Illustrate the above information in a Venn-diagram.
(ii) If 360 people liked both types of songs, how many people were surveyed?
f) In a survey of some farmers of a community, 70% of them are found cultivating
rice, 60% cultivating wheat, 20% are not cultivating both the crops, and 450
farmers are found cultivating both the crops.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the total number of farmers participated in the survey.
(iii) Find the number of farmers who are cultivating rice only.
g) Out of some students who appeared in an examination, 80% passed Mathematics,
75% passed in English, and 5% failed in both subjects. If 300 of them had passed
in both subjects, how many students had appeared in the examination. Find by
using a Venn-diagram.
h) In a survey of 15,000 students of different schools, 40 % of them were found to
have tuition class before the SEE examination. Among them, 50 % studied only
Mathematics, 30 % only Science, and 10 % studied none of these two subjects.
(i) Represent the above information in a Venn-diagram,
(ii) How many students studied Mathematics as well as Science?
i) In an examination 45 % students passed in Science only, 25 % passed in English
only and 5 % students failed in both subjects. If 200 students passed in English,
find the total number of students by using Venn-diagram.
j) In an examination, 50 % examinees got grade 'A' in Mathematics, 75 % got grade
'A' in Science, and 35 students got grade 'A' in both the subjects. If none of the
examinees got other than grade 'A' in both the subjects,
(i) represent the given information in a Venn-diagram.
(ii) Find the number of examinees who got grade A in Science only.
5. a) In a survey of a group of people, it was found that 65% of them liked comedy
movies, 63% liked action movies, 33% liked both types of movies, and 150 people
did not like both types of movies.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of people participated in the survey.
(iii) Find the number of people who liked only one type of movies.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 15 Vedanta Excel in Mathematics - Book 10
Set
b) In an examination, 80 % examinees passed in English, 70 % in Mathematics,
60 % passed in both the subjects, and 45 examinees failed in both subjects.
(i) Draw a Venn-diagram to represent the above information.
(ii) Find the number of examinees who passed only one subject.
(iii) Find the number of students who failed Mathematics.
c) In an examination, 56 % examinees failed in Science, 54 % failed in Nepali and 25
students failed in both the subjects. If none of the examinees passed in both the
subjects,
(i) find the number of examinees who passed in Science only.
(ii) Find the number of examinees who passed in Nepali only.
(iii) Represent the above information in a Venn-diagram.
6. a) 75 students in a class like tea or coffee or both. Out of them 10 like both the drinks.
The ratio of the number of students who like tea to those who like coffee is 2 : 3.
(i) Find the number of students who like tea.
(ii) Find the number of students who like coffee only.
(iii) Represent the above information in a Venn-diagram.
b) In s survey of 825 farmers, 125 of them were not using any type of fertilizer in their
vegetable farming. 150 of them were using chemical as well as organic fertilizers.
The ratio of the number of only chemical fertilizer users to that of only organic
fertilizer users is 5 : 6.
(i) Find the number of chemical fertilizer users.
(ii) How many of them were using only one type of fertilizer?
(iii) How many of them were not using organic fertilizer?
(iv) Illustrate the above information in a Venn-diagram.
c) In a group of students, the ratio of the number of students who liked music and
sports is 9 : 7. Out of which 25 liked both the activities, 20 liked music only, and
15 liked none of the activities.
(i) Represent the above information in a Venn-diagram.
(ii) Find the total number of students in the group.
d) In a survey of 80 people, it was found that 60 liked oranges only and 10 liked both
oranges and apples. The number of people who liked oranges is five times the
number of people who liked apples. By using a Venn-diagram, find the number of
people who liked apples only and who didn't like any of these fruits.
e) Out of 120 students appeared in an examination, the number of students who
passed in Mathematics only is twice the number of students who passed in Science
only. If 50 students passed in both subjects and 40 students failed in both subjects,
(i) find the number of students who passed in Mathematics.
(ii) Find the number of students who passed in Science.
(iii) Show the result in a Venn-diagram.
Vedanta Excel in Mathematics - Book 10 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Creative section - B
7. a) In an election of a municipality, A and B were two candidates for the post of the
mayor and 25,000 voters were in the voter list. Voters were supposed to cast the
vote for a single candidate. 12,000 people cast vote for A, 10,000 people cast for B,
and 1,000 people cast vote even for both the candidates
(i) Show these information in a Venn-diagram.
(ii) How many people didn't cast the vote?
(iii) How many votes were valid?
b) There are 1200 students in a school. They are allowed to cast vote either only for
X or for Y as their school prefect. 50 of them cast vote for both X and Y and 24 did
not cast the vote. The candidate Y won the election with the majority of 56 more
votes than X.
(i) How many students cast the vote?
(ii) How many valid votes are received by X candidate?
(iii) Show the result in Venn-diagram.
c) There are 650 voters in the voter list of a Trade Union Election. They are allowed
to cast vote either only for A or for B as the president of the union. 15 people did
not cast the vote and 212 voters cast vote for both the candidates. The candidate
A won the election with the majority of 325 voters.
(i) Show these information in a Venn-diagram.
(ii) How many people of the voter list cast the vote?
(iii) Find the number of valid votes.
8. a) Of the 84 farmers who attended a meeting at a Ruler Municipality, 35 volunteered
to supervise the use of harmful pesticides in vegetable farming, and 11 volunteered
both in supervising and in counselling not to use harmful pesticides. If the number
of farmers who volunteered for counselling was 1.5 times the number of farmers
who neither volunteered in supervision nor in counselling, how many of the
farmers volunteered for counselling? [Hint: n(C) = 1.5 × n(S C) ]
b) A marketing firm determined that of 200 household surveyed, 80 used neither
brand A nor brand B soap, 60 used only brand A soap, and for every household that
used both brands of soap, 3 used only brand B soap. How many of 200 households
surveyed used both brands of soap?
c) Each person who attended a company meeting was either a stockholder in the
company, an employee of the company, or both. If 62% of those who attended
the meeting were stockholders and 47% were employees, what percent were
stockholders who were not employees? Find it by drawing a Venn-diagram.
d) Due to the heavy rainfall during a few monsoon days, 140 households were
victimized throughout the country. In the first phase, Nepal government has
decided to provide the support of either food, shelter or both to a few victimized
households. 80 households got food support, 70 got shelter and 50 households got
the support of food and shelter both. The government has managed the budget of
Rs 10,000 per household for food, Rs 25,000 per household for shelter and
Rs 35,000 per household for food and shelter.
(i) Calculate the amount of budget for food only.
(ii) Calculate the amount of budget for shelter only.
(iii) Calculate the total amount of budget allocated.
(iv) How many households were remained to get support in the first phase?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 17 Vedanta Excel in Mathematics - Book 10
Set
2. Cardinality relation of union of three sets
If A, B, and C are any three overlapping sets in a universal set U,
n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)
Proof
n(A B C) = n[A (B C)]
= n(A) + n(B C) – n[A (B C)]
= n(A) + n(B) + n(C) – n(B C) – n[(A B) (A C)]
= n(A) + n(B) + n(C) – n(B C) – n[(A B) + n(A C) – n(AB C)]
= n(A) + n(B) + n(C) – n(B C) – n(A B) – n(A C) + n(A B C)
= n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)
U Proved
AB
n (A) n (B)
n (only A)
n (only B)
n (only A C) n (only A B)
C n (A B C)
n (only C) n (only B C)
n (C)
Let’s learn a few more relations from the cardinality relation of union of three sets.
(i) n(A B C) = n(A B C) – n(A) – n(B) – n(C) + n(A B) + n(B C) + n(A C)
If the cardinal value of the complement of (A B C) is given, then
(ii) n(U) = n(A B C) + n(A B C)
(iii) no(A) = n(A) – n(A B) – n(A C) + n(A B C)
(iv) no(B) = n(B) – n(A B) – n(B C) + n(A B C)
(v) no(C) = n(C) – n(A C) – n(B C) + n(A B C)
Worked-out examples
Example 1: A, B, and C are subsets of a universal set U. If n(U) = 180, n(A) = 95,
n(B) = 65, n(C) = 50, n(A B) = 25, n(B C) = 15, n(A C) = 20 and
n(A B C) = 5, illustrate this information in a Venn-diagram and find:
(i) n(only A) (ii) no(B) (iii) no(C) (iv) no(A B)
(v) n( only B C) (vi) no(A C)
(vii) n(A B C ).
Solution:
Here, n(U) = 180, n(A) = 95, n(B) = 65, n(C) = 50, n(A B) = 25,
n(B C) = 15, n(A C) = 20 and n (A B C) = 5
Vedanta Excel in Mathematics - Book 10 18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Venn-diagram Hints:
AB U At first, insert n(A B C) = 5,
25
55 20 30 Then no(A B) = n(only A B) = 25 - 5 = 20,
15 5 10 Then no(B C) = n (only B C) = 15 - 5 = 10,
Then no(A C) = n (only A C) = 20 - 5 = 15,
20 Then no(A) = n (only A) = 95 – 20 – 5 – 15 = 55,
C Then no(B) = n (only B) = 65 – 20 – 5 – 10 = 30,
Then no(C) = n (only C) = 50 – 15 – 5 – 10 = 20
From the Venn-diagram, From rules
n(only A) = 55 no(A) = n(A) – n(A B) – n(A C) + n(A B C)
no(B) = n(B) – n(A B) – n(B C) + n(A B C)
no(B) = 30 no(C) = n(C) – n(A C) – n(B C) + n(A B C)
no(A B) = n(A B) – n(A B C)
no(C) = 20 no(B C) = n(B C) – n(A B C)
no(A C) = n(A C) – n(A B C)
no(A B) = 20
no(B C) = 10
no(A C) = 5
n(A B C) = n(U) – n((A B C)]
= 180 – (55 + 30 + 20 + 5 + 20 + 10 + 15) = 180 – 155 = 25
Alternative process
n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)
= 95 + 65 + 50 – 25 – 15 – 20 + 5 = 215 – 60 = 155
? n(A B C) = n(U) – n(A B C) = 180 – 155 = 25
Example 2: In a survey, 650 students were asked what future careers they prefer. 235
of them replied they prefer to be a Doctor, 210 a Pilot, 175 an Educator, 100
Doctor as well as pilot, 70 Doctor as well as Educator, 80 pilot as well as
Educator and 40 replied they prefer all three careers.
(i) Draw a Venn-diagram to illustrate the above information
(ii) Find how students do not prefer any of the three careers.
Solution:
Let A, B, and C be the sets of students who prefer to be a Doctor, a Pilot and an Educator
respectively.
Here, n(U) = 650, n(A) = 235, n(B) = 210, n(C) = 175
n(A B) = 100, n(A C) = 70, n(B C)= 80 and n(A B C) = 40
(i) Illustration in a Venn diagram AB U
105 60 70
30 40 40
65
C 240
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 19 Vedanta Excel in Mathematics - Book 10
Set
(ii) n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(A C) – n(B C) + n(A B C)
= 235 + 210 + 175 – 100 – 70 – 80 + 40 = 410
? n (A B) C)= n(U) – n(A B C) = 650 – 410 = 240
Hence, 240 students do not prefer any of the three careers.
Example 3: In a survey of a group of people, it was found that 60 of them have business,
45 have government jobs, 125 have farming, 27 have business only, 15 have
government jobs only, 10 have business and government jobs only, 5 have
government jobs and farming only.
(i) Draw a Venn-diagram to illustrate the above information
(ii) Find how many people have all three professions.
(iii) How many people were there in the survey?
Solution:
Let B, G, and F be the sets of people who have business, government jobs, and farming
respectively.
Here, n(B) = 60, n(G) = 45, n(F) = 125, no(B) = 27, no(G) = 15,
no(B G) = 10, no(G F) = 5
Let the number of people who have all three professions be x.
(i) B U (ii) From the Venn-diagram, n(G) = 10 + x + 5 + 15
G
27 10 15 or, 45 = x + 30
x5 or, x = 15
F
? The number of people who have all three professions = n(B G F) = x = 15
(iii) Again, the number of people who have business and farming only = no(B F)
= n(B)– (27 + 10 + x)
= 60 – 37 – 15 = 8
Also, the number of people who have farming only = no(F) = n(F) – (5 + x + 8)
= 125 – 13 – 15 = 97
? The total number of people, n(U) = n(B G F) = 27 + 10 + 15 + 15 + 5 + 97 + 8 = 177
EXERCISE 1.2
General section
1. a) A, B, and C are the subsets of a universal set U. Draw a Venn-diagram and insert
the cardinality of the following information.
n(U) = 50, n(A) = 18, n(B) = 21, n(C) = 20, n(A B) = 7, n(B C) = 5,
n (A C) = 8 and n(A B C) = 3.
Vedanta Excel in Mathematics - Book 10 20 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
b) In the given Venn-diagram, A, B, and C represent the sets of tourists who visited
Bardiya, Khaptad, and Rara respectively. If n(U) = 200, find
(i) n(A), (ii) n(B), (iii) n(C) A BU
(iv) n(A B C) (v) n(A B C ) (vi) n(A B C) 38 25 32
(vii) n(only A) 20 5 15
(x) n(A B) (viii) no(B) (ix) no(C)
(xi) n(B C) (xii) n(A C) 40
(xiii) no(A B) (xiv) no(B C) (xv) no(A C) n(C)=60% C
C U
c) The given Venn-diagram shows the set of students
C, F, and B who play cricket, football, and basketball Fn(F)=30%
respectively. If C F B = 100%, n(C F) = 10%,
n(F B) = 10%, and n(C B) = 15%, find the
percentage of the students who play all three types B n(B)=40%
of games.
Creative section - A
2. a) If n(A) = 50, n(B) = 48, n(C) = 42, n(A B) = 12, n(B C) = 8, n(C A) = 9,
n(A B C) = 5, and n(U) = 125, find the value of n(A B C) and
n(A B C ). Present the above information in a Venn-diagram.
b) Given that, n(U) = 150, n(P) = 48, n(Q) = 56, n(R) = 44, no(P) = 17, no(Q) = 20,
no(P Q) = 10, no(Q R) = 8. Find
(i) n(P Q R) (ii) no(P R) (iii) no(R) (iv) n(P Q R)
Illustrate the above information in a Venn-diagram.
3. a) The survey of a group of students showed that 50 liked tea, 40 liked coffee, 35 liked
milk, 20 liked coffee as well as tea, 18 liked tea as well as milk, 12 liked coffee as
well as milk, and 7 liked all three. If every student liked at least one drink, how many
students were asked this question? Solve it by drawing a Venn-diagram.
b) In an examination, out of 180 students, 65 succeeded in English, 75 in Mathematics,
60 in Nepali, 30 in English as well as in Mathematics, 25 in English as well as
in Nepali, 40 in Nepali as well as in Mathematics, and 10 succeeded in all three
subjects.
(i) Draw a Venn-diagram showing the above information.
(ii) Find also the number of students who didn't succeed in all three subjects.
c) Of the total candidates in an examination, 45% students passed in Maths, 55% in
Science and 60% in Computer. 20% students passed in both Maths and Science, 25%
in Science and Computer, and 20% in Computer and Maths. If every student passed
at least one subject,
(i) draw a Venn-diagram to show the above information.
(ii) Calculate the percent of students who passed in all the three subjects.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 21 Vedanta Excel in Mathematics - Book 10
Set
4. a) In a survey of 900 tourists who arrived in Nepal during 'Visit Nepal 2020', 450
preferred to go trekking, 300 preferred rafting, 400 preferred forest safari, and 100
preferred none of these activities. If 200 preferred trekking and rafting, 110 preferred
trekking and safari, 100 preferred rafting and safari,
(i) how many of them preferred all of these activities?
(ii) Show the above information in a Venn-diagram.
b) In a survey, people were asked what types of movies they like. It was found that,
75 liked Nepali movies, 60 liked English, 40 liked Hindi, 35 liked Nepali and English,
30 liked Nepali and Hindi, 20 liked Hindi and English, 10 liked all three, and 25
people were found not interested in any types of movies.
(i) How many people did not like only Hindi films?
(ii) How many people did not like only Nepali or Hindi films?
c) In a group of students, 20 study Economics, 18 study History, 21 study Science, 7
study Economics only, 10 study Science only, 6 study Economics and Science only
and 3 study Science and History only.
(i) Represent the above information in Venn-diagram
(ii) How many students study all the subjects?
(iii) How many students are there altogether?
d) In a survey of a group of people, it was found that 30 of them have business, 35 have
services, 25 have farming, 12 have business only, 15 have services only, 10 have
business and services only and 6 have services and farming only.
(i) Draw a Venn-diagram to illustrate the above information
(ii) Find how many people have all three occupations?
(iii) How many people were in the survey?
Creative section - B
5. a) Each student in a class of 32 plays at least one game: cricket, football, or basketball.
20 play cricket, 18 play basketball and 25 play football. 9 play cricket and basketball,
13 play football and basketball, and 5 play all three. Find the number of students
who play:
(i) cricket and football
(ii) cricket and football but not basket ball
(iii) Show the information in a Venn-diagram.
b) In a competition, a school awarded medals in different categories. 40 medals in
sports, 15 medals in dance, and 21 medals in music. If the total of 55 students
got medals and only 6 students got medals in all the three categories, how many
students received medals in exactly two of these categories?
Hint: no(S D) = n(S D – 6, no(S M) = n(S M) – 6, no(D M) = n(D M) – 6
Then find no(S D) + no(S M) + no(D M)
Vedanta Excel in Mathematics - Book 10 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Set
Project Work
6. Make the groups of your friends and conduct a survey inside your classroom: how many
of your friends like Cricket, Football, and Cricket as well as Football. Then, tabulate the
data as shown in the table given below:
Total No. of No. of students No. of students No. of students who like
students who like cricket who like football cricket as well as football
a) Find the number of students in the following cases by using the cardinality relations
of two sets.
(i) students who like only cricket (ii) students who like only football
(iii) students who like cricket or football (iv) students who like cricket and football
(v) students who like neither cricket nor football.
b) Compare these results with the surveyed data.
c) Draw a Venn-diagram to show the collected data.
7. Conduct a survey inside your classroom and collect data about how many of your friends
like tea, coffee, milk, tea and coffee, tea and milk, coffee and milk and tea, coffee as well
as milk. Then tabulate the data and compute the following numbers by using cardinality
relation of three sets.
a) Number of friends who like tea, coffee and milk.
b) Number of friends who do not like any of these three drinks.
c) Number of friends who like only
(i) tea and coffee (ii) tea and milk (iii) coffee and milk.
d) Number of friends who like only (i) tea (ii) coffee (iii) milk
e) Show your data in a Venn-diagram
Objective Questions
Tick the correct alternatives.
1. If A is a subset of U, which one of the following relation is always true?
(A) A U = U (B) A U = U (C) A A = I (D) A I = I
2. If A and B are two overlapping sets, then which of the following relations is incorrect?
(A) n (A B) = n (A) + n (B) – n (A B) (B) n (A B) = nn0(A(A) )++nn0(0B()B) + n (A B)
(C) n (AB) = n0 (A) + n (B) – n (A B) (D) n (A B) =
3. If A is a subset of B such that n (A) = 10 and n (B) = 18 then n (only B) is equal to
(A) 28 (B) 18 (C) 10 (D) 8
4. If n (A) = 11 and n (B) = 18, what is the possible maximum value of n (AB)?
(A) 0 (B) 11 (C) 18 (D) 29
5. Out of 30 students of a class, 15 like to play volleyball, 20 like to play basketball and each
student like to play at least one of the game. How many students like to play volleyball and
basketball both?
(A) 35 (B) 20 (C) 15 (D) 5
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 23 Vedanta Excel in Mathematics - Book 10
Set
6. Out of 77 districts of Nepal, 27 districts have shared their boarder with India, 15 districts
have shared their boarder with China and 37 districts have not shared their boarder with
India and China both. How many districts have shared their boarder with China but not
with India?
(A) 15 (B) 13 (C) 25 (D) 38
7. In a group of 25 people, 10 favored tea only, 7 favored coffee only, and 3 favored none of
the drinks. What percent of people favored both the drinks?
(A) 35% (B) 25% (C) 20% (D) 15%
8. In a school if 9 teachers teach in basic level only, 6 teachers teach in secondary level only,
and 3 teachers teach in both basic and secondary levels, the ratio of the number of teachers
who teach in basic level to the number of teachers who teach in secondary level is:
(A) 3:2 (B) 4:3 (C) 3:1 (D) 2:1
9. In a survey it was found that 30 people enjoy Nepali movie, 10 people do not enjoy Nepali
movies and 15 do not enjoy Hindi movie. How many people enjoy Hindi movie?
(A) 25 (B) 20 (C) 15 (D) 40
10. In a survey of a community, 55% of the people like yoga, 65% like jugging and 10 % like
neither yoga nor jugging. What percent of people like only one?
(A) 30% (B) 25% (C) 35% (D) 60%
11. In a group of 60 students, 20 passed in both Mathematics and Science and 30 passed in
only one subject. How many students failed in both the subjects?
(A) 40 (B) 30 (C) 10 (D) 50
12. In a group of children, 25 like sweet and 15 like balloon. If the number of children who like
sweet only is twice the number of children who like balloon only, how many children like
sweet as well as balloon?
(A) 5 (B) 10 (C) 20 (D) 35
13. Some women were asked what they would like: sari or kurtha or lehenga. If 15 women
responded as only one, 10 responded as only two, 3 women responded as all these three
and 2 responded as none of these cloths, how many women were asked the questions?
(A) 10 (B) 15 (C) 25 (D) 30
14. Which of the following Venn-diagram represents the interrelationship among fruits (F),
apple (A), and banana (B)?
F F F FA B
AB BA AB
(A) (B) (C) (D)
15. Which of the following Venn-diagram represents the interrelationship among cloths (C),
red (R) and flower (F)?
CR F C R F
RF FC CR
(A) (B) (C) (D)
Vedanta Excel in Mathematics - Book 10 24 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Unit Tax and Money Exchange
2
2.1 Taxation - review
A tax is a compulsory financial charge or other levy imposed upon a tax payer by a
state. Let's discuss the following questions and learn about different types of taxes.
(i) Suppose the annual income of an individual is Rs 4,60,000. How much social
security tax does he/she pay to the state at the rate of 1%?
(ii) The annual income of a married couple is Rs 5,20,000. If 10% income tax is
levied on the annual income of Rs 4,50,000 to Rs 5,50,000, how much income
tax does the couple pay to the state?
(iii) The annual income of a Private School is Rs 1,50,00,000. How much tax should
be paid by the school in a year at the rate of 1% Education Service Charge?
(iv) When you purchased a mobile set you paid Rs 6,780 including Value Added Tax
(VAT) at the rate of 13%. How much sales tax did you pay as VAT?
These are some of the various cases of taxation. The government does not have its
own money. Its receipts come from individual income taxes, corporate income taxes,
estate and gift taxes, education service taxes, excise tax, and so on. The government
uses these taxes to support and develop transportation, communication, education,
health services, law and order, culture, civil service, trade and industry, and so on.
The Inland Revenue Department (IRD) under the Ministry of Finance of Government
of Nepal is responsible for the administration of Value Added Tax, income tax and
excise duty.
2.2 Value Added Tax (VAT) - review Deurali Electronics
Tahachal-13, Kathmandu
Ph. No. 9841240225
Let's study the adjoining bill given by a
shopkeeper to a customer.
(i) How much is the selling price of the Mobile 6,000 6000
mobile without VAT?
(ii) What is the VAT rate and amount of VAT Rs. In words: ........................................... Total 6000
in the bill? ...S..i.x...t.h..o..u..s..a..n..d...o..n..l..y...................................
............%
(iii) How much price should the customer ................................................................
pay with VAT? Taxable Amount
Ownership Tax
Sub Total 6000
13 % VAT 780
Grand Total 6780
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 25 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Value Added Tax (VAT) is a sales tax. The VAT rate is given in percent and the rate is
decided by the concerned authority of the government. The VAT rate may vary from
country to country. Even in the same country, it may be changed from time to time.
For example, when VAT was introduced in Nepal for the first time on 16 November,
1997, the VAT rate was 10% and it is now 13%.
There are certain goods and services for which VAT is exempted. For example,
educational items, social welfare services, etc. are VAT-free goods and services.
VAT is levied on the actual selling price.
VAT amount = Rate of VAT (in %) × Selling price.
S.P. with VAT = S.P. + VAT % of S.P. or (100 + VAT)% of S.P.
S.P. without VAT= S.P. with VAT
(100 + VAT)%
Worked-out examples
Example 1: Calculate the VAT amount on the selling price of Rs 2,400 at the rate of 13%.
Solution:
Here, the amount of VAT = VAT% of S.P.
= 13% VAT of Rs 2,400 = Rs 312
Hence, the required VAT amount is Rs 312.
Example 2: The selling price of a mobile is Rs 7,500. How much should a customer pay
for it with 13% VAT?
Solution:
Here, the selling price of mobile = Rs 7,500 and VAT rate = 13%
Now, the cost of the mobile with VAT = S.P. + VAT% of S.P. Direct process
= Rs 7,500 + 13% of Rs 7,500 S.P. with VAT=113% of S.P.
= Rs 7,500 + 13 × Rs 7,500 = 113 × Rs 7,500
100 100
= Rs 8,475
= Rs 8,475
Hence, the customer should pay Rs 8,475 for the mobile.
Example 3: A family had dinner in a restaurant. If the cost of the dinner was Rs 1,500,
how much did the family pay with 10% service charge and 13% VAT?
Solution:
Here, cost of the dinner = Rs 1,500
Service charge = 10%
VAT rate = 13%
Now, the cost of the dinner with service charge = S.P. + 10% of S.P.
= Rs 1,500 + 10% of Rs 1,500 = Rs 1650
Again, the cost of the dinner with service charge and VAT = Rs 1650 + 13% of Rs 1650
= Rs 1,864.50
Therefore, the family should paid Rs 1,864.50.
Vedanta Excel in Mathematics - Book 10 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Example 4: Mrs. Shrestha purchased a watch for Rs 7,360 with 15% VAT. Find the cost of
the watch without VAT?
Solution: Direct process
Here, the cost of the watch with 15% VAT = Rs 7,360
Let the cost of the watch without VAT = Rs x Cost of watch without VAT
Now, x + 15% of x = Rs 7,360
= cost of watch with VAT
(100 + VAT)%
or, 115x = Rs 7,360
100 = Rs 7,360 = Rs 6,400
115%
or, x = Rs 6,400
Hence, the cost of the watch without VAT is Rs 6,400. = Rs 6,400
Example 5: If the cost of an article with VAT is Rs 4,746 and without VAT is Rs 4,200, find
the VAT rate.
Solution:
Here, the cost of the article with VAT = Rs 4,746
And, the cost of the article without VAT = Rs 4,200
Now, the amount of VAT = Rs 4,746 – Rs 4,200 = Rs 546
? Rate of VAT = VAT amount × 100%
cost without VAT
Rs 546
= Rs 4200 × 100% = Rs 13%
Hence, the required VAT rate is 13%.
Example 6: A trader bought an electric oven for Rs 16,000 and sold at a profit of 20% to
a customer with 13% VAT. How much did the customer pay for the oven?
Solution: Direct process
Here, C.P. of the oven = Rs 16,000. S.P. of the oven = 120% of C.P.
Profit percent = 20% = 120 × Rs 16,000
100
S.P. of the oven = Rs 16,000 + 20 × Rs 16,000
= Rs 19,200 100 = Rs 19,200
Now, S.P. of the oven with 13% VAT= S.P. + 13% of S.P. Direct process
= Rs 19,200 + 13 × Rs 19,200 S.P. with VAT = 113% of S.P.
= Rs 21,696 100
= 113 × Rs 19,200
100
Hence, the customer paid Rs 21,696 for the oven. = Rs 21,696
Example 7: A shopkeeper bought a television set for Rs 24,000 and fixed its price to
make 15% profit. If the television was sold for Rs 31,188 with VAT, calculate
the rate of VAT. Direct process
Solution: S.P. of television
Here, S.P. for the shopkeeper = Rs 24,000 + 15% of Rs 24,000 = 115% of C.P.
= Rs 24,000 + Rs 3,600 = 115 × Rs 24,000
100
= Rs 27,600 = Rs 27,600
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 27 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Also, S.P. with VAT = Rs 31,188
? VAT amount = Rs 31,188 – Rs 27,600 = Rs 3,588
Again, rate of VAT = VAT amount × 100%
S.P.
= Rs 3,588 × 100% = 13%
Rs 27,600
Hence, the required rate of VAT is 13%.
Example 8: Sunayana purchased a fancy item for Rs 4,800 and sold it for Rs 6,780 with
13% VAT. Find her profit or loss percent.
Solution:
Here, C.P. of the fancy item = Rs 4,800
S.P of the item with VAT = Rs 6,780
Rate of VAT = 13%
Let the S.P. of the item without VAT be Rs x. Direct process
? x + 13% of x = Rs 6,780 S.P. without VAT = S.P. with VAT
113%
13x
or, x+ 100 = Rs 6,780 = Rs 6,780
113%
or, 113x = Rs 6,780
100 = Rs 6,000
or, x = Rs 6,000
Thus, S.P. without VAT = Rs 6,000
Now, profit = S.P. – C.P. Direct process
= Rs 6,000 – 4,800 = Rs 1,200 Profit percent = S. P. – C. P. × 100%
C. P.
Also, profit percent = Actual profit × 100% = Rs 6,000 – Rs 4,800 × 100%
C.P. Rs 4,800
= Rs 1,200 × 100% = 25% = Rs 1,200 × 100%
Rs 4,800 Rs 4,800
Hence, the required profit percent is 25%. = Rs 25%
Example 9: Mr. Pariyar bought a camera and sold it at a profit of 20% to Mr. Thakali. If
Mr. Thakali paid Rs 34,500 for it with 15% VAT, at what price did Mr. pariyar
buy the camera?
Solution:
Here, S.P. of the camera with VAT = Rs 34,500
Rate of VAT = 15%
Let, S.P. of the camera without VAT be Rs x. Direct process
? x + 15% of x = Rs 34,500 S.P. of the camera without VAT
or, 115x = Rs 34,500 = S.P. with VAT = Rs 34,500
100 115% 115%
or, x = Rs 30,000 = Rs 30,000
Vedanta Excel in Mathematics - Book 10 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Thus, S.P. of the camera without VAT = Rs 30,000.
Also, the profit of Mr. Pariyar = 20% Direct process
And, C.P. of the camera for Mr. Pariyar = ? C.P. of the camera = S.P. without VAT
120%
Now, C.P. + 20% of C.P. = Rs 30,000 = Rs 30,000
or, 120%
or, 120 C.P. = Rs 30,000
100 = Rs 25,000
C.P. = Rs 25,000
Hence, Mr. Pariyar bought the camera for Rs 25,000.
Example 10: A wholesaler sold a Honda generator for Rs 2,40,000 to a retailer. The
retailer spent Rs 3,000 for transportation and Rs 2,000 for the local tax.
If the retailer sold it at a profit of 12% to a customer, how much did the
customer pay for it with 13% VAT?
Solution:
Here, C.P. for the retailer = Rs 2,40,000
Also, C.P. for the retailer including transportation cost and the local tax
= Rs 2,40,000 + Rs 3,000 + Rs 2,000 = Rs 2,45,000
Again, S.P. for the retailer = C.P. + 12% of C.P.
= Rs 2,45,000 + 12 × Rs 2,45,000 = Rs 2,74,400
100
Now,cost of the generator for the customer = Rs 2,74,400 + 13% of 2,74,400 = Rs 3,10,072
Hence, the customer paid Rs 3,10,072 for the generator.
Example 11: A dealer purchased some building materials for Rs 3,60,000. She sold them
Solution: at 5% profit to a supplier. The supplier spent Rs 7,500 for transportation
and Rs 4,500 for the local tax and sold at a profit of 10% to a customer.
How much did the customer pay for the materials with 15% VAT?
Here, S.P. for the wholesaler at 5% profit = Rs 3,60,000 + 5% of Rs 3,60,000
= Rs 3,78,000
C.P. for the supplier = Rs 3,78,000
C.P. for the supplier with transportation cost and local tax
= Rs 3,78,000 + Rs 7,500 + Rs 4,500
= Rs 3,90,000
S.P. for the supplier at 10% profit = Rs 3,90,000 + 10% of Rs 3,90,000
= Rs 4,29,000
Now, C.P. for the customer with 15% VAT = Rs 4,29,000 + 15% of Rs 4,29,000
= Rs 4,93,350
Therefore, the customer paid Rs 4,93,350 for the building materials.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 29 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
EXERCISE 2.1
General Section
1. a) If R% be the rate of VAT and Rs x be the selling price, write the formula to find
amount of VAT.
b) If Rs x be the selling price and Rs y be the amount of VAT, write the formula to find
VAT percent.
c) If Rs P be the selling price and R% be the VAT rate, write the formula to find
selling price with VAT.
2. a) Find the selling price with VAT from the table given below:
No. S.P. without VAT VAT% S.P. with VAT
(i) Rs 900 13% .............................
(ii) Rs 2,700 13% .............................
(iii) Rs 17,100 15% .............................
b) Find the selling price without VAT from the table given below.
No. S.P. with VAT VAT% S.P. without VAT
(i) Rs 1,568 12% .............................
(ii) Rs 6,215 13% .............................
(iii) Rs 24,840 15% .............................
3. a) The cost of a fan is Rs 1,600. If Mrs. Khadka purchased it with 13% VAT, how
much did she pay for it?
b) A family had dinner in a restaurant. If the cost of the dinner was Rs 2,100, how
much did the family pay with 10% service charge and 13% VAT?
4. a) The cost of a rice cooker with 13% VAT is Rs 4,068. Find its cost without VAT.
b) Mr. Magar purchased a mobile set for Rs 11,155 with 15% VAT exclusive. Find the
cost of the mobile without VAT and also calculate the VAT amount.
c) Mrs. Maharjan bought a refrigerator for Rs 26,442 with 13% VAT. How much did
she pay for the VAT?
5. a) If the cost of a watch with VAT is Rs 5,130 and without VAT is Rs 4,500, find the
VAT rate.
b) Malvika purchased a fancy bag for Rs 7,119 with VAT. If its cost without VAT is
Rs 6,300, calculate the rate of VAT.
Creative Section - A
6. a) A retailer purchased an electric bicycle for Rs 35,000 and sold at a profit of 16% to
a customer. How much did the customer pay for it with 13% VAT?
b) A dealer bought digital watches at Rs 5,500 per piece and fixed the price of each
watch to make 20% profit. How much should a customer pay for it with 13% VAT?
7. a) A trader purchased a laptop for Rs 45,000 and marked its price to make 24% profit.
If he/she sold it for Rs 63,054 with VAT, calculate the rate of VAT.
Vedanta Excel in Mathematics - Book 10 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
b) A wholesaler purchased a few number of vacuum cleaners for Rs 14,500 each
and sold to customers at 20% profit. If a customer purchased a cleaner for
Rs 19,836 with VAT, find the VAT percent.
8. a) A shopkeeper bought a few number of pairs of shoes at Rs 3,000 per pair. At what
profit percent did she/he sell each pair of shoes, if a customer paid Rs 4,407 with
13% VAT?
b) A trader bought some electric blenders for Rs 4,000 per piece. He sold each blender
to customers for Rs 4,332 with 14% VAT. Find his profit or loss percent.
9. a) A shopkeeper bought a sunglasses and sold it at a profit of 20% to a customer. If
the customer paid Rs 21,696 for it with 13% VAT, at what price did the shopkeeper
buy it?
b) A retailer sold an electric item to a customer at a loss of 10%. The customer
purchased it for Rs 25,425 including 13% VAT. Calculate the cost price of the
electric item to the retailer.
Creative Section - B
10. a) A retailer purchased construction materials for Rs 2,40,000 from a dealer. She
spent Rs 3,500 for transportation and Rs 1,500 for the local tax. If she sold it at a
profit of Rs 24,500 to a customer, how much did the customer pay for it with 13%
VAT?
b) A supplier purchased a photocopy machine for Rs 2,00,000 and spent Rs 1,500 for
transportation and Rs 500 for local tax. He sold it to a customer at 10% profit. At
what price did the customer purchase the machine with 13% VAT?
c) A wholesaler purchased a washing machine for Rs 60,000 and sold it to a retailer
at 10% profit. The retailer spent Rs 2,400 for transportation and Rs 1,600 for local
tax. Then she sold it to a customer at 12% profit. How much did the customer pay
for it with 13% VAT?
2.3 Marked Price, Discount and VAT
The price which is tagged on an article is called its marked price (M.P.). When a
shopkeeper reduces the marked price of any article and sells it to customers, the
reduced amount is called discount. Discount is usually given as a certain percent of
marked price.
Thus,
Discount amount = discount percent of M.P.
Selling price (S.P.) = M.P. – Discount amount or M.P. – D% of M.P.
Discount percent = Discount amount × 100 %
M.P.
In the case of finding selling price with VAT, at first, discount is to be deducted from
the given marked price (M.P.) to find selling price (S.P.). Then, VAT is levied on the
selling price.
Thus,
S.P. = M.P. – D% of M.P.
And, S.P. with VAT = S.P. + VAT% of S.P.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 31 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Worked-out examples
Example 1: The marked price of a laptop is Rs 1,20,000. What is the price of the laptop
after 10 % discount and 13 % VAT included in its price?
Solution:
Here, M.P. of the laptop = Rs 1,20,000
Discount percent (D%) = 10 %
VAT = 13 %
Now, Direct process
S.P. = (100 – 10)% of M.P.
S.P. of the laptop = M.P. – D% of M.P.
= 90% of Rs 1,20,000
= Rs 1,20,000 – 10% of Rs 1,20,000 = Rs 1,08,000
S.P. with VAT = 113% of S. P.
= Rs 1,08,000 = 113% of Rs 1,08,000
= Rs 1,22,040
Again, S.P. with VAT = S.P. + VAT % of S.P.
= Rs 1,08,000 + 13 % of Rs 1,08,000
= Rs 1,22,040
Hence, the required price of the laptop is Rs 1,22,040.
Example 2: A shopkeeper allows a discount of 20 % and still makes a profit of 20 % by
selling a fan. If the marked price of the fan is Rs 1,800, find his profit.
Solution:
Here, M.P. of the fan = Rs 1,800 Direct process
Discount percent (D%) = 20 % S.P. = (100 – 20)% of M.P.
Profit percent (P%) = 20 % = 80 u Rs 1,800
100
Now, S.P. of the fan = M.P. – D% of M.P. = Rs 1,440
= Rs 1,800 – 20% of Rs 1,800
= Rs 1,440
Again, at 20% of profit, Then, profit = 20% of C.P.
120% of C.P. = S.P. = 20 × Rs 1,200
100
or, 120 × C.P. = Rs 1,440
or, 100 C.P. = Rs 1,200 = Rs 240
Hence, his profit is Rs 240.
Example 3: A retailer fixed the price of his/her articles 25% above the cost price. If
she sold an article allowing 10% discount, find her profit percent.
Solution: Direct process:
Let the C.P. of the article be Rs x. M.P. = 125 % of C.P.
Then, the M.P. of the article = Rs x + 25% of Rs x 125
= 100 u Rs x
= Rs x + 25 u Rs x = Rs 5x = Rs 5x
100 4 4
Vedanta Excel in Mathematics - Book 10 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Now, S.P. of the article = M.P. – discount % of M.P. Direct process:
= Rs 5x – 10 u Rs 5x S.P. = 90 % of M.P.
4 100 4
= 90 Rs 5x
= Rs 9x 100 u 4
8
= Rs 9x
Again, profit = S.P. – C.P. 8
= Rs 9x – x = x
8 8
? Profit percent = Actual profit u 100% = x u 100% = 12.5%
C.P. 8×x
Hence, her profit percent is 12.5%.
Alternative process
Let the C.P. of the article be Rs 100
Then, the M.P. of the article = Rs 100 + Rs 25 = Rs 125
Now, the S.P. of the article = M.P. – discount % of M.P. = Rs 125 – 10 u Rs 125 = Rs 112.50
100
Again, profit = S.P. – C.P. = Rs 112.50 – Rs 100 = Rs 12.50
?Profit percent = Actual profit u 100% = 12.50 u 100% = 12.5%
C.P. 100
Example 4: A dealer bought a refrigerator for Rs 11,515. After allowing a discount
Solution:
of 16% on its marked price, she gains 20%. Find the marked price of the
refrigerator.
Here, C.P. of the refrigerator= Rs 11,515
Profit = 20%
? S.P. of the refrigerator = (100 + 20)% of C.P. = 120 × Rs 11,515 = Rs 13,818
100
Again, discount = 16%
? (100 – 16)% of M.P. = S.P.
or, 84 × M.P. = Rs 13,818
100
or, M.P. = Rs 16,450
Therefore, the marked price of the refrigerator is Rs 16,450.
Example 5: Find the single discount which is equivalent to two successive discounts of
15% and 10%.
Solution:
Let, M.P. = x
1st discount = 15%
? 1st S.P. = (100 – 15)% of M.P. = 85 × x = Rs 17x
Again, 100 20
2nd discount = 10%
?
2nd S.P. = (100 – 10)% of 1st S.P. = 90 × 17x = Rs 153x
100 20 200
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 33 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Now, single discount = M.P. – 2nd S.P.
= Rs x – Rs 153x = Rs 47x
200 200
Single discount percent = single discount × 100% = 47x × 100% = 23.5%
M.P. 200x
Hence, the required single discount is 23.5%.
Example 6: The marked price of a mobile is 40% above the cost price. When it is sold
Solution: allowing 30% discount on the marked price, there is a loss of Rs 240. What
is the marked price of the mobile?
Let the C.P. of the mobile be Rs x and M.P. is 40% above C.P.
Then, M.P. of the mobile = 140% of C.P. = 140 u Rs x = Rs 7x
100 5
Now, S.P. of the mobile = M.P. – discount % of M.P.
Direct process:
7x 30 7x
= Rs 5 – 100 u Rs 5 S.P. = (100 – 30)% of M.P.
= Rs 49x = 70 × Rs 7x = 49x
50 100 5 5
Again, loss = C.P. – S.P.
or, Rs 240 =x – 49x
50
x
or, Rs 240 = 50
or, x = Rs 12,000
Now, M.P. = Rs 7x = Rs 7 × 12,000 = Rs 16,800
5 5
Therefore, the marked price of the mobile is Rs 16,800.
Example 7: After allowing 10% discount on the marked price of a radio, 15% VAT is
charged on it. Now, its price becomes Rs 6,210. Calculate the amount of
discount.
Solution:
Let the marked price (M.P.) of the radio be Rs x.
Here, discount percent (D%) =10 %
VAT = 15% Direct process
Now, S.P. = M.P. – D% of M.P.
S.P. = 90 % of M.P.
=x – 10% of x = 9x = 9 u x = Rs 9x
10 10 10
Also, S.P. with VAT = S.P. + VAT% of S.P. Direct process
= 9x + 15% of 9x S.P. with VAT
10 10
9x 207x
= 207x = 115 % of 10 = 200
200
Vedanta Excel in Mathematics - Book 10 34 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
According to the question, Answer checking:
M.P. = Rs 6,000 and D% = 10%
207x = 6,210 S.P. = 90% of Rs 6,000 = Rs 5,400
200 S.P. with VAT = 115% of Rs 5,400
or, x = Rs 6,000 = Rs 6,210 which is
given in the question.
? M.P. = Rs 6,000
Now, discount = 10% of M.P. = 10 × Rs 6,000
100
= Rs 600
So, the required amount of discount is Rs 600. Again,
Alternative process Let M.P. of the watches be Rs y.
Here, S.P. with 15% VAT = Rs 6,210 Then, S.P. = M.P. – D% of M.P.
Let S.P. without VAT = Rs x or, 5,400 = y – 10% of y
Then, x + 15% of x = Rs 6,210 or, y = Rs 6,000
or, x = Rs 5,400 i.e. M.P. = Rs 6,000
i.e. S.P. = Rs 5,400
Again, discount = 10% of M.P. = 10% of Rs 6,000 = Rs 600
Example 8: A watch was sold on its marked price at a gain of 20%. But allowing 5%
discount, there would have been a gain of Rs 490. Find the cost price of the
watch.
Solution: Answer checking:
Let the M.P. of the watch be Rs x. C.P. = Rs 3,500
Then, the (S.P.)1 of the watch is also Rs x M.P. = S.P.1 = 120% of Rs 3,500
= Rs 4,200
Also, at 20% profit, 120% of C.P. = (S.P.)1
At 5% discount,
or, 120 u C.P. = Rs x
100 S.P.2 = 95% of Rs 4,200
5x = Rs 3,990
or, C.P. = Rs 6 ....... (i)
Again, (S.P.)2 = M.P. – discount % of M.P. ? Profit =S.P.2 – C.P.
= Rs 3,990 – Rs 3,500
= Rs x – 5 u Rs x = Rs 19x
100 20 = Rs 490 which is given
in the question.
When profit is Rs 490, C.P. = (S.P.)2 – Rs 490
or, C.P. = Rs 19x – Rs 490 ....... (ii)
20
From equations (i) and (ii),
Rs 19x – Rs 490 = Rs 5x
20 6
or, Rs 19x – Rs 5x = Rs 490
20 6
or, x = Rs 4,200
? C.P. = Rs 5x = 5 × 4,200 = Rs 3,500
6 6
Hence, the cost price of the watch is Rs 3,500.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 35 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Example 9: A television was sold at a gain of 20% after allowing a discount of 20%
on its marked price. Had it been sold after allowing 25% discount, there
would have been a gain of Rs 1,450. Find the marked price of the television.
Solution:
Let, the M.P. of the television be Rs x and its C.P. be Rs y. Direct process:
Now, S.P. in 20% discount = Rs x – 20% of Rs x = Rs 4x S.P. = 80% of M.P.
5
= 80 × x = Rs 4x
? C.P. = S.P. – profit % of C.P. 100 5
y = 4x – 20% of y Direct process:
5
or, y = 4x – y C.P. = S.P.
5 5 120%
or, 6y = 4x y = Rs 4x × 100 = Rs 2x
5 5 5 120 3
? y = Rs 4x = Rs 2x ………….. (i) Direct process:
6 3 S.P. = 75% of M.P.
Again, S.P. in 25% discount = M.P. – discount % of M.P.
= Rs x – 25% of Rs x = Rs 3x = 75 × Rs x = Rs 3x
4 100 4
? C.P. = S.P. – profit
y = Rs 3x – Rs 1450 ………… (ii)
4
From equations (i) and (ii)
3x – Rs 1450 = 2x
4 3
3x 2x
or, 4 – 3 = Rs 1450
or, x = Rs 17,400
Hence, the marked price of the television is Rs 17,400.
Example 10: After allowing 20% discount on the marked price and then levying 10%
VAT, a radio was sold. If buyer had paid Rs 320 for VAT, how much would
he have got the discount?
Solution:
Let, the marked price of the radio be Rs x. 4x
5
S.P. of radio = M.P. – discount % of M.P. = Rs x – 20% of Rs x = Rs
Now, the given amount of VAT = Rs 320 Answer checking:
M.P. = Rs 4,000 and D% =20%
or, 10% of S.P. = Rs 320 S.P. = 80% of Rs 4,000 = Rs 3,200
or, 10 u Rs 4x = Rs 320
100 5
or, x = Rs 4,000
VAT amount = 10% of Rs 3,200
? M.P. of the radio is Rs 4,000. = Rs 320 which is
Again, discount = 20% of M.P. given in the question.
= 20 u Rs 4,000 = Rs 800
100
Hence, he had got the discount of Rs 800.
Vedanta Excel in Mathematics - Book 10 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Example 11: A shopkeeper allowed 10% discount and sold a rice cooker for Rs 3,051
with 13% VAT and made a profit of 20%. By what percent is the discount to
be reduced to increase the profit by 4%?
Solution: Direct process
Let, S.P. without VAT be Rs x.
S.P. without VAT = S.P. with VAT
Here, discount percent = 10% 113%
3051
S.P. with VAT = Rs 3,051 = 113% = Rs 2,700
VAT = 13% M. P. = S.P. without VAT
90%
2700
Now, (100 + 13)% of x = Rs 3,051 = 90% = Rs 3,000
or, 113 × x = Rs 3,051 C.P. = S.P. without VAT
100 120%
2700
or, x = Rs 2,700 = 120% = Rs 2,250
? S.P. without VAT = Rs 2,700 New S.P. = 124% of C.P.
Again, (100 – 10)% of M.P. = S.P. without VAT = 124 × Rs 2250
100
90
or, 100 × M.P. = Rs 2,700 = Rs 2790
or, M.P. = Rs 3,000 New discount = Rs 3000 – Rs 2790
Also, profit = 20% = Rs 210
? (100 + 20)% of C. P. = S.P. without VAT New discount percent = 210 × 100%
= Rs 2,700 3000
120 = Rs 2,250
or, 100 × C.P. = 7%
or, C.P. ? Reduction in discount percent
Now, profit = 20% + 4% = 24% = 10% – 7% = 3%
? New S.P. = (100 + 24)% of C.P. = 124 × Rs 2,250 = Rs 2,790
100
And, new discount = M.P. – new S.P. = Rs 3,000 – Rs 2,790 = Rs 210
Then, new discount percent = New discount × 100% = 210 × 100% = 7%
M.P. 3000
? Reduction in discount percent = 10% – 7% = 3%
Hence, the discount is to be reduced by 3%.
Example 12: A business person hired a room in a shopping mall at Rs 40,000 rent
per month and started a business of electronic items. She invested
Rs 25,00,000 to purchase different electronic items in the first phase
and labelled the price of each item 40% above the cost price. She then
allowed 10% discount on each item and sold to customers. Her monthly
miscellaneous expenditure was Rs 18,000 and the items of worth 10% of
the investment remained as stocks after three months. Find her net profit
or loss percent.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 37 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
Solution:
Here, the amount of investment = Rs 25,00,000
Stocks after three months = 10% of Rs 25,00,000 = Rs 2,50,000
? The investment excluding stocks = Rs 25,00,000 – Rs 2,50,000 = Rs 22,50,000
Now, M. P. of the items = 140% of Rs 22,50,000 = Rs 31,50,000
Discount percent = 10%
? S.P. of the items = 90% of M.P. = 90 × Rs 31,50,000 = Rs 28,35,000
100
? Gross profit = Rs 28,35,000 – Rs 22,50,000 = Rs 5,85,000
Again, the rent of room in 3 months = 3 × Rs 40,000 = Rs 1,20,000
Miscellaneous expenditure in 3 months = 3 × Rs 18,000 = Rs 54,000
? Total expenditure= Rs 1,20,000 + Rs 54,000 = Rs 1,74,000
Now, net profit = Gross profit – total expenditure
= Rs 5,85,000 – Rs 1,74,000
= Rs 4,11,000
Then, net profit percent = net profit × 100% = 4,11,000 × 100% = 18.27%
investment 22,50,000
Hence, her net profit percent is 18.27%.
EXERCISE 2.2
General section
1. a) If marked price is Rs x and amount of discount is Rs y, write the formula to find S.P.
b) If marked price is Rs a, selling price after discount is Rs b, write the formula to
find the amount of discount.
c) When the amount of discount and marked price are given, write the formula to
find the rate of discount.
d) When the marked price and the selling price after discount are given, write the
formula to find discount percent.
2. a) If M.P. = Rs 990, discount percent = 10%, find S.P.
b) If M.P. = Rs 1,500, S.P. = Rs 1,275, find discount percent.
c) If S.P. = Rs 1,480, discount percent = 20 %, find M.P.
d) If S.P. = Rs 1,200, VAT percent = 13 %, find S.P. with VAT.
e) If S.P. = Rs 2,100, S.P. with VAT = Rs 2,415, find the rate of VAT.
f) If S.P. with VAT = Rs 1,921, VAT percent = 13 %, find S.P. without VAT.
3. a) The cost of an article is Rs 860. If a shopkeeper sells the article by allowing 10 %
discount, find the selling price of the article.
b) The selling price of a fan is Rs 1,476. If the fan was sold at 10 % discount from the
marked price, find the marked price.
Vedanta Excel in Mathematics - Book 10 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
c) A shopkeeper sold a watch for Rs 2,736 allowing 20 % discount. Find its marked
price.
d) What percent of discount should be given in a doll costing Rs 180 so that a customer
has to buy it for Rs 160?
e) Radha purchased a cosmetic item at 5 % discount. If she got the discount amount of
Rs 74, at what price did she purchase the item?
f) Find the net selling price of a shirt marked at Rs 1,200 when VAT of 13 % is levied.
g) When the VAT rate is 10 % a customer pays Rs 2,816 to buy a mobile set. Find the
cost of mobile without VAT.
h) While paying a bill, a man paid Rs 189.80 as the amount of VAT at 13 %. How
much was the actual bill amount?
4. a) The marked price of a mobile is Rs 12,600 and 10 % discount is allowed.
(i) Find the discount amount. (ii) Find the selling price of the mobile.
(iii) How much is its price with 13 % VAT?
b) The marked price of a watch is Rs 4,000 and 20 % discount is allowed.
(i) Find its selling price.
(ii) How much should a customer pay for it with 13 % VAT?
Creative section - A
5. a) The marked price of a lady bag is Rs 4,800. If a customer is given 10 % discount,
how much does he/she pay with 15 % VAT?
b) The marked price of an electric heater is Rs 3,000. What is the price of the heater
if 13 % VAT is levied after allowing 12 % discount on it?
6. a) The marked price of a woolen sweater is Rs 1,750. If the shopkeeper allows 20 %
discount and makes a profit of Rs 150, at what price did he purchase the sweater?
b) The marked price of a calculator is Rs 1,800. The shopkeeper allows 10 %
discount and still makes 8 % profit. At what price did the shopkeeper purchase
the calculator?
c) Mr. Rai bought a radio for Rs 2,000 and fixed its price so that after giving 20 %
discount he made 10 % profit. Find the fixed price of the radio.
d) A retailer bought a watch for Rs 2,500 and he labelled its price 20 % above the cost
price. If he allows 10 % discount to a customer, find his profit percent.
e) The marked price of an article is Rs 2,800 which is 40 % above the cost price. If it
is sold by allowing 20 % discount, what is the profit percent?
f) Mr. Jha purchased a bicycle costing Rs 5,600 from a dealer at 5 % discount and
sold at a profit of 10 %. If he had sold it at 5 % discount, find its marked price.
7. a) A trader marks the price of his goods 40 % above the cost price and allows
20% discount. If his purchase price of an item is Rs 6,000, how much should a
customer pay for it levying 13 % VAT?
b) The cost price of an electric fan is Rs 2,800. If the shopkeeper marks its price 30%
above the cost price and sells it at 10 % discount, how much should a customer
pay for it with 15 % VAT?
c) The marked price of an article is Rs 4,500. After allowing some percent of discount
and levying 10 % VAT it is sold at Rs 4,400, find the discount percent.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 39 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
d) The marked price of an electric item is Rs 2,400 and the shopkeeper allows 20%
discount. After levying VAT, if a customer pays Rs 2,208 for it, find the VAT
8. a) percent.
b)
A grocer fixed the price of his goods 25 % above the cost price. If he/she sold a box
c) of noodles allowing 5 % discount, find his profit percent.
d) Rabi Sahu fixed the marked price of his radio to make a profit of 30 %. Allowing
15 % discount on the marked price, the radio was sold. What percent profit did he
e) make?
f)
g) The marked price of a watch is 30 % above the cost price. When it is sold allowing
9. a) 20 % discount on it, there is a gain of Rs 150. Find the marked price and the cost
b) price of the watch.
c)
A trader fixed the price of cosmetic items 30 % above the cost price. When he/she
d) sold an item at 25 % discount, there was a loss of Rs 15. Find the cost price and
marked price of the item.
10. a)
b) The selling price of an article is 20 % less than its marked price and the marked
price is 30 % above the cost price. Find the profit percent.
c)
d) The marked price of a radio is 25 % above the selling price and the cost price is
e) 30 % less than its marked price. Find the discount percent and gain percent.
Mrs. Sharma fixed the price of a pen to make a profit of 10 %. But, she sold it
allowing a discount of Rs 7.50 and lost 5 %. At what price did she purchase the
pen?
When an article is sold at a discount of 10 %, a profit of Rs 8 is earned. If the same
article is sold without allowing a discount, there is a profit of Rs 20. Find the cost
price of the article.
A watch was sold on its marked price at a gain of 20 %. But allowing 5 % discount,
there would have been a gain of Rs 140. Find the cost price of the watch.
A shopkeeper sold an article at 20 % discount and made a loss of Rs 90. If he had
sold it at 5 % discount, he would have gained Rs 90. Find the cost price and the
marked price of the article.
After allowing a discount of 20 % on its marked price, an article was sold at a gain
of 20 %. Had it been sold after allowing 25 % discount, there would have been a
gain of Rs 125. Find the marked price of the article.
A bicycle is sold at Rs 9,040 after allowing 20 % discount and imposing 13 % VAT.
Find the marked price of the bicycle.
After allowing 15 % discount on the marked price of an article 13 % VAT was
levied on the remaining amount. Then, the price of the article becomes Rs 13,447.
Find the marked price of the article.
The marked price of an article is Rs 4,000. If the price of the article including
13 % VAT is Rs 3,616, find the discount percent given in it.
A colour TV is sold at Rs 20,700 after 10 % discount with 15 % VAT. Find the VAT
amount.
After allowing 10 % discount on the marked price of an article and then 15 %
VAT is charged, its price becomes Rs 16,720. How much amount was given as
discount?
Vedanta Excel in Mathematics - Book 10 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
11. a) A shopkeeper purchased a bicycle for Rs 5,000 and marked its price a certain
percent above the cost price. Then, he sold it at 10% discount. If a customer paid
Rs 6,356.25 with 13% VAT to buy it, how many percent is the marked price above
the cost price?
b) The price of an article is fixed a certain percent above the cost price and sold it at
5% discount. If the cost price of the article is Rs 16,000 and sold it for Rs 20,064
with 10% VAT by how many percent is the marked price above the cost price?
12. a) When an article was sold at a discount of 10%, a customer paid Rs 9,153 with
13% VAT. If 8% profit was made in this transaction by how many percent was the
marked price above the cost price?
b) A retailer marked the price of a mobile a certain percent above the cost price. Then,
he allowed 20% discount to make 12% profit. If the mobile was sold for Rs 5,062.40
with 13% VAT, by what percent is the marked price above the cost price?
13. a) After allowing 25 % discount on the marked price and then levying 10 % VAT, a
cycle was sold. If the discount amount was Rs 750, how much VAT was levied on
the price of the cycle?
b) A tourist buys a Nepali flag at a discount of 15 % but pays 10 % VAT. If she pays
Rs 170 for VAT, calculate the discount amount.
c) A person buys an article at a discount of 13 % and pays 16 % VAT. If he/she pays
Rs 261 for VAT, find the marked price of the article and also the amount paid by
him/her to buy the article.
d) After allowing 20% discount on the marked price of a mobile, 15% VAT was levied
and it was sold. If the difference between the selling price with VAT and selling
price after discount is Rs 1,800, find the marked price of the mobile.
Creative section - B
14. a) In the peak season of winter days, a retailer marked the price of an electric heater
as Rs 4000 and 10% discount was given to make 20% profit. But in the summer
days she increased the discount percent to get only 12% profit from the same type
of heater. How much did she/he increase the discount percent?
b) The marked price of a mobile set is Rs 9,600 and 40% discount is allowed to make
20% profit. By what percent is the discount to be reduced to increase the profit by
10%?
c) A dealer marked the price of a digital watch as Rs 6,000. She then allowed 20%
15. a) discount to make 20% profit. If she increased the discount to 25%, by how much
was the profit percent decreased?
A retailer allowed 4% discount on his goods to make 20% profit and sold a
refrigerator for Rs 10,848 with 13% VAT. By how much is the discount to be
increased so that he can gain only 15%?
b) A supplier sold a scanner machine for Rs 41,400 with 15% VAT after allowing
10% discount on it's marked price and gained 20%. By how much is the discount
percent to be reduced to increase the profit by 4%?
c) Mrs. Gurung allowed 10% discount on her fancy items to make 25% profit and
sold a lady bag for Rs 5,085 with 13% VAT. Due to the excessive demands of her
items, she decreased the discount percent by 2%. By how much was her profit
percent increased?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 41 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
16. a) Mrs. Dhital makes a profit of 50% of the cost of her investment in the transaction
b) of her cosmetic items. She further increases her cost of investment by 25% but the
c) selling price remains the same. How much is the decrease in her profit percent?
Mr. Yadav makes a profit of 20% in the transaction of his electrical items. He
d) decreases his cost of investment by 4% but the selling price remains the same.
How much is the increase in his profit percent?
A customer goes to a shop to buy a laptop. The marked price of the laptop is
Rs 57,500 excluding 15% VAT. The customer bargains with shopkeeper and
convince him for Rs 57,500 including VAT as the final cost of the laptop. Find the
amount reduced by the shopkeeper.
A retailer hired a room in a shopping mall at Rs 45,000 rent per month and started
a business of garments. He spent Rs 20,00,000 to purchase different garment
items in the first phase and marked the price of each item 30% above the cost
price. Then, he allowed 10% discount on each item and sold to customers. His
monthly miscellaneous expenditure was Rs 15,000 and the items of worth 10%
of the investment remained as stocks after two months. Find his net profit or loss
percent.
Project Work
17. a) Make groups of your friends. Collect different types of goods purchasing bills.
b) Study about the marked price rate of discount, rate of VAT or other rates of taxes
c) mentioned in the bills. Prepare the reports and present in your class.
Collect electricity bill, water bill, telephone bill, etc. of your house. Study about
the rate of rebate, rate of VAT, rate of penalty charges or other rates of taxes
mentioned in the bills.
Visit the available website. Search and collect the information about various VAT
rates of different countries and compare with VAT rate of our country.
2.4 Currency exchange- introduction Exchange Rates Fixed by Nepal Rastra Bank
A currency exchange is a business that the legal Currency Unit Buying/Rs. Selling/Rs.
right to exchange one currency for another to its
customers. Currency exchange of physical money Indian Rupee 100 160.00 160.15
is usually done over a counter at a teller station.
International trade requires an organised system Open Market Exchange Rates
for exchanging money. The exchanging rate
between two currencies is used for this purpose. (For the purpose of Nepal Rastra Bank)
The exchanging rate is either the bank selling or
the bank buying rate. Currency Unit Buying/Rs. Selling/Rs.
An exchanging rate is the price of one nation's
currency in terms of another nation's currency. U.S. Dollar 1 114.58 115.18
The table given alongside is a display of exchange
rates in a commercial bank on a day. European Euro 1 127.53 128.20
Most of the world's currencies, including the euro
(Eur), the US dollar (USD), the Canadian dollar UK Pound Sterling 1 149.82 150.61
(CAD), the Australian dollar (AUD), and the British
pound (GBP), are floating, or variable. This means Swiss Franc 1 117.69 118.30
their values and their exchanging rates depend on
the international money market. Australian Dollar 1 79.45 79.89
Canadian Dollar 1 88.18 88.64
Singapore Dollar 1 84.86 85.30
Japanese Yen 10 10.60 10.66
Chinese renminbi (Yuan) 1 15.75 15.84
Saudi Arabian Riyal 1 30.54 30.70
Qatari Riyal 1 31.47 31.63
Thai Baht 1 3.80 3.82
UAE Dirham 1 31.20 31.36
Malaysian Ringgit 1 27.93 28.08
South Korean Won 100 9.82 9.87
Swedish Kroner 1 12.13 12.19
Danish Kroner 1 17.06 17.15
Hong Kong Dollar 1 14.73 14.81
Kuwaity Dinar 1 378.03 380.01
Bahrain Dinar 1 303.89 305.48
Note: Under the present system the open market exchange
rates quoted by different banks may differ. (The above
mentioned exchange rates are on 2076-09-20)
Vedanta Excel in Mathematics - Book 10 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Worked-out examples
Example 1: Find the difference of selling and buying rates of US $ 500 in Nepali rupee.
(Buying rate of $1 = 114.28 and selling rate of $1 = Rs 114.88)
Solution:
The buying rate of US$ 1 = Rs 114.28
The buying rate of US$ 500 = 500 × Rs 114.28 = Rs 57,140
Also, the selling rate of US$ 1 = Rs 114.88
the selling rate of US$ 500 = 500 × Rs 114.88 = Rs 57,440
? The difference due to the selling and buying rates = Rs 57,440 – Rs 57,140 = Rs 300
Example 2: The exchange rate of USD ($) and Nepali rupees (NPR) on a day is
$ 1 = NPR 113.65.
a) How many dollars can be exchanged for NPR 77,282?
b) How many rupees can be exchanged for $ 1,500?
Solution:
a) Here, NPR 113.65 = $ 1 b) $ 1 = NPR 113.65
$ 1,500 = 1,500 × NPR 113.65
NPR 1 = $ 1 × 77,282 = NPR 1,70,475
NPR 77,282 = $ 1131.65
113.65
= $ 680
Example 3: Study the following exchange rate table:
Country Currency Exchange rate
United Kingdom Pounds (£) 1 £ = NPR 147.40
United States Dollars ($) 1 $ = NPR 113.85
In Nepal the cost of a television is Rs 1,45,000. In England the same television
costs £ 1,050 and in the USA $ 1,220. In which country is the television
cheaper?
Solution: = NPR 1,45,000
The cost of the television in Nepal
The cost of the television in England is NPR = 1,050 × NPR 147.40 = NPR 1,54,770
The cost of the television in USA is NPR = NPR 1,220 × NPR 113.85 = NPR 1,38,897
Hence, the television is cheaper in United State.
Example 4: Use the exchange rates given in the table and solve the following problems.
USD ($) USD ($) GBP (£) CAD ($) EUR ( ) AUD ($)
GBP (£) 1 0.80 1.34 0.90 1.30
CAD ($) 1 1.67 1.12 1.62
EUR ( ) 1.25 0.60 1 0.67 0.97
AUD ($) 0.75 0.90 1.48 1 1.44
1.10 0.62 1.03 0.69 1
0.76
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 43 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
a) Convert USD 2,000 into GBP and EUR
b) Convert EUR 4,000 into AUD and CAD.
Solution:
a) From the table,
US$ 1 = GBP£ 0.80 US$ 1 = ( ) 0.90
? US$ 2,000 = 2,000 × £ 0.80 ? US$ 2,000 = 2000 × 0.90
= £ 1,600 = 1,800
b) From the table,
1 = AUD$ 1.44 1 = CAD$ 1.48
? 4,000 = 4,000 × AUD$ 1.44 ? 4,000 = 4,000 × CAD$ 1.48
= AUD $ 5,760 = CAD$ 5,920
Example 5: A trader purchased 500 pieces of Nepali carpet at Rs 6,500 per piece. He
exported them to United Kingdom with 5% export tax. If he sold them at
£ 65 per piece in UK, calculate his profit or loss percent. (£ 1 = NPR 145.00)
Solution:
Here, C.P. of 500 piece of carpets = 500 × Rs 6,500
= Rs. 32,50,000
Also, C.P. with 5% export tax = Rs 32,50,000 + 5% of Rs 32,50,000
= Rs 34,12,500
Again, S.P. of 500 pieces of carpet in UK = 500 × £ 65 = £ 32,500
Now, £ 1 = Rs 145.00
£ 32,500 = 32,500 × Rs 145.00
= Rs 47,12,500
? Profit = Rs 47,12,500 – Rs 34,12,500
= Rs 13,00,000
And, profit percent = Actual profit × 100%
C.P.
= Rs 13,00,000 × 100% = 38.1%
Rs 34,12,500
Hence, his profit percent is 38.1%.
2.5 Money exchange by using chain rule
Chain rule is an alternative method of unitary method and ratio and proportion
method to find the value of unknown variable in a compound proportion. Study the
following examples and learn about this method.
If A = B and
B = C, then
C =A
? A×B×C =B×C×A
Vedanta Excel in Mathematics - Book 10 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
Example 6: If US $ 1 = NPR 113.00 and GBP £ 1 = NPR 146.00, convert $ 730 into pound.
Solution: Now, using chain rule, we get,
Here, $1 = Rs 113 1 × 146 × x = 113 × 1 × 730
=£1
Rs 146 = $ 730 113 × 730
146
£x or, x = = £ 565
Hence, $ 730 is equal to £ 565 according to the given exchange rate.
EXERCISE 2.3
General Section
The exchange rates of foreign currencies on a day in January 2020 is given below in the
table. Use the exchange rates and solve the following problems.
Country Currency Exchange rate
India Rupee ( ) 100 = NPR 160.00
United States Dollar ($) $ 1 = NPR 114.50
European Union EURO ( ) 1 = NPR 127.50
Untied Kingdom Pound Sterling (GBP) £ 1 = NPR 150.30
Australia AUD ($) Australian Dollar AUD $ 1 = NPR 79.60
Canada CAD ($) Canadian Dollar CAD $ 1 = NPR 88.00
Singapore SGD ($) SGD $ 1 = NPR 84.50
Japan Yen (Y) 10Y = NPR 10.50
Saudi Arab Riyal (SAR) SAR 1 = NPR 30.50
Qatar Qatar Riyal (QAR) QAR 1 = NPR 31.45
Malaysia Ringgit (RM) RM 1 = NPR 28.00
South Korea Won (KRW) KRW 100 = NPR 9.00
Hongkong Hongkong Dollar (HKD$) 1 = NPR 14.00
China Yuan ( )
1 = 16.45
1. Convert the following currencies into Nepali Currency (NPR)
a) IN 17,550 b) US$ 990 c) GBP£ 630 d) SAR 3,240
e) QAR 2,880 f) KRW 50,000 g) RM 2,460 h) AU$ 1,230
2. Convert NPR 1,80,000 into a) INR b) USD c) GBP d) RM
3. Use the exchange rates given in the table and solve the following problems.
USD ($) USD ($) GBP (£) CAD ($) EUR ( ) AUD ($)
GBP (£) 1 0.80 1.34 0.90 1.30
CAD ($) 1.25 1 1.67 1.12 1.62
EUR ( ) 0.75 0.60 1 0.67 0.97
AUD ($) 1.10 0.90 1.48 1 1.44
0.76 0.62 1.03 0.69 1
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 45 Vedanta Excel in Mathematics - Book 10
Tax and Money Exchange
a) Convert USD 5000 into (i) GBP (ii) CAD (iii) EUR (iv) AUD
b) Covert GBP 3,000 into (i) USD (ii) CAD (iii) EUR (iv) AUD
c) Convert EUR 2,000 into (i) USD (ii) GBP (iii) CAD (iv) AUD
d) Convert CAD 4,000 into (i) USD (ii) GBP (iii) EUR (iv) AUD
e) Convert AUD 6,000 into (i) USD (ii) GBP (iii) EUR (iv) CAD
4. a) Hari K.C. works as a supervisor in Malaysia. His monthly salary is RM 3,250. If
the exchange rate is RM 1 = NPR 28.50, how much is his monthly salary in Nepali
currency?
b) Bishu Rai is a pre-primary teacher in Australia. She earns AUD$ 58 per
hour. She teaches 5 hours per day except Sunday. The exchange rate of
AU$ 1 = NPR 80.50.
(i) Find her earning per week in Nepali rupees.
(ii) How much Nepali rupee does she earn in 4 weeks?
c) Sunayana wants to buy an iPad that costs US$ 850, with the exchange rate
currently at US$ 1 = NPR 114. She estimates that the exchange rate drops to
NPR 112 in a month.
(i) How much will the iPad cost in Nepali rupees, if she buy it now?
(ii) How much will she save if the exchange rate drops to NPR 112?
(iii) How much will she lose if the exchange rate moves to NPR 115.50?
d) Manoj Regmi is going to visit Thiland for his family trip. He estimated to exchange
US $ 4,000 in a bank. If the bank charges 1.5% commission for exchanging the
money, how much Nepali rupees is required for him? (US$ 1 = NPR 114.50)
e) Nirmal Shrestha bought some EURO ( ) for NPR 3,20,000 at the exchange rate of
1 = NPR 128 to visit a few European countries. Unfortunately, because of his Visa
problem, he cancelled his trip. Within a week Nepali rupee is devaluated by 2%.
He again exchanged his EURO to Nepali rupee after a week. How much did he gain
or lose?
f) A businessman exchanged Rs 5,50,000 into US dollars at the rate of $ 1 = NPR 110.
After a few days, Nepali currency was revaluated by 10% and he exchanged his
dollars into Nepali currency again. Calculate his gain or loss.
5. a) In Nepal the cost of a mobile set is Rs 1,20,000. In England the same set costs
£ 950 and in America US$ 1,200. In which country is the mobile set cheaper?
(£ 1 = NPR 145.00, $ 1 = NPR 114.00)
b) Mrs. Magar wants to buy a book online. She finds a publisher in London selling
the book for £ 15. This publisher is offering free transportation on the product.
She then finds the same book from a publisher in New York for $17 with a
transportation fee of $2. Which publisher should she buy the book from?
(Exchange rate: £1 = NPR 140 and $ 1 = NPR 113.00)
Vedanta Excel in Mathematics - Book 10 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Tax and Money Exchange
6. a) Mr. Gurung bought 100 g of gold in Hong Kong for KHD$ 38,000 and bought to
Nepal paying 25% custom duty. If he sold the gold with 13% VAT in Nepal, how
much Nepali rupee did he get? (HKD$ 1 = NPR 14.00)
b) A merchant purchased 500 pieces of 'Nepali Pasmina' at Rs 2,700 per piece. He
exported them to USA with 5% export tax. If he sold them at $ 35 per piece in USA,
calculate his profit or loss percent. ($ 1 = NPR 112.00)
c) Laxmi Yadav bought a television for RM 2,000 and sold to her friend at 10% profit
in Malaysia. Her friend brought it to Nepal and sold at a profit of 25%. Find the
selling price of the television in Nepal. (RM 1 = NPR 28.00)
7. a) If 260 dollars can be exchanged for 200 pounds and 1 pound is exchanged for
Rs 148, how many dollars can be exchanged for Rs 59,200?
b) If $ 5 = £ 4 and Rs 342 = $ 3, how many pounds (£) will be equal to Rs 8,550?
c) The exchange rate of 10 Japanese Yen on a certain day was Rs 10.25. On the same
day, if $ 3 was exchanged for Rs 345, how many Yens could be exchanged for
$ 123?
d) If the exchange rate of £ 1 is Rs 147.00 and the exchange rate of US $ 1 is
Rs 113.00, how many dollars can be exchanged for 100 sterling pounds?
e) If Rs 2,280 = $ 20 and Rs 2,190 = £ 15, how many dollars can be exchanged for
£ 50?
f) If QAR 20 =$ 5.40 and $ 5 = Rs 560, how many QAR can be exchanged for
Rs 50,000?
g) If the exchange rate of RM 1 is Rs 28 and the exchange rate of $ 1 is Rs 113, find the
exchange rate of dollars for RM 100.
Project Work
8. a) Collect the information from the daily newspapers or from internet about today's
exchange rates of different currencies in terms of Nepali currency. Then find the
exchange rate between the following currencies:
(i) US ($) and GBP (£) (ii) US ($) and AUD ($) (iii) GBP (£) and EUR ( )
(iv) EUR ( ) and US ($) (v) AUD ($) and GBP (£) (vi) US ($) and CAD ($)
b) Collect the information from the daily newspapers or from internet about the cost
of oil per barrel and cost of gold per 10 gram in the international market. Calculate
the selling price of these materials in Nepali market according to the present rates
of custom duty, VAT and other rates of taxes.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 47 Vedanta Excel in Mathematics - Book 10
Unit Compound Interest
3
3.1 Principal and interest - review
Interest is the charge for the privilege of borrowing money from a lender. When we
borrow a loan from a bank, we should pay interest for the use of borrowed money
under the certain condition. Similarly, if we deposit money in a bank, the bank pays
us interest. The borrowed or deposited sum is the principal.
There are three major variables which are used in the calculation of interest: principal,
rate of interest, and duration of time. Interest is expressed as a rate percent per year.
There are two types of calculation of interest.
(i) Calculation of simple interest and (ii) Calculation of compound interest.
Simple interest is calculated always from the original principal at the given rate
percent for any interval of time. We calculate simple interest by using the following
formula, P uTu R
100
Simple interest (I) =
Where, P is the principal; T is the time; and R is the rate of interest.
3.2 Compound interest
When the interest of a principal for one year is added to the principal and the interest
for the next year is calculated from the new principal, it is called compound interest.
In this way, in the case of simple interest, the principal always remains the same;
however, the principal is increased every year in case of compound interest.
Now, let’s derive the formula to calculate compound interest.
Let the original principal = Rs P
the rate of interest = R% p.a.
the time interval = T years
Now, the interest for the 1st year = P u T u R% = P u 1 u R% = PR%
? The amount in the 1st year = P + PR% = P (1 + R%)
Again, the principal for 2nd year = P (1 + R%)
The interest for 2nd year = P (1 + R%) u 1 u R% = P (1 + R%) R%
Then, the amount in 2nd year = P (1 + R%) + P (1 + R%) R%
= P (1 + R%) (1 + R%) = P (1 + R%)2
Similarly, the amount in 3rd year = P (1 + R%)3
the amount in 4th year = P (1 + R%)4
Vedanta Excel in Mathematics - Book 10 48 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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