Vedanta Excel in Mathematics Book 9 Final (2080)_compressed

Book 9 MATHEMATICS MATHEMATICS vedanta Excel in Approved by the Government of Nepal, Ministry of Education, Science and Technology, Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material Vedanta Publication (P) Ltd. j]bfGt klAns];g k|f= ln= Vanasthali, Kathmandu, Nepal +977-01-4982404, 01-4962082 [email protected] www.vedantapublication.com.np vedanta

Book 9 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 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. MATHEMATICS MATHEMATICS vedanta Excel in First Edition : B. S. 2077 (2020 A. D.) Second Edition (Updated): B. S. 2078 (2021 A. D.) Third Revised and Updated Edition: B. S. 2079 (2022 A. D.) Fourth Edition (Updated): B. S. 2080 (2023 A. D.) Price: Rs 563.00 Printed Copies: 35,000.00

Publisher's words Vedanta, the emergent publication house, is committed to cater quality school-level textbooks and reference materials that are well-researched and thoughtfully prepared. We make the utmost endeavours to ensure the readers that our products are error-free. We believe, the processes of updating, upgrading and modification of textbooks are the most significant aspects of quality education. Therefore, our textbooks which are completely based on the latest curriculum developed by Curriculum Development Centre (CDC), Nepal are also updated, upgraded and modified as per time. The textbooks of Vedanta Excel in Mathematics series are originally written by Mr. Hukum Pd. Dahal, who has been writing mathematics text books since B.S. 2052 (1995 A.D.) to till the date. Besides the Excel in Mathematics series, Mr. Dahal has written three more series: Elementary Mathematics, Maths in Action and Speedy Maths. Recently, according to the new curriculum developed by CDC, Nepal, Vedanta Excel in Mathematics Book 9 is updated, upgraded and modified by Mr. Tara Bdr. Magar, who is also involving as the editor of all the textbooks of the series. -Vedanta Publication (P) Ltd.

Forewords The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical teaching learning activities and methodologies extracted from Teachers' training, workshops, seminars, and symposia. It is an innovative and unique series in the sense that the contents of each textbooks of the series are written and designed to fulfill the need of integrated teaching learning approaches. The series has focused on promoting the fundamental mathematical skills: Social skill, Logical skill, Computational skill and Problem solving skill of the learners. 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. Furthermore, the evaluations of all levels of learning achievements (knowledge, understanding, application and higher ability) based on a single problem are incorporated in every exercise, according to the revised and updated specification grid developed by CDC, Government of Nepal. 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 most of the exercises in the textbooks of each grade provide some ideas to connect the learning of mathematics to the real life situations.

The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities to integrate mental maths and manual maths simultaneously. Moreover, the problems in ‘Section A’ judge the level of achievement of knowledge and understanding, and diagnose the learning difficulties of the learners. The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an important role of ‘placement evaluation’ which is in fact used by a teacher to judge the level of prior knowledge and understanding of every learner to select their teaching learning strategies. The socially communicative approach by language and literature in every textbook, especially in primary level of the series, plays a vital role as a ‘textual-parents’ to the young learners and helps them overcome maths anxiety. Furthermore, as per the need of ICT in teaching and learning mathematics, we have included 'Vedanta ICT Corner' to fulfil this requirement in all classes. 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 the previous series, We are highly grateful to the Principals, HODs, Mathematics teachers and experts, PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN, UNIPS and independent clusters of many other Schools of Nepal, for providing us with opportunities to participate in workshops, Seminars, Teachers’ training, Interaction programme, and symposia as the resource person. Such programmes helped us a lot to investigate the teaching-learning problems and to research the possible remedies and reflect to the series. We are extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, England who provided us with very valuable suggestions about the effective methods of teachinglearning mathematics and many reference materials. We are thankful to Dr. Komal Phuyal for editing the language of the series. Moreover, we gratefully acknowledge all Mathematics Teachers throughout the country who encouraged us and provided us with the necessary feedback during the workshops/interactions and teachers’ training programmes in order to prepare the series in this shape. We are profoundly grateful to the Vedanta Publication (P) Ltd. for publishing this series. We would like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha, and Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during the preparation of the series. Also, we are heartily thankful to Mr. Pradeep Kandel, the Computer and Designing Senior Officer of the publication house for his skill in designing the series in such an attractive form. -Authors

Contents S.N Chapter Page No. 1. Set 9 - 30 1.1 Set – Looking back, 1.2 Set- Review, 1.3 Notation and specification of sets, 1.4 Methods of describing a set, 1.5 Cardinal number of sets 1.6 Types of sets, 1.7 Relationship between sets, 1.8 Universal set, 1.9 Venn-diagrams, 1.10 Set operations, 1.11 Cardinality relationships of two sets 2. Taxation 32 - 54 2.1 Tax - Introduction, 2.2 Income Tax, 2.3 Tax withholding, 2.4 Marked price (M.P.) and Discount, 2.5 Value Added Tax (VAT) 3. Commission, Bonus and Dividend 55 - 67 3.1 Commission – Introduction, 3.2 Bonus, 3.3 Dividend 4. Household Arithmetic 68 - 90 4.1 Introduction, 4.2 Electricity bill, 4.3 Telephone bill, 4.4 Water bill, 4.5 Calculation of taxi fare in a taximeter 5. Mensuration (I): Area 92 - 113 5.1 Area – Looking back, 5.2 Mensuration, 5.3 Perimeter and area of plane figures-review, 5.4 Area of triangle, 5.5 Area of 4 walls, floor and ceiling 6. Mensuration (II): Prism 114 - 126 6.1 Prism – Looking back, 6.2 Area and volume of solids, 6.3 Prisms, 6.4 Surface area and volume of triangular prism 7. Mensuration (III): Cylinder and Sphere 127 - 147 7.1 Cylinder – Looking back, 7.2 Cylinder, 7.3 Area of cylinder, 7.4 Volume of cylinder, 7.5 Curved surface area, total surface area and volume of a hollow cylinder, 7.6 Half cylinder or semi-cylinder, 7.7 Area and Volume of a Sphere, 7.8 Hemisphere and great circle, 7.9 Surface area and volume of a hollow hemispherical object, 7.10 Area and Volume of cylinder having hemispherical ends, 7.11 Cost estimation 8. Sequence and Series 149 - 167 8.1 Sequence - Introduction, 8.2 General term of sequence, 8.3 Series, 8.4 Sigma Notation, 8.5 Partial Sum, 8.6 Arithmetic Sequence, 8.7 Terms and Common Difference of an A.P., 8.8 Geometric Sequence, 8.9 Terms and Common Ratios of a G.P. 9. Factorisation of Algebraic Expressions 168 - 181 9.1 Factorisation-Looking back, 9.2 Factors and Factorisation - Review Assessment - I Assessment - II Assessment - III

10. Highest Common Factor (H.C.F.) and 182 - 192 Lowest Common Multiples (L.C.M.) 10.1 H.C.F. and L.C.M.-looking back, 10.2 H.C.F. of algebraic expressions, 10.3 L.C.M. of algebraic expressions 11. Indices 193 - 201 11.1 Indices – review, 11.2 Laws of Indices 12. Simultaneous Linear Equations 202 - 218 12.1 Simultaneous equations - review, 12.2 Method of solving simultaneous equations, 12.3 Application of simultaneous linear equations 13. Geometry - Triangle 220 - 249 13.1 Triangle-Looking back, 13.2 Types of triangles - review, 13.3 Median and altitude of a triangle, 13.4 Properties of triangles, 13.5 Triangle inequality property, 13.6 Congruent triangles, 13.7 Conditions of congruency of triangles, 13.8 Similar triangles- review 14. Geometry - Parallelogram 250 - 262 14.1 Special types of Quadrilateral 15. Geometry - Construction 263 - 270 15.1 Quadrilaterals, 15.2, Construction of Rhombus, 15.3 Construction of scalene quadrilaterals, 15.4 Construction of trapezium 16. Geometry - Circle 271 - 287 16.1 Circle and it’s various parts, 16.2 Theorems related to chords of a circle 17. Statistics (I): Classification and Graphical Representation of data 289 - 310 17.1 Statistics – Looking back, 17.2 Statistics, 17.3 Types of data, 17.4 Frequency table, 17.5 Grouped and continuous data, 17.6 Cumulative frequency table, 17.7 Graphical representation of data: Histogram, Frequency polygon, Ogive (Cumulative frequency curve) 18. Statistics (II): Measure of Central Tendency 311 - 324 18.1 Central tendency – Looking back, 18.2 Arithmetic mean, 18. 3 Median, 18.4 Quartiles, 18.5 Mode, 18.6 Range 19. Probability 325 - 336 19.1 Probability – Looking back, 19.2 Probability-Introduction, 19.3 Probability scale, 19.4 Probability of an event, 19.5 Empirical probability (or Experimental probability) 20. Trigonometry 338 - 349 20.1 Trigonometry - Introduction, 20.2 Trigonometric ratios, 20.3 Relation between trigonometric ratios, 20.4 Values of trigonometric ratios of some standard angles Revision and Practice Time 351 - 363 Answers 364 - 379 Model Questions 381 - 384 Assessment - IV Assessment - V Assessment - VI Assessment - VII

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 9 Vedanta Excel in Mathematics - Book 9 Classwork - Exercise 1.1 Sets – Looking back 1. Let A = {2, 3, 5, 7}. Insert the appropriate symbol or in the blank space. a) 2 …. A b) 4 …. A c) 10 …… A d) 7 … A 2. Re-write the following sets in roster form. a) A = {the letters in the word ‘PARALLELOGRAM’}, A = {...............................................................................} b) B = {the prime numbers less than 10}, B = {...............................................} c) C = {x: x = 4n, n ∈ N and n<5}, C = {..............................................} 3. Identify and write whether the following sets are empty (null), unit (singleton), finite or infinite. a) A = {the prime numbers between 90 and 100} ................................... b) B = {x: x is a perfect square number} ................................... c) C = {all leap years between 2076 and 2080} ................................... d) D = {y: y is a common factor of 36 and 48} ................................... 1.2 Sets - Review Let’s take a collection of multiples of 5 between 20 and 50. The members of this collection are definitely 25, 30, 35, 40, 45. These members are distinct objects when considered separately. However, when they are considered collectively, they form a single set of size five, written {25, 30, 35, 40, 45}. It is a set of multiples of 5 between 20 and 50. Here, any multiple of 5 between 20 and 50 is definitely the member of the set. Therefore, a set is a collection of ‘well-defined and distinct objects’. The German mathematician and logician Georg Cantor (1845 –1918) created modern set theory between the years 1874 to 1897. Today, it is used in almost every branch of Mathematics. Unit 1 Sets

Vedanta Excel in Mathematics - Book 9 10 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Sets 1.3 Notation and specification of sets We usually denote sets by capital letters and the members or elements of the sets are enclosed inside the braces { } separated with commas. The table given below shows a summary of the use of symbols in sets. Here, we are taking any two sets, i.e., N = {1, 2, 3, 4, 5} and E = {2, 4, 6, 8}, to discuss about the use of symbols. Symbol Name Example Explanation { } Set N = {1, 2, 3, 4, 5} E = {2, 4, 6, 8} The members of the sets are enclosed inside braces. ∈ Membership 2 ∈ N, 5 ∈ N, 4 ∈ N, 6 ∈ E, 8 ∈ E The symbol ‘∈‘ denotes the membership of an element of the given set. ∉ Non-membership 6 ∉ N, 8 ∉ N, 3 ∉ E, 5 ∉ E The symbol ‘∉‘ denotes the non-membership of an element to the given set. ⊂ Proper subset {1, 2} ⊂ N, {4, 6, 8} ⊂ E A set that is contained in another set. ⊆ Improper subset {2, 4, 6, 8} ⊆ E, it means {2, 4, 6, 8} ⊂ E and {2, 4, 6, 8} = E A set which is contained in and equal to another set. ⊃ Super set N ⊃ {1, 2}, E ⊃ {4, 6, 8} Set N includes {1, 2,} and Set E includes {4, 6, 8}. 1.4 Methods of describing a set We usually describe a set by three methods: description, listing (or roster), and set-builder (or rule) methods. Method Example Explanation Description P is a set of prime numbers less than 10. Words description of common properties of elements of a set. Listing (or roster) P = {2, 3, 5, 7} The distinct elements of a set are listed inside curly brackets { }. Set-builder (or rule) P = {x : x ∈ prime numbers, x < 10} A variable is used to describe the common properties of the elements of a set by using symbols.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 11 Vedanta Excel in Mathematics - Book 9 Sets 1.5 Cardinal number of sets Let’s take a set A = {2, 3, 5, 7, 11, 13, 17, 19}. Since, it has 8 elements, the cardinal number of the set A is 8. The number of elements contained by a set is called its cardinal number. If A be a given set, it’s cardinal number is represented as n(A). For example: If A = {1, 2, 3, 4, 6, 12}, then n(A) = 6. 1.6 Types of sets On the basis of the of elements contained in sets, we classify sets into four types: empty (or null) set, singleton (or unit) set, finite set, and infinite set. Type of sets Examples Explanation Empty or null set The set of whole numbers less than 0. W = { } or φ It does not contain any element. It is denoted by empty braces { } or by φ(Phi) Unit or singleton set The set of even numbers between 3 and 5, E = {4} It contains only one element. Finite set W = {0, 1, 2, 3, ..., 100} It contains finite number of elements. Infinite set W = {0, 1, 2, 3, ...} It contains infinite number of elements. Facts to remember 1. {0} and {∅} are not null sets rather these are singleton sets. 2. The empty set ∅, has no element. So, it’s a finite set. 3. The sets of natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), irrational numbers (P) and real numbers (R) are infinite sets. 1.7 Relationship between sets On the basis of the types of elements contained in two or more sets, the types of their relationship can be defined in the following ways. Type of relationship Examples Explanation Equal sets A = {क, ख, ग, घ, ङ} B = {घ, ख, ङ, क, ग} ∴ A = B They have exactly the same elements. Equivalent sets P = {2, 3, 5, 7} Q = {1, 4, 9, 16} ∴ P ~ Q They have equal number of elements. Overlapping sets M = {2, 4, 6, 8, 10} N = {1, 2, 5, 10} ∴ M and N are overlapping sets. They have at least one common element. 2 and 10 are the common elements of sets M and N. Disjoint sets X = {अ, आ, इ, ई} Y = {च, छ, ज, झ} ∴ X and Y are disjoint sets. They do not have any common element. I understood! {0} and {∅} are not null sets rather these are singleton sets.

Vedanta Excel in Mathematics - Book 9 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Sets 1.8 Universal set Let’s consider a set of natural numbers less than 20. From this set, we can make many other subsets such as set of even numbers less than 20, set of odd numbers less than 20, set of prime numbers less than 20, and so on. In this case, the set of natural numbers less than 20 is considered as a universal set and denoted by U. Thus, a set under the consideration from which many other subsets can be formed is known as a universal set. A universal set is the set of all the elements of any group under consideration. 1.9 Venn-diagrams We can represent sets and set operations by using diagrams like rectangle, circle, or oval shape. The idea of representation of sets in diagrams was first introduced by Swiss Mathematician Euler. It was further developed by the British Mathematics John Venn. So, such diagrams are famous as Venn Euler diagrams or simply Venn-diagrams. In Venn-diagram, the universal set is represented by rectangle and its subsets are represented by circles or ovals inside the rectangle. Let’s study the following set relationship by using Venn-diagrams. A A B A B U U A is subset of U A and B are disjoint sets A and B are overlapping sets U A and B are overlapping, B and C are overlapping, but A and C are disjoint sets. A, B and C are overlapping sets. A B C U A and B are overlapping, A and C are overlapping, but B and C are disjoint sets. A C B U A B C U 1.10 Set operations Like the operations of addition, subtraction, multiplication, division on numbers in Arithmetic, there are four types of operations on sets: 1. Union of sets 2. Intersection of sets 3. Difference of sets 4. Complement of a set These operations are well known as set operations. Now, let’s learn about these sets operations by using Venn-diagrams. 1. Union of sets Let’s study the following examples and investigate the idea about union of sets.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 13 Vedanta Excel in Mathematics - Book 9 Sets a) Suppose that the sport and cultural committees are recently formed with a few number of students of class 9. The set of students who are involving in the sport committee is S = {Raju, Pinky, Ganesh, Lakpa, Diya, Biswant, Rahul} and the set of students who are involving in cultural committee is C = {Salina, Diya, Suntali, Ojaswi, Rahul, Smrity}. If a joint meeting of these committees is held and a new committee is formed, who will be the members of the joint committee? Obviously, the list of the students of the joint committee is {Raju, Pinky, Ganesh, Lakpa, Diya, Biswant, Binaya, Rahul, Salina, Suntali, Ojaswi, Smrity}. Of course, it is the union of two committees S and C. b) Let’s take any two sets A and B, where A = {1, 2, 3, 4, 5}, B = {1, 3, 5, 7}. Now, the union of sets A and B denoted as A ∪ B = {1, 2, 3, 4, 5, 7}. Thus, the union of two or more sets is made by grouping their elements together in a single set. In the case of overlapping sets, the common elements are listed only once while making the union. The shaded regions in the following Venn-diagrams represent the union of the given sets. A B B A U U U A ∪ B A and B are disjoint sets A ∪ B A and B are overlapping sets A ∪ B A is subset of B A B A ∪ B ∪ C A, B and C are disjoint sets. A B C U A ∪ B ∪ C A, B and C are overlapping sets. C A ∪ B ∪ C A and B are overlapping, A and C are overlapping, but B and C are disjoint sets. A B C U U Facts to remember 1. The union of two sets A and B, denoted by A ∪ B, is the set of all elements that belong to either set A, or set B, or to both sets A and B. In set-builder form, union of sets A and B is defined as: A ∪ B = {x : x ∈ A or x ∈ B} 2. A ∪ A = A 3. A ∪ ∅ = A 4. A ∪ U = U 5. A ⊆ (A ∪ B) and B ⊆ (A ∪ B) 6. A ∪ B = A if B ⊆ A and A ∪ B = B if A ⊆ B

Vedanta Excel in Mathematics - Book 9 14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Worked-out Examples Example 1: If A = {x: x is a prime number less than 20} and B = {y: y is a factor of 30} are the subsets of a universal set U, find A ∪ B and show it in a Venndiagram. Solution: Here, A = {x: x is a prime number less than 20} = {2, 3, 5, 7, 11, 13, 17, 19} B = {y: y is a factor of 30} = {1, 2, 3, 5, 6, 10, 15, 30} Now, A ∪ B = {2, 3, 5, 7, 11, 13, 17, 19} ∪ {1, 2, 3, 5, 6, 10, 15, 30} = {1, 2, 3, 5, 6, 7, 10, 11, 13, 15, 17, 19, 30} The shaded region represents A ∪ B. Example 2: If F = {x: x is a multiple of 5, 20<x<50} and T = {y: y is a multiple of 10, 20<y<50}, find F ∪ T and show it in a Venn-diagram. Solution: Here, F = {x: x is a multiple of 5, 20<x<50} = {25, 30, 35, 40, 45} T = {y: y is a multiple of 10, 20<y<50} = {30, 40} Now, F ∪ T = {25, 30, 35, 40, 45} ∪ {30, 40} = {25, 30, 35, 40, 45} The shaded region represents F ∪ T. 2. Intersection of sets Let’s study the following examples and investigate the idea about intersection of sets. a) The lists of a few animals that live on land (L) and in water (W) are shown in the table given below. Land (L) Water (W) cat, duck, horse, frog, rabbit, cow, crocodile, elephant, tiger fish, frog, crocodile, duck, turtle, octopus, sea-horse Now, the list of animals that live both on land and in water is {duck, frog, crocodile} The, new set so formed is the intersection of sets of animals that live in land (L) and the set of animals that live in water (W). b) Let’s take any two sets P and Q, where P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 7, 9}. Now, the intersection of sets P and Q denoted as P ∩ Q = {3, 5, 7}. Thus, the intersection of two or more sets is formed just by listing their common element/s in a separate set. A B 7 1 6 2 3 5 11 10 15 30 13 17 19 U F T 25 30 40 35 45 U Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 15 Vedanta Excel in Mathematics - Book 9 The shaded regions in the following Venn-diagrams represent the intersection of the given sets. P U P ∩ Q ∩ R P, Q and R are overlapping sets. Q R P Q U P ∩ Q P and Q are overlapping sets P Q U P ∩ Q Q is subset of P Facts to remember 1. The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that belong to both the sets A and B. In set-builder form, intersection of sets A and B is defined as: A ∩ B = {x : x ∈ A and x ∈ B} 2. A ∩ A = A 3. A ∩ f = f 4. A ∩ U = A 5. (A ∩ B) ⊆ A and (A ∩ B) ⊆ B 6. A ∩ B = A if A ⊆ B and A ∩ B = B if B ⊆ A Example 3: If A = {odd numbers less than 10} and B = {prime numbers less than 15} are the subsets of a universal set U, find A ∩ B and show it in a Venn-diagram. Solution: Here, A = {odd numbers less than 10} = {1, 3, 5, 7, 9} B = {prime numbers less than 15} = {2, 3, 5, 7, 11, 13} Now, A ∩ B = {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7, 11, 13} = {3, 5, 7} The shaded region represents A ∩ B. Example 4: If M= {x: x = 100n, n∈N and n<5} and N = {y: y= 200n, n∈N and n<3}, find M ∩ N and show it in a Venn-diagram. Solution: Here, M = {x: x = 100n, n∈N and n<5} = {100, 200, 300, 400} N= {y: y = 200n, n∈N and n<3} = {200, 400} Now, M ∩ N = {100, 200, 300, 400} ∩ {200, 400} = {200, 400} The shaded region represents M ∩ N. 3. Difference of sets a) The sets of teachers teaching in basic level (B) and secondary level (S) of a school are shown in the Venn-diagram. Let's make the following sets. (i) The set of teachers who teach in both the levels. A B 1 2 3 5 7 9 11 13 U M N 100 200 400 300 U B S Bishnu Bina Januka Surendra Ramesh Nischal Sunita Kamala Ambika Hari U Sets

Vedanta Excel in Mathematics - Book 9 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur (ii) The set of teachers who teach in either basic or secondary level. (iii) The set of teachers who teach in basic level only. (iv) The set of teachers who teach in secondary level only. b) Let A = {1, 2, 3, 4, 5} and B = {2, 3, 5, 7, 11} be any two sets. Then, the difference between the sets A and B denoted as A – B = {1, 4}. Also, the difference between the sets B and A denoted as B – A = {7, 11}. Thus, the difference of two sets A and B denoted by ‘A – B’ is the set of the elements of only A which do not belong to B. Similarly, the difference of two sets B and A denoted by B – A is the set of the elements of only B which do not belong to A. In set-builder form, the difference of sets A and B is defined as: A – B = {x: x∈A, but x∉B} Also, the difference of sets B and A is defined as: B – A = {x: x ∈B, but x ∉A} In this way, A – B = A – (A ∩ B) ←Removing the common elements form A B – A = B – (A ∩ B) ←Removing the common elements form B The shaded regions in the following Venn-diagrams represent the difference of the given sets. B – A A and B are disjoint sets A B U A B U A – B B is subset of A B A U B – A A is subset of B A – B A and B are overlapping sets A B U B – A A and B are overlapping sets A B U A – B A and B are disjoint sets A B U The union of A – B and B – A is called symmetric difference between the sets A and B. It is denoted by A ∆ B. Thus, A ∆ B = (A – B) ∪ (B – A) The shaded regions in the following Venn-diagram represent the symmetric difference between the given sets. A B U B A U A B U A – B B – A B ∆ A ∪ = Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 17 Vedanta Excel in Mathematics - Book 9 Facts to remember 1. A – A = ∅ 2. A – ∅ = A 3. A – B = A and B – A = B if A and B are disjoint sets. 4. In general, A – B ≠ B – A but A – B = B – A only when A = B 5. A D B = B D A Example 5: Let P = {x: x is a positive even number and x≤12}, Q = {y: y is a factor of 18} and R = {z: z = 3n, n∈W and n ≤ 2} are three sets. Find : (i) P – Q (ii) Q – P (iii) Q – R (iv) R – P Also, represent the operations by shading in Venn-diagrams. Solution: Here, P ={x: x is a positive even number and x≤12} = {2, 4, 6, 8, 10, 12}, Q = {y: y is a factor of 18} = {1, 2, 3, 6, 9, 18} and R = {z: z = 3n, n∈W and n ≤ 2} = {1, 3, 9} Now, (i) P – Q = {2, 4, 6, 8, 10, 12} – {1, 2, 3, 6, 9, 18} = {4, 8, 10, 12} The shaded region represents P – Q. (ii) Q – P ={1, 2, 3, 6, 9, 18}–{2, 4, 6, 8, 10, 12} = {1, 3, 9, 18} The shaded region represents Q – P. (iii) Q – R ={1, 2, 3, 6, 9, 18}–{1, 3, 9} = {2, 6, 18} The shaded region represents Q – R. (iv) R – P ={1, 3, 9}–{2, 4, 6, 8, 10, 12} = {1, 3, 9} The shaded region represents R – P. 4. Complement of a set a) If the universal set, U is the set of all students of class 9 and A is the set of students who are present in class on Sunday in a certain week. Then, we can list out the students of class 9 who are absent on that day and we can form a new set of absentees. b) Let A = {1, 3, 5, 7, 9} be a subset of a universal set U = {1, 2, 3, …, 10}. Then, the complement of the set A denoted as A = {2, 4, 6, 8, 10}. Thus, the complement of set A is the difference of U and A. The complement of set A can also be denoted by A' or Ac or A. P Q 4 1 2 6 8 10 12 9 3 18 U P Q 4 1 2 6 8 10 12 9 3 18 U Q R 1 3 9 2 6 18 U R P U 1 2 6 10 12 4 8 3 9 Sets

Vedanta Excel in Mathematics - Book 9 18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur In set-builder form, the complement of a set A is defined as: A = {x : x∈U, but x∉A} which is U – A. Similarly, the set builder forms of the complements of union and intersection of set A and set B are: A ∪ B = {x: x∈U, but x ∉ (A or B)} or, {x: x∈ U, but x∉A∪B} A ∩ B = {x: x∈U, but x ∈(A and B)} or, {x: x∈ U, but x∉A∩B} The shaded regions in the following Venn-diagrams represent the complement of the given sets. A ∩ B A B U A – B A B U A ∪ B A B U A U A Facts to remember 1. If A be a subset of a universal set U, the complement of A, denoted by A or A' or Ac is the set of the elements of U which do not belong to the set A. 2. A = U – A = A 3. U = U – U = ∅ 4. ∅ = U – ∅ = U 5. A ∪ A = U 6. A ∩ A = ∅ 7. A – B = A ∩ B and B – A = A ∩ B 8. A ∪ B = A ∩ B and A ∩ B = A ∪ B Example 6: Let A = {2, 4, 6, 8, 10} and B = {2, 3, 5, 7} are the subsets of the universal set of U = {1, 2, 3, 4, ... 10} . Compute the following set operations with Venn-diagrams. (i) A ∪ B and A ∪ B (ii) A ∩ B and A ∩ B (iii) A – B and A – B (iv) B – A and B – A Solution: Here, A = {2, 4, 6, 8, 10}, B = {2, 3, 5, 7} and U = {1, 2, 3, 4, … 10} (i) A ∪ B = {2, 3, 4, 5, 6, 7, 8, 10} A ∪ B = U – (A ∪ B) = {1, 9} (ii) A ∩ B = {2} A ∩ B = U – (A ∩ B) = {1, 3, 4, 5, 6, 7, 8, 9, 10} U A 4 6 3 5 7 8 10 B A ∪ B 2 1 9 A ∪ B U A B 2 1 9 4 6 8 10 3 5 7 4 6 3 5 7 8 10 1 9 U A B A ∩ B A ∩ B U A B 2 1 9 4 6 8 10 3 5 7 2 https://www.geogebra.org/m/dbgxdh3p Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 19 Vedanta Excel in Mathematics - Book 9 (iii) A – B = A – (A ∩ B) = {4, 6, 8, 10} A – B = U – (A – B) = {1, 2, 3, 5, 7, 9} (iv) B – A = B – (A ∩ B) = {3, 5, 7} B – A = U – (B – A) = {1, 2, 4, 6, 8, 9, 10} Example 7: S is a subset of a universal set U. If U = {x: x ∈N, x≤10} and S = {y: y is a perfect square number }, find: (i) S (ii) S ∪ S (iii) S ∩ S (iv) S Solution: Here, U = {x: x ∈N, x≤10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and S = {y: y is a perfect square number } = {1, 4, 9} Now, (i) S = U – S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 4, 9} = {2, 3, 5, 6, 7, 8, 10} (ii) S ∪ S = {1, 4, 9} ∪ {2, 3, 4, 5, 6, 7, 8, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}=U (iii) S ∩ S = {1, 4, 9} ∩ {2, 3, 4, 5, 6, 7, 8, 10} = ∅ (iv) S = U – S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {2, 3, 5, 6, 7, 8, 10} = {1, 4, 9} = S Example 8: P and Q are the subsets of a universal set U. If U = {–4, –3, –2, –1, 0, 1, 2, 3, 4}, P = {–4, –2, 0, 2, 4} and Q = {–2, –1, 0, 1, 2}, verify that P ∪ Q = P ∩ Q . Solution: Here, U = {–4, –3, –2, –1, 0, 1, 2, 3, 4}, P = {–4, –2, 0, 2, 4}, and Q = {–2, –1, 0, 1, 2} Now, P∪Q = {–4, –2, 0, 2, 4}∪{–2, –1, 0, 1, 2}={–4, –2, –1, 0, 1, 2, 4} Also, P ∪ Q = U – (P ∪ Q) = {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–4, –2, –1, 0, 1, 2, 4} = {–3, 3} … (i) Again, P = U – P = {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–4, –2, 0, 2, 4} = {–3, –1, 3} Q = U – Q = {–4, –3, –2, –1, 0, 1, 2, 3, 4} – {–2, –1, 0, 1, 2} = {–4, –3, 3, 4} ∴P ∩ Q = {–3, –1, 3} ∩ {–4, –3, 3, 4} = {–3, 3} … (ii) From (i) and (ii), P ∪ Q = P ∩ Q verified Example 9: Let A = {x: x is a factor of 18 less than 10} and B= {y: y is a multiple of 3, y ≤ 15} are the subsets of a universal set U = {1, 2, 3,…, 15}. (i) List out the elements of A ∪ B. (ii) Illustrate the above information in a Venn-diagram. (iii) Prove: A ∆ B = (A ∪ B) – (A ∩ B) (iv) State the relation between the sets A ∆ B and (A ∩ B)′ with reason. 4 6 3 5 7 8 10 1 9 U A B A – B A – B U A B 2 1 9 4 6 8 10 3 5 7 2 4 6 3 5 7 8 10 1 9 U A B B – A B – A U A B 2 1 9 4 6 8 10 3 5 7 2 Sets

Vedanta Excel in Mathematics - Book 9 20 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Solution: Here, U = {1, 2, 3, …, 15}, A = {1, 2, 3, 6, 9}and B = {3, 6, 9, 12, 15} (i) A ∪ B = {1, 2, 3, 6, 9} ∪ {3, 6, 9, 12, 15} = {1, 2, 3, 6, 9, 12, 15} (ii) Illustrating the above information in a Venn-diagram: (iii) Now, A – B = {1, 2, 3, 6, 9} – {3, 6, 9, 12, 15} = {1, 2} B – A = {3, 6, 9, 12, 15} – {1, 2, 3, 6, 9} = {12, 15} We have, A ∆ B = (A – B) ∪ (B – A) = {1, 2, 12, 15} … (i) Again, A ∩ B = {1, 2, 3, 6, 9} ∩ {3, 6, 9} = {3, 6, 9} ∴ (A ∪ B) – (A ∩ B) = {1, 2, 3, 6, 9, 12, 15} – {3, 6, 9} = {1, 2, 12, 15} … (ii) From (i) and (ii), we get A ∆ B = (A ∪ B) – (A ∩ B) verified. (iv) A ∆ B = {1, 2, 12, 15} (A ∩ B)′ = U – (A ∩ B) = {1, 2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15} Since, all the elements of A ∆ B are also the elements of (A ∩ B)′. Thus, A ∆ B is a proper subset of (A ∩ B)′. Example 10: P, Q, and R are the subsets of the universal set U. If U = {whole numbers less than 12}, P = {natural numbers less than 6}, Q = {odd numbers less than 12}, and R = {prime numbers} , answer the following questions. (i) What is the cardinal number of U? (ii) Write the set P ∩ Q ∩ R in listing method. (iii) Find (P ∪ Q) – R and show it in a Venn-diagram by shading. (iv) If the subset Q were the set of perfect square numbers less than 10, what would be the elements of (P ∪ Q) ∩ R? Solution: Here, U = {0, 1, 2, 3, …, 11}, P = {1, 2, 3, 4, 5}, Q = {1, 3, 5, 7, 9} and R = {2, 3, 5, 7, 11} (i) U has 12 elements. So, the cardinal number of U is 12 , i.e., n (U) = 12 (ii) P ∩ Q ∩ R = {1, 2, 3, 4, 5}∩ {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7, 11} = {3, 5} (iii) Now, P ∪ Q = {1, 2, 3, 4, 5}∪{1, 3, 5, 7, 9} = {1, 2, 3, 4, 5, 7, 9} Also, (P ∪ Q) – R = {1, 2, 3, 4, 5, 7, 9} – {2, 3, 5, 7, 11} = {1, 4, 9} U A B 4 1 3 12 10 11 13 14 15 6 5 2 7 8 9 https://www.geogebra.org/m/zbtapnzy Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 21 Vedanta Excel in Mathematics - Book 9 Again, (P ∪ Q) – R = U – [(P ∪ Q) – R] = {0, 1, 2, …, 11} – {1, 4, 9} = {0, 2, 3, 5, 6, 7, 8, 10, 11} The shaded region represents (P ∪ Q) – R (iv) Q = {perfect square numbers} = {1, 4, 9} P ∪ Q = {1, 2, 3, 4, 5} ∪{1, 4, 9} = {1, 2, 3, 4, 5, 9} ∴ (P ∪Q) ∩ R = {1, 2, 3, 4, 5, 9} ∩ {2, 3, 5, 7, 11} = {2, 3, 5} EXERCISE 1.1 General section 1. Let P and Q are the subsets of a universal set U. Write the set operations defined by the following set-builder forms. a) {x: x ∈ P or x∈Q} b) {x: x∈P and x ∈ Q} c) {x: x∈P but x ∉Q} d) {x: x ∈ Q but x∉P} e) {x: x∈U but x ∉ P} f) {x: x∈U but x ∉Q} g) {x: x ∈ U but x ∉P or Q} h) {x: x∈U but x ∉P and Q} i) {x: x∈U but x ∉P – Q} 2. Write the set operations represented by shaded regions shown in the following Venn-diagrams. c) U A B a) U A B d) U A B b) U A B U A B e) U X Y f) g) U A B C h) U P Q R 3. a) A and B are the subsets of the universal set U. From the given diagrams, list the elements of the following set operations. (i) A ∪ B and A ∪ B (ii) A ∩ B and A ∩ B (iii) A – B and A – B (iv) B – A and B – A b) P, Q and R are the subsets of the universal set U. List the elements of the following set operations from the given diagram. (i) P ∪ Q ∪ R (ii) P ∩ Q ∩ R (iii) P ∪ Q ∪ R (iv) P ∩ Q ∩ R (v) (P ∪ Q) ∩ R (vi) (P ∩ Q) ∪ R P U Q R 1 2 11 5 7 4 6 0 8 10 9 3 U A B 5 7 4 8 10 2 6 1 3 9 U P Q R 7 11 13 14 8 10 3 5 1 9 15 12 6 2 4 Sets

Vedanta Excel in Mathematics - Book 9 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 4. a) If A = {n, e, p, a, l} and B = {b, h, u, t, a, n}, find (i) A∪B (ii) A∩B (iii) A – B (iv) B – A. Also, represent them in Venn-diagrams. b) Let P = {x: x ∈N and x<10} and Q= {y: y is a factor of 8}, find (i) P ∪ Q (ii) P ∩ Q (iii) P – Q (iv) Q – P. Also, show these operations in Venn-diagrams. c) If M = {x: x is an odd number between 10 and 20} and Q= {y: y is a prime number between 15 and 25}, find the following set operations and show in Venn-diagrams. (i) M ∪ N (ii) M ∩ N (iii) M – N (iv) N – M 5. a) Assuming that A and B are two overlapping sets, draw two separate Venn-diagrams to verify A ∪ B = B ∪ (A – B) by shading. b) Let P and Q are two overlapping sets. Draw two separate Venn-diagrams of P – Q and P – (P ∩ Q) and verify P – Q = P – (P ∩ Q) by shading. 6. a) If A = {1, 2, 4, 8} and B = {4, 6, 8, 10}, find (A – B) ∪ (B – A). b) Find the symmetric difference between the following sets. (i) A = {m, a, t, h} and B = {m, i, n, d, e, r} (ii) P = {2, 3, 5, 7, 11} and Q={1, 3, 5, 11} 7. a) If U = {1, 2, 3, …, 10}, A ={1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, find the following sets. (i) A (ii) B (iii) A ∪ B (iv) A ∩ B (v) A ∪ B (vi) A ∩ B (vii) A (viii) B b) If U = {1, 2, 3, …, 15} and A ={2, 4, 6, 8, 10, 12, 14}, find: (i) A (ii) A ∪ A (iii) A ∩ A (iv) U Creative section 8. a) If A = {letters in the word ‘mathematics’} and B = {letters in the word ‘science’}, verify that: (i) A ∪ (A ∩ B) = A ∩ (A ∪ B) (ii) A – (A ∩ B) = (A ∪ B) – B (iii) A ∪ (B– A) = B ∪ (A – B) (iv) (A ∪ B) – (A ∩ B) = (A – B) ∪ (B – A) b) If U = {0, 1, 2, …., 10}, A = {2, 3, 5, 7}and B = {1, 3, 5, 7, 9}, verify the following operations. (i) A ∪ B = A ∩ B (ii) A ∩ B = A ∪ B c) If a universal set U = {x : x ∈ N, x ≤ 10}, A = {y : y = 2n, n ∈ N, n < 5} and B = {z : z = 3n, n ∈ N, n<4 }, prove that: (i) A – B = B – A (ii) A D B = A D B 9. a) If P = {1, 2, 3, 4, 5, 6}, Q = {2, 4, 6, 8}, and R = {3, 6, 9, 12}, explore the relationship between the following operations. (i) P ∪ (Q ∩ R) and (P ∪ Q) ∩ (P ∪ R) (ii) P ∩ (Q ∪ R) and (P ∩ Q) ∪ (P ∩ R) Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 23 Vedanta Excel in Mathematics - Book 9 b) Let, A={1, 2, 3, 4, 6, 8, 12, 24}, B={3, 6, 9, 12, 15, 18} and C = {2, 3, 5, 7, 11, 13, 17}, verify the following relations. (i) A – (B ∪ C) = (A – B) ∩ (A – C) (ii) A – (B ∩ C) = (A – B) ∪ (A – C) (iii) A ∩ (B – C) = (A ∩ B) – (A ∩ C) (iv) A ∩ (B ∆ C) = (A ∩ B) ∆ (A ∩ C) c) A, B and C are the subsets of a universal set U. If U = {x: x∈N, x≤12}, A = {odd numbers less than 10}, B = {prime numbers less than 12}, and C = {prime numbers less than 6}, verify the following operations. (i) A ∩ B ∩ C = A ∪ B ∪ C (ii) A ∪ B ∪ C = A ∩ B ∩ C 10. a) A and B are the subsets of a universal set U. If U = {x: x is a natural number, x ≤ 10}, A = {y: y is an odd number less than 10} and B= {z: z is a prime number less than 10}, answer the following questions. (i) List the elements of A ∩ B. (ii) What is the cardinal number of A – B? (iii) Find A ∪ B and illustrate it in a Venn-diagram by shading. (iv) What is the relation between the sets A – B and A? Give reason. b) Let U = {x: x is a whole number, x ≤ 12} is a universal set, P = {y: y is a factor of 6} and Q= {z: z is a multiple of 3 less than 15} are the subsets of U. (i) List the elements of P ∪ Q. (ii) What is the cardinal number of Q – P? (iii) Find P – Q and show it in a Venn-diagram by shading. (iv) Mention the type of the set P ∩ P with reason. 11.a) If A = {1, 3, 5, 7, 9, 11}, B = {1, 2, 3, 4, 5, 6, 7} and C = {3, 6, 9, 12, 15} are the subsets of the universal set U = {1, 2, 3, ..., 15}, answer the following questions. (i) Find A ∩ B ∩ C in listing method. (ii) List the elements of A – (B ∩ C). (iii) Find A ∪ (B – C) and illustrate it in a Venn-diagram by shading. (iv) Write the relation between the sets A ∩ B and A ∩ B ∩ C with reason. b) X, Y, and Z are the subsets of A universal set U. If U = {a, b, c, …, j}, X = {a, b, c, d}, Y = {c, a, f, e}, and Z = {c, d, e, g, h} , answer the following questions. (i) List the elements of X ∪ Y ∪ Z. (ii) List the elements of X ∪ (Y – Z). (iii) Find X ∩ (Y ∪ Z) and illustrate it in a Venn-diagram by shading. (iv) Write the relation between the sets X ∪ Y ∪ Z and X ∪ Y with reason. c) Let U ={students of a class with roll numbers from 1 to 20} and its subsets are: A = {students of even roll numbers involving in sports}, B = {students of roll numbers multiples of 3 involving in music} and C = {students of roll numbers factors of 24 involving in dance} Sets

Vedanta Excel in Mathematics - Book 9 24 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur (i) List out the elements of each set. (ii) Show the above information in a Venn-diagram. (iii) Verify that: A – (B ∪ C) = (A – B) ∩ (A – C) (iv) How many students are not involving in all three activities? 12. a) If A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} are two disjoint sets verify that n(A ∪ B) = n(A) + n(B). b) If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 6, 12} are two overlapping sets, show that n(A∪B) = n(A) + n(B) – n(A∩B) Project work and activity section 13. Conduct a survey among your at least 20 friends whether they like to play volleyball or basketball or both. Represent the set of students who like to play volleyball by V and the set of students who like to play basketball by B. Draw Venn-diagrams and show each of the following set operations by shading. (i) Set of students who like to play volleyball or basketball or both, V ∪ B. (ii) Set of students who like to play volleyball as well as basketball, V ∩ B. (iii) Set of students who like to play volleyball but not basketball, V – B. (iv) Set of students who like to play basketball but not volleyball, B – V (v) Set of students who like to play none of these games, V ∪ B . (vi) Set of students who like to play only one of these games, V ∆ B. 14. Suppose, A and B are overlapping sets and the subsets a universal set U. Draw the separate Venn-diagrams in the chart paper and shade to verify the following operations. (i) (A – B) ∪ (B – A) = (A∪B) – (A∩B) (ii) A ∪ B = A ∩ B (iii) A – B = A ∪ B ∪ B 1.11 Cardinality relationships of two sets The cardinality of a set is a measure of the number of members of the set. For example, the set A = {1, 3, 5, 7, 9} contains 5 members, and therefore A has a cardinality of 5. The number of members of a set is called its cardinality. Certain relationships can be generalised by taking the cardinalities of different sets. (i) Cardinality relations of union of two disjoint sets Let U = {a, b, c, d, e, f, g, h, i, j} is a universal set. A = {a, b, c, d} and B = {e, f, g, h, i} are the subsets of U. Now, A ∪ B = {a, b, c, d, e, f, g, h, i} Here, n (U) = 10, n (A) = 4 and n (B) = 5 Now, n (A ∪ B) = 9 = 4 + 5 = n (A) + n (B) https://www.geogebra.org/m/mxm83pss Vedanta ICT Corner Please! Scan this QR code or browse the link given below: A B a e b f c d g h i j U Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 25 Vedanta Excel in Mathematics - Book 9 Also, n (A ∪ B) = 1 = 10 – 9 = n (U) – n (A ∪ B) Thus, if A and B are any two disjoint subsets of a universal set U, then n (A ∪ B) = n (A) + n (B) n (A ∪ B) = n (U) – n (A ∪ B) (ii) Cardinality relations of union of two overlapping sets Let A = {a, b, c, d, e} and B = {c, h, i, e, f} are two subsets of a universal set U = {a, b, c, d, e, f, g, h, i, j} Now, A ∪ B = {a, b, c, d, e, f, h, i} A ∩ B ={c, e} A ∪ B = {g, j} only A = A – B = {a, b, d} only B = B – A = {f, h, i} Here, n (U) = 10, n (A) = 5, n (B) = 5 and n (A ∩ B) = 2 Now, n (A ∪ B) = 8 = 5 + 5 – 2 = n (A) + n (B) – n (A ∩ B ) Also, n (A ∪ B) = 2 = 10 – 8 = n (U) – n (A ∪ B) Again, n (only A) = n(A – B) = nO(A) = 3 = 5 – 2 = n(A) – n(A ∩ B) n (only B) = n(B – A) = nO(B) = 3 = 5 – 2 = n(B) – n(A ∩ B) Thus, if A and B are any two overlapping subsets of a universal set U, n (A ∪ B) = n (A) + n (B) – n (A ∩ B) n (A ∪ B) = n (U) – n (A ∪ B) n (A ∩ B) = n (A) + n (B) – n (A ∪ B) n (only A) = no (A) = n(A) – n(A ∩ B) n (only B) =no (B) = n(B) – n(A ∩ B) Furthermore, if the universal set U contains only the members of A and B, n (A ∪ B) = 0 and n (U) = n (A ∪ B) Worked-out Examples Example 1: From the adjoining Venn-diagram, find the cardinal numbers of the following sets. a) n (P) b) n (Q) c) n (P ∪ Q) d) n (P ∩ Q) e) n (P ∪ Q) f) n (P) g) n (Q) h) no (P) i) no (Q) Solution: a) n (P) = 5 b) n (Q) = 6 c) n (P ∪ Q) = 9 U A B a c d e f i h n (A) n (B) n (only A) n (A ∩ B) n (only B) g j b n (A ∪ B) U P Q a d e f g h i j k c l b Sets

Vedanta Excel in Mathematics - Book 9 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur d) n (P ∩ Q) = 2 e) n (P ∪ Q) = 3 f) n (P ) = n (U) – n (P) = 12 – 5 = 7 g) n (Q) = n (U) – n (Q) = 12 – 6 = 6 h) no (P) = n(P) – n(P ∩ Q) = 5 – 2 = 3 i) no (Q) = n(Q) – n(P ∩ Q) = 6 – 2 = 4 Example 2: A and B are the subsets of a universal set U. If n (U) = 95, n (A) = 64, n (B) = 56, n (A ∩ B) = 30, find a) n (A ∪ B) b) n (A ∪ B) c) no (A) d) no (B) Solution: Here, n (U) = 95, n (A) = 64, n (B) = 56 and n (A ∩ B) = 30 a) Now, n (A ∪ B) = n (A) + n (B) – n (A ∩ B) = 64 + 56 – 30 = 90 b) n (A ∪ B) = n (U) – n (A ∪ B) = 95 – 90 = 5 c) no (A) = n (A) – n (A ∩ B) = 64 – 30 = 34 d) no (B) = n (B) – n (A ∩ B) = 56 – 30 = 26 Example 3: If n(A – B) = 24, n(A ∪ B) = 80, and n(A ∩ B) = 20, then find n(B). Illustrate this information in a Venn-diagram. Solution: Here, n(A – B) = n0 (A) = 24, n(A ∪ B) = 80 and n(A ∩ B) = 20 Now, n(A ∪ B) = n0 (A) + n(A ∩ B) + n0 (B) or, 80 = 24 + 20 + n0 (B) or, n0 (B) = 36 Again, n(B) = n0 (B) + n(A ∩ B) = 36 + 20 = 56 Example 4: If n (U) = 120, n (A) = 63, n (B) = 54 and n (A ∪ B) = 99, complete the following: (i) Find n (A ∪ B), n (A ∩ B), no (A) and no (B). (ii) Illustrate the information in a Venn-diagram. Solution: Here, n (U) = 120, n (A) = 63, n (B) = 54 and n (A ∪ B) = 99 (i) Now, n (A ∪ B) = n (U) – n ( A ∪ B) = 120 – 99 = 21 Again, n (A ∪ B) = n (A) + n (B) – n (A ∩ B) or, 99 = 63 + 54 – n (A ∩ B) or, n (A ∩ B) = 117 – 99 = 18 Also, no (A) = n (A) – n (A ∩ B) = 63 – 18 = 45 And no (B) = n (B) – n (A ∩ B) = 54 – 18 = 36 A B 24 20 36 U A B 45 18 36 21 (ii) Illustration in Venn-diagram Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 27 Vedanta Excel in Mathematics - Book 9 Example 5: In a survey of a few number of farmers, it is found that 50 farmers use chemical fertilizers, 65 use bio-fertilizers, and 35 use both chemical and bio-fertilizers. Find the number of farmers who use chemical or bio-fertilizers. Solution: Let C and B be the sets of the farmers who use chemical and bio-fertilizers respectively. Here, n(C) = 50, n(B) = 65 and n(C ∩ B) = 35 Now, n (C ∪ B) = n(C) + n(B) – n(C ∩ B) = 50 + 65 – 35 = 115 – 35 = 80 Hence, 80 farmers use chemical or bio-fertilizers. Example 6: In a group of 150 students who like at least one profession: educator or doctor. 80 of them like educator and 95 like doctor. By drawing Venn-diagram, find (i) How many students like both the professions? (ii) How many students like only educator? Solution: Let E and D be the sets of the students who like educator and doctor respectively. Let n(E ∩ D) = x, (i) Venn-diagram Example 7: In a survey, 140 cinema lovers were asked what they would like comedy or action movies. 72 of them said comedy movies and 53 said action movies. However, 45 people said they did not like any type of movies. (i) Find the number of people who like both comedy and action movies. (ii) Find the number of people who like only one type of movie. (iii) Illustrate the result in Venn-diagram. Solution: Let C and A be the sets of the people who like comedy and action movies respectively. Here, n (U) = 140, n (C) = 72, n (A) = 53 and n (C ∪ A ) = 45 (i) Now, n (C ∪ A) = n (U) – n (C ∪ A ) = 140 – 45 = 95 U E D 80-x x 95-x From the Venn-diagram, n (E ∪ D) = 80 – x + x + 95 – x or, 150 = 175 – x or, x = 175 – 150 = 25 Hence, 25 of them like both the professions. (ii) no (E) = 80 – x = 80 – 25 = 55 Sets

Vedanta Excel in Mathematics - Book 9 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Again, n (C ∪ A) =n (C) + n (A) – n (C ∩ A) or, 95 = 72 + 53 – n (C ∩ A) or, n (C ∩ A) = 125 – 95 = 30 Hence, 30 people like both comedy and action movies. (ii) Also, the number of people who like comedy movies only = no (C) = n (C) – n (C ∩ A) = 72 – 30 = 42 And, the number of people who like action moves only = no (A) = n (A) – n (C ∩ A) = 53 – 30 = 23 ∴ The number of people who like only one type of movie = 42 + 23 = 65 EXERCISE 1.2 General section 1. a) A and B are any two disjoint sets. If n(A) = x and n(B) = y, find n(A ∪ B). b) If n(A) = p, n(B) = q, and n (A ∩ B) = r, show this information in a Venn-diagram and show that n (A ∪ B) = n(A) + n(B) – n(A ∩ B). c) If X and Y are two overlapping subsets of a universal set U, write the relation between n(U), n (X ∪ Y), and n (X ∪ Y ). d) If A and B are two overlapping subsets of a universal set U, write the relation between n(A), n(A ∩ B) and no (A). 2. From the adjoining Venn-diagram, find the cardinal numbers of the following sets: a) n (U) b) n (A) c) n (B) d) n (A ∪ B) e) n (A ∩ B) f) n (A ∪ B ) g) n (A) h) n (B) i) no (A) j) no (B) 3. a) If n (U) = 65, n (A) = 28, n (B) = 45, and n (A ∩ B) = 20, find (i) n (A ∪ B) (ii) n (A ∪ B ) (iii) no (A) (iv) no (B) b) P and Q are the subsets of a universal set U. If n (P) = 55 %, n (Q) = 50 %, and n (P ∪ Q ) = 15 %, find: (i) n (P ∪ Q) (ii) n (P ∩ Q) (iii) n (only P) (iv) n (only Q) c) X and Y are the subsets of a universal set U. If n (U) = 88, no (X) = 35, no (Y) = 30, and n (X ∩ Y) = 10, find: (i) n (X) (ii) n (Y) (iii) n (X ∪ Y) (iv) n (X ∪ Y ) Creative section 4. a) In a survey of 600 people in a village of Dhading district, 400 people said they can speak Tamang language, 350 said Nepali language and 200 of them said they can speak both the languages. (i) Draw a Venn-diagram to illustrate the above information. (iii) Venn-diagram U C A 42 23 45 30 U A B 4 1 3 6 2 5 7 8 9 Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 29 Vedanta Excel in Mathematics - Book 9 (ii) How many people can speak Tamang language only? (iii) How many people can speak Nepali language only? (iv) How many people cannot speak any of two languages? b) In a survey of 1500 people, 775 of them like Nepal Idol, 975 liked Comedy Champion, and 450 people liked both the shows. (i) Show the above information in a Venn-diagram. (ii) How many people did not like both the shows? c) In a group of 250 music lovers, 135 of them like folk songs, and 150 like modern songs. By drawing a Venn-diagram, find: (i) how many people like both the songs? (ii) How many people like only folk songs? d) In a survey of a group of farmers, it was found that 80 % farmers have crops farming, 30% farmers have animals farming, and every farmer has at least one farming. (i) Represent the information in a Venn-diagram. (ii) What percent of farmers had both farming? (iii) What percent of farmers had only animals farming? 5. a) In a group of 500 students, 280 like bananas, 310 like apples, and 55 do not like both the fruits. (i) Find the number of students who like both the fruits. (ii) Find the number of students who like only one fruit. (iii) Show the result in a Venn-diagram. b) In a survey of 900 students in a school, it was found that 600 students liked tea, 500 liked coffee, and 125 did not like both drinks. (i) Draw a Venn-diagram to illustrate the above information. (ii) Find the number of students who like both drinks. (iii) Find the number of students who like one of these drinks only. 6. a) In a group of 75 students, 20 liked football only, 30 liked cricket only and 18 did not like any of these two games? (i) How many students liked at least one game? (ii) Find the number of students who liked both the games. (iii) How many of them liked football? (iv) How many of them liked cricket? (v) Represent the result in a Venn diagram. b) In a survey of 750 tourist who visited Nepal during 'Visit Nepal 2020', it was found that 260 tourists visited Pokhara but not Sauraha, 240 visited Sauraha but not Pokhara and 125 of them did not visit both places. (i) Find the number of tourists who visited Pokhara or Sauraha. (ii) Find the number of tourist who visited Pokhara and Sauraha. (iii) How many tourists visited Pokhara. (iv) Show the result in a Venn-diagram. Sets

Vedanta Excel in Mathematics - Book 9 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Project work and activity section 7. Conduct a survey inside your classroom and collect the data about how many of your friends like football, cricket, and both football and cricket. Then, tabulate the data and find the following numbers by using cardinality relation of two sets: a) Number of friends who like football and cricket. b) Number of friends who do not like any of these two games. c) Number of friends who like only (i) football (ii) cricket d) Show your data in a Venn-diagram 8. Make one verbal problem reflecting to the real life situations by using each of the following sets of known and unknown variables: a) n(U) = 310, n(A) = 120, n(B) = 180, n(A ∩ B) = 40, n(A ∪ B) = ? b) n(U) = 190, n(A) = 105, n(B) = 115, n(A ∪ B) = 15, n(A ∩ B) = ? c) no (A) = 72, n(B) = 100, n(A ∩ B) = 36, n(A ∪ B) = 28, n(U) = ? Now, solve your verbal problems and illustrate the results in Venn-diagrams. OBJECTIVE QUESTIONS Let’s tick (√) the correct alternative. 1. In set builder form, A∪B is written as (A) {x: x ∈ A or x∈B} (B) {x: x ∈ A and x ∈B} (C) {x: x∈A, but x∉B} (D){x: x ∈ B, but x ∉A} 2. The set operation defined by {x: x∈P and x∈Q} is (A) P∩Q (B) P ∪ Q (C) P – Q (D) (P ∩ Q)′ 3. If A = {x : x ∈N, x < 5} and B = {y: y ∈ N, 3 < y < 7} then A ∪ B is (A) {4} (B) {1, 2, 3} (C) {5, 6} (D) {1, 2, 3, 4, 5, 6} 4. Which of the following statements is true? (A) A ∪ B = A ∩ B (B) A ∩ B =A – B (C) A ∪ B = A – B (D) A ∩ B ⊆ A 5. If A ⊂ B then A ∩ B is equal to (A) A (B) B (C) A D B (D) A ∪ B 6. If U = {1, 2, 3, 4, 5} and A = {2, 3} then A'' is (A) {} (B) {1, 4, 5} (C) {2, 3} (D) {1, 3, 5} 7. Which of the following relations is not correct? (A) A ∪ B = B ∪ A (B) A ∩ B = B ∩ A (C) A – B = B – A (D) A ∪ ∅ = A 8. Which of the following relations is correct? (A) A ∩ ∅ = ∅ (B) A ∪ ∅ = ∅ (C) A ∪ U = A (D) A ∩ U = U 9. Which of the following relations is correct? (A) ∅' = ∅ (B) A ∪ A' = ∅ (C) A ∩ A = U (D) A ∪ A' = U 10. Which one of the following relation is always true on the operations of sets? (A) A ∪ B = A ∪ B (B) A ∩ B = A ∩ B (C) A ∩ B = A ∪ B (D) A ∩ B = A – B Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 31 Vedanta Excel in Mathematics - Book 9 Assessment –I 1. U = {x: x is a whole number, x ≤ 10} is a universal set. A = {y: y is an odd number} and B = {z: z is a factor of 12} are the subsets of U, answer the following questions. (a) Write A – B in listing method. (b) Show the relationship of U, A and B in a Venn-diagram. (c) Verify that: A ∪ B = A ∩ B (d) Write the relation between the sets A ∪ B and A ∩ B with reason. 2. U = {x: 5 ≤ x ≤ 15, x∈ N} is a universal set. The sets X = {factors of 10} and Y = {multiples of 5}, are the subsets of U, answer the following questions. (a) What is the cardinal number of universal set U? (b) Find X ∪ Y in listing method. (c) Prove that (X – Y) ∪ (Y – X) = (X∪ Y) – (X∩Y) (d) What type of set X ∩ X? Give reason. 3. H and T are the subsets of a universal set U where U = {x: x ≤ 10, x∈ N}, H = {y: y=2n, n ∈ W, n ≤ 3} and T= {z: z =3n, n ∈ W, n ≤ 2}, answer the following questions. (a) List the elements of H and T. (b) Find H and T in listing method. (c) Prove that n (H ∪ T) = n (H) + n (T) – n (H ∩ T) (d) Establish the relation between H – T and T – H. 4. A, B and C are the subsets of a universal set U. If U = {x: x ≤ 10, x∈ W}, A = {even numbers}, B = {prime numbers} and C = {factors of 12}, answer the following questions. (a) Write A ∪ B ∪ C in listing method. (b) Show the above information in a Venn-diagram. (c) Prove that: A – (B ∩ C) = (A – B) ∪ (A – C) (d) Is A ∪ B a subset of A? Give reason. 5. P, Q, and R are the subsets of the universal set U. If U = {whole numbers less than 12}, P = {natural numbers less than 6}, Q = {odd numbers}, and R = {prime numbers}, answer the following questions. (a) What is the cardinal number of U? (b) Write the set P ∪ Q ∪ R in listing method. (c) Find P ∩ (Q ∪ R) and show it in a Venn-diagram by shading. (d) If the subset Q were the set of perfect square numbers less than 10, what would be the elements of (P ∪ Q) – R?

Vedanta Excel in Mathematics - Book 9 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2.1 Tax – Introduction Let's study a few examples of taxes. a) Bimal paid Rs 5,000 for bluebook renewal tax of his bike. b) Nikita paid 1% of her yearly income as social security tax. c) Anil paid 25% tax for his normal business. d) Mr. Chaudhary sold a television with 13% VAT. e) Discuss with your family members and the various types of taxes paid by your family. f) Discuss with your friends about the various types of taxes paid by their families. The government does not have its own money. Its receipts come from individual income taxes, corporate income taxes, estate and gift taxes, social insurance taxes, excise taxes, etc. The taxes we pay are used by government for transport, education, health, law and order, culture, media and sport, trade and industry, environment, etc. The Inland Revenue Department (IRD) under the Ministry of Finance of the Government of Nepal is responsible for the administration of Value Added Tax, Income Tax, and Excise Duty. Facts to remember 1. A tax is a compulsory financial charge or some other type of levy imposed upon taxpayer by a government authority in order to fund various public expenditures. 2. Tax is a compulsory contribution to state revenue, levied by the government on citizen’s income and business profits, or added to the cost of some goods, services, and transactions. Here, among these taxes, we shall discuss about the income tax and Value Added Tax (VAT). Unit 2 Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 33 Vedanta Excel in Mathematics - Book 9 2.2 Income Tax An income tax is a tax imposed on individuals or taxpayers as per their annual income. The rates of income tax vary with the specified limits of taxable incomes. The specified limit of income to levy different rates of tax is decided by the government on the basis of minimum needs of individuals. The minimum needs of an individual are justified according to his/her personal details, e.g. marital status, number of dependents, etc. 1% Social Security Tax is levied on the annual income up to the certain limit. In this way, to calculate the income tax, at first the social security taxable income is deducted from the total income to find the taxable income above the minimum limit. Then, the tax is levied as per the rate of tax in percent. Taxable income = Total income – Social security taxable income Income tax = Rate of tax (in %) × Taxable income The table given below shows the taxable income and the tax rate in percent for the unmarried individuals and married couples based on Nepal Income Tax Rates 2079/79 (Source: www.nbsm.com.np). Tax Rates for Natural person for the Fiscal Year 2079/80 (2022/23) (Only for Employment Income) Assessed as Individuals Assessed as Couples Particulars Taxable Income (Rs) Tax Rate Taxable Income (Rs) Tax Rate First Tax slab Next Next Next Balance Exceeding 5,00,000 2,00,000 (5,00,001 to 7,00,000) 3,00,000 (7,00,001 to 10,00,000) 10,00,000 (10,00,001 to 20,00,000) 20,00,000 1% 10% 20% 30% 36% 6,00,000 2,00,000 3,00,000 9,00,000 20,00,000 1% 10% 20% 30% 36% (Only for Proprietorship firm) Assessed as Individuals Assessed as Couples Particulars Taxable Income (Rs) Tax Rate Taxable Income (Rs) Tax Rate First Tax slab Next Next Next Balance Exceeding 5,00,000 2,00,000 (5,00,001 to 7,00,000) 3,00,000 (7,00,001 to 10,00,000) 10,00,000 (10,00,001 to 20,00,000) 20,00,000 0% 10% 20% 30% 36% 6,00,000 2,00,000 3,00,000 9,00,000 20,00,000 0% 10% 20% 30% 36% https://www.geogebra.org/m/bzvyteuh Vedanta ICT Corner Please! Scan this QR code or browse the link given below: (6,00,001 to 8,00,000) (8,00,001 to 11,00,000) (11,00,001 to 20,00,000) (6,00,001 to 8,00,000) (8,00,001 to 11,00,000) (11,00,001 to 20,00,000) Taxation

Vedanta Excel in Mathematics - Book 9 34 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Tax rebate income: S.N. Title Limitations 1. Deposit in Employee provident Fund (PF) Up to Rs. 3 lakhs or one-third of assessable income whichever is lower 2. Deposit in Social Security Fund (SSF) Up to Rs. 5 lakhs or one-third of assessable income whichever is lower 3. Deposit in Citizen Investment Trust (CIT) 4. Premium of life insurance Annual premium of life insurance or up to Rs. 40,000 whichever is lower 5. Medical insurance premium Premium of health insurance or up to Rs. 20,000 whichever is lower 6. Remote area allowances (A, B, C, D, E) (A) Rs. 50,000 (B) Rs. 40,000 (C) Rs. 30,000 (D) Rs. 20,000 (E) Rs. 10,000 7. Religious functions 10% of income or Rs.10,00,000 whichever is lower 8. Foreign allowances 75% of foreign allowances of employee working in diplomatic mission in foreign countries. 9. Disable person Additional 50% of tax bindings deduction from his/her taxable income 10. Medical expenses 15% of actual expense on approved medical expenses or Rs. 750 whichever is lower. 11. Woman having only remuneration income In case of individual woman, 10% rebate on income tax Note: We can find the detail about taxation in the website http://www.ird.gov.np. 2.3 Tax Withholding Tax withholding, is an income tax paid to the government by the payer of the income rather than by the recipient of the income. The tax is withheld or deducted from the income due to the recipient. The withholding taxes are charged on payments of service fee, interest, dividends, royalties, rent or even the sale of real estate. The interest paid by a bank or financial institution to any natural person for deposit which is sourced in Nepal and is not related to operation of a business, then 5% final withholding tax applies. Simple interest When we borrow money from a bank, we should pay interest to the bank. When we deposit money in a bank, the bank pays interest to us. The interest which is calculated from the original borrowed (or deposited) sum is called simple interest. Let’s review the following terms which are needed to calculate simple interest. (i) Principal (P) – It is the deposited or borrowed sum of money. (ii) Rate of interest (R) – It is the interest of Rs 100 for 1 year. It is expressed as the percent per year (or per annum, p.a.) Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 35 Vedanta Excel in Mathematics - Book 9 (iii) Time (T) – It is the duration for which principal is deposited or borrowed. (iv) Interest (I) – It is the simple interest on Rs P at R% p.a. in T years. (v) Amount (A) – It is the total sum of principal and interest. So, A = P + I Facts to remember (i) If I is the simple interest on the principal Rs. P for the duration of T year at the rate of R% per annum, then I = PTR 100 (ii) The account holder is charged 5% tax on interest paid by the bank or any financial institutions as withholding tax. Worked-out Examples Example 1: Mr. Thakur is the sole proprietor of a barber shop. His annual income is Rs 6, 40, 000. Study the following tax rates and answer the questions. Assessed as Individuals Assessed as Couples Tax rates Up to Rs 5,00,000 Up to Rs 6,00,000 0% Rs 5,00,001 – Rs 7,00,000 Rs 6,00,001 – Rs 8,00,000 10% Rs 7,00,001 – Rs. 10,00,000 Rs 8,00,001 – Rs 11,00,000 20% a) How much social security tax should he pay? b) Calculate the annual income tax if he is married. c) If he were unmarried, how much more income tax be paid? Solution: a) For the case of sole proprietorship firm, social security tax is free. So, he should not pay it. b) If Mr. Thakur is married, Rs 6,40,000 should be divided in to tax slabs as Rs 6,40,000 = Rs 6,00,000 + Rs 40,000 Now, total income tax =10% of Rs 40,000 = Rs 4,000 Hence, the annual income tax is Rs 4,000. c) If Mr. Thakur is unmarried, Rs 6,40,000 should be divided in to tax slabs as Rs 6,40,000 = Rs 5,00,000 + Rs 1,40,000 Now, total income tax = 10% of Rs 1,40,000 = Rs 14,000 Difference between taxes due to marital status = Rs 14,000 – Rs 4,000 = Rs 10,000 Hence, he would pay Rs 10,000 more income tax if he were unmarried. Rs 6,40,000 Rs 6,00,000 Rs 40,000 (0%) (10%) Rs 6,40,000 Rs 5,00,000 Rs 1,40,000 (0%) (10%) Taxation

Vedanta Excel in Mathematics - Book 9 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Example 2: Siddhika is an unmarried employee in a commercial bank. Her monthly salary is Rs 40,500. She has to pay 1% social security tax on her income up to Rs 5,00,000 and 10% income tax on Rs 5,00,001 to Rs 7,00,000. If she gets 15 months’ salary in a year and 10% rebate on her income tax, answer the following questions. a) What is income tax? b) Find her annual income. c) How much annual income tax should she pay? Solution: a) The tax imposed by the government on the annual income of individuals is called income tax. b) Monthly income = Rs 40,500 ∴Annual income = 15 month's salary = 15 × Rs 40,500 = Rs 6,07,500 c) Splitting Rs 6,07,500 in to tax slabs as Rs 6,07,500 = Rs 5,00,000 + Rs 1,07,500 Now, total income tax = 1% of Rs 5,00,000 + 10% of Rs 1,07,500 = Rs 5,000 + Rs 10,750 = Rs 15,750 Hence, the annual income tax is Rs 15,750. But, she gets 10% rebate on her income tax. The annual income tax to be paid by her = Rs 15,750 – 10% of Rs 15,750 = Rs 15,750 – Rs 1,575 = Rs 14,175 Hence, she should pay Rs 14,175 as her annual income tax. Example 3: A company hired Mr. Chaudhary as the managing director. He is a married man and his details are given below. Basic salary: Rs 80,000 Festival expense: Basic salary of 1 month Dearness allowance: Rs 2,000 per month Citizen Investment Trust: 13% of the basic salary Premium of life insurance: Rs 50,000 Based on this information, answer the following questions. a) What is the annual basic salary of Mr. Chaudhary? b) Calculate his taxable income. c) Find the total income tax. Solution: a) Monthly basic salary = Rs 80,000 ∴His annual basic salary = 12 × Rs 80,000 = Rs 9,60,000 Rs 6,07,500 Rs 5,00,000 Rs 1,07,500 (1%) (10%) For couples Tax rates Up to Rs 6 lakh 1% Next Rs 2 lakh 10% Next Rs 3 lakh 20% Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 37 Vedanta Excel in Mathematics - Book 9 b) Calculating total assessable income Particulars Amount Annual basic salary Rs 9,60,000 Annual dearness allowance 12 × Rs 2,000 = Rs 24,000 Festival expense =One month’s basic salary Rs 80,000 Total assessable income Rs 10,64,000 Also, calculating tax rebate amount Heading Tax rebate amounts Remarks Citizen Investment Trust 13% of Rs 9,60,000 = Rs 1,24,800 Life insurance Maximum limitation = Rs 40,000 Whichever lower Actual premium = Rs 50,000 Total tax rebate amount Rs 1,24,800 + Rs 40,000 = Rs 1,64,800 ∴Taxable income = Total assessable income – Tax rebate amount = Rs 10,64,000 – Rs 1,64,800 = Rs 8,99,200 c) Splitting Rs 8,99,200 in to tax slabs as Rs 8,99,200 = Rs 6,00,000+Rs 2,00,000+Rs 99,200 Now, total income tax = 1% of Rs 6,00,000 + 10% of Rs 2,00,000 + 20% of Rs 99,200 = Rs 6,000 + Rs 20,000 + Rs 19,840 = Rs 45,840 Hence, the annual income tax is Rs 45,840. Example 4: Dharmendra Sah is an unmarried secondary level Science teacher in a school. His monthly salary is Rs 47,988 with dearness allowance of Rs 2,000. He gets Dashain allowance which is equivalent to his basic salary of one month. He contributes 10% of his basic salary in Employee's Provident Fund (EPF) and he pays Rs 30,000 as the premium of his life insurance. Given that 1% social security tax is levied upon the income of Rs 5,00,000, 10% and 20% taxes are levied on the next incomes of Rs 2,00,000 and up to Rs 3,00,000 respectively. Answer the following questions. a) What is his monthly basic salary? b) Find his taxable income. c) Find his total income tax to be paid. Solution: a) Monthly basic salary = Rs 47,988 – Rs 2,000 = Rs 45,988 Help line Taxable income = Rs 8,99,200 Slabs Rates Rs 6,00,000 1% Rs 2,00,000 10% Rs 99,200 20% Taxation

Vedanta Excel in Mathematics - Book 9 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur b) Calculating total assessable income Particulars Amount Annual basic salary 12 × Rs 45,988 = Rs 5,51,856 Annual dearness allowance 12 × Rs 2,000 = Rs 24,000 Festival expense =One month’s basic salary Rs 45,988 Additional provident fund = 10% of Rs 5,51,856 Rs 55,185.60 Total assessable income Rs 6,77,029.60 Also, calculating tax rebate amount Heading Tax rebate income Remarks Provident Fund One-third of assessable income = 1 3 × Rs 6,77,029.60 = Rs 2,25,676.53 Whichever Maximum limitation = Rs 3,00,000 is least Actual contribution in PF = 20% of Rs 5,51,856 = Rs 1,10,371.20 Life insurance Maximum limitation = Rs 40,000 Whichever is Lower Actual premium = Rs 30,000 Total tax rebate amount = Rs 1,10,371.20 + Rs 30,000 = Rs 1,40,371.20 ∴Taxable income = Total assessable income – Tax rebate amount = Rs 6,77,029.60 – Rs 1,40,371.20 = Rs 5,36,658.40 c) Splitting Rs 5,36,658.40 in to tax slabs as Rs 5,36,658.40= Rs 5,00,000 + Rs36,658.40 Now, Total income tax = 1% of Rs 5,00,000 + 10% of Rs 36,658.40 = Rs 5,000 + Rs 3665.84 = Rs 8,665.84 Hence, the annual income tax is Rs 8,665.84 EXERCISE 2.1 General section 1. a) The annual income of a sole proprietor of a grocery shop is Rs 10,00,000. If the tax is exempted up to Rs 6,00,000, what is his/her taxable income? b) The yearly income of an individual is Rs 4,44,000 with Rs 24,000 remote area allowance. What is his/her taxable income? c) The yearly income of an officer is Rs 5,50,000. If he pays Rs 25,000 as premium of his life insurance in the year, what is his taxable income? Rs 5,36,658.40 Rs 5,00,000 Rs 36,658.40 (1%) (10%) Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 39 Vedanta Excel in Mathematics - Book 9 d) The monthly income of a government servant is Rs 77,280 and he gets the festival expense of one month’s salary, what is his taxable income? 2. a) The monthly salary of an individual is Rs 25,450. If 1% social security tax is charged upto the annual income of Rs 5,00,000, calculate the income tax paid by the individual. b) 1 % social security tax is charged upto the yearly income of Rs 6,00,000 to a married couple. If the monthly income of a couple is Rs 45,000, how much tax should the couple pay in a year? Creative section - A 3. Inland Revenue Department (IRD) has fixed the following rates of income tax for Proprietorship firm. Use it to calculate the income taxes. For an individual For couple Income slab Tax rate Income slab Tax rate Up to Rs 5,00,000 0% Up to Rs 6,00,000 0% Rs 5,00,001 to Rs 7,00,000 10% Rs 6,00,001 to Rs 8,00,000 10% Rs 7,00,001 to Rs 10,00,000 20% Rs 8,00,001 to Rs 11,00,000 20% Rs 10,00,001 to Rs 20,00,000 30% Rs 11,00,001 to Rs 20,00,000 30% a) Mr. Baral has a stationery shop. His annual income is Rs 6,40,000. If he is unmarried, how much income tax should he pay? b) Mr. Yadav is still unmarried but he is the proprietor of a furniture factory. He earned Rs 8,00,000 last year, how much income tax did he pay last year? c) Mrs Adhikari is the proprietor of boutique training centre. If her annual income is Rs 6,75,000, how much income tax does she pay? d) Mr. Manandhar is a married person. He has a registered computer repair service centre. He earned Rs 9,25,000 in this year. How much tax is charged on his income? 4. Study the given income tax rates fixed by IRD and workout the following problems. For an individual For couple Income slab Tax rate Income slab Tax rate Up to Rs 5,00,000 1% Up to Rs 6,00,000 1% Rs 5,00,001 to Rs 7,00,000 10% Rs 6,00,001 to Rs 8,00,000 10% Rs 7,00,001 to Rs 10,00,000 20% Rs 8,00,001 to Rs 11,00,000 20% Rs 10,00,001 to Rs 20,00,000 30% Rs 11,00,001 to Rs 20,00,000 30% More than Rs 20,00,000 36% More than Rs 20,00,000 36% Taxation

Vedanta Excel in Mathematics - Book 9 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur a) The monthly income of an unmarried civil officer is Rs 43,600 and one month’s salary is provided as Dashain expense. (i) What do you mean by income tax? (ii) What is his annual income? (iii) How much income tax should he pay in a year? b) The monthly salary of a married couple is Rs 48,000 plus a festival expense of Rs 30,000. (i) Find the annual income of the couple. (ii) Calculate the income tax paid by the couple in a year. c) Mrs Gurung is a bank Manager in a development bank. Her monthly salary is Rs 50,000. If her annual income is equivalent to her 15 month’s salary, find her income tax in a year. d) The monthly salary of an individual employee of an INGO is Rs 1,80,000. Calculate the income tax paid by the individual in a year. 5. a) Mrs Thakuri deposited Rs 2,00,000 in her saving account in a bank for 3 years. The bank pays her the simple interest at the rate of 10% p.a. (i) Find the simple interest in 3 years. (ii) How much net interest did she get if 5% of interest was charged as income tax? b) Mr. Thapa deposits Rs 50,000 in a bank at the rate of 8% p.a. for 4 years (i) Find the simple interest. (ii) How much net interest does he get if 5% of interest is charged as income tax? c) In the beginning of BS 2077, Dolma deposited Rs 1,20,000 in her account at the rate of 9% p.a. If she paid 5% of her interest as income tax, how much total amount did she receive in the beginning of BS 2080? d) On the occasion of daughter’s 14th birthday, Dharmendra deposits Rs 25,000 in his daughter's bank account at the rate of 6% p.a. If 5% of the interest is charged as income tax, how much amount will she withdraw on her 16th birthday? 6. a) Mrs Majhi deposited a certain amount in her bank account at the rate of 6.5% p.a. If she paid 5% of her interest as income tax and received Rs 4940 net interest after 4 years, how much money was deposited by her? b) Madan Bahadur deposited a sum of money at his bank account at the rate of 10% p.a.. After 5 years, he received Rs 1900, the net interest when 5% of the total interest was charged as income tax. Find, how much sum was deposited by him? Creative section-B 7. a) Jenisha is an unmarried employee in a commercial bank. Her monthly salary is Rs 45,000. She has to pay 1% social security tax on her income up to Rs 5,00,000 and 10% income tax on Rs 5,00,001 to Rs 7,00,000. She gets 15 months’ salary in a year. She pays Rs 30,000 as the annual premium of life insurance and gets 10% rebate on her income tax, answer the following questions. (i) What is her annual income? (ii) Find her taxable income. (iii) How much tax will be rebated to her? (iv) How much annual income tax should she pay? Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 41 Vedanta Excel in Mathematics - Book 9 b) Swornima is an unmarried nurse in a hospital. Her monthly basic salary is Rs 48,000. She has to pay 1% social security tax on her income up to Rs 5,00,000 and 10% income tax on Rs 5,00,001 to Rs 7,00,000. She gets 1 months’ salary as the Dashain allowance. She deposits 10% of her basic salary in Citizen Investment Trust (CIT) and gets 10% rebate on her income tax. Answer the following questions. (i) What is her annual income? (ii) How much tax is rebated to her? (iii) How much annual income tax should she pay? c) Mr. Khatiwada is an unmarried secondary level mathematics teacher in a community school. His details are given below. Basic salary: Rs 45,990 Dearness allowance: Rs 2,000 per month Festival expense: Basic salary of 1 month Deposit in citizen investment trust: 10% of the basic salary Premium of life insurance: Rs 30,000 Based on this information, answer the following questions. (i) What is the annual basic salary of Mr. Khatiwada? (ii) Calculate his taxable income. (iii) How much income tax should he pay in this year? 8. a) The monthly basic salary of the married Chief of Army Staffs (COAS) General is Rs 79,200 with Rs 2,000 dearness allowance. He gets Dashain allowance which is equivalent to his basic salary of one month. He contributes 10% of his basic salary in Employee's Provident Fund (EPF) and he pays Rs 50,000 as the premium of his life insurance. Given that 1% social security tax is levied upon the income of Rs 6,00,000, 10% and 20% taxes are levied on the next incomes of Rs 2,00,000 and up to Rs 3,00,000 respectively. Answer the following questions. (i) What is his monthly basic salary? (ii) Find his taxable income. (iii) Find the total income tax paid by him. b) The monthly basic salary of the married Chief Secretary of Nepal Government is Rs 74,000 with Rs 2,000 dearness allowance. He gets 1 month’s basic salary as festival allowance. He contributes 10% of his basic salary in Employee's Provident Fund (EPF) and he pays Rs 45,000 as the premium of his life insurance. Given that 1% social security tax is levied upon the income of Rs 6,00,000, 10% and 20% taxes are levied on the next incomes of Rs 2,00,000 and up to Rs 3,00,000 respectively. Answer the following questions. (i) What is his monthly basic salary? (ii) Find his taxable income. (iii) Find his total income tax. For individual Tax rates Up to Rs 5 lakh 1% Next Rs 2 lakh 10% Next Rs 3 lakh 20% Taxation

Vedanta Excel in Mathematics - Book 9 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Project work and activity section 9. a) Let's ask the monthly salary of your Mathematics, Science, and English teachers. Then, complete the table given below. Name of teachers Marital status Monthly Salary Festival expense Provident fund Insurance Now, calculate the annual income tax paid by each subject teacher. b) If your parents are involving in any government or private service, let's ask their monthly salary, bonus, provident fund, insurance, etc. Then, calculate the income tax paid by them in a year. 2.5 Marked Price (M.P.) and Discount Have you ever seen the price tagged on fancy items, the price printed on books, packets of noodles, the price on catalogue of television, refrigeration, etc? The price on the label of an article or product is called the marked price (M. P.). It is also called list price. This is the price at which the article is intended to be sold. However, there can be some reduction given on this price and the actual selling price of the article may be less than the marked price. The amount of reduction in the marked price is called discount. Discount is usually given as a certain percent of marked price. The formulae given below are useful to workout the problems of marked price and discount. Discount amount = discount percent of M.P. (D% of M.P.) Selling price = M.P. – D% of M.P. Discount percent (D%) = Discount amount M.P. × 100% 2.6 Value Added Tax (VAT) Let's study the adjoining bill given by a shopkeeper to a customer. (i) How much is the selling price of the mobile without VAT? (ii) What is the VAT rate and amount of VAT in the bill? (iii) How much price should the customer pay with VAT? Value added tax (VAT) is an indirect tax which is charged at the time of consumption of goods and services. Therefore, it is a consumption tax paid by a consumer while purchasing goods or services. 6000 Rs. In words: ........................................... ................................................................. ................................................................ ............% Taxable Amount Ownership Tax Sub Total 13 % VAT 6000 780 Grand Total 6780 Total 6000 Mobile Six thousand only. 6,000 Deurali Electronics Tahachal-13, Kathmandu Ph. No. 9841240225 Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 43 Vedanta Excel in Mathematics - Book 9 The VAT rate is given in percent and it is decided by the concerned authority of the government of a country. The VAT rate may vary from country to country and even in a country, it may be changed from time to time. For example, when VAT was introduced for the first time in Nepal on 16 November, 1997, the rate of VAT was 10%, but it is 13% now. VAT is one of the sources of government revenue of a country for its administrative expenses, welfare, development expenses, and so on. VAT is levied on the actual selling price of goods. VAT amount = Rate of VAT × selling price (S.P.) S.P. with VAT = S.P. + VAT% of S.P. If goods are sold by giving a certain discount, at first the amount of discount is to be deducted from the given marked price to find actual selling price. Then, VAT is levied on the actual selling price. i.e., Actual S.P. = M.P. – D% of M.P. S.P. with VAT = S.P. + VAT% of S.P The process of calculating actual selling price is shown in the following diagram. Cost Price (C.P.) + Transportation cost + Local tax + Service charge + Commission Marked Price (M.P.) + Insurance Actual S.P. + Profit – Discount – Loss The purchaser should not pay VAT on tax-exempt goods and services. The following goods and services are VAT exempted in Nepal. (a) Goods and services of basic needs which include rice, pulses flour, fresh fish, meat, eggs, fruits, flowers, edible oil, piped water, wood fuel etc. (b) Basic agricultural products are also tax-exempt, for example, paddy, wheat, maize, millet, cereals and vegetables. (c) The expense of buying goods and services required to grow basic agricultural products are tax-exempt. This includes live animals, agricultural inputs including machinery, manure, fertilizer, seeds, and pesticides. (d) Social welfare services including medicine, medical services, veterinary services and educational services. (e) Goods made for the use of disabled persons. (f) Air Transport. (g) Educational and cultural goods and services such as books and other printed materials, radio and television transmissions, artistic goods, cultural programmes, non-professional sporting events and admissions to educational and cultural facilities. (h) Personal services are also tax-exempt. These are services provided, for example, by actors and other entertainers, sportsmen, writers, translators and manpower supplies agents. (i) Exemption from VAT is also extended to the purchase and renting of land and buildings (j) Financial and insurance services. (k) Postage and revenue stamps, bank notes, cheque books. Taxation

Vedanta Excel in Mathematics - Book 9 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Worked-out Examples Example 1: Calculate the VAT amount on the selling price of Rs 3,600 at the rate of 13%. Solution: Here, the amount of VAT = VAT% of S.P. = 13% VAT of Rs 3,600 = Rs 468 Hence, the required VAT amount is Rs 468. Example 2: The selling price of a mobile is Rs 7,500. How much should a customer pay for it with 13% VAT? Solution: Here, the selling price of mobile = Rs 7,500 and VAT rate = 13% Now, the cost of the mobile with VAT = S.P. + VAT% of S.P. = Rs 7,500 + 13% of Rs 7,500 = Rs 7,500 + 13 100 × Rs 7,500 = Rs 8,475 Hence, the customer should pay Rs 8,475 for the mobile. Example 3: The marked price of a bicycle is Rs 4,400 and the shopkeepers levies 13% VAT on it. If you give Rs 5,000 to the shopkeeper while buying it, what change will the shopkeeper return to you? Solution: Here, M.P. of a bicycle = S.P. of bicycle = Rs 4,400 [ No discount is allowed] Rate of VAT = 13%, and S.P. with VAT = ? Now, S.P. with VAT = S.P. + VAT% of S.P. = Rs 4,400 + 13 100 × Rs 4,400 = Rs 4,972 Again, Rs 5000 – Rs 4972 = Rs 28 Hence, the shopkeeper will return Rs 28 to me. Example 4: A family had dinner in a restaurant. If the cost of the dinner was Rs 1,500, how much did the family pay with 10% service charge and 13% VAT? Solution: 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 Direct process S.P. with VAT=113% of S.P. = 113 100 × Rs 7,500 = Rs 8,475 Direct process S.P. with VAT = (100 + 13)% of S. P. = 113% of Rs 4,400 = 113 100 × Rs 4,400 = Rs 4,972 https://www.geogebra.org/m/xjgvxst3 Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 45 Vedanta Excel in Mathematics - Book 9 Again, the cost of the dinner with service charge and VAT = Rs 1650 + 13% of Rs 1650 = Rs 1,864.50 Therefore, the family should paid Rs 1,864.50. Example 5: Mrs. Shrestha purchased a watch for Rs 7,360 with 15% VAT. Find the cost of the watch without VAT? Solution: Here, the cost of the watch with 15% VAT = Rs 7,360 Let the cost of the watch without VAT = Rs x Now, x + 15% of x = Rs 7,360 or, 115x 100 = Rs 7,360 or, x = Rs 6,400 Hence, the cost of the watch without VAT is Rs 6,400. Example 6: If the cost of an article with VAT is Rs 4,746 and without VAT is Rs 4,200, find the VAT rate. Solution: Here, the cost of the article with VAT = Rs 4,746 And, the cost of the article without VAT = Rs 4,200 Now, the amount of VAT = Rs 4,746 – Rs 4,200 = Rs 546 ∴ Rate of VAT = VAT amount cost without VAT × 100% = Rs 546 Rs 4200 × 100% = Rs 13% Hence, the required VAT rate is 13%. Example 7: A trader bought an electric oven for Rs 16,000 and sold at a profit of 20% to a customer with 13% VAT. How much did the customer pay for the oven? Solution: Here, C.P. of the oven = Rs 16,000. Profit percent = 20% S.P. of the oven = Rs 16,000 + 20 100 × Rs 16,000 = Rs 19,200 Now, S.P. of the oven with 13% VAT= S.P. + 13% of S.P. = Rs 19,200 + 13 100 × Rs 19,200 = Rs 21,696 Hence, the customer paid Rs 21,696 for the oven. Example 8: A shopkeeper bought a television set for Rs 24,000 and fixed its price to make 15% profit. If the television was sold for Rs 31,188 with VAT, calculate the rate of VAT. Solution: Direct process Cost of watch without VAT = cost of watch with VAT (100 + VAT)% = Rs 7,360 115% = Rs 6,400 = Rs 6,400 Direct process S.P. of the oven = 120% of C.P. = 120 100 × Rs 16,000 = Rs 19,200 Direct process S.P. with VAT = 113% of S.P. = 113 100 × Rs 19,200 = Rs 21,696 Taxation

Vedanta Excel in Mathematics - Book 9 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Here, S.P. for the shopkeeper = Rs 24,000 + 15% of Rs 24,000 = Rs 24,000 + Rs 3,600 = Rs 27,600 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 S.P. × 100% = Rs 3,588 Rs 27,600 × 100% = 13% Hence, the required rate of VAT is 13%. Example 9: The marked price of a mobile set is Rs 18,000 and 15 % discount is allowed. How much should a customer pay for it with 13 % VAT? Solution: Here, M.P. of the mobile set = Rs 18,000 Discount percent = 15 % S.P. of the mobile set = M.P. – 15 % of M.P. = Rs 18,000 – 15 100 × Rs 18,000 = Rs 15,300 Now, S.P. with VAT = S.P. + VAT % of S.P. = Rs 15,300 + 13 % of Rs 15,300 = Rs 17,289 Hence, the customer should pay Rs 17,289 for the mobile set. Example 10: Mr. Dorje paid Rs 31,188 for a washing machine with a discount of 20 % including 13 % VAT. What is the marked price of the washing machine? Solution: Let the marked price of the washing machine be Rs x. Now, S.P. of the washing machine = M.P. – discount % of M.P. = x – 20 % of x = 4x 5 Again, S.P. with VAT = 4x 5 + 13 % of 4x 5 = 113x 125 Also, the given S.P. = Rs 31,188 or, 113x 125 = Rs 31,188 or, x = Rs 34,500 Hence, the marked price of the washing machine is Rs 34,500. Alternative process Let,S.P. without VAT be Rs x. Direct process S.P. of television = 115% of C.P. = 115 100 × Rs 24,000 = Rs 27,600 Direct process S.P. = (100 – 15)% of M. P. = 85% of Rs 1,800 = Rs 15,300 Direct process S.P. with VAT = (100 + 13)% of S. P. = 113% of Rs 15,300 = Rs 17,289 https://www.geogebra.org/m/k7kkap33 Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 47 Vedanta Excel in Mathematics - Book 9 Then, S.P. without VAT = S.P. with VAT – 13 % of x or, x = Rs 31,188 – 13x 100 or, x + 13x 100 = Rs 31,188 or, x = Rs 27,600 Again, let M.P. be Rs y. Then, M.P. – discount % of M.P. = S.P. (without VAT) or, y – 20 % of y = Rs 27,600 or, y = Rs 34,500 Hence, the required marked price is Rs 34,500. Example 11: Sunayana purchased a fancy item for Rs 4,800 and sold it for Rs 6,780 with 13% VAT. Find her profit or loss percent. 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. ∴ x + 13% of x = Rs 6,780 or, x + 13x 100 = Rs 6,780 or, 113x 100 = Rs 6,780 or, x = Rs 6,000 Thus, S.P. without VAT = Rs 6,000 Now, profit = S.P. – C.P. = Rs 6,000 – 4,800 = Rs 1,200 Also, profit percent = Actual profit C.P. × 100% = Rs 1,200 Rs 4,800 × 100% = 25% Hence, the required profit percent is 25%. Example 12: Pashupati supplier purchased some building materials for Rs 5,55,000 and sold them at 5% profit to Lumbini supplier. The Lumbini supplier spent Rs 7,000 for transportation and Rs 4,250 for the local tax and sold at a profit of 10% to a customer. How much did the customer pay for the materials with 13% VAT? Solution: Here, for Pashupati supplier, C.P. of the building materials = Rs 5,55,000 Direct process 113 % of S.P. = Rs 31,188 or, S.P. = Rs 31,188 113% S.P. = Rs 27,600 Again, (100 – 20)% of M.P. = S.P. or, 80 100 × M.P. = Rs 27,600 or, M.P. = Rs 34,500 Direct process S.P. without VAT = S.P. with VAT 113% = Rs 6,780 113% = Rs 6,000 Direct process Profit percent = S. P. – C. P. C. P. × 100% = Rs 6,000 – Rs 4,800 Rs 4,800 × 100% = Rs 1,200 Rs 4,800 × 100% = Rs 25% Taxation

Vedanta Excel in Mathematics - Book 9 48 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur ∴ S.P. at 5% profit = C.P. + 5% of C.P. = Rs 5,55,000 + 5% of Rs 5,55,000 = Rs 5,82,750 For Lumbini supplier, C.P. of the materials =S.P. of Pashupati supplier = Rs 5,82,750 C.P. with transportation cost and local tax = Rs 5,82,750 + Rs 7,000 +Rs 4,250 = Rs 5,94,000 ∴Actual S.P. = Rs 5,94,000 + 10% of Rs 5,94,000 = Rs 6,53,400 Now, C.P. for customer with VAT = S.P. for Lumbini supplier with 13% VAT = Rs 6,53,400+13% of 6,53,400= Rs 7,38,342 Hence, the customer paid Rs 7,38,342 for the materials with VAT. Example 13: 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: Let, S.P. without VAT be Rs x. Here, discount percent = 10% S.P. with VAT = Rs 3,051 VAT = 13% Now, (100 + 13)% of x = Rs 3,051 or, 113 100 × x = Rs 3,051 or, x = Rs 2,700 ∴ S.P. without VAT = Rs 2,700 Again, (100 – 10)% of M.P. = S.P. without VAT or, 90 100 × M.P. = Rs 2,700 or, M.P. = Rs 3,000 Also, profit = 20% ∴ (100 + 20)% of C. P. = S.P. without VAT or, 120 100 × C.P. = Rs 2,700 or, C.P. = Rs 2,250 Now, profit = 20% + 4% = 24% ∴ New S.P. = (100 + 24)% of C.P. = 124 100 × Rs 2,250 = Rs 2,790 And, new discount = M.P. – new S.P. = Rs 3,000 – Rs 2,790 = Rs 210 Then, new discount percent = New discount M.P. × 100% = 210 3000 × 100% = 7% ∴ Reduction in discount percent = 10% – 7% = 3% Hence, the discount is to be reduced by 3%. Direct process S.P. without VAT = S.P. with VAT 113% = 3051 113% = Rs 2,700 M. P. = S.P. without VAT 90% = 2700 90% = Rs 3,000 C.P. = S.P. without VAT 120% = 2700 120% = Rs 2,250 New S.P. = 124% of C.P. = 124 100 × Rs 2250 = Rs 2790 New discount = Rs 3000 – Rs 2790 = Rs 210 New discount percent = 210 3000 × 100% = 7% ∴ Reduction in discount percent = 10% – 7% = 3% Taxation

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 49 Vedanta Excel in Mathematics - Book 9 Example 14: 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. a) Calculate the investment excluding stocks. b) Find the selling price of items after discount then, calculate gross profit. c) Find the net profit or loss in three months and express it in percent. Solution: a) 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 b) 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. = 90 100 × Rs 31,50,000 = Rs 28,35,000 ∴ Gross profit = Rs 28,35,000 – Rs 22,50,000 = Rs 5,85,000 c) 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 investment × 100% = 4,11,000 22,50,000 × 100% = 18.27% Hence, her net profit percent is 18.27%. EXERCISE 2.2 General section 1. a) If R% be the rate of VAT and Rs x be the selling price, write the formula to find amount of VAT. b) If Rs x be the selling price and Rs y be the amount of VAT, write the formula to find VAT percent. c) If Rs P be the selling price and R% be the VAT rate, write the formula to find selling price with VAT. d) If marked price (M.P.) = Rs x, discount = Rs y and VAT = Rs z, what is the selling price including VAT? Taxation

Vedanta Excel in Mathematics - Book 9 50 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2. a) Find the VAT amount from the table given below. S.N. Particulars S.P. without VAT VAT rate VAT amount (i) Mobile set Rs 22,000 13% ………… (ii) Camera Rs 35,000 13% ………… (iii) Television Rs 40,000 10% ………… b) Find the selling price without VAT from the table given below. S.N. Particulars VAT amount VAT rate S.P. without VAT (i) Radio Rs 585 13% ………… (ii) Bicycle Rs 975 13% ………… (iii) Laptop Rs 9,900 15% ………… 3. a) The selling price of a watch is Rs 3,000 and 13% VAT is imposed while selling it. (i) What will be the VAT amount on it? (ii) How much should a customer pay for it with VAT? b) A shopkeeper sells a tablet costing Rs 15,000 with 13% value added tax. (i) What will be the VAT amount on it? (ii) How much should a customer pay for it with VAT? c) Mrs. Khadka went to an electrical shop to buy a refrigerator. She found that the catalogue price of a refrigerator as Rs 45,500 and she bought it with 13% VAT. (i) How much amount of VAT was levied on it? (ii) How much amount did she pay with VAT? (iii) If she gave a bundle of 52 thousand rupees notes, what change did the shopkeeper return to her? 4. a) The cost of a rice cooker with 13% VAT is Rs 4,068. Calculate: (i) its cost without VAT (ii) VAT amount. b) Mr. Magar purchased a mobile set for Rs 11,155 with 15% VAT inclusive. (i) Find the cost of the mobile without VAT (ii) Calculate the VAT amount. c) Mrs. Maharjan bought a refrigerator for Rs 26,442 with 13% VAT. (i) How much did she pay for it excluding VAT? (ii) How much did she pay for the VAT? 5. a) If the cost of a watch with VAT is Rs 5,085 and without VAT is Rs 4,500, find: (i) VAT amount. (ii) the rate of VAT. b) Malvika purchased a fancy bag for Rs 7,119 with VAT. If its cost without VAT is Rs 6,300, calculate: (i) the VAT amount (ii) the rate of VAT. c) If the cost of a computer with VAT is Rs 67,800 and without VAT is Rs 60,000, find the VAT rate. Taxation