Oasis Math 9

Approved by Government of Nepal, Ministry of Education, Curriculum Development Centre,
Sanothimi, Bhaktapur as an additional material.

9

Author
Shyam Datta Adhikari

M.Sc. (Maths), T.U.

9

Name : ............................................................
Class : ....................... Roll No. : ................
Section : ..........................................................
School : ............................................

Publisher
Oasis Publication Pvt. Ltd.
Copyright
The Publisher
Edition
B.S. 2073 (2016 AD)
B.S. 2074 (2017 AD)
B.S. 2075 (2018 AD)
B.S. 2078 (2021 AD) (Completely revised)

Contributors
Man Bahadur Tamang
Laxmi Gautam
Shuva Kumar Shrestha
Purushottam Bhatta
Prakash Ghimire

Layout
Oasis Desktop

Ramesh Bhattarai

Printed in Nepal

Preface

Oasis School Mathematics has been designed in compliance with the
latest curriculum of the Curriculum Development Center (CDC), the
Government of Nepal with a focus on child psychology of acquiring
mathematical knowledge and skill. The major thrust is on creating an
enjoyable experience in learning mathematics through the inclusion
of a variety of problems which are closely related to our daily life.
This book is expected to foster a positive attitude among children
and encourage them to enjoy mathematics. A conscious attempt has
been made to present mathematical concepts with ample illustrations,
assignments, activities, exercises and project work to the students in
a friendly manner to encourage them to participate actively in the
process of learning.

I have endeavored to present this book in a very simple and interesting
form. Exercises have been carefully planned. Enough exercises have
been presented to provide adequate practice.

I have tried to include the methods and ideas as suggested by the
teachers and subject experts who participated in the seminars, and
workshops conducted at different venues. I express my sincere
gratitude to my friends and well wishers for their valuable suggestions.

I am extremely grateful to Megh Raj Adhikari, Madan Kumar
Shrestha, Shanta Kumar Sen (Tamang), Man Bahadur Tamang, Laxmi
Gautam and Yadav Siwakoti for their invaluable suggestions and
contributions.

Sincere gratitude to Managing Director Mr. Harish Chandra Bista for
his invaluable support and cooperation in getting this series published
in this shape.

In the end, constructive and practical suggestions of all kinds for
further improvement of the book will be appreciated and incorporated
in the course of revision.

Shyam Datta Adhikari
Author

March 2019

9

Contents

Sets 1-18

Unit : 1 Sets................................................................................... 2
1.1. Warm-up Activities............................................................ 2
1.2 Operation on Sets................................................................ 3
1.3 Cardinality of Two Sets..................................................... 9

Arithmetic 19-82

Unit : 2 Profit and Loss, Discount......................................... 20

2.1 Warm-up Activities............................................................ 20

2.2 Profit and Loss.................................................................... 20

2.3 Discount.............................................................................. 31

Unit : 3 Commission, Bonus, Taxation and Dividend.......... 38

3.1 Warm-up Activities............................................................ 38

3.2 Commission........................................................................ 38

3.3 Bonus................................................................................... 43

3.3 Taxation............................................................................... 45

3.4 Dividend.............................................................................. 57

Unit : 4 Household-Arithmetic................................................. 63

4.1 Warm-up Activities............................................................ 63

4.2 Electricity Bill...................................................................... 63

4.3 Water Bill.............................................................................. 69

4.4 Telephone Bill...................................................................... 74

4.5 Taxi Fare............................................................................... 77

Mensuration 83-124

Unit : 5 Mensuration........................................................................... 84
5.1 Warm-up Activities............................................................ 84
5.2 Perimeter and Area of Plane Figures............................... 84
5.3 Cost Estimation................................................................... 90

Contents

5.4 Area of Pathways......................................................................... 98

5.5 Prism.............................................................................................. 104

5.6 Area of 4 Walls............................................................................. 109

5.7 Cost Estimation............................................................................ 113

5.8 Area and Volume of Walls and Estimations of

Number of Bricks, Cost of Making Walls................................. 117

Algebra 125-178

Unit : 6 Factorisation................................................................................. 126

6.1 Warm-up Activities...................................................................... 126

6.2 Factorisation of Various types of Expression........................... 126

6.3 Factorisation of the Expression of the type a2-b2..................... 128

6.4 Factorising the Expression of the type a4+a2b2+b4................... 132

6.5 Factorisation of the Expression of the a3+b3 and a3-b3..................... 133

6.6 Factorisation of the Expression of the type ax2+bx+c......................... 135

Unit : 7 Indices................................................................................................. 138

7.1 Warm-up Activities...................................................................... 138

7.2 Indices............................................................................................ 138

7.3 Exponential Equation.................................................................. 144

Unit : 8 Ratio and Proportion....................................................................... 148

8.1 Ratio............................................................................................... 148

8.2 Proportion..................................................................................... 152

Unit : 9 Simultaneous Equations in Two-Variables................................. 159

9.1 Warm-up Activities...................................................................... 159

9.2 Simultaneous Linear Equations in two

Variables x and y.......................................................................... 159

9.3 Substitution Method.................................................................... 159

9.4 Elimination Method..................................................................... 161

9.5 Graphical Method of Solving Simultaneous Equation........... 166

Unit : 10 Quadratic Equations................................................................... 169

10.1 Warm-up Activities...................................................................... 169

10.2 Quadratic Equation...................................................................... 169

Geometry 179-264

Unit : 11 Triangles........................................................................................ 180
11.1 Warm-up Activities...................................................................... 180
11.2 Triangles........................................................................................ 181
11.3 Theorems on the Relation between Three Sides
and Angles of a Triangle.............................................................. 186

Contents

11.4 Congruent Triangles.................................................................... 189
11.5 Types of Triangles........................................................................ 194
11.6 Similarity........................................................................................ 201
11.7 Pythagoras Theorem..................................................................... 212
Unit : 12 Quadrilateral and Parallelogram................................................. 210
12.1 Warm-up Activities...................................................................... 210
12.2 Quadrilateral................................................................................. 210
12.3 Mid-point Theorem..................................................................... 232
Unit : 13 Circle.............................................................................................. 237
13.1 Warm-up Activities...................................................................... 237
13.2 Circle.............................................................................................. 237
Unit : 14 Construction................................................................................. 231
14.1 Warm-up Activities...................................................................... 251
14.2 Construction of Quadrilaterals.................................................. 251
14.3 Construction of Some Special Types of Quadrilaterals...................... 254

Trigonometry 265-283

Unit : 15 Trigonometry................................................................................ 266
15.1 Introduction.................................................................................. 266
15.2 Relation among the Trigonometric Ratios................................ 272
15.3 Trigonometric Ratios of Some Standard Angles..................... 276

Statistics 284 - 315

Unit : 16 Statistics........................................................................................ 285
16.1 Warm-up Activities...................................................................... 285
16.2 Collection of Data......................................................................... 285
16.3 Graphical Representation of Data.............................................. 292
16.4 Measures of Central Tendency................................................... 301
16.5 Partition Values............................................................................ 310

Probability 316 - 326

Unit : 17 Probability.................................................................................... 317
17.1 Introduction.................................................................................. 317
17.2 Some Terms Related to Probability............................................ 317
17.3 Theoretical Probability and Empirical Probability................... 319
Specification Grid ............................................................................... 327
Model Test Paper.................................................................................. 328

Sets

6Estimated Teaching Hours

Contents
• Review
• Operations on set
• Simple problems on sets using Venn diagram
Expected Learning Outcomes
At the end of this unit, students will be able to develop the
following competencies:
• To find the union, intersection, difference and

complement of sets
• To represent above operations in Venn diagrams
• To find the cardinal number of given sets
• To solve simple problems on two sets using

Venn diagram
Teaching Materials
• A4 size paper, Tracing paper, Chart paper, etc.

Oasis School Mathematics-9 1

Unit

1 Sets

1.1 Warm-up Activities
Discuss the following in your class.

Identify whether the given collections can be specified or not?
• Collection of vowel letters.
• Collection of prime numbers less than 10.
• Collection of honest people in the society.
List the elements of above collections.
Is it possible to list the elements of all collections?
The first collection is {a, e, i, o, u}
The second collection is {2, 3, 5, 7}
But it is difficult to specify the third one.
It is not well defined.
Hence, the collection of well defined objects is the set.

Notation of set

The set is denoted by capital letters A, B, C..., X, Y, Z … etc.
The members of set are mentioned by small letters a, b, c, x, y, z, etc. within curly

brackets { }.
Symbols used in set

∪ Union
∩ Intersection
A Complement of A
⊂ Proper subset
⊆ Subset
∈ Belongs to or a member of
∉ Doesn't belong to or not a member of

Universal set and subset

Given figure shows whether the students of class IX like tea, milk, both or none.
Study the given figure and answer the questions given below.

2 Oasis School Mathematics-9

• How many students are there? U
• Name the students who like tea. TM
• Name the students who like milk.
• What are the notations of a tea and milk? • ram • Suntali
• Which two students do not like both? • Salman
• How many subsets can be formed from the set T?
• John • Dristi • Numa
•Lakpa • Nabida

• Lochan

• David

Altogether there are 10 students.
Collection of these 10 students is the universal set. It contains all the elements of 'T'

and 'M'.
i.e. 'T' and 'M' are the subsets of universal set 'U'.
Hence the set under consideration which contains all the sets of its surroundings is

the universal set.
All the sets which can be formed from the given universal set are the subsets of

universal set.

1.2 Operation in Sets

Union of Sets

Let A = {a, b, c}, B = {d, e, f}, then the set containing all the elements
of A as well as B is called union of two sets A and B.
It is denoted by (A ∪ B).

∴ A ∪ B = {a, b, c, d, e, f}

Similarly, if A = {x: x is a prime factor of 12} and

B = { x : x is a prime factor of 15}
Then, the set containing all prime factors of both 12 as well as 15 is denoted by,

A ∪ B = { x : x is either prime factor of 12 or 15}

In roster form:

A = { 2, 3 }

B = { 3, 5 }

A ∪ B = { 2, 3, 5 }

Thus, let A and B be any two sets then their union is denoted by A ∪ B and is defined
by A ∪ B = { x : x ∈ A or x ∈ B}.

A BU A U U
B A

B

(A∪B when A and B are overlapping) (A∪B when A and B are disjoint) (A∪B when B is the subset of A)

Oasis School Mathematics-9 3

Intersection of Sets

Let A = {1, 2, 3, 4}, B = { 1, 3, 5, 9, 12}. Then, there is a set of common elements of A
and B which is called intersection of A and B. i.e. A ∩ B = { 1, 3}.

Similarly, let A = {a, b, c }, B = { s, v, u}

then, A ∩ B = { }= φ ( There is no common element in A and B. )
∴
Thus, let A and B be any two sets then their intersection is denoted by A ∩ B and∴
defined by;

A ∩ B = { x : x ∈ A and x ∈ B}

A BU A U U
B A

B

(A∩B when A and B are overlapping) (A∩B when A and B are disjoint) (A∩B when B is the subset of A)

Difference of Sets

Let A = { 1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, then the difference of A and B is denoted by
A – B. A – B = {1, 2} [ ∵ set of elements of A but not the element of B]

Similarly, B – A = {5, 6, 7} ( 5, 6, 7 ∈ B but ∉ A )

Thus let 'A' and 'B' be any two sets then their difference is denoted by A – B and
B – A and defined by A – B = {x : x ∈ A and x ∉ B}

B – A = {x : x ∈ B and x ∉ A}

UU UU
A B A B AB A

B

A–B B–A A–B A–B
when A and B are disjoint sets when B ⊂ A
Representation in the Venn diagram

Complement of Sets

Let A = {2, 3, 4} is a subset of universal set

U = { 2, 3, 4, 6, 9}.

Then the complement of A is the set of elements from
U but not the elements of A.

A = {x: x ∈ U, x ∉A}

So, A = AC = A' = U – A = {2, 3, 4, 6, 9} – { 2, 3, 4} = {6, 9}

From above example, we conclude that A is the set of all elements of U but not the
elements of A. i.e. we can write A = U – A

4 Oasis School Mathematics-9

U
A

Representation in the Venn diagram

Symmetric Difference

Let A = {1, 2, 3}, B = {3, 4, 5}
A – B = {1, 2, 3 } – {3, 4, 5} = { 1, 2 }
and, B – A = { 3, 4, 5} – {1, 2, 3} = {4, 5}

Now, (A – B) ∪ (B – A) = {1, 2} ∪ {4, 5} = {1, 2, 4, 5} is called symmetric difference of
A and B.

Thus, symmetric difference between two sets is defined by A ∆ B = { x : x ∈ A or B
but x ∉ A ∩ B}
U
AB

A∆B
Representation in the Venn diagram

Operations in Three Sets

U A U U
AB B AB

C C C
A∪B∪C A∩B∩C A∩B

U U U
AB AB AB

C C C
B∩C A∩C A∪B∪C

Exercise 1.1

1. If U = {1, 2, 3, …. 10 }, A = { 2, 3, 5, 7 } and B = { 2, 4, 6, 8, 10 }, find
(a) A∩B (b) A ∪ B (c) B – A (d) A – B
(e) A (f) B (g) A ∪ B (h) A ∩ B (i) (A–B) ∪ (B–A)

Oasis School Mathematics-9 5

2. What sets do the shaded regions in the following diagrams represent?

(a) U (b) U (c) A U
B
A BA B

(d) U (e) U (f) A U
B
A BA B

3. From the adjoining diagram, write the elements of the following sets.

(a) A (b) B (c) A ∪ B U
(d) A ∩ B AB
(g) A ∩ B 1 35
(e) A – B (f) B – A 10 6
2 7
(h) (A–B) ∪ (B–A) (i) A ∪ B 4

98

4. Draw the given diagram in your copy and shade the region represented by the

following sets separately.

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

(d) B (e) A ∪ B (f) A ∩ B

(g) A – B (h) B – A (i) A ∪ B

(j) A ∩ B (k) A – B (l) B – A

5. If U = { x : 1 ≤ x ≤ 12, x ∈ N }, A = { 1, 3, 9, 10 }, B = {3, 4, 6, 11, 12 }, find

(a) A ∪ B (b) (A ∪ B). (c) A∩ B (d) (A ∩ B).

(e) A ∩ B (f) A ∪ B

6. What sets do the shaded part of each of the following diagrams represent?

(a) U (b) U (c) U
B
A BA BA

C C C

(d) U (e) U (f) U
B B
A A BA
C C C


6 Oasis School Mathematics-9

7. If U = {1, 3, 5, 7, 9, 11, 13, 15}, A = {1, 3, 5, 13}, B = {9, 11, 13, 15} and C = {1, 7, 9}, find
the set represented by

(a) (A∪B)∪C (b) ( A∩B )∩C (c) (A – B)∩C

(d) (B∪C)∩A (e) (A∪B)∩C (f) A ∩(B –C)

8. Draw the given diagram in your copy and shade the region represented by

following sets separately. U
B
(a) A (b) A∪B (c) A∪B∪C A

(d) A∩B∩C (e) (A –B)∩C (f) A∩(B∪C)

C

9. From the given Venn diagram, list the following sets. AU

(a) A (b) B (c) C 1 7 3 B
25 9 4
(d) A∩B (e) B∩C (f) A∪B
6 10
(g) A∪B∪C (h) (A – B)∩C (i) A∪B∪C 8

14 11 12 13
C

10. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4, 5, 6} B = { 3, 4, 5, 6, 7 } and
C = { 4, 5, 6, 7, 8}, find

(a) A ∪ B, (b) A ∪ C (c) A ∩ B (d) A ∩ C

(e) A – C (f) C – B (g) A∩( B∪C ) (h) A∪ ( B ∩ C).

Also represent all the sets in Venn diagram.

11. (a) If A = {2, 3, 4, 5}, B = { 4, 5, 6, 7, 8} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, prove that:

(i) A ∪ B = B ∩ A (ii) A ∩ B = B ∪ A

(b) If U = {a, b, c, d, e, f, g, h}, A = { a, b, c, d }, B = {b, c, e, f} and C = {a, e, h},
verify the following:

(i) A ∪ B = B ∪ A (ii) A ∩ B = B ∩ A

(iii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (iv) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(v) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (vi) (A – B) ∩ C = ( A ∩ C) – (B ∩ C)

Answer

1. (a) {2} (b) {2,3,4,5,6,7,8,10} (c) {4,6,8,10} (d) {3,5,7}

(e) {1,4,6,8,9,10} (f) {1,3,5,7,9} (g) {1,9} (h) {1,3,4,5,6, 7,8,9,10}

(i) {3,4,5,6,7,8,10} 2. (a) A∪B (b) A∩B (c) A – B (d) A∪B
(e) (A –B)∪(B –A) (f) A

3. (a) {1, 2, 3, 4, 10} (b) {3, 4, 5, 6, 7} (c) {1, 2, 3, 4, 5, 6, 7, 10} (d) {3, 4}

(e) {1, 2, 10} (f) {5, 6, 7} (g) {1, 2, 5, 6, 7, 8, 9, 10} (h) {3, 4, 8, 9} (i) {8, 9}

Oasis School Mathematics-9 7

Answer U (b) U (c) U (d) A U
4. (a) B B
BA BA
A

(e) U (f) A U (g) A U (h) A U
B B B
A B
U
(i) U (j) A U (k) A U (l) A B
B B
A B

5. (a) {1, 3, 4, 6, 9, 10, 11, 12} (b) {2, 5, 7, 8} (c) {3}

(d) {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12} (e) {2, 5, 7, 8} (f) {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12}

6. (a) A∪B∪C (b) A∩B c) B∩C (d) A∩C (e) A∪B∪C (f) A∪B
7. (a) {1,3,5,7,9,11,13,15} (b) { } c) {1} d) {1,13} (e) {1. 9} (f) {13}

8. (a) U (b) U (c) U
B
A BA BA

(d) C C C

A U (e) U (f) U
B
BA BA

CC C

9. (a) {1,2,5,6,7,8,9} (b) {3,4,7,9,10} (c) {6, 8, 9, 10, 11, 12} (d) {7, 9} (e) {9, 10}

(f) {1,2,3,4,5,6,7,8,9,10} (g) {1,2,3,4,5,6,7,8,9,10,11,12} (h) {6,8} (i) {13,14}

10. (a) A .1 .3 U (b) .1 .3 U (c) A .1 .3 U (d) .1 .3 U
.2 B .2 .2 .2 B
A B BA

.4 .5 .4 .5 .4 .5 .4 .5
.6 .7 .9 .6 .7 .9 .6 .7 .6 .7

.8 C .8 C .9 .8 C .9 .8 C

(e) A .1 .3 U (f) A .1 .3 U (g) .1 .3 U (h) .1 U
.2 .2 .2 B .2 B
B BA A .3

.4 .5 .4 .5 .4 .5
.6 .7
.6 .7 .6 .7 .4 .5
.6 7
.9 .8 C .9 .8 C .9 .8 C .9 .8 C

11. Consult your teacher.

8 Oasis School Mathematics-9

1.3 Cardinality of Two Sets

Cardinal Number
Let A = {a, b, c, d}, B = { 1, 2 }, C = { 0 } and φ are some sets. There are four elements

in set A, two elements in set B, one element in set C and no element in set φ .

In short, we write n(A) = 4, n(B) = 2, n(C) = 1 and n(φ) = 0, which are called
cardinal numbers of sets. Thus, the number of elements present in the set is called
cardinal number of the set.

Cardinality of two sets

Let 'A' and 'B' be any two overlapping sets, such that n(A) = a, n(B) = b and
n(A∩B) = x.

Then, from the Venn diagram, U
n(A∪B) = a – x + x + b – x AB
n(A∪B) = a + b – x
n(A∪B) = n(A) + n(B) –n(A∩B) (a-x) x (b-x)

If two sets are disjoint, then n(A∪B) = n(A) + n(B)

Some important results on cardinality of two sets

(i) n(A∪B) = n(A) + n(B) – n(A∩B)

(ii) n(A∪B) = n(A) + n(B) [if A and B are disjoint sets]

(iii) n0(A) = n(A) – n(A∩B) (iv) n0(B) = n(B) – n(A∩B)

(v) n(A∪B) = n0(A) + n0(B) + n(A∩B) n(A∪B) A U
n(A) B
(vi) n (A∪B) = n(U) – n(A∪B)
n(B)

(vii) n0(A) = n(A∪B) – n(B) n0(A) n0(B)

(viii) n0(B) = n(A∪B) – n(A) n(A∩B)
(ix) n(U) = n(A) + n(B) – n(A∩B) + n(A∩B)

Note: n0(A) = n(A–B) and n0(B) = n(B – A)

Remember !

Verbal form Set notations

number of elements in either A or B or both n(A∪B)

number of elements in at least one of A or B n(A∪B)

number of elements in both sets A and B n(A∩B)

number of elements only in set A n0(A) or n(A–B)
number of elements in A but not in B n0(A) or n(A–B)
number of elements only in set B n0(B) or n(B–A)
number of elements in B but not in A n0(B) or n(B–A)
number of elements neither in A nor in B n(A∪B)

number of elements in exactly one set n0(A) + n0(B)

Oasis School Mathematics-9 9

Worked Out Examples

Example: 1
If A = { x : x is a multiple of 3 less than 16}, B = { x :x is odd number less than 16 },
verify the following.

(i) n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )

(ii) n ( A – B ) = n ( A ) – n ( A ∩ B )

(iii) n ( A ∪ B) = n0( A ) + n0( B ) + n ( A ∩ B)
Solution:

A = { x : x is a multiple of 3 less than 16 }

or, A = { 3, 6, 9, 12, 15 }

B = { x : x is odd number less than 16 }

or, B = { 1, 3, 5, 7, 9, 11, 13, 15 }

Here, A ∩ B = { 3, 6, 9, 12, 15 } ∩ { 1, 3, 5, 7, 9, 11, 13, 15 }

= {3, 9, 15},

A∪B = { 3, 6, 9, 12, 15} ∪ {1, 3, 5, 7, 9, 11, 13, 15}

= { 1, 3, 5, 6, 7, 9, 11,12, 13, 15 }

A–B = { 3, 6, 9, 12, 15 } – { 1, 3, 5, 7, 9, 11, 13, 15 }

= { 6, 12 }

We can represent the above sets in the Venn diagram. U
AB
Now, n ( A ∪ B ) = 10 •6 •3 •1 •5

n(A) = 5 •12 •9 •7 •11
•15 •13

n (B) = 8

n(A∩B) = 3

n0 ( A ) = 2
n0 ( B ) = 5
(i)
n ( A∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )

L.H.S. = n ( A ∪ B ) = 10

R.H.S. = n ( A ) + n ( B) – n ( A ∩ B)

= 5 + 8 – 3 = 10

∴ L.H.S = R.H.S.

(ii) n(A–B) = n(A)–n(A∩B)
L.H.S. = n ( A – B ) = n0(A) = 2
R.H.S. = n (A) – n ( A ∩ B ) = 5 – 3 = 2
∴ L.H.S. = R.H.S.

10 Oasis School Mathematics-9

(iii) n (A ∪ B) = n0 (A) + n0 (B) + n (A ∩ B)
L.H.S. = n(A∪B) = 10

R.H.S. = n0(A) + n0(B) + n(A∩B)
= 2 + 5 + 3 = 10

Example: 2

If n(U) = 140, n(A) = 80, n(B) = 70, n(A∩B) = 50, find n(A∪B), n(A∪B ), n0(A) and n0(B).
Solution:

Here, n(A) = 80, n(B) = 70, n(A∩B) = 50, n(U) = 140, n(A∪B) = ?, n(A∪B ) = ?,

n0(A) = ?, n0(B) = ? n(A∪B) = n(A) + n(B) –n(A∩B)
We have,

= 80 + 70 – 50 U
AB
Now, = 100
n(A∪B ) = n(U) – n(A∪B) 30 50 20

= 140 – 100

= 40 40

Again, n0(A) = n(A) – n(A∩B) = 80 – 50 = 30
n0(B) = n(B) – n(A∩B) = 70 – 50 = 20

Example: 3

If n(U) = 200, n(A) = 120, n(B) = 140, n(A∪B ) = 30, using Venn diagram find the
value of n(A∩B).

Solution:

Given, n(U) = 200, n(A) = 120, n(B) = 140

n(A∪B ) = 30, n (A∩B) = ?

Let, n(A∩B) = x. U
Now, AB

120-x x 140-x

30

From the Venn diagram,

n(U) = 120 – x + x + 140 – x + 30
or, 200 = 290 – x
or, x = 290 – 200

x = 90

∴ n(A∩B) = 90.

Oasis School Mathematics-9 11

Example: 4

In an examination it was found that 85% students passed in Maths, 75% passed in

Science. If 65% passed in both subjects, find the percentage of students who failed in

both subjects.

Solution: Note:

Let 'M' and 'S' be the set of students who passed in Maths and If every information is
Science respectively. given in percentage then
it is better to suppose
Let n(U) = 100 n(U) = 100.

Then, n(M) = 85, n(S) = 75, n(M∩S) = 65

n(M∪S ) = ? Alternative method
We have,
n(U) = 100, n(M) = 85, n(S) = 75, n(M∩S)
n(U) = n(M) + n(S)–n(M∩S) + n(M∪S ) = 65,
U
M
n(M∪S ) = ? S

or, 100 = 85 + 75 – 65 + n(M∪S ) Let n(M∪S )= x 85–65 65 75–65
From the Venn diagram, =20 =10

x

or, 100 = 160 – 65 + n(M∪S ) n(U) = 20+65+10+x

or, 100 = 95 + n(M∪S ) 100 = 95+x
or n(M∪S ) = 100 – 95 = 5
or, x = 100–95

or, x = 5

∴ 5% students failed in both subjects. or, n(M∪S ) = 5

Note ∴ 5% students failed in both subjects.

Students are requested to solve the questions by using Venn-diagram rather than using
formula.

Example: 5
In a survey of 150 people, it has been found that 90 people like to drink tea, 80 people
like to drink coffee. If 40 people do not like to drink both,

(i) find the number of people who like to drink at least one.

(ii) find the number of people who like to drink both. Alternative method
Let n (T∩ C) = x
(iii) find the number of people who like to drink only one.
U
(iv) show this information in Venn diagram. TC

Solution: 90–x x 80–x

Let 'T' and 'C' be the sets of people who like to drink tea 40
and coffee respectively.
Here, n(U) = 150, n(T) = 90, n (C)= 80, n(T∪C ) = 40 From the Venn diagram,
n(U) = 90 – x + x + 80 – x + 40
We have, or, 150 = 210 – x = 110
or, x = 210 – 150
(i) n(T ∪ C) = n(U) – n(T∪C ) or, x = 60
= 150 – 40 = 110 ∴ n(T∩C) = 60

12 Oasis School Mathematics-9

∴ The number of people who like to drink
at least one = 110

(ii) We have, n(T∪C) = n(T) + n(C) – n(T∩C)

110 = 90 + 80 – n(T∩C)

or, n(T∩C) = 170 – 110

or, n(T∩C) = 60. Again,
n(T∪ C) =90 – x + x + 80 – x
∴ The number of people who like to drink = 170 – x
both = 60 = 170 – 60 = 110

(iii) We have, n0(T) = n(T) – n(T∩C) Again,
n0(T) = 90 – x = 90 – 60 = 30
= 90 – 60 n0 (C) = 80 – x = 80 – 60 = 20
n(only one) = n0 (T) + n0 (C)
= 30 = 30 + 20 = 50

n0(C) = n(C) – n(T∩C) U
TC
= 80 – 60
n(only one) = 20 30 60 20
= n0(T) + n0(C)
= 30 + 20 40
= 50

∴ The number of people who like to drink
only one = 50

(iv) T U
C

30 60 20
40

Example: 6

In a survey among the people in the community, it was found that 80% people liked
oranges and 60% liked mangoes, 50% liked both and 40 people did not like both of
them. Using Venn diagram, find the number of people on the survey.
Solution:

Let 'O' and 'M' be the set of people who liked orange and mango respectively.

Let n(U) = x, 80 8x
100 10
Then n(O) = 80% of x = × x =

n(M) = 60% of x = 80 × x = 6x .
100 10

n(O∩M)= 50% of x = 50 × x = 5x .
100 10

n(O∪M)= 40

Oasis School Mathematics-9 13

Now, plotting the above informations in the Venn diagram

From the Venn diagram, U
MS
3x + 5x + x + 40 = x
10 10 10 6x –x51x0
8x – 5x 5x 10 10
3x 5x x 10 10 10 =
or, x– 10 – 10 – 10 = 40 3x
= 10
40
10x–3x–5x–x
or, 10 x = 40
or, 10 = 40

or, x = 400

∴ Total number of people on the survey = 400.

Exercise 1.2

1. (a) If A = {2, 3, 4, 5}, B = { 4, 5, 6, 9, 10}, U = { 1, 2, 3, … 10}, find (i) n(A), (ii) n(B),
(iii) n(U), (iv) n(A ∩ B), (v) n(A ∪ B), (vi) n(A – B), (vii) n(B – A), (viii) n(A ∪ B),
(ix) n(A ∩ B)

(b) If P = {x : 3 ≤ x ≤ 9, x ∈ N} and Q = { x : x is a multiple of 3 which is less than 16},
find

(i) n(P) (ii) n(Q) (iii) no (P)

(iv) no (Q) (v) n(P ∪ Q) (vi) n(P ∩ Q)

2. (a) If A = { a, b, c, d}, B = { b, c, e, f, g }, U = {a, b, c, d, e, f, g, h}, verify that:

(i) n (A ∪ B) = n(A) + n(B) – n(A ∩ B) (ii) n (A – B) = n(A) – n(A ∩ B)

(iii) n( B – A) = n(B) – n(A∩B) (iv) n(A∪B) = no(A) + no(B) + n(A∩B)

(v) n0(A) = n(A∪B) – n(B) (vi) n0(B) = n(A∪B) – n(A)

(b) From the given Venn diagram, find the value of: A U
B
(i) n (A – B) (ii) n(B – A) (iii) n(A∩ B) •8 •1 •4
•3 •2
(iv) n(A∪B) (v) n(A∪B) (vi) n(A∩B) •9
•6 •5 •10

•7

3. (a) If n(A) = 20, n(B) = 30 and n(A∩B) = 10, find

(i) no(A) (ii) no(B) (iii) no(A) + no(B) + n(A∩B)

(iv) n(A) + n(B) – n(A∩B) (v) Are the results of (iii) and (iv) equal?

(b) If no (A) = 25, no (B) = 20, n(A∩B) = 15, find

(i) n(A) (ii) n(B) (iii) no(A) + no (B) + n(A∩B)

(iv) n(A) + n(B) – n (A∩B) (v) compare the result of (iii) and (iv).

4. (a) If A and B are two disjoint sets and n(A) = 30, n(B) = 40, find n(A∪B).

(b) If A and B are disjoint sets and n(A∪B) = 40, n(A) = 25, find n(B).

14 Oasis School Mathematics-9

5. (a) From the given diagram, find the value of: U
AB
(i) x (ii) no (A)
(iii) no (B) 60–x x 70–x
(iv) n(A ∪ B) 30

n(U) = 120

(b) From the given diagram, find the value of: P U
Q
(i) x (ii) n(P)

(iii) n(Q) (iv) n(P∪Q) 30 x 50

30

n(U) = 120

6. (a) If n(U) = 200, n(M) = 120, n(E) = 110, n(M∪E ) = 150, using Venn diagram find
the value of (i) n(M∩E) (ii) n(M∪E).

(b) If A and B are the subsets of universal set U, such that n(U) = 50, n(A) = 28, n(B) = 22,
n(A∩B) = 12, find n(A∪B) and n(A ∩ B). Illustrate the above information in the
Venn digram.

(c) If n(P) = 60, n(Q) = 70, n(P∩Q) = 40, find

(i) no(P) (ii) no(Q) (iii) n(P∪Q)

(iv) Show the above information in the Venn diagram.

(d) If n(A) = 100, n(B) = 60, n(A∪B) = 120, n(U) = 130, find (i) n(A∩B),

(ii) n(A∪B) (iii) n(A – B) (iv) n (B – A)

7. (a) If no(A) = 30, no(B) = 25, n(A∩B) = 18, find the n(A∪B).
(b) If no(A) = 45, no(B) = 50, n(A∪B) = 120, and n(U) = 150,

(i) show the information in Venn diagram.

(ii) find n(A∩B), n(A), n(B) and n(A∪B).

8. (a) In a language class of 85 students, 60 study German and 45 study French. If
20 study none of these two languages, using Venn diagram, find how many
students study both of the languages?

(b) In a survey among the people of internet lovers, it was found that 150 people
used Facebook, 90 people used Instagram and 50 people did not use any of them.
If 200 people used at least one of them, using Venn diagram.

(i) find the number of people who used both Facebook and Instagram.

(ii) find the total number of people in the survey.

(c) In a survey of 400 students, 300 like Mathematics, 270 like Science and 80
students do not like any of these subjects. Find

Oasis School Mathematics-9 15

(i) how many students like at least one subject?

(ii) how many students like both subjects?

(iii) how many students like only one subject?

(iv) show the above information in the Venn diagram?

(d) In a survey of a group of people, it was found that 170 people like noodles and
120 like Mo : Mo, 64 like both and 72 do not like any of these.

(i) Illustrate the above information in the Venn diagram.

(ii) Find the total number of people surveyed.

(iii) Find the number of people who like only one of these.

(e) In a survey of a group of people they like either to watch movie or to listen to
music or both. 150 people like to watch movie and 160 people like to listen to
music. If 120 people like to watch movie as well as to listen to music,

(i) find the number of people under the survey.
(ii) find the number of people who like to watch movie only.

(iii) find the number of people who like to listen to the music only.

(iv) show the above information in the Venn diagram.

9. (a) In a survey among a group of people it was found that 80 people like football
only and 70 people like cricket only. If 35 people like both and 25 do not like any
of them, by drawing Venn diagram, find the number of people in the survey.

(b) In a group of 200 people, 60 like milk only, 50 like curd only and 10 like none of
these two. Using Venn diagram find the number of people who like both.

10. (a) In an examination, 60% students passed in English, 70% passed in Nepali and
15 % failed in both subjects. Find the number of students passed in both subjects.

(b) In an examination 20% students failed in Science and 30% failed in Maths. If 60%
students passed in both subjects, find the number of students failed in both subjects.

11. (a) In a survey, 80% people like Coke and 85% like Pepsi, 75% people like both the
drinks. If 45 people do not like both the drinks, find the total number of people
in the survey. Show the given information in Venn diagram.

(b) In an examination, 60% students passed in Mathematics, 75% passed in English
and 55% passed in both subjects. If 60 students failed in both subjects, using
Venn diagram, find the total number of examinees.

(c) 40% of the students of a school play football, 30% play volleyball and 50% play
neither. If 36 students play both football and volleyball, use Venn diagram and find,

(i) the number of students in the school.
(ii) number of students who play only football.

16 Oasis School Mathematics-9

(iii) number of students who play only one game.

12. (a) In a survey of 56 students, it was found that the number of students who like
football and cricket is in the ratio 5 : 4. If 7 like both games and 9 like neither of
the games then by drawing Venn diagram,

(i) find how many students like football.

(ii) find how many students like cricket only.

(b) In a survey among 65 people about their favourite fruits, it was found that
the number of people who like mango is twice the number of people who like
orange. If 10 people like both fruits and 15 do not like any of them, using Venn
diagram find the number of people who like (i) Mango only, (ii) Orange.



Answer

1. (a) (i) 4 (ii) 5 (iii) 10 (iv) 2 (v) 7 (vi) 2 (vii) 3
(ii) 5
(viii) 3 (ix) 8 (b) (i) 7 (ii) 3 (iii) 4 (iv) 2 (v) 9 (vi) 3
(iv) 40
2. (a) Consult your teacher (b) (i) 3 (b) 15 (iii) 2 (iv) 8 (v) 2 (iv) 8
(iv) 90
3. (a) (i) 10 (ii) 20 (iii) 40 (iii) 90 (v) equal (b) (i) 40 (ii) 35 (iii) 60

(iv) 60 (v) equal 4. (a) 70 5. (a) (i) 40 (ii) 20 (iii) 30 (iv) 90

5. (b) (i) 10 (ii) 40 (iii) 60 6. (a) (i) 80 (ii) 50

(b) (i) 38, (ii) 38 (c) (i) 20 (ii) 30 (d) (i) 40 (ii) 10 (iii) 60 (iv) 20

7. (a) 73 (b) (ii) 25, 70, 75, 30 8. (a) 40 (b) (i) 40 (ii) 250 (c) (i) 320 (ii) 250 (iii) 70 (d) 298,

162, (e) 190, 30, 40, 9. (a) 210, (b) 80

10. (a) 45% (b) 10% 11. (a) 450 (b) 300 (c) 180, 36, 54 12. (a) 30, 17 (b) (i) 30 (ii) 20

Do you know!

The concept Venn diagram was first
introduced by John
of set was first Venn in 1880 A.D. Venn
introduced
by German diagram is used in many
fields, including set
Mathemati- theory, probability, logic,
cian George
Cantor. statistics and Computer (1834AD - 1923AD)
(1845AD - 1918 AD) science.

Project work

Take a survey among 20 people in your locality about their interest in fruits whether
they like mango, orange, both or none of them. List each sets. Find their cardinal
number and present the result in the Venn-diagram.

Oasis School Mathematics-9 17

Assessment Test Paper

Attempt all the questions. Full Marks: 20

Group – "A" [4×2=8]

1. (a) If A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10}, find A∪B and A – B.

(b) From the given Venn diagram, •f U

Find (i) A (ii) n(A) A •a •q •x B
•b
(iii) B (iv) n(B) •r •y
•p •z
•d •e

2. (a) If U = {x : 1 ≤ x ≤ 10}, A = a set of odd numbers less then 10, B = a set of prime
numbers less than 10, find A – B, n(A – B) and show this relation in the Venn
diagram.

(b) What does the shaded part of each of the given figures represent?

U A UB
AB

CC

Group – "B" [3 × 4=12]
3. If n(A) = 50, n(B) = 60 , n(A∪B) = 100, then find (i) n(A∩B) (ii) no(A) (iii) no(B).
4. In a group of 70 people, 37 like tea, 52 like milk and each person likes at least one of

the two drinks. Using Venn diagram find,
(i) how many like both tea and milk?
(ii) how many like tea only?
(iii) how many like milk only?
5. In a group of students, 50% like Mathematics, 70% like Science, 10% do not like any

of these subjects and 120 like both. By using Venn diagram, find the total number of
students.

18 Oasis School Mathematics-9

Arithmetic

36Estimated Teaching Hours

Contents

• Profit and Loss
• Discount
• Commission
• Taxation
 Income Tax
 Education Service Tax
 VAT
• Dividend
• Household Arithmetic
 Electricity Bill
 Water Bill
 Telephone Bill
 Taxi Fare

Expected Learning Outcomes

At the end of this unit, students will be able to develop the
following competencies:

• To solve the simple problems on profit and loss
• To calculate the income tax of an individual and married couple
• To calculate the education service charge to be paid by an educational

institution
• To calculate the commission to be received by a middle man or salesman

while selling a land or articles
• To calculate the amount of dividend obtained by a shareholder of a

company
• To calculate the amount of value added tax to be paid by a customer on the bill
• To read the water bill, telephone bill and to calculate the amount to be paid
• To calculate the taxi-fare at the given rate

Teaching Materials

• Sample of Electricity Bill, Water Bill, Telephone Bill, etc.

Oasis School Mathematics-9 19

Unit

2 Profit and Loss

2.1 Warm-up Activities

Discuss the following in your class and draw out the conclusion.
– A man bought an article for Rs. 400 and sold it for Rs. 500.
What is the cost price of the article?
What is the selling price of the article?
Is their profit or loss?
Which value should be greater to make profit?
– A man bought an article for Rs. 9000, he paid Rs. 1000 for transportation charge

and sold it for Rs. 9,500.
What is the cost price of the article?
Is their profit or loss?
– Where the transportation charge Rs. 1000 is to be added? Whether on

CP or on SP.
– An article is marked Rs. 6,000. It is sold for Rs. 5400.
What is the marked price of the article?
What is the selling price of the article?
– On which condition MP and SP of an article are equal?

2.2 Profit and Loss

The term profit and loss is generally used by the businessmen in the transaction of
goods while buying or selling them. A person buys goods at a certain cost which
is called its cost price, in short C.P. and the cost at which an article is sold called its
selling price, in short S.P.

If the cost price is less than the selling price, there will be profit and if the cost price
is more than the selling price, there will be loss in the transaction. Hence, we can
relate the terms profit, loss, C.P. and S.P. as follows:

(i) If C.P. < S.P., then profit = S.P. – C.P.
(ii) If S.P. < C.P., then loss = C.P. – S.P.

Note:

(i) Overhead expenses like transportation, maintenance and repair and other expenditures
are always added with the cost price to get net cost price.

(ii) Profit or loss is calculated with the help of S.P. and C.P. of the same number of articles.

20 Oasis School Mathematics-9

Profit and Loss Percentage

A businessman always desires to expand the business by studying the market value
of goods in terms of his/her profit percentage. Profit or loss percentage always
gives an exact idea about the nature of demand and supply. Just by knowing the
profit or loss, one can't predict the future prospect of the business.

For example, if a calculator is bought for Rs. 500 and sold for Rs. 700, the profit is Rs. 200.
Also if a geometrical box is bought at Rs. 100 and sold for Rs. 300, the profit is again Rs.
200.

But, profit percent in first case = Rs.200 ×100 % = 40%
Rs.500

While, profit percent in second case = Rs.200 ×100 % = 200%
Rs.100

Hence, a businessman would better go for the business of second type.

Note:

Profit percent or loss percent is always calculated out of the given cost price
unless and until it is stated otherwise.

Hence, we can write,

Profit % = Profit × 100% ........ (i)
C.P.

Loss % = Loss × 100% ........ (ii)
C.P.

Again, if profit % is given, profit = Profit % of CP ........ (iii)

S.P. = C.P. + Profit % of C.P. ........ (iv)

If loss % is given, loss = Loss% of CP ........ (v)

S.P. = C.P. – Loss % of C.P. ........ (vi)

From (iii) and (iv) we can derive:

(i) S.P = (100 + P %) × C.P. (ii) S.P = (100 - L %) × C.P.
100 100

(iii) C.P. = S.P. × 100 (iv) C.P. = S.P. × 100
(100 + P%) (100 – L%)

From these, we can derive that:

C. P. = 100 and C. P. = 100
S. P. 100+P% S. P. 100 – L%

Oasis School Mathematics-9 21

Worked Out Examples

Example: 1

An article which is bought for Rs. 1500 is sold for Rs. 2500. Find the profit or loss percent.

Solution: C.P. = Rs. 1500
Here,

S.P. = Rs. 2500

Since, S.P. > C.P., there is profit

We have, Profit = S.P. – C.P.

= Rs. 2500 – Rs. 1500 = Rs. 1000

We have, Profit
C.P.
Profit percent = × 100%

= 1000 × 100%
1500

= 200 %
3
2
= 66 3 %

Example: 2

Nadeem bought a second hand motorcycle for Rs. 36,000 and he spent Rs. 4,000 for
repairing. After a few days he sold it for Rs. 39,000. Find his profit or loss percent.

Solution:

Here, C.P. of the motorcycle = Rs. 36,000

Repairing cost = Rs. 4,000

∴ Net C.P. = Rs. 36,000 + Rs. 4,000

= Rs. 40,000

S.P. of the motorcycle = Rs. 39,000

Since, C.P. > S.P., there is loss

and, Loss = C.P. – S.P.

= Rs. 40,000 – Rs. 39,000

= Rs. 1,000

We have, Loss percentage = Loss × 100%
C.P.

= RsR. s1.04000,0×01000

= 5 % = 2.5%
2

22 Oasis School Mathematics-9

Example: 3

If the cost price of 5 articles is equal to the Alternative method
selling price of 4 articles, find the profit and loss

percentage. Let, C.P. of 5 articles = S.P. of 4
Solution: articles = x

Let, C.P. of 1 article = Rs. x Then, C.P. of 1 article = x
∴ C.P. of 5 articles = Rs. 5x 5
Again, by question, x
S.P. of 1 article = 4

S.P. of 4 articles = C.P. of 5 articles Profit = S.P. – C.P.

i.e. S.P. of 4 articles = Rs. 5x = x – x = x
4 5 20
∴ S.P. of 1 article = Rs. 5x
4 Profit
We have, profit % = C.P. × 100%
N ow, comparing the S.P. and C.P. of 1 article, we x

have S.P. > C.P., Hence, there is profit = 20 × 100% = 25%
x
We have, Profit = S.P. – C.P. 5

= 5x – x
4
x
= 4

We have,p rofit percent = Profit × 100%
C.P.

x

= 4
x
x 1
= 4 × x × 100 % = 25%

Example: 4

Find the S. P. of an article which was bought for Rs. 600 and sold at a profit of 20%.

Solution:

Here, C. P. = Rs. 600

Profit % = 20%

S. P. = ? Alternative method-I
We have,
We have, S. P. = C. P. + P% of C. P.

= 600 + 20% of 600 S.P. = (100 + P%) C.P.
= 600 + 20 × 600 = 100

100 (100 + 20) × 600
= 600 + 120 = 100

= Rs. 720 Rs. 120 × 6

∴ S. P. of the article = Rs. 720. = Rs. 720

Oasis School Mathematics-9 23

Example: 5 Alternative method-II
We have,
Find the cost price of an article which was sold at
S.P. 100+P%
Rs. 1700 at a loss of 15%. C.P. = 100

Solution: S.P. 100+20
600 100
Here, S. P. = Rs. 1700 or, =

Loss % = 15% or, S.P. = 120
600 100
C. P. = ?
or, S.P. = Rs. 120 × 6

We have, S. P. = C. P. – L% of C. P. ∴ S.P. = Rs. 720

or, 1700 = C. P. – 15 × C. P.
or, 1700 100
or, 1700
= C. P. – 3 C.P.
20

= 17CP
20

or, 1700 × 20 = 17 C. P.

or, C.P. = 1700 × 20
∴ C.P. 17

= Rs. 2000.

Alternative method-I Alternative method-II

We have C.P. = S.P. × 100 We have,
(100 – L%)
S.P. = 100–L%
C.P. 100
= 1700 × 100
100 – 15 or, 1700 = 100–15
C.P. 100
= 1700 × 100
85 or, 1700 = 85
C.P. 100
= Rs. 2000
or, 85 C.P. = 1700 × 100
Alternative method-II or, C.P. = 1700×100

Since, L % = 15% 85
∴ C.P. = Rs. 2000
When S. P. = Rs. 85, C. P. = Rs. 100

When S. P. = Re. 1, C. P. = Rs. 100
85
When S. P. = Rs. 1700, C. P. = Rs. 100 × 1700
85
Rs. 100
∴ C. P. = 85 × 1700

= Rs. 2000.

24 Oasis School Mathematics-9

Example: 6

Two transistors are bought paying Rs. 2500. While selling at the same rate, it is found
that there is profit of 20% in one and loss of 20% in another. Find the cost price of each
transistor.

Solution: Alternative method

Let, the C.P. of first transistor = Rs. x Let, the CP of first transistor = Rs. x

Then, CP of second transistor

Profit = 20% = 2500 – x

We have, S.P. = C.P. + P% of C.P. For first transistor,

CP = x, P% = 20%

= x+ 20% of x We have,
C.P. 100
= x+ 20 ×x S.P. = 100+P%
100
x5 = 6x x = 110
= x+ 5 SP 120

Here, the C.P. of second transistor = Rs. (2500 – x) x = 5
SP 6

Loss = 20% SP = 6x
5
S.P. = C.P. – L% of C.P.
For second transistor,

= (2500 – x) – 20% of (2500 – x) CP = (2500–x), L% = 20%

= (2500 – x) – 20 × (2500 – x) CP = 100
100 SP 100 – L%

= (2500 – x) – 1 × (2500 – x) 25000–x = 5
5 SP 4

= 4 (2500 – x) 5SP = 4(2500–x)
5
SP = 4 (2500–x)
S.P. of first transistor = S.P. of second transistor 5

6x 4 Since SP of both transistor same,
5 5
or, = (2500 – x) Now, 6x = 4 (2500–x)
5 5
or, 6x + 4x = 10000
6x = 10000 – 4x

or, 10x = 10,000 10x = 10000

or, x = 10,000 = 1000 x = 1000.
10

i.e. C.P. of first transistor is Rs. 1000 and CP of first transistor = x, CP of
second transistor = 2500 –x
C.P. of second transistor is = Rs. 2500 – Rs. 1000
= 2500 –1000 = Rs. 1500
= Rs. 1500

Oasis School Mathematics-9 25

Example: 7

A shopkeeper sold a mobile set for Rs. 7200 at a loss of 20%. At what price should he
sell to get the profit of 25%?

Solution:

Here, S.P. of a mobile set = Rs. 7200

Loss% = 20% Alternative method

We have, S.P. = C.P. – L% of C.P. Here, S. P. = Rs. 7200

L% = 20%

7200 = C.P. – 20% of C.P. We have

20 C.P. = 100
100 S.P. 100–L%
or, 7200 = C.P. – × C.P.
or, 7200 or, C.P. = 100
or, 7200 = C.P. – CP 7200 100-20
5
or, C.P. = 100
= 4C5P 7200 80

or, C.P. = 100×7200
80
or, 4C.P. = 7200 × 5
or, C.P. = Rs. 9000
7200×5
or, C.P. = 4 Again, in the second case,

∴ C.P. = Rs. 9000 C.P. = Rs. 9000, P% = 25%, S. P. = ?

C.P. = 100
S.P. 100+P%

Again, in the next case, or, 9000 = 100
C.P. = Rs. 9000, P rofit % = 25%, S.P. = ? S.P. 100+25
We have,
S.P. = C.P. + P% of C.P. or, 9000 = 100
S.P. 125

or, S.P. = 9000×125
100

= 9000 + 25% of 9000 or, S.P. = Rs. 11.250

= 9000 + 25 9000 ∴ Required selling price = 11.250
100
×

= 9000 + 1 × 9000
4

= 9000 + 2250

= Rs. 11,250
∴ Required selling price = Rs. 11,250.

26 Oasis School Mathematics-9

Example: 8
Amit sold an article at a profit of 15%. If he had sold it for Rs. 81 less, his loss would
have been 12%. Find the cost price of the article.
Solution:

Let, the C.P. of article = Rs. x

Profit percent = 15%

We have, S.P. = C.P. + P% of C.P.

= x + 15% of x

= x + 15 × x = 115x = 23x
100 100 20

Now, if the S.P. would be Rs. 81 less, then Alternative method

New S.P. of article = Rs.  23x – 81 Let C. P. = x, P% = 15%
 20
We have,
Loss percent = 12%
C.P. = 100
We have, S.P. 100+P%

S.P. = C .P. – L% of C.P. or, x = 100
S.P. 115

or, 23x – 81 = x – 12% of x or, S.P. = 111050x = 23x
or, 20 20

23x – 81 = x – 12 x In the second case,
20 100
× ( )If S.P. = 23x – 81 L% = 12%
20
or, 23x – 81 = x– 3x We have,
or, 20 25
C.P. = 100
23x – 81 = 22x S.P. 100–L%
20 25
or, x = 100
23x 22x 23x 100–12
20 – 25 = 81 20
= 81
or, 115x – 88x or, 20x = 100
or, 100 23x–1620 88

20x 25
27x or, 23x–1620 = 22
100
or, = 81 or, 440x = 575x–25×1620

or, x = 3 or, 440x – 575x = –25 × 1620
∴ = Rs. 300
100 or, 137x = 25 × 1620
x
or, x= 25 × 1620
135
∴ x = Rs. 300.

Oasis School Mathematics-9 27

Exercise 2.1

1. (a) Write the formula to calculate profit if S.P. and C.P. are given.

(b) Write the formula to calculate profit percent if C.P. and S.P. are given.
(c) Write the formula to calculate profit percent if profit and C.P. are given.
(d) On which condition there is loss? Write the formula to calculate loss if C.P.

and S.P. are given.
(e) Write the formula to calculate loss percent if C.P. and S.P. are given.
(f) Write the formula to calculate loss if loss percent and C.P. are given.
(g) Write the formula to calculate profit if profit percent and C.P. are given.
(h) Write the relation between C.P. and S.P. if profit percent is given.

(i) Write the relation between C.P. and S.P. if loss percent is given.

2. (a) An article purchased for Rs. 1000 is sold for Rs. 1200. Find the profit and its
percentage.

(b) Manish sold an article for Rs. 2000 which he had bought for Rs. 2500. Find his
loss and its percentage.

(c) Susmita bought a T.V. set for Rs. 30,000. After using it for a few months she
sold it for Rs. 27,000. Find her loss percentage.

3. (a) A cosmetic shopkeeper bought 1 gross lip guard for Rs. 2880 and sold each of
them at a rate of Rs. 25. Find the profit or loss percentage.

(b) A man bought 2 dozen of copy for Rs. 288 and sold at a rate of Rs. 15 per copy.
Find his profit or loss percent.

4. (a) A man bought a second hand television for Rs. 12,000 and he spent Rs. 2,000
for repairing. After a few days he sold it for Rs. 16,500. Find his profit or loss
percent.

(b) Ritesh bought a mobile set for Rs. 5000 and spent Rs. 1500 for its maintenance.
If he sold it for Rs. 6000, find his profit or loss percent.

(c) Ram sold a computer for Rs. 18,000 after spending Rs. 1,500 for its maintenance.
If he bought it for Rs. 17,000, find his profit or loss percent.

(d) Saurav sold a mobile set for Rs. 15000 after spending Rs. 2000 for its repairing.
If he bought it for Rs. 10000, find his profit or loss percent.

5. (a) Find the profit percent when 6 pens are sold at the cost price of 9 pens.

(b) If the cost price of 10 articles is equal to the selling price of 6 articles, find the
profit percentage.

(c) A man makes a profit of cost price of 4 pens on selling 15 pens. Find his profit
percent.

28 Oasis School Mathematics-9

(d) A man gains the selling price of 5 books by selling 25 books. What is his gain
percent?

6. (a) 4 lemons are bought for Re. 1 and sold 3 for Re. 1. Find profit or loss percent.

(b) 6 lemons are bought for Rs. 5 and sold 5 at Rs. 6. Find profit or loss percent.

7. (a) Ankit bought a table for Rs. 1500. Find the selling price of the table in order to
gain 15% profit.

(b) Kanchan bought apples at the cost of Rs. 10,000. Selling all of them, she got
50% profit. Find their selling price.

(c) A mobile set which is bought for Rs. 20,000 is sold in order to gain 17.5%
profit. Find the selling price.

(d) Find the cost price of an article which was sold for Rs. 720 at a profit of 20%.

(e) Sarada sold an electric iron for Rs. 1,900 at a loss of 5%. Find the cost price of
the iron.

(f) Sunayana sold an article for Rs. 2300 at a profit of 15%. Find the cost price of
the article.

8. (a) Pratik sold a second hand motorbike for Rs. 42,500 after spending Rs. 3500
for repairing. If by doing so he gained 15% profit, find the price he paid while
buying.

(b) A house is purchased for Rs. 50,00,000. After spending some amount for its
maintenance it is sold for Rs. 60,50,000 in order to make a profit of 10%, find
the maintenance cost.

(c) A man sold a car for Rs. 7,20,000 after spending some money for its maintenance
at a profit of 20%. If he purchased the car at Rs. 5,00,000. Find the maintenance
cost of the car.

9. (a) Kiran bought two school bags for Rs. 1000. While selling at the same rate, he
found that there was a profit of 20% in one and a loss of 10% in another. Find
the cost price of each bag.

(b) Nikhil bought two radios for Rs. 800. He sold them to gain 10%, profit on one
and 10% loss on the other. Calculate his gain or loss percent in this transaction
if the selling prices of both radios are the same.

(c) A shopkeeper bought two watches for Rs. 400. He sold them to gain 5% on
one and lose 5% on the other. Calculate his final gain or loss percent if the
selling prices of both watches are the same.

(d) Kavi bought two bicycles for Rs. 6000. He sold the first bicycle at a profit of
20% and the second one at a loss of 10%. If he gained 2% profit on his total
outlay, at what price did he purchase each of the two bicycles?

10. (a) A sold two radios for Rs. 1980 each. He made a profit of 10% on one and loss
of 10% on the other. Find his gain or loss percent in his total outlay.

Oasis School Mathematics-9 29

(b) Kriti sold two mobile sets for Rs. 3600 each. She made a profit of 20% on one
and loss of 10% on the other. Find her gain or loss percent in her total outlay.

11. (a) Smriti sold an article for Rs. 2,000 and found that there was a loss of 20%. At
what price should she have sold in order to have 10% as a profit?

(b) Asim got a profit of 5% while selling a chair for Rs. 525. What profit percent
would he get by selling the same for Rs. 600?

(c) A shopkeeper sold a shirt for Rs. 1,080 at a loss of 10%. At what price should
he have sold to gain 10% profit, on it ?

12. (a) Sandip bought 1 quintal of oranges paying Rs. 1200. He sold 40kg at a rate of

Rs. 20/kg, another 40kg at the rate noeftRpsr.o1f5it/okfg3.3At13w%h?a t rate should he sell
the remaining in order to make the

(b) A man bought 40 pens for Rs. 1,600. He sold 10 pens for Rs. 40 each and 15

pens at Rs. 45 each. At what rate must he sell the rest to gain 15% profit on his

outlay ?

(c) Aruna bought 1,000 duck eggs of which 100 were spoilt. If she gained 2% by
selling the rest at a rate of Rs. 9.18 each, at what rate did she buy them?

(d) A fruit seller bought 800 oranges for Rs. 2,500. 100 oranges were rotten. He
sold the remaining oranges and gained 12% profit. Find the selling price of
each orange.

13. (a) Subash sold an article at a profit of 15%. If he sold it for Rs. 81 less, his loss
would have been 12%. Find the cost price of the article.

(b) A shopkeeper sold a radio at a loss of 10%. Had he sold it for Rs. 150 more he
would have made a profit of 20%. What was the cost price of the radio?

(c) Bikram sold an article at a loss of 4%. Had it been sold for Rs. 39 more, the
profit would have been 9%. What is the cost price of the article?

14. (a) P sold a laptop to Q making a profit of 10%. Q sold the same to R making a
profit of 15%. If P had bought the laptop for Rs. 40,000, find how much R paid
for it.

(b) Lakpa bought a mobile set for Rs. 5000 and sold it to Sonam making a profit
of 10%. Sonam sold the same to Pemba making a profit of 15%. Find the cost
price of the mobile for Pemba.

(c) A sold a cycle to B at a profit of 10%. B sold the same cycle to C at a profit of 20%. If
C has sold it for Rs. 3300 thereby earning a profit of 25%, find the cost price for A.

(d) A sold a bicycle at 10% profit to B. B used the bicycle for some days then she
sold it to C at 15% loss. If C bought the bicycle for Rs. 7480, then at what price
was the bicycle bought by A?

30 Oasis School Mathematics-9

Answers

1. Consult your teacher 2. (a) Rs. 200, 20% (b) Rs. 500, 20% (c) 10%
6 9 26
3. (a) 25% profit (b) P% = 25% 4. (a) 17 7 =%6p6 r32of%it ( b ()c)7P1%3 % loss (c) 2 37 % loss
(d) 25% profit 5. (a) = % (d) 25% profit
P% = 50% (b) P% 26 2
1 3

6. (a) P% = 33 3 % (b) P % = 44% 7. (a) Rs. 1,725 (b) Rs. 15,000

(c) Rs. 23,500 (d) Rs. 600 (e) Rs. 2000 (f) Rs. 2000

8. (a) Rs. 33,456.52 (b) Rs. 5,00,000 (c) Rs. 100,000

9. (a) 428.57, Rs. 571.43 (b) Loss 1% (c) Loss 0.25 % (d) Rs. 2,400, Rs. 3,600
6

10 (a) 1% Loss (b) 2 7 % Profit 11. (a) Rs. 2,750 (b) 20% (c) Rs.1,320
12. (a) Rs. 10 per kg (b) Rs. 51 (c) Rs. 8.10 (d) Rs. 4

13. (a) Rs. 300 (b) Rs. 500 (c) Rs. 300 14. (a) Rs. 50,600 (b) Rs. 6,325 (c) Rs. 2,000 (d) Rs. 8,000

2.3 Discount

Sandesh wants to buy a pair of shoes and Asmita wants to
buy a Sari.

The tagged price of shoes is Rs. 2500 and the tagged price
of Sari is Rs. 8,000.

There is a festival discount offer of 25%.
What is the amount of discount in each article?
How much Sandesh paid for the shoes?
How much Asmita paid for the Sari?

Let's discuss it.

The tagged price in each article are their marked price.

amount of discount in shoes = 25% of 2500

= 25 × 2500 = Rs. 625
100

The price of shoes = Rs. (2500 – 625) = Rs. 1875

Again, discount in Sari = 25% of 8000

= Rs. 25 × 8000 = Rs. 2000
100

The price of Sari = Rs. 8000 – Rs. 2000 = Rs. 6000

Hence, we can observe the price tag which is kept along with the article while
purchasing goods from the shop or department store. The price of the goods so
kept using the tag or the list or mark is called marked price or the quoted price or
the labeled price or the list price.

i.e. M.P. ≥ S.P.

The concession or deduction over the marked price of an article while at the time of
selling is called discount. It is normally expressed in terms of percentage of its marked

Oasis School Mathematics-9 31

price.
Hence, Discount = Discount % of M.P.

Selling price

The price of the article after allowing discount from its M.P. is called its selling price.

Hence, S. P. = M. P. – Discount

= MP – Discount % of MP

Note:

Note: If there is no discount then M. P. = S.P.

We can relate the terms C.P., S.P., Profit, Loss, MP and Discount as shown.

MP – Discount S.P. – Profit CP
+ Loss


Remember !

Discount = Discount % of M.P.

S.P. = M.P. – Discount

= M.P. – Discount % of M.P.

Discount percent = Discount × 100%
M.P.

Worked Out Examples

Example: 1

If a discount of 10% is offered on the goods whose M.P. is Rs.1000, find its S.P.
Solution:

Here, M.P. = Rs. 1000

Discount = 10% of M.P. = 10% of Rs. 1000

= 11000 × Rs. 1000

= Rs. 100

∴ S.P. = M.P. – discount = Rs. 1000 – Rs. 100 = Rs. 900

32 Oasis School Mathematics-9

Example: 2

Pratima sold a book for Rs. 135 after a discount of 10% on M.P. Find its M.P.

Solution: Alternative method

Let, M.P. = Rs. x 10
100
Discount = 10% of x = ×x= x S.P. = M.P. – D% of M.P.
10
10
Now, S.P. = Rs. 135 135 = M.P. – 100 × M.P.

We have, 90M.P.
100
M.P. – Discount = S.P. or, 135 =

or x– x = Rs. 135 or, 135 = 9M.P.
10 10

or, 91x0 = Rs. 135 190 or, M.P. = 135 × 10 = Rs. 150.
or, x = Rs. 135 × 9

∴ M.P. = Rs. 150

Example: 3

The M.P. of an article is 25% above the C.P. If it is sold after allowing a discount of 10%,
the trader will get a profit of Rs. 500. Find the S.P. of the article.

Solution:

Let the C.P. of the article be x

∴ M.P. = x + 25% of x

= x + 25 × x = 5x
100 4

Hence, Discount = 10% of M.P.

= 10 × 54x = x
100 8

∴ S.P. = M.P. – Discount

= 54x – x
8

= 10x8– x = 98x

Now, S.P. – C.P. = Profit

9x – x = Rs. 500
8 = Rs. 500
x
8
x = Rs. 4000

∴ S.P. = 98x = 9 × Rs. 4000
8

= Rs. 4500

Oasis School Mathematics-9 33

Example: 4

A shopkeeper selling an article at a discount of 20% loses Rs. 200. If he allows 10%
discount, he gains Rs. 150. Find the marked price and cost price of the article.

Solution:

Let, M. P. = Rs. x

S.P. = M. P. – Discount % of M.P.

= x – 20% of x

= x – 20 × x
100

= x– x
5
4x
= 5

Loss = Rs. 200

∴ C.P. = S. P. + Loss

or, C.P. = 4x + 200 ………… (i)
5
In second case,

S.P. = M. P. – Discount % of M.P.

= x – 10% of x

= x – 10 × x
100

= x – x
10
= 91x0

Profit = Rs. 150

Now, C.P. = S. P. – Profit

C.P. = 9x – 150 ………… (ii)
10

From (i) and (ii)

4x + 200 = 9x – 150
5 10

or, 9x – 4x = 200 + 150
10 5

or, 9x – 8x = 350
or, 10 x = 350
10
or, x = 3500

∴ M.P. = Rs. 3500

34 Oasis School Mathematics-9

Again, C.P. = 4x + 200
We have, 5

= 4 × 3500 + 200
5

= 2800 + 200

= 3,000

∴ C.P. = Rs. 3,000

Exercise 2.2

1. (a) Write the formula to calculate discount if M.P. and S.P. are given.
(b) Write the formula to calculate discount if discount percent and M.P. are given.
(c) Write the formula to calculate discount percent if discount and M.P. are given.
d) Write the relation among MP, S.P. and discount percentage.

2. Find the discount and the discount percent in each of the following cases.

(a) M.P. = Rs. 450, S. P. = Rs. 400

(b) M.P. = Rs. 1200, S.P. = Rs. 1150

3. (a) If M.P. = Rs. 2,000, Discount % = 20%, find S.P.

(b) If M.P. = Rs. 18,500, Discount % = 15%, find S.P.

(c) If S.P. = Rs. 990, Discount % = 10%, find M.P.

(d) If S.P. = Rs. 3,400, Discount % = 15%, find M.P.

(e) If discount% = 20% Discount = Rs. 120, find M.P.

(f) If discount % = 25% Discount = Rs. 1200, find MP and SP.

4. (a) A radio which is marked to sell for Rs. 300 is finally sold for Rs. 250. Find the
discount and its percentage.

(b) There is a festival discount of 20% in shirts. Find the price to be paid by a
customer for which the marked price is Rs. 800.

(c) The marked price of a suitcase is Rs. 2000. If 15% discount is offered, find the
price to be paid by the customer.

(d) Roshan sells a dictionary for Rs. 765 after a discount of 10%. Find its marked
price.

(e) The selling price of a piece of land is Rs. 2,70,000 after a discount of 25%. Find
its original price.

Oasis School Mathematics-9 35

5. (a) Marked price of an article is Rs. 10,000. What is the cost price of the article if it
is sold at a profit of 10% after allowing 12% discount?

(b) Marked price of a mobile set is Rs. 24000. A discount of 12.5% is given to the
customer. If the profit percent is 10%, find the cost price of the article.

(c) A man sold an article at a profit of 20% after allowing 16% discount on the
marked price. If its marked price is Rs. 10,000, find the cost price of the article.

6. (a) Sarbottam bought a cycle for Rs. 6400. If he got a profit of 25% after allowing
discount of 20% while selling, find the marked price.

(b) Bibek bought a bag for Rs. 1200. After allowing a discount of 13%, he sold it
and got a profit of 6%. Find the marked price.

(c) After allowing 12% discount on the marked price of an article, there is a profit
of 6%. If the cost price of the article is Rs. 264, calculate the marked price of the
article.

(d) What marked price should a shopkeeper fix for the bicycle bought for Rs.
3600, so that there is a profit of 20% even if it is sold at 20% discount?

7. (a) Tobika bought a mobile set for Rs. 2500 and labelled its price 20% above the
cost price. If she allows 10% discount to the customer, find her profit percent.

(b) Vibhuti bought a laptop of Rs. 50,000 and labelled its price 20% above its cost
price. If she allows 20% discount while selling, find her profit or loss percent.

(c) A shopkeeper marks the price of his goods 10% above the cost price and gives
the discount of same percent to the customer. Find his profit or loss percent.

8. (a) A book is marked to sell at a profit of 10%. If it is sold for Rs. 5 less, there
20
would be a profit of 3 %. Find the cost price.

(b) The marked price of a radio is fixed to make a profit of 25%. If it is sold after

allowing a discount of Rs. 800, there will be a loss of 20%. Find the cost price.

9. (a) The marked price of an article is fixed to be 50% above the cost price. If 10%
discount is offered on it, the selling price reduces into Rs. 2700. Find the
marked price as well as its cost price.

(b) A shopkeeper marked the price of an article 40% above the cost price. After
allowing a discount of 15% on its marked price, it was sold at a gain of Rs. 950.
Find the marked price of the article.

(c) The marked price of a watch was 30% above the cost price. When it was sold
allowing 20% discount on it, there was a gain of Rs. 150. Find the marked price
and cost price of the watch.

10. (a) A man sold an article on its marked price at a gain of 20%. But allowing 5%
discount, there would have been a gain of Rs. 140. Find the cost price of the
article.

36 Oasis School Mathematics-9

(b) A man sold an article on its marked price at a gain of 25%. But allowing Rs. 600
discount there would have been a profit of Rs. 200 only. Find the cost price and
the marked price of the article.

(c) An article, after allowing a discount of 20% on its marked price, was sold at a
gain of 20%. Had it been sold after allowing 25% discount, there would have
been gain of Rs. 125. Find the cost price of the article.

(d) A shopkeeper selling an article at a discount of 20%, loses Rs. 600. If he allows
10% discount, he gains Rs. 150. Find the marked price and the cost of the article.

(e) A mobile set, after allowing a discount of 10% on its marked price, was sold at
a gain of 20%. Had it been sold after allowing 20% discount, there would have
been a profit of Rs. 350. Find the cost price of the mobile set.

11. (a) The selling price of a T.V. set is Rs. 9000 after allowing 10% discount. If thereby
a shopkeeper makes a profit of 12.5%, find by what percent M.P. is greater
than C.P.?

(b) Anmol sold an article for Rs. 1,500 after allowing a discount of 25%. If he made
a profit of 20%, find the marked price and cost price. Also find by what percent
marked price is greater than cost price ?

12. (a) The marked price of an article is 20% above the selling price and the cost
price is 40% below the marked price. Find the profit percent and the discount
percent.

(b) The marked price of an article is 30% above its selling price and the cost price
is 35% less than the marked price. Find the discount percent and profit percent.

Answers

1. Consult your teacher
1 1
2. (a) Rs. 50, 11 9 % (b) Rs. 50, 4 6 % 3. (a) Rs. 1,600 (b) Rs. 15,725 (c) Rs. 1100

(d) Rs. 4,000 (e) Rs. 600, (f) Rs. 4800, Rs. 3600

2
4. (a) Rs. 50, 16 3 % (b) Rs. 640 (c) Rs. 1,700 (d) Rs. 850 (e) Rs. 3,60,000

5. (a) Rs. 8000 (b) Rs. 19090.91 (c) Rs. 7000 6. (a) Rs. 10,000 (b) Rs. 1462.07
(c) Rs. 318 (d) Rs. 5400 7. (a) Rs. 8% profit (b) 4% loss (c) 1% loss 8. (a) Rs. 150
(b) Rs. 1777.78 9. (a) Rs. 3000, 2000 (b) Rs. 7000 (c) Rs. 4,875, Rs. 3,750

10. (a) Rs. 1000 (b) Rs. 3200, Rs. 4000 (c) Rs. 1000 (d) Rs. 75,00, Rs. 6600 (e) Rs. 5,250
1
11. (a) 25% (b) Rs. 2000, Rs. 1250, 60% 12. (a) 38.89%, 16.67% (b) 23 13 %, 18 58 %
169

Project work

Visit any shop/departmental store. Ask the cost price, see marked price and note
the selling price of different items. Calculate profit or loss and their percentage.
Also calculate the discount and its percentage. Draw out the conclusion, which
article is more profitable to sell.

Oasis School Mathematics-9 37

Unit Commission, Bonus,
Taxation and Dividend
3

3.1 Warm-up Activities

Discuss the following questions in your class and draw out the conclusion.

– What is the value of 25% of Rs. 5,00,000?
– What percent of Rs. 2,000 is Rs. 400?
– What is commission?
– How is government collect the money for its different expenses?
– On the bill, some extra charge is added on the selling price, what is that extra

charge called?
– Lochan has some shares in a bank. After a year he gets some extra amount, what

is that extra amount called?

– Ramesh works as the middle man to sell the land, how can he earn the money?

3.2 Commission

Harka Bahadur is an agent who establishes connection between buyer and landowner.
He takes certain percentage of the selling price of land from the landowner.

Whatever money he gets from the landowner is his commission.

In this example, Harka Bahadur is a middle man or a broker.

Dolma works at a Department Store. She draws Rs. 10,000 monthly. In addition to
this if she sells the goods exceeding Rs. 1,00,000, she gets 5% of the sale value. This
extra income is her commission.

Biren is a middle man who makes the contact with the customer with electric shop
and he gets 3% of total sale from the owner. This amount is his commssion.

How to calculate the commission?

If a landowner sold a land for Rs. 25,00,000, he has to pay some commission to his
agent or broker if the rate of commission is 5%, then the amount of commission is

5% of 25,00,000.
= 1050 × 25,00,000 = Rs. 1,25,000

∴ Commission = Commission rate of S.P.

From this relation we can derive that

Rate of commission = Commission × 100%
Total sale

38 Oasis School Mathematics-9

Worked Out Examples

Example: 1

A sales agent gets commission of 5 % on the sale of land costing Rs. 50,00,000. Find the
commission given to the agent.

Solution:

Selling price of the land = Rs. 50,00,000

Commission = 5 % of Rs. 50,00,000

= 1500 × Rs. 50,00,000
= Rs. 2,50,000

Hence, amount of commission = Rs. 2,50,000.

Example: 2

Namuhang works in a department store. His monthly salary is Rs. 12,000 and he gets
the commission of 2% of his total sale. If his monthly sale is Rs. 2,65,000, find his
monthly income.

Solution:

Monthly salary of Namuhang = Rs. 12,000

His total sale = Rs. 2,65,000

His commission = 2% of Rs. 2,65,000

= 2 × Rs. 2,65,000
100

= Rs. 5,300

His total income = His salary + His commission
= Rs.12,000 + Rs. 5,300
= Rs. 17,300.

Example: 3

Monthly salary of Shankar is Rs. 15,000. He gets some commission from his total sale.
If his monthly income is Rs. 21,000, and his total sale is Rs. 3,00,000, find the rate of
commission.

Solution:

His monthly salary = Rs. 15,000

His total income = Rs. 21,000

His commission = Rs. 21,000 – Rs. 15,000

= Rs. 6,000

Oasis School Mathematics-9 39

His total sale = Rs. 3,00,000

We have, Rate of commission = Commission × 100%
Total sale

= 6000 × 100%
3,00,000

= 2%

Example: 4

A business firm pays an agent the commission as follows from the daily sales. Calculate
the commission received by the agent.

Sales Commission Rate
Sales upto Rs. 15,000 1 %

Sales between Rs. 15,000 and Rs. 25,000 2 %
Sales above Rs. 25,000 3 %

If the sales amount are: (a) Rs. 12,000 (b) Rs. 20,000 (c) Rs. 35,000

Solution:

a) Sales amount = Rs. 12,000

C ommission rate for the amount less than Rs. 15,000 is

= 1 % of Rs. 12,000

b) = 1010 × Rs. 12000 = Rs. 120
Total sale = Rs. 20,000

= Rs.15,000 + Rs. 5000

Commission from first Rs. 15,000 = 1% of Rs. 15,000

= 1010 × Rs. 15,000

= Rs. 150

Commission from next Rs. 5000 = 2 % of Rs. 5000

= 1200 × Rs. 5000

= Rs. 100

∴ Total commission = Rs. 150 + Rs. 100

= Rs. 250

c) Total sales amount = Rs. 35,000

= Rs. 15,000 + Rs. 10,000 + Rs. 10,000

Commission for 1st Rs. 15,000 = 1 % of Rs. 15,000

40 Oasis School Mathematics-9

= 1010 × Rs. 15,000
= Rs. 150

Commission for next Rs. 10,000 = 2 % of Rs. 10,000

= 1200 × Rs. 10,000 = Rs. 200

Commission for remaining Rs. 10,000 = 3 % of Rs. 10,000

= 3 × Rs. 10,000 = Rs. 300
∴ 100

Total commission = Rs. 150 + Rs. 200 + Rs. 300 = Rs. 650.

Exercise 3.1

1. (a) Write the formula to calculate commission if commission rate and total sale are
given.

(b) Write the formula to calculate rate of commission if commission and total sale
are given.

2. Calculate the amount of commission with given rate and selling price.
(a) Land : S . P. = Rs. 40,50,000, Rate = 2%
(b) House: S.P = Rs. 70,60,000, Rate = 3%
(c) Other items: S.P. = Rs. 15,000, Rate = 2%
3. (a) Ram Lal sold a land for Rs. 18,00,000. How much commission does he have to

pay to his agent if the rate of commission is 5%?

(b) A broker gets a commission of 3% on the sale of the land costing Rs. 12,00,000.
Find the commission given to the broker.

(c) Ramesh sold his share from a printing press through an agent. He paid 3%
commission to the agent. If he sold his share for Rs. 4,50,000, find

(i) commission received by the broker. (ii) net amount received by Ramesh.

(d) A business firm gives commission of 5 % on the total sales of Rs. 2,00,000. Find
the commission amount. Also find how much does the firm get from the sales?

4. (a) If an agent gets a commission of Rs. 1000 from the sales amount of Rs. 40,000,
find the commission rate.

(b) A broker gets Rs. 20,000 as a commission from the sale of a piece of land which
costs Rs. 80,00,000. Find the rate of commission.

(c) Santosh gets a commission of Rs. 1,00,000 on the sale of a house for Rs. 75,00,000.
Find the rate of commission.

Oasis School Mathematics-9 41

(d) Monthly salary of a salesman is Rs. 8000. His monthly income is Rs. 12,000. Find
the rate of commission if his monthly sale is Rs. 4,00,000.

5. (a) What amount of sales yields a commission of Rs. 2,40,000 at the rate of 3%?

(b) Raju Shahu sells a piece of land and gives commission to the agent at the rate of
5%. If the agent gets Rs. 1,50,000, find the amount of sales of the land.

(c) A salesman receives a commission of 7% on his sales. What would be the amount
of his sales in order to receive Rs. 5,075 commission?

6. (a) A sales agent gets a monthly salary of Rs. 8,000 plus 5 % commission of the total
sales amount Rs. 2,00,000. Find the net income of the agent.

(b) A salesgirl gets a monthly salary of Rs. 10,000 and gets 5% commission of the
total sales Rs. 1,50,000. Find the net income of the sales girl.

(c) The monthly salary of a salesman is Rs. 12,000 and he also gets a commission of
2% on the total monthly sale. Calculate his monthly income if he makes a sale of
Rs. 2,50,000 in a month.

7. (a) A girl is paid a monthly salary of Rs. 5,000 and 2% commission for the monthly
sales. If she receives Rs. 8500 at the end of the month, find the monthly sales
amount.

(b) A boy working in a departmental store has monthly salary Rs. 8,000. He also gets
a commission of 1% on his total monthly sale. At the end of the month if he gets
Rs. 8,750, find the total monthly sale of the departmental store.

(c) Monthly salary of a salesman is Rs. 12,000. He gets 1.5% commission on his total
sale. If his monthly income is Rs. 15,000, find the total amount of sale.

8. Following table shows the commission rate for different range of sales.

Sales Rate of Commission
Upto Rs. 1,00,000 1%
Rs. 1,00,000 – Rs. 3,00,000 1.5 %

Rs. 3,00,000 and above 2%

Find the amount of commission, if the total sales per month is

(a) Rs. 150000 (b) Rs. 2,50,000 (c) Rs. 3,50,000.



Answer

1. Consult your teacher. 2. (a) Rs. 81,000 (b) Rs. 2,11,800 (c) Rs.300

3. (a) Rs. 90,000 (b) Rs. 36,000 (c) Rs. 13,500, Rs. 4,36,500 (d) Rs. 10,000, Rs. 1,90,000

4. (a) 2.5% (b) 0.25% (c) 1.33% (d) 1% 5. (a) Rs. 80,00,000

(b) Rs. 30,00,000 (c) Rs. 72,500 6. (a) Rs. 18,000 (b) Rs. 17,500 (c) Rs. 17,000

7. (a) Rs. 1,75, 000 (b) Rs. 75,000 (c) Rs. 2,00,000 8. (a) Rs. 1750 (b) Rs. 3,250 (c) Rs. 5000

42 Oasis School Mathematics-9

3.3 Bonus

When any business firm or organization generates profit, the management of the
organization may decide to provide certain amount of profit to their employees as
an incentive in that year. Such additional allowance provided to the employees other
than the monthly salary or income is called "Bonus". Bonus is generally expressed in
terms of percentage of the profit.

Bonus = Bonus% of Total Profit

Bonus Rate = Bonus × 100%
Total Profit

Worked Out Examples

Example: 1

A bonus of 1% was given to the staff of a company which gained Rs. 2,50,000 last year.
Find the amount of bonus.
Solution:

Here, Total Profit = Rs. 2,50,000

Bonus = 1 % of Rs. 2,50,000

= 1 × Rs. 2,50,000
100

= Rs. 2,500

Example: 2

Each employee of a five star hotel is given a bonus of 0.5%. If an employee gets Rs. 5,000
as bonus, find the total profit made by the hotel during that year.

Solution:

Here,

Amount of bonus = Rs. 5000

Let, Total profit = Rs. x

∴ 0.5 % of Rs. x = Rs. 5000
or,
0.5 × Rs. x = Rs. 5,000
or, 100
5
∴ x 1000 × Rs. x = Rs. 5,000

= 5000 × 1000
5
= Rs. 10,00,000

∴ Total profit in that year = Rs. 10,00,000

Oasis School Mathematics-9 43

Example: 3

A company distributes bonus on the basis of salary as given below:

Salary per month Bonus
Rs. 10,000 - Rs. 15,000 50 % of salary
Rs. 15,000 - Rs. 20,000 40 % of salary.
Rs. 20,000 - and above 30 % of salary.

If the total profit of the company is Rs. 20,00,000, find the bonus given to the employees
having following salary.

(a) Rs. 12,000 (b) Rs. 16,000 (c) Rs. 25,000

Solution: Basic salary = Rs. 12,000
(a)

∴ Bonus = 50% of Rs. 12,000

= 50 × Rs. 12,000
100

= Rs. 6,000

(b) Basic salary = Rs. 16,000

∴ Bonus = 40 % of Rs. 16,000

= 40 × Rs. 16,000
100

= Rs. 6,400

(c) Basic salary = Rs. 25,000

∴ Bonus = 30 % of Rs. 25,000

= 30 × Rs. 25,000
100

= Rs. 7,500.

Exercise 3.2

1. (a) Write the formula to calculate bonus if total profit and bonus are given.
(b) Write the formula to calculate bonus percentage if bonus and total profit are

given.
2. (a) The management of a cottage industry decided to pay the bonus of 50% of total

profit amounts Rs. 2,50,000 among 20 employees. Find the bonus given to each.
(b) Anuj is paid a bonus of 0.3 % from the total profit of Rs. 30,00,000 made by the

firm, where he works. Find the bonus given to him.

44 Oasis School Mathematics-9