6
Author
Shyam Datta Adhikari
M.Sc. (Maths), T.U.
66
Name : ............................................................
Class : ....................... Roll No. : ................
Section : ..........................................................
School : ............................................
Publisher
Oasis Publication Pvt. Ltd.
Copyright
The Publisher
Edition
First Edition : 2078 B.S.
Contributors
Man Bahadur Tamang
Shuva Kumar Shrestha
Laxmi Gautam
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 Man Bahadur Tamang, Laxmi Gautam,
Sunil Kumar Chaudhary, Ram Prasad Sapkota, Saroj Neupane and
Yadav Siwakoti for their invaluable suggestions and contributions.
Sincere gratitude to Managing Director Oasis Publication 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 2021
Contents
Sets
1. Sets......................................................................................... 2-15
1.1 Introduction......................................................................... 2
1.2 Methods of Describing Sets............................................... 6
1.3 Types of Sets........................................................................ 9
1.4 Set Relations........................................................................ 11
Arithmetic
2. Real Number........................................................................ 17-51
2.1 Natural Numbers (Review)............................................... 17
2.2 Order of Operation of Simplification............................... 18
2.3 Test of Divisibility............................................................... 22
2.4 Multiples of Factors........................................................... 25
2.5 Prime and Composite Numbers....................................... 32
2.6 Prime Factorization............................................................ 35
2.7 Highest Common Factor (HCF)....................................... 38
2.8 Lowest Common Multiples (LCM).................................. 42
2.9 Square and Square Roots................................................... 46
3. Integer................................................................................... 56-61
3.1 Introduction of Integers..................................................... 52
3.2 Operation on Integers........................................................ 56
4. Fractions and Decimals...................................................... 62-99
4.1 Equivalents Fractions......................................................... 62
4.2 Comparison of Fractions................................................... 64
4.3 Fundamental Operations on Fractions............................ 69
4.4 Multiplication of Fractions................................................ 75
4.5 Division of a whole number by a fraction...................... 79
4.6 Simplification of Fractions................................................. 84
4.7 Decimals............................................................................... 87
4.8 Division of Decimals.......................................................... 92
4.9 Rounding off Decimal Numbers...................................... 95
Commercial Arithmetic
5. Percentage...................................................................... 101-107
5.1 Introduction.................................................................. 101
5.2 Calculation of Percentage........................................... 105
6. Profit and Loss............................................................... 108-113
6.1 Profit and Loss............................................................. 108
6.2 Profit and Loss Percent............................................... 111
7. Unitary Method............................................................ 114-120
7.1 Introduction.................................................................. 114
Mensuration
8. Mensuration.................................................................. 122-149
8.1 Distance......................................................................... 122
8.2 Perimeter....................................................................... 127
8.3 Area............................................................................... 131
8.4 The Unit of Area.......................................................... 133
8.5 Area of Rectangle and Squares.................................. 137
8.6 Volume.......................................................................... 142
Algebra
9. Indices............................................................................. 151-153
9.1 Introduction................................................................. 151
9.2 Multiplication of Indices............................................ 152
10. Algebraic Expressions.................................................. 154-173
10.1 Algebraic Terms and Expressions............................. 154
10.2 Addition and Subtraction of Algebraic Terms........ 159
10.3 Addition and Subtraction of Algebraic Expressions 162
10.4 Multiplication of Algebraic Expressions.................. 166
10.5 Multiplication of a Binomial by a Monomial.......... 170
10.6 Division of Algebraic Expressions............................ 172
11. Equation, Inequality and Graphs.............................. 174-186
11.1 Mathematical Statements........................................... 174
11.2 Equation........................................................................ 176
11.3 Word Problems and Equation...................................... 179
11.4 Inequations or Inequality.............................................. 181
Geometry
12. Line and Line Segment.................................................. 188-204
12.1 Line and Line Segment (Review)................................. 188
12.2 Parallel Lines and Interesting Lines............................ 190
12.3 Measurement and Construction of Given Line Segment 194
12.4 Construction of Perpendicular Bisector of a Given Line
Segment Using Compass.............................................. 199
13. Angles............................................................................... 205-225
13.1 Fundamental Concepts................................................. 205
13.2 Types of Angles.............................................................. 207
13.3 Construction of an Angle of Given Measurement.... 211
13.4 Pair of Angles.................................................................. 218
14. Triangle and Quadrilateral............................................ 227-251
14.1 Triangles.......................................................................... 227
14.2 Quadrilateral................................................................... 233
14.3 Construction of Regular Polygons............................... 244
14.4 Circles.............................................................................. 247
15. Solid Figures.................................................................... 252-259
15.1 Introduction.................................................................... 252
Co-ordinates
16. Co-ordinates..................................................................... 261-266
16.1 Co-ordinates System...................................................... 261
Symmetry, Tessellation, Pattern and Designs
17. Symmetry, Tessellation, Pattern and Designs............ 268-277
17.1 Symmetry........................................................................ 268
17.2 Tessellation...................................................................... 272
17.3 Designs Using Circles and Polygons........................... 274
Statistics
18. Statistics............................................................................ 279-290
18.1 Collection of Data........................................................... 279
18.2 Bar Graph........................................................................ 283
Model Test Paper.............................................................................. 291-294
15Estimated Teaching Hours
Contents
• Definition and formation of set
• Notation of set
• Method of describing sets
Expected Learning Outcomes
At the end of this unit, students will be able to develop the following
competencies:
• To define and to form the set
• To use set notation to describe its various relation
• To describe the set by various methods
• To define and identify the various types of sets
Teaching Materials
• Model of Venn diagram
Oasis School Mathematics – 6 1
Unit
1 Sets
1.1 Introduction
Let's observe the following examples:
Here is a motorcycle, a car, a bus
and a truck.
All of these are vehicles.
∴ It is a set of vehicles.
Here is a mango, a banana, an orange
and an apple.
All of these are fruits.
∴ It is a set of fruits.
Here is a number 2, a number 4, a number 6 and 24 6
a number 8. All these are the even numbers less than 9. 8
∴ It is a set of even numbers less than 9.
Well defined collections
Let's make the list of odd numbers less than 10. 13 5
It is easy to make its collection. 79
∴ It is well defined.
Let's make the list of intelligent students of class VI.
It is difficult to make its collection. How to measure the
∴ It is not well defined. intelligence?
There collections are not well defined: It is really different.
• collection of intelligent students
• collection of fat boys
• collection of beautiful girls
• collection of tall people, etc.
2 Oasis School Mathematics – 6
Definition of Set 2 46
8 10
Let's observe the following examples:
It is a collection of even numbers less than 11.
It is the collection of seven days of a week. SWuneddanyeFsrdidaMyayonTdhuaSyrastduarTdyuaeysday
It is the collection of vowel alphabets. a ei
ou
All these collections are well defined.
Hence, the set is the well defined collection of objects.
Notation of Set Cow Dog
Goat Cat
Let's take a set of animals
Here, cow, goat, dog, cat are the elements of set.
The set is generally denoted by capital letters A, B, C, D,
......... etc.
Its elements can be written insides the pair of curly braces { } and are separated by commas.
If "A" be the set of animals, then
A = {cow, goat, dog, cat}.
Elements of Set Write Read
Let's take a set of vowel alphabets. a∈V 'a' belongs to V
V = {a, e, i, o, u}.
Here, 'a' is a member of set V. a∉V 'a' does not belong to V
∴ a ∈ V i.e. 'a' belongs to V
Again, 'e' is a member of set V
∴ e ∈ V i.e. e belongs to V, b is not an element of 'V'
∴ b ∉ V i.e. b does not belong to V.
Note:
• ∈ (epsilon) is a Greek letter.
• ∈ stands for "belongs to" or "is an element of".
• ∈ stands for "does not belong to" or "is not an element of".
Oasis School Mathematics – 6 3
Worked Out Examples
Example: 1
Cross one element of the following sets which doesn't belong to. Name the rest sets.
Sunday Monday
Tuesday Wednesday
Thursday Bhadra Friday
Saturday
Solution:
Sunday Monday
Here, Bhadra is a month of Nepalese calendar and remaining Tuesday Wednesday
are the days of week. Thursday Bhadra Friday
Hence, it is the set of seven days of the week. Saturday
Example: 2 A set of seven days of a week
Identify whether the following collections are well defined or not.
(a) A set of girls who study in class VI.
(b) A set of teachers having loud voice.
Solution:
(a) A set of girls who study in class VI is well defined because we can specify the
girls who study in class VI.
(b) A set of teachers having loud voice is not well-defined because it is difficult to
specify such teachers who have loud voice.
Example: 3
Write the following in set notation
(a) 2 belongs to the set of natural number N.
(b) 1 does not belong to the set N.
5
Solution: 1
5
(a) 2∈ N (b) ∉ N
Example: 4
Insert an appropriate symbol ∈ or ∉ in the blank spaces
(a) 4 ... {1, 2, 3, 4} (b) 6 ... {1, 2, 3, 4}
Solution:
(a) Here, 4 is an element of given set.
Thus, 4 ∈ {1, 2, 3, 4}
(b) Here, 6 is not an element of the given set. So, it is not an element of the set.
Thus, we write it as 6 ∉ {1, 2, 3, 4}
4 Oasis School Mathematics – 6
Exercise 1.1
1. Cross the odd one and name the set.
(a) 2 4 8 (b) N epal I ndia
6 10 13 Bhutan China Bangladesh
Maldives Pakistan Sri Lanka
(c) 369 (d) L ion Tiger (e) January June
12 15 17
Cow
July Sunday
Bear Elephant
2. Identify whether the followings are well-defined or not.
(a) A collection of odd numbers from 1 to 10.
(b) A collection of such students of class VI who write slowly.
(c) A collection of English months that begin with J.
(d) A collection of prime numbers less than 10.
(e) A collection of intelligent students of a class.
3. Write the following in set notation.
(a) 'a' belongs to the set A.
(b) '3' does not belong to the set B.
(c) '4' is an element of the set of real numbers R.
(d) '2' is not an element of the set {1, 3, 5, 7}
(e) 'a' belongs to the set {a, e, i, o, u}
4. Write the following set notations in words.
(a) x ∈ B (b) b ∉ S (c) 3 ∈ {1, 2, 3, 4, 5}
(d) 1 ∉N (e) 4 ∉ {1, 3, 5, 7}
2
5. Insert an appropriate symbol ∈ or ∉ in the blank spaces.
(a) e ... {a, e, i, o, u} (c) cow ... {cow, cat, dog, goat}
(b) 3 ... { 4, 5, 6, 7} (d) cat ... {tiger, lion, dear, jackal}
(e) orange ... {banana, apple, orange, mango} (f) ∆ ... { , ○, }
6. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Insert an appropriate symbol ∈ or ∉ in the
blank spaces.
(a) 2 ... A (b) 10 ... A (c) 5 ... A (d) 12 ... A
Oasis School Mathematics – 6 5
7. State whether the following statements are true or false.
(a) a ∈ a set of vowels (b) 3 ∈ a set of multiples of 5
(c) 4 ∉ a set of odd numbers (d) Yamuna ∉ {Gita, Sita, Sunita, Rita}
8. State whether the following statements are true or false.
(a) If S be a set of SAARC countries
(i) Nepal ∈ S (ii) India ∉ S
(iii) China ∈ S (iv) Bhutan ∈ S
(v) Pakistan ∉ S (vi) England ∈ S
(b) If C = set of capitals of SAARC countries, then
(i) Kathmandu ∈ C (ii) London ∈ C
(iii) Delhi ∉ C (iv) Male ∈ C
(v) Pokhara ∉ C
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Answers
Consult your teacher.
1.2 Methods of Describing Sets
Look at the following example:
This is the collection of prime numbers which are less than 10.
Let's represent this in different ways. 235
If this set is A, then 7
A = {2, 3, 5, 7} → Elements are listed within a pair of curly braces { }.
This method is called the listing method.
A = A set of prime numbers less than 10
Common property of elements is described by
an statement.
This method is called Description method
A = {x : x is a prime number less than 10}
Common property is described symbolically
by a rule.
This method is called set builder method.
6 Oasis School Mathematics – 6
Note:
The change of order of writing the elements of a set does not change the set. For
example: {a, e, i, o, u} = {e, a, i, o, u} = {i, a, e, u, o}
The repetition of elements is considered as a single element.
For example: The set {a, b, c, a, d, a} is the same as the set {a, b, c, d}.
Note: The symbol colon (:) is read as "such that".
Let's take an example. Let A be a set of odd numbers less than 10, then:
Description method Listing method Set builder method
A = a set of o dd umbers A = {1, 3, 5, 7,9} A = { x : x is an odd number
less than 10 less than 10}
B = a set of factors of 8 B = {1, 2, 4, 8} B = {x : x is a factor of 8}
Worked Out Examples
Example: 1
Write the following set by listing method:
A = {x: x is an even number, less than 10}.
Solution:
A = {x: x is an even number, x ≤ 10}
= {2, 4, 6, 8}.
Example: 2
Write the letters of the word 'mathematics' by listing and set builder method.
Solution:
Let A be the set of letters in the word 'mathematics'
A = {m, a, t, h, e, i, c, s} Listing method
A = {x : x is a letter in the word 'mathematics'} Set builder method.
Example: 3
Represent the following sets by description method:
(a) A = {a, e, i, o, u}
(b) B = {January, February, March}
Solution:
(a) A is a set of vowels of the English alphabet.
(b) B is a set of first three English months.
Oasis School Mathematics – 6 7
Exercise 1.2
1. List the elements of the following sets within braces { } by listing method
(tabulation method).
(a) A set of the days of a week.
(b) A set of odd numbers between 10 and 20.
(c) {x : x is an even number less than 10}.
(d) A set of odd numbers less than 7.
(e) A set of first 4 numbers divisible by 5.
(f) A set of the letters of the word 'algebra'.
2. Describe the following sets in description.
(a) A = {x, y, z}
(b) B = {2, 4, 6, 8}
(c) C = {2, 3, 5, 7, 11, 13, 17, 19}
(d) D = {Sunday, Monday, Tuesday}
(e) E = {1, 2, 3, 4, 5, 6, 7, 8, 9}
3. Write the following sets in the set builder form.
(a) A = {5, 10, 15, 20, 25}
(b) B = {2, 4, 6, 8, 10}
(c) P = {2, 3, 5, 7, 11}
(d) S = {1, 4, 9, 16, 25}
(e) V = {a, e, i, o, u}
4. (a) From the set of numbers from 1 to 20, make five different sets. Show each of them
by listing method and set builder form.
(b) From the set of alphabets 'a' to 'z', make five different collections and show each
of them by listing method and set builder form.
Answers
Consult your teacher.
Activity
Collect the name of 15 different objects from the surrounding. Make 5 different
sets and present that in your classroom.
8 Oasis School Mathematics – 6
1.3 Types of Sets
Finite set
Look at the following examples:
The set of vowels or A = {a, e, i, o, u} is finite, because there are 5 elements in it.
A = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} is a finite
set, because there are 7 days in a week.
A set having finite number of elements is called a finite set. In a finite set, elements
can be counted.
Infinite set
Look at the following examples:
A = set of whole numbers = {0, 1, 2, 3, .... } is an infinite set, because we cannot
count the elements of whole numbers.
B = set of stars in the sky is an infinite set because we cannot count stars in the sky.
A set having an unlimited number of elements is called an infinite set. In an infinite
set, elements of set are countless.
Cardinality of set
Let A = {1, 2, 4, 6, 7} and B = {2, 4, 6, 8}
Let's count the number of elements in A, A has 5 elements.
∴ Cardinal number of A, n(A) = 5
Similarly, number of elements in B = 4
∴ Cardinal number of B, n(B) = 4.
∴ The number of distinct element in the finite set is the cardinal number of that set.
Null set (Empty set)
Look at the following examples:
Let A = {Triangles with four sides}. Then,
A = φ or { }; because there is no triangle with four sides.
Let B = {Boys studying in Padma Kanya college}
B = { }; or φ because no boys are there in Padma Kanya College.
A set having no element is called a null set or an empty set.
Singleton set
Look at the following examples:
A = a set of highest peak in the world.
Oasis School Mathematics – 6 9
Then,
A = {Mount Everest}
It has only one element.
∴ A is a singleton set.
Again,
B = a set of even numbers which are prime.
Then, B = {2}
∴ B is a singleton set.
Hence, set having only one element is called a singleton set.
Remember !
• Sets having finite number of elements are finite sets.
• Sets having unlimited number of element are infinite sets.
• A set having only one element is a singleton set.
• A set having no element is a null set.
• Number of elements in a set is its cardinal number.
Exercise 1.3
1. Identify whether the following sets are finite or infinite.
(a) A = { 1, 3, 5, 7} (b) B = {2, 4, 6, 8, 15, 17}
(c) C = {2, 4, ..., 20, 22, ...} (d) D = {1, 2, 3, 4, ... }
(e) E = A set of prime numbers less than 20
(f) F = A set of even numbers
(g) G =A set of numbers having 5 in the ones place
2. Find the cardinal number of the following sets.
(a) A = {a, b, c}
(b) N = {1, 2, 3, 4, 5}
(c) C = {x:x is an odd number less than 10}
(d) F = {apple, orange, mango}
(e) E = A set of even numbers less than 15.
(f) P = A set of prime numbers less then 10.
(g) L = A set of the letters of the word COMMERCE.
(h) M = A set of the letters of the word MATHEMATICS.
3. List the elements of each of the following sets and identify whether the following
sets are finite or infinite. If the sets are finite, find their cardinal number.
(a) A set of odd numbers less than 15.
10 Oasis School Mathematics – 6
(b) A set of even numbers between 20 and 35.
(c) A set of even numbers greater than 10.
(d) A set of numbers less than 50 having 6 in the ones place.
4. State whether the following sets are null sets or not.
(a) { 0 }
(b) φ
(c) A = {men over 300 years old}.
(d) People more than 10 feet tall.
(e) Even numbers less than 10.
(f) A set of odd numbers divisible by 2.
(g) A set of whole numbers between 4 and 6.
5. State with reason whether the following sets are singleton sets or not.
(a) {4} (b) { } (c) {0} (d) {φ}
(e) A set of longest river in the word.
(f) A set of even numbers.
õõõ
Answers
Consult your teacher.
1.4 Set Relations
Equivalent sets
Look at the given example:
Let, A = {2, 3, 5} and B = {a, b, c}.
Here, set A has 3 elements and set B has also 3 elements
Then n(A) = 3 and n(B) = 3
Since, n(A) = n(B), sets A and B are equivalent sets.
Two finite sets A and B are said to be equivalent if the number of elements of both
sets are equal i.e. if n(A) = n(B).
It is denoted by A~B.
Equal sets
Look at the following examples:
Let A = {1, 2, 3, 4} and B = a set of natural numbers less than 5, then
Oasis School Mathematics – 6 11
A = {1, 2, 3, 4} and B = {1, 2, 3, 4}
Here, both sets have same elements. Then set A = set B.
Two sets A and B are said to be equal, if the elements of both the sets are same.
i.e. the elements of both the sets are identical.
The symbol used for equality of sets is the usual sign "=" "equal to".
Note
Equal sets are always equivalent. The equivalent sets may not necessarily be equal.
Worked Out Examples
Example: 1
Identify whether the given pair of sets are equal or equivalent.
A = a set of odd numbers less than 10.
B = {a, e, i, o, u}
Solution:
Here,
A = a set of odd numbers less than 10 .
Then A = {1, 3, 5, 7, 9}
∴ n(A) = 5.
Again, B = {a,e, i, o, u}
∴ n(B) = 5.
Since the elements are not identical but n(A) = n(B), A and B are equivalent sets.
Example: 2
Let A = a set of the letters of the words LESS and B = a set of the letters
of the word SELL.
(i) List the elements of both sets 'A' and 'B'.
(ii) Identify whether they are equal or equivalent sets.
Solution:
A = a set of the letters of the word LESS, then A = {L, E, S}
Again, B = a set of the letters of the word SELL, then B = {S, E, L}
Here, the elements of both sets A and B are identical.
Hence, A and B are equal sets.
12 Oasis School Mathematics – 6
Exercise 1.3
1. Identify whether each pair of sets given below are equal sets, equivalent sets
or neither.
(a) A = {3, 5, 7}, B = {5, 3, 7}
(b) A = a set of natural numbers less than 5, B = a set of letters of the word APPLE
(c) A = {2, 2, 7, 1}, B = {1, 2, 7} }
, ○,(d) A = {a, b, c, d}, B = {
(e) P = {letters of the word 'tea'}, R = {letters of the word 'eat'}
(f) A = {M, I, S, H, P}, B = {letters in the word MISSISSIPPI}
(g) A = {letters of the word 'remember'}, B = {letters of the word 'member'}
2. (a) If A = a set of natural number less than 10
B = {x : x is natural number greater than 10 and less than 20}
(i) Find n(A) and n(B)
(ii) Are A and B equivalent sets? Justify your answer.
(iii) Are A and B equal sets? Justify your answer.
(b) A = {2, 3, 5, 7} and B = {x : x is a prime number less than 10}.
(i) Find n(A) and n(B)
(ii) Are 'A' and 'B' equivalent sets?
(iii) Are 'A' and 'B' equal sets? Justify your answer.
3. If A = {1, 2, 3, 4}, B = {a, b, c, d, e}, C = {w, x, y, z}, D = a set of odd
numbers less than 10, E = { }, F = {4, 3, 2, 1} and G = {letters of the word
'mathematics'},
(i) find n(A), n(B), n(C), n(D), n(E), n(F) and n(G)
(ii) which pair of sets are equal?
(iii) which pairs of sets are equivalent?
Answers
1. Consult your teacher.
2. (a) (i) n(A) = 9, n(B) = 9 (ii) Yes (iii) No. (b) (i) n(A) = 4, n(B) = 4 (ii) Yes (iii) No.
3. (a) (i) n(A) = 4, n(B) = 5, n(C) = 4, n(D) = 5, n(E) = 0, n(F) = 4, n(G) = 8
(ii) A and C (iii) B and D, A, C and F
Oasis School Mathematics – 6 13
Objective Questions
Choose the correct alternatives.
1. If A = {2, 3, 5, 7, 11} then the set 'A' also be written as
(i) A = {x : x is a prime number less than 12}
(ii) A = {x : x is an odd number less than 12}
(iii) A = {x : x is an even number less than 12}
2. If V = {a, e, i, o, u} then which of the following relation is not true?
(i) a ∈ V (ii) b ∈ V (iii) c ∉ V
3. Which of the following set is a null set?
(i) φ (ii) {φ} (iii) a set of the largest Lake
4. Which of the following set is a singleton set?
(i) φ (ii) {φ} (iii) the set of birds in the zoo.
5. If A = {x : x is the letter of the word CALCULAS}, then cardinal number of
set A is :
(i) 8 (ii) 4 (iii) 5
6. What is the cardinal number of singleton set?
(i) 0 (ii) 1 (iii) 2
7. What is the cardinal number of null set?
(i) 0 (ii) 1 (iii) 2
8. A = the set of the letters of word TITLE, B = the set of the letters of the word
ELITE, then
(i) A and B are equal sets
(ii) A and B are equivalent sets
(iii) A and B are neither equal nor equivalent set
9. If A = {a, e, i, o, u}, B = {a, b, c, d, e} then A and B are:
(i) equal sets (ii) equivalent sets (iii) none
10. If A = a set of longest river of the world then A is
(i) null set (ii) singleton set (iii) infinite set
14 Oasis School Mathematics – 6
Assessment Test Paper
Attempt all the questions. Full marks – 15
Group 'A' [5×1=5]
1. (a) If A = {2, 4, 6, 8}, write the set A by description method.
(b) If A = {a, b, c, d, e} write a ∈ A and p ∉ A in word.
(c) If A = a set of natural number, less than 5, write set A by listing method.
(d) If P = a set of the letters of the word EVEREST then write P by listing method.
(e) If A = a set of vowel alphabets, then, identify whether b ∈ A and a ∈ A are true or
not.
Group 'B' [5×2=10]
2. (a) Write the given set by set builder method and listing method, A = a set of odd
numbers less than 15.
(b) Identify whether the given collections are sets or not.
(i) Collect of the letters of the word BAGMATI
(ii) Collection of students having good habit
(c) Identify whether the following statements are true or false.
(i) b ∈ set of consonant
(ii) 4 ∈ a set of prime number
3. (a) If A = {5, 10, 15, 20, 25}, B = {a, e, i, o, u}
(i) Find the value of n(A) and n(B)
(ii) Identify whether 'A' and 'B' are equal sets or equivalent sets. Justify your
answer.
(b) Identify whether the given sets are null or singleton.
(i) A set of people having height 10 fit.
(ii) A set of largest planet of solar system.
Oasis School Mathematics – 6 15
Arithmetic
43Estimated Teaching Hours
Contents
Real Numbers
• Place value system • Four fundamental operations on mathematics
• Whole number • Prime Number
• Factors and Multiples • H.C.F. and L.C.M.
• Prime factorisation
• Square and square root
Integer • Operations on integers
• Concept of integers
Fractions and Decimals
• Equivalent fractions • Comparison of unlike fractions
• Fundamental operations on fractions • Simplification of fraction
• Review on decimals
• Multiplication and Division on decimals
• Rounding off decimal numbers
Commercial Arithmetic
• Percentage • Profit and loss • Unitary method
Expected Learning Outcomes
At the end of this unit, students will be able to develop the following competencies:
• To the concept of natural number and whole number
• To keep the very large number in place value chart
• To form the greatest and the smallest number from the given digits
• To test the divisibility of numbers • To identify the prime numbers
• To factorise the given numbers • To find H.C.F. and L.C.M of given numbers
• To find square, square root, cube and cube root of given numbers
• To operate the integers • To compare the fractions
• To add, subtract multiply and divide fractions • To simplify the decimals
• To solve the simple problems on percentage, ratio and proportion, profit and loss
and simple interest
Teaching Materials
• flash cards, place value chart, graph sheet, cryons, etc.
16 Oasis School Mathematics – 6
Unit
2 Real Number
2.1 Natural Numbers (Review)
Look at the following figures:
How many pencils? How many apples? How many marbles ?
The answer of the first question is 2, second and third are 3 and 4 respectively.
Here, we use counting numbers to get the answers. Such counting numbers 1, 2, 3,
4 ..... are the natural numbers.
Hence, the numbers used for counting are the natural numbers.
Zero
Consider the following example;
There are 3 apples in a plate. If a boy eats all 3 apples,
how many apples are left on the plate?
There is no apple on the plate.
How many students in your class have the height of 10 ft?
None of the students has the height of 10 ft.
Answer of both questions cannot be represented by the natural numbers.
Answer of each question is represented by zero symbolically '0'. i.e. Zero means
absence of the item.
Whole numbers:
The numbers 1, 2, 3, 4, ………… are the natural numbers or the counting numbers.
If one more number zero '0' is added to the list of natural number, the set becomes
the set of whole numbers. So the set of numbers 0, 1, 2, 3, 4 …………… are the
whole numbers.
Oasis School Mathematics – 6 17
Remember !
• 1, 2, 3, 4 ………… are the natural numbers.
• 0, 1, 2, 3, 4 ……… are the whole numbers.
• The smallest natural number is 1.
• The smallest whole number is 0.
• Every natural number is also a whole number.
• There is no specific largest whole number.
• There is no specific largest natural number.
Note:
Numerals are the symbols which are written for the numbers.
Number name: One Numeral: 1
Exercise 2.1
1. Answer the following questions.
a. Which is the smallest natural number?
b. Which is the smallest whole number?
c. Which is the largest natural number?
d. Which is the largest whole number?
e. Are all natural numbers whole numbers also? Justify your answer.
f. Name a number which is a whole number but not a natural number.
2. Answer the following questions.
a. If N = {1, 2, 3, 4 ....}, what is N called?
b. If W = {0, 1, 2, 3, 4, ...}, what is W called?
Answers
Consult your teacher.
2.2 Order of Operation of Simplification
Some mathematical problems contain mixed operations addition, subtraction,
multiplication, division and different brackets. When more than one operation
is involved, we have to use the following rules for the simplification of given
expression of integers.
This rule is known as 'BODMAS'
18 Oasis School Mathematics – 6
Steps ______
• Remove bar , brackets ( ), { }, [ ] in order by
simplifying all the operations within it or (B) 4th in order
2nd in order
• Perform the operation involving 'of' (O)
1st in order
• Perform the operation involving division (D) 3rd in order
• Perform the operation involving multiplication (M)
• Perform the operation involving addition (A)
• Perform the operation involving subtraction (S)
Worked Out Examples
Example 1
Simplify: 18-4×5+6÷2
Solution:
Here, 18 - 4 × 5 + 6 ÷ 2
= 18 - 4 × 5 + 3 [operating division]
= 18 - 20 + 3 [operating multiplication]
= 21 - 20 [operating addition]
= 1 [operating subtraction]
Example: 2
Simplify: 60 ÷ [150 ÷ {6 + 3 (17 - 14)}]
Solution:
Here, 60 ÷ [150 ÷ {6 + 3 (17 - 14)}]
= 60 ÷ [150 ÷ {6 + 3(3)}]
= 60 ÷ [150 ÷ {6 + 9}]
= 60 ÷ [150 ÷ {15}]
= 60 ÷ [ 150 ÷ 15]
= 60 ÷ [10]
= 60 ÷ 10 = 6
Example: 3
Simplify: 6 ÷ [30 ÷ {20 – (7 –2)}]
Solution:
Here, 6 ÷ [30 ÷ {20 – (7 – 2)}]
= 6 – [30 ÷ {20 – (7 – 2)}]
Oasis School Mathematics – 6 19
= 6 ÷ [30 ÷ {20 – 5}]
= 6 ÷ [30 ÷ 15]
= 6÷2 = 3
Example: 4
Convert the following statement into mathematical sentence and simplify:
The difference of 50 and 8 is divided by the sum of 4 and 3.
Solution:
(50 – 8) ÷ (4 + 3)
= 42 ÷ 7 = 6.
Example: 5
Convert the following statement into mathematical sentence and simplify:
6 is added to the 3 times the difference of 14 and 6.
Solution:
6 + 3 (14 – 6)
= 6 + 3 × 8
= 6 + 24
= 30
Exercise 2.2
1. Simplify:
(a) 7 + 6 – 3 (b) 80 – 50 + 15
(c) -35 + 40 – 8 (d) -56 – 24 + 42 + 30
2. Simplify:
(a) 7 × 4 – 4 (b) 9 × 5 – 5 + 4
(c) 18 + 6 × 2 – 7 (d) -8 + 9 + 8 × 3
3. Simplify:
(a) 12 – 6 ÷ 3 (b) 30 + 8 ÷ 2 – 4
(c) 16 ÷ 4 × 5 (d) 88 ÷ 11 × 6 ÷ 3
4. Simplify:
(a) 40 ÷ 4 × 5 – 1 (b) 30 + 24 ÷ 6 + 8 × 4 – 15
(c) 18 + 18 ÷ 2 + 6 × 2 + 12 ÷ 4 (d) 12 + 6 × 3 + 2 × 2 ÷ 2 – 9
20 Oasis School Mathematics – 6
5. Simplify: (b) 35 ÷ (3 + 4)
(a) 15 + (5 – 3) (d) 6 + (4 ÷ 2) – 10
(c) (15 + 17) ÷ (6 + 2)
(b) 15 +{5 + (15 ÷ 3 + 2)}
6. Simplify: (d) 30 + {– 12 – (16 – 49 ÷ 7)}
(a) 10 – {12 – (30 ÷ 2)}
(c) 15 – {6 + (4 – 2)} + 4
7. Simplify: (b) 6 – [30 ÷ {20 – (10 – 5)}]
(a) 14 ÷ [3 + 16 ÷ {2 + 8 ÷ (6 – 2)}] (d) 8 + [20 ÷ {12 – (16 × 2 + 8) ÷ 5}]
(c) 16 – [14 – {6 ÷ (8 – 2)}]
8. Convert the following statements into number sentence and simplify.
(a) 5 is multiplied to the difference between 10 and 6.
(b) 16 is added to the product of 17 and 4.
(c) 10 is subtracted from 15 times 5.
(d) 12 is added to 6 times difference between 7 and 6.
(e) 10 is divided by 5 and multiplied by 3.
(f) Divide the sum of 12 and 6 by the product of 6 and 3.
(g) Add 9 multiplied by 3 to the sum of 6 and 2.
Answers (b) 45 (c) –3 (d) – 8 5. (a) 17 (b) 5 (c) 4 (d) –2 (e) 6
(b) 44 (c) 23 (d) 25 6. (a) 13 (b) 27 (c) 11 (d) 9
1. (a) 10 (b) 30 (c) 20 (d) 16 7. (a) 2 (b) 4 (c) 3 (d) 13
2. (a) 24 (b) 51 (c) 42 (d) 23 8. (a) 20 (b) 84 (c) 65 (d)18
3. (a) 10 (g) 35
4. (a) 49 (f) 1
Revision Work
Start with the number given and perform the operations in order shown by arrows.
(a)
(b)
Oasis School Mathematics – 6 21
Use +, –, × or ÷ in the box to get the given result.
(a) 5 6 3 = 10 Way: 5 × 6 ÷ 3 = 10
(b) 9 6 8 = 7
(c) 2 5 3 = 13
(d) 6 2 3 = 1
(e) 3 5 7 = 8
(f) 18 3 4 = 10
Use + or – in the gap of one, two or three digits to get the result 100.
1 2 3 4 5 6 7 8 9 = 100
Example: 1 2 3 + 4 5 – 6 7 + 8 – 9 = 100
Try to get 100 by other way also.
2.3 Test of Divisibility
Look at the following examples 20 ÷ 2 = 10, 20 ÷ 4 = 5, 20 ÷ 5 = 4
In all of above, remainder = 0, 20 is divisible by 2, 4 and 5.
∴ A number is said to be divisible by another number if the remainder is zero while
dividing the dividend by the divisor.
To find out, without actual division whether the given number is divisible by given
numbers or not there are certain tests. Here are some rules of divisibility of the numbers
from 2 to 11.
Divisibility by 2
A number is divisible by 2 if the digit in the place of ones is zero or even number.
Test
Numbers Digit in the ones place Divisible by 2
278 8 (even) Yes
1963 3 (odd) No
7459 9 (odd) No
7450 0 (Zero) Yes
Divisibility by 3
A number is divisible by 3, if the sum of the digits of the given number is
a multiple of 3.
22 Oasis School Mathematics – 6
Numbers Test Divisible by 3
123 Yes
946 Sum of the digits No
2769 1 + 2 + 3 = 6 (a multiple of 3) Yes
1039 9 + 4 + 6 = 19 ( not a multiple of 3) No
2 + 7 + 6 + 9 = 24 (a multiple of 3)
1 + 0 + 3 + 9 = 13 (not a multiple of 3)
Divisibility by 5
A number is divisible by 5, if the digit in the place of ones is '0' or '5'.
Test
Numbers Digit in the place of ones Divisible by 5
110 0 Yes
3685 5 Yes
3864 4 No
1693 3 No
Divisibility by 6
A number is divisible by 6, if it is divisible by both 2 and 3. In other word a number
is divisible by 6 if it is even and divisible by 3. Divisible Divisible
Test by 3 by 6
Numbers Digit in the Divisible Sum of the digits
place of ones by 2
736 6 (even) Yes 7 + 3 + 6 = 16 No No
(not divisible by 3)
3336 6 (even ) Yes 3 + 3 + 3 + 6 = 15 Yes Yes
(divisible by 3)
1503 3 (odd) No 1 + 5 + 0 + 3 = 9 Yes No
(divisible by 3)
1397 7 (odd) No. 1 + 3 + 9 + 7 = 20 No. No
(not divisible by 3)
Divisibility by 7
A number is divisible by 7, if the difference between twice the last digit and the
number formed by other digits is divisible by 7.
Oasis School Mathematics – 6 23
Number Test Difference Divisible
by 7
224 Twice the number formed Yes
434 last digit by other digits
1035 Yes
363 2×4=8 22 22 – 8 = 14 (Divisible by 7)
No
2×4=8 43 43 – 8 = 35 (Divisible by 7)
No
2 × 5 = 10 103 103 – 10 = 93 (not divisible by 7)
2×3=6 36 36 – 6 = 30 (not divisible by 7)
Divisibility by 11
A number is divisible by 11, if the difference of sum of the digits in odd places and
the digits in the even places is 'O'.
Test
Numbers Sum of the digits at odd places Sum of the dig- Difference Divisible by 11
121 (from the right) its at even places 2–2=0 Yes
(from the right)
1+1=2
2
12364 1+3+4=8 2+6=8 8–8=0 Yes
1376277 7 + 2 + 7 + 1 = 17 7 + 6 + 3 = 16 17 – 16 = 0 No
1331 1+3=4 3+1=4 4–4=0 Yes
Exercise 2.3
1. What digit should be there in the ones place, to be a number divisible by 2?
2. Which of the following numbers are divisible by 2?
(a) 182 (b) 363 (c) 1469 (d) 520 (e) 3631
3. What is the condition of divisibility of a number by 3?
4. Which of the following numbers are divisible by 3?
(a) 343 (b) 861 (c) 783 (d) 231 (e) 71
5. What digit should be there in the ones place to be a number divisible by 5?
6. Which of the following numbers are divisible by 5?
(a) 43 (b) 565 (c) 1000 (d) 877 (e) 125
7. What type of number is divisible by 7?
24 Oasis School Mathematics – 6
8. Which of the following numbers are divisible by 7?
(a) 63 (b) 123 (c) 147 (d) 224 (e) 331
(e) 644314
9. On which condition, a number is divisible by 11?
10. Which of the following numbers are divisible by 11?
(a) 71412 (b) 48295 (c) 14909 (d) 97526
Answers
1. Consult your teacher 7. Consult your teacher.
2. (a) yes (b) no (c) no (d) yes (e) no 8. (a) yes (b) no (c) yes (d) yes (e) no
3. Consult your teacher. 9. Consult your teacher
4. (a) no (b) yes (c) yes (d) yes (e) no 10. (a) yes (b) no (c) no (d) yes
5. Consult your teacher (e) yes
6. (a) no. (b) yes (c) yes (d) no (e) yes
2.4 Multiples and Factors
Multiples:
Look at the following example; circle the numbers which are divisible by 2.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Hence, 2, 4, 6, 8, 10, 12 etc. are the multiples of 2. Dividend
Again, look at one more example, is the multiple of
3 × 1 = 3 3÷3 = 1
3 × 2 = 6 6÷3 = 2 divisor
3 × 3 = 9 9÷3 = 3
3 × 4 = 12 12÷3 = 4
3 × 5 = 15 15÷3 = 5
3 × 6 = 18 18÷3 = 6
Here, 3 divides 3, 6, 9, 12, 15, 18 exactly.
∴ The numbers 3, 6, 9, 12, 15, 18, etc. are the multiples of 3.
From the above example, it is clear that if a number divides another exactly, the
second is multiple of the first.
Oasis School Mathematics – 6 25
Remember !
If 4 × 5 = 20; 20 is the multiple of both 4 and 5.
For example:
If M(4) denotes the set of the multiples of 4 less than 30 then.
M(4) = {4, 8, 12, 16, 20, 24, 28}
(Listing method)
Factors:
Look at the following examples,
2×3=6 and 5 × 6 = 30
Hence, 2 and 3 are the factors of 6 . Hence, 5 and 6 are the factors of 30.
Again, let's see the following examples:
20 ÷ 4 = 5 4 is a factor of 20.
24 ÷ 6 = 4 6 is factor of 24.
18 ÷ 3 = 6 3 is a factor of 18.
Let's find the possible factors of 24.
24 ÷ 1 = 24 1 is a factor of 24
24 ÷ 2 = 12 2 is a factor of 24.
24 ÷ 3 = 8 3 is a factor of 24.
24 ÷ 4 = 6 4 is a factor of 24.
24 ÷ 6 = 4 6 is a factor of 24.
24 ÷ 8 = 3 8 is a factor of 24.
24 ÷ 12 = 2 12 is a factor of 24.
24 ÷ 24 = 1 24 is a factor of 24.
Therefore the possible factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 .
Hence, a factor of a number is that number, which divides the given number exactly.
Remember ! • 3 is a factor of 18.
3 × 6 = 18 • 6 is a factor of 18.
• 18 is the multiple of 3
Factor × Factor = Multiple •18 is the multiple of 6.
26 Oasis School Mathematics – 6
Example:
If F(8) is the set of factor of 8, then
F(8) = {8, 4, 2, 1}.
Note:
• Every number is a factor of itself.
• Every number is the multiple of itself.
• 1 is the factor of every number.
Activity on factors.
Take 8 pieces of square paper of size 1 cm × 1cm
Arrange these 8 pieces to form a shape of rectangular or a square.
It is a rectangle of size 8 cm × 1 cm.
8 cm
1 cm
4 cm
It is a rectangle of size 4 cm × 2 cm. 2 cm
2 cm
It is a rectangle of size 2cm × 4 cm
8 cm
Oasis School Mathematics – 6 27
1cm
It is a rectangle of size 1 cm × 8 cm.
8 cm
Guess, what are the factors of 8?
Factors of 8 are ................................. .
Worked Out Examples
Example 1
Convert the multiplication fact 3×5=15 into division facts.
Solution:
Given,
3 × 5 = 15
Then,
15 ÷ 3 = 5
And, 15 ÷ 5 = 3.
Example 2
Convert the division fact 20 ÷ 5 = 4 into multiplication fact.
Solution:
Given,
20 ÷ 5 = 4
Then,
5 × 4 = 20
Example 3
Obtain 4 statements from the fact 8 × 4 = 32.
Solution:
28 Oasis School Mathematics – 6
Given,
8 × 4 = 32
Then,
32 is a multiple of 8
32 is a multiple of 4
8 is a factor of 32.
4 is a factor of 32.
Example 4
Obtain 4 statements from the given division fact 21 ÷ 7 = 3.
Solution:
Given, 21 ÷ 7 = 3
Then,
21 is a multiple of 7.
21 is a multiple of 3.
3 is a factor of 21.
7 is a factor of 21.
Example 5
Find all the multiples of 6 which are less than 40.
Solution:
6 × 1 = 6 < 40
6 × 2 = 12 < 40
6 × 3 = 18 < 40
6 × 4 = 24 < 40
6 × 5 = 30 < 40
6 × 6 = 36 < 40
6 × 7 = 42 > 40
∴ The multiples of 6 less than 40 are 6, 12, 18, 24, 30 and 36.
Example 6
Write down all the possible factors of 18.
Solution:
18 = 18 × 1 18 and 1 are the factors of 18.
18 = 9 × 2 9 and 2 are the factors of 18.
18 = 6 × 3 6 and 3 are the factors of 18.
∴ Possible factors of 18 are 1, 2, 3, 6, 9 and 18.
Oasis School Mathematics – 6 29
Example 7
If F(20) be the set of factors of 20, find the set F(20).
Solution:
20 = 20 × 1
20 = 10 × 2
20 = 5 × 4
The possible factors of 20 are 1, 2, 4, 5, 10 and 20.
∴ F(20) = {1, 2, 4, 5, 10, 20}
Example 8
If M(7) be the multiples of 7 from 20 to 40; find M(7).
Solution:
Multiple of 7 from 1 to 50 are 7, 14, 21, 28, 35
Since M(7) is the multiples of 7 from 20 to 40
Then, M(7) = {21, 28, 37}.
Exercise 2.4
1. Convert each of the following multiplication fact into two division facts.
(a) 2 × 4 = 8 (b) 7 × 3 = 21 (c) 13 × 4 = 52
2. Convert each of the following division fact into multiplication fact.
(a) 24 ÷ 6 = 4 (b) 10 ÷ 2 = 5 (c) 33 ÷ 3 = 11
3. Obtain four statements from the given multiplication fact.
(a) 7 × 3 = 21 (b) 6 × 4 = 24 (c) 7 × 5 = 35
(d) 9 × 4 = 36 (e) 9 × 5 = 45
4. Find four statements from this given division fact.
(a) 15 ÷ 5 = 3 (b) 24 ÷ 8 = 3 (c) 8 ÷ 4 = 2
(d) 20 ÷ 5 = 4 (e) 35 ÷ 5 = 7
5. Answer the following questions.
(a) Which number is a factor of every number?
(b) What are the factors of 2?
(c) What are the first two multiples of 6?
(d) What are the factors of 4?
30 Oasis School Mathematics – 6
6. Find all the possible factors of:
(a) 12 (b) 18 (c) 10 (d) 24 (e) 30 (f) 45
7. Write down the first five multiples of the following numbers:
(a) 7 (b) 9 (c) 10 (d) 5 (e) 13
8. (a) If F(20) be the set of all possible factors of 20, express F(20) by listing method.
(b) If F(24) be the set of all possible factors of 24, express F(24) by listing method.
(c) If F(12) be the set of factors of 12, find F(12).
9. (a) If M(5) be the set of first five multiples of 5, express M(5) by listing method.
(b) If M(4) be the set of the multiples of 4 from 30 to 50, find M(4).
(c) If M(9) be the set of the multiples of 9 from 30 to 80, find M(9).
10. Complete the following table.
Numbers Factors
1 1
2 1, 2
3 1, 3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Oasis School Mathematics – 6 31
Answers
1. Consult your teacher. 2. Consult your teacher. 3. Consult your teacher.
4. Consult your teacher. 5. Consult your teacher.
6. (a) 1, 2, 3, 4, 6 and 12 (b) 1, 2, 3, 6, 9 and 18 (c) 1, 2, 5 and 10
(d) 1, 2, 3, 4, 6, 8, 12 and 24 (e) 1, 2, 3, 5, 6, 10, 15 and 30 (f) 1, 3, 5, 9, 15 and 45
7. (a) 7, 14, 21, 28 and 35 (b) 9, 18, 27, 36 and 45 (c) 10, 20, 30, 40 and 50 (d) 5, 10, 15, 20
and 25 (e) 13, 26, 39, 52 and 65
8. (a) F(20) = {1, 2, 4, 5, 10, 20}
(b) F(24) = {1, 2, 3, 4, 6, 8, 12, 24} (c) F(12) = {1, 2, 3, 4, 6, 12}
9. (a) M(5) = {5, 10, 15, 20, 25} (b) M(4) = {32, 36, 40, 44, 48} (c) M(9) = {36, 45, 54, 63, 72}
10. Consult your teacher.
2.5 Prime and Composite Numbers Number Factors
1 1
Let's find the factors of 12. 2
3 1, 2
Here, 12 = 1 × 12 = 2 × 6 = 3 × 4 4 1, 3
5 1, 2, 4
= 6 × 2 = 12 × 1 6 1, 5
7 1, 2, 3, 6
We see that 12 can be divided by each of 1, 2, 8 1, 7
3, 4, 6 and 12 itself. These numbers are factors 9 1, 2, 4, 8
of 12. Thus, the set of factors of 12 is given by: 10 1, 3, 9
11 1, 2, 5
F(12) = {1, 2, 3, 4, 6, 12} 12 1, 11
13 1, 2, 3, 4, 6, 12
Let's observe the given table. 14 1, 14
15 1, 2, 7, 14
From the table alongside, we have seen that, 1 1, 3, 5, 15
is the factor of all the numbers.
The numbers 2, 3, 5, 7, 11, 13, ... have only two
factors 1 and the number itself. These numbers
are prime numbers.
Hence, numbers which can only be divided by 1
and the number itself are called prime numbers.
The numbers which have more than two factors
are the composite numbers.
32 Oasis School Mathematics – 6
Let us learn more about the prime and composite numbers from the given table.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
• The number 1 is neither prime nor composite. So encircle it.
• The number 2 is a prime number; do not encircle it but encircle all the multiples of 2.
• The number 3 is a prime number; do not encircle it but encircle all the multiples of 3.
• The number 5 is a prime number; do not encircle it but encircle all the multiples of 5.
• The number 7 is a prime number; do not encircle it but encircle all the multiples of 7.
Thus the numbers which are not encircled, are all prime numbers. This method of finding
the prime numbers was first introduced by the Greek Mathematician Eratosthenes. So it is
known as sieve of Eratosthenes.
Remember !
Two prime numbers with the gap of only one number are twin prime numbers.
Example: 5 and 7 are twin prime numbers.
• 1 is neither a prime nor a composite number.
• Prime numbers have two factors, 1 and the numbers itself.
• 2 is only one prime number, which is even.
• Numbers with more than 2 factors are composite.
Oasis School Mathematics – 6 33
Do you know!
The concept of prime number was introduced by the great Mathematician G.H. Hardy.
Worked Out Examples
Example 1:
Write the set of prime numbers less than 10.
Solution:
The prime numbers less than 10 are 2, 3, 5 and 7
Hence the set of prime numbers less than 10 is { 2, 3, 5, 7}
Exercise 2.6
1. Answer the following questions.
(a) What types of numbers are prime numbers?
(b) What types of numbers are composite numbers?
(c) Which number is neither a prime nor a composite number?
(d) Which even number is a prime number?
(e) What are the two factors of a prime numbers?
2. Write 'T' for true and 'F' for false statements.
(a) All prime numbers are odd.
(b) All odd numbers are prime.
(c) 2 is a prime and even number.
(d) Every even number except 2 is composite.
(e) Prime numbers have only two factors.
(f) Every natural number except 1 is either prime or composite.
3. Write every element of following sets by listing method.
(a) Set of prime numbers from 1 to 15.
(b) Set of composite numbers from 1 to 15.
4. State whether the following numbers are prime or composite.
(a) a number greater than 3 which is exactly divisible by 3.
(b) a number whose two factors are 1 and the number itself.
(c) a number which has 5 factors.
(d) a two digit number which is even.
34 Oasis School Mathematics – 6
5. Find out whether the following numbers are prime or composite
(a) 67 (b) 42 (c) 85 (d) 43 (e) 97
(f) 91 (g) 38 (h) 77 (i) 39 (j) 83
Answers (e) True (f) True.
1. Consult your teacher 2. (a) False (b) False (c) True (d) True
3. (a) P (15) = {2, 3, 5, 7, 11, 13} (b) C (5) = {4, 6, 8, 9, 10, 12, 14, 15}
4. Consult your teacher 5. Consult your teacher
2.6 Prime Factorisation
Prime Factors
Let's find the factors of a composite number.
24 = 24 × 1
24 = 12 × 2
24 = 8 × 3
24 = 6 × 4
∴ The possible factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Among these, 2 and 3 are the prime numbers. So they are the prime factors
of 24.
Hence, the factors of a number which are only the prime are the prime factors.
Prime Factorisation
Let's find the factors of a composite number 24
24 = 2 ×12 [12 in not a prime number]
=2×2×6 [6 is not a prime number]
=2×2×2×3 [all the factors are prime number]
∴ 24 = 2 × 2 × 2 × 3
Thus the factorisation of a number in the factors which are only the prime
is called a prime factorisation.
The methods of obtaining prime factors are:
(i) by making factor tree
(ii) by successive division method.
By making factor tree:
Let's find the prime factors of 140 by using factor tree:
Oasis School Mathematics – 6 35
140
2 × 70 70 is not a prime number
2 × 2 × 35 35 is not a prime number
2 × 2 × 5 × 7 Every number is prime
∴ 140 = 2 × 2 × 5 × 7
Hence the prime factors of 140 are 2, 5 and 7.
Successive division method:
Let's take a composite number and factorise it by using successive division method.
2 108 108 ÷ 2 = 54 A pply the test of divisibility rule to factorise a
2 54 54 ÷ 2 = 27 number.
3 27 27 ÷ 3 = 9
39 9 ÷ 3 = 3 T he successive division is carried out by dividing
the given number with respective prime number
3 until the last dividend is prime.
∴ 108 = 2 × 2 × 3 × 3 × 3
Hence the prime factors of 108 are 2 and 3.
Worked Out Examples
Example: 1
Find the prime factors of 72 by:
(i) factor tree method (ii) division method.
Solution:
(i) By factor tree method:
72
2 × 36 36 is not prime.
2 × 2 × 18 18 is not prime.
2 × 2 × 2 ×9 9 is not prime.
2×2×2×3× 3 each factor is prime
∴ 72 = 2 × 2 × 2 × 3 ×3.
Hence the prime factors of 72 are 2 and 3.
36 Oasis School Mathematics – 6
(ii) By division method:
∴ 172 = 2 × 2 × 2 × 3 × 3
Hence the prime factors of 72 are 2 and 3.
Example 2
(i) Factorise 45 and 60.
(ii) Find the product of common factors of these two numbers.
(iii) Does this product divide 45 and 60 exactly?
Solution:
(i) 3
3
∴ 45 = 3 × 3 × 5 ∴ 60 = 2 × 2 × 3 × 5
(ii)
45 = 3 × 3 × 5
60 = 2 × 2 × 3 × 5
∴ Common factors are 3 and 5.
Thus, the product of these two common factors = 3 × 5 = 15
(iii) 15 45 3 15 60 4
–45 – 60
×× ××
Yes, the product 15 divides the numbers exactly without leaving remainder.
Exercise 2.7 (b) 36
1. Complete the given factor tree:
(a) 24
2× ×
2× × ××
2× × × ×× ×
Oasis School Mathematics – 6 37
2. Find the prime factors of the following numbers by factor tree method.
(a) 24 (b) 36 (c) 64 (d) 81 (e) 144
3. Find the prime factors of the following numbers by division method.
(a) 27 (b) 56 (c) 90 (d) 375 (e) 625
(f) 2025 (g) 2500
4. (i) Factorise 48 and 72, (ii) Find the product of common factors of these two
numbers, (iii) Does this product divide 48 and 72 respectively?
Answers
1. Consult your teacher. 2. (a) 2 × 2 × 2 × 3 (b) 2 × 2 × 3 × 3 (c) 2 × 2 × 2 × 2 × 2 × 2
(d) 3 × 3 × 3 × 3 (e) 2 × 2 × 2 × 2 × 3 × 3 3. (a) 3 ×3 × 3 (b) 2 × 2 × 2 × 7 (c) 2 × 3 × 3 × 5
(d) 3 × 5 × 5 × 5 (e) 5 × 5 × 5 × 5 (f) 3 × 3 × 3 × 3 × 5 × 5 (g) 2 × 2 × 5 × 5 × 5 × 5
4. (i) 47 = 2×2×2×2×3, 72 = 2×2×2×3×3, (ii) 24 = 2×2×3×3, (iii) Yes.
2.7 Highest Common Factor
I. Highest common factor
Take two numbers 12 and 18
Let, A = Factors of 12 = {1, 2, 3, 4, 6, 12}
B = Factors of 18 = {1, 2, 3, 6, 9, 18}
Common factor of A and B.
Among these common factors, the greatest factor is 6.
∴ Highest Common factor = 6
The highest common factor (H.C.F.) of two or more than two numbers is the highest
number that divides each of them exactly. It is also called the greatest common
divisor (G.C.D.).
Activity
Write all the numbers from 1 to 40.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
• Take any two numbers 24 and 36.
• Make the list of factors of 24.
• Similarly, make the list of factors of 36.
• Circle all the factors of 24.
38 Oasis School Mathematics – 6
• Tick all the factors of 36.
• List the numbers having circle and tick both.
• Select the largest number from the list that number is the H.C.F.
Methods of finding H.C.F.
The methods of finding H.C.F. of two or more than two numbers are:
a. Set of factors method Steps:
b. Prime factor method
• Find the factors of each of the
Set of factors method given numbers.
Let's find the H.C.F. of 15 and 25 • Express the factors in set notation.
using this method. • Find the set of common factors.
• Select the greatest factor.
The set of factors of 15 = {1, 3, 5, 15} • The greatest factor thus obtained
The set of factors of 25 = {1, 5, 25} is H.C.F.
The common factors of both set = {1, 5}
∴ The greatest common factor is 5.
∴ H.C.F. = 5
Prime factorisation method
Let's find the H.C.F. of 24 and 18 using this method
Here, 2 18
∴ Prime factors of 24 = 2 × 2 × 2 × 3 39
Again ,
3
Prime factors of 18 = 2 × 3 × 3 Steps:
Now,
• Break down each of the number
into its prime factors
Common factors of 24 and 18 are 2 and 3
∴ H.C.F. = 2 × 3 = 6. • Take out the common prime
factors
• Multiply all the common factors
• Which is the H.C.F. of given
numbers.
Note:
1. The two numbers which do not have any common prime factors except 1
are called co-prime numbers.
For examples: 15 and 16 are co-prime numbers because they have no
common factor except 1.
2. The H.C.F. of two co-prime numbers is always 1.
Thus, H.C.F. of 15 and 16 = 1
Oasis School Mathematics – 6 39
Worked Out Examples
Example: 1
Using set of factors method, find the H.C.F. of 36 and 24.
Solution:
Here, Set of factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Set of factors of 24 = {1, 2, 3, 4, 6, 8, 12, 24}
Common factors of 36 and 24 = {1, 2, 3, 4, 6, 12}
Largest common factor = 12
∴ H.C.F. = 12
Example: 2
Using prime factorisation method, find the H.C.F. of 24 and 30.
Solution:
Here,
Here,
24 = 2 × 2 × 3
30 = 2 × 3 × 5
∴ H.C.F. = 2 × 3 = 6
Example: 3
Find the greatest number which divides 36 and 60 exactly.
Solution:
Here, the greatest number which divides 36 and 60 exactly is their H.C.F.
∴ To find the H.C.F. of 36 and 60
2 36 2 60 36 = 2 × 2 × 3 × 3
2 18 2 30 60 = 2 × 2 × 3 × 5
39 3 15
3 5
H.C.F. = 2 × 2 × 3 = 12
∴ Required number = 12.
40 Oasis School Mathematics – 6
Example: 4
Find the greatest number of boys among them 30 pencils and 20 copies can
be distributed equally.
Solution:
Here,
2 30 2 20
3 15 2 10
5 5
∴ 30 = 2 × 3 × 5
20 = 2 × 2 × 5
∴ H.C.F of 30 and 20 = 2× 5 = 10
Hence, among 10 boys, 30 pencils and 20 copies can be distributed equally.
Exercise 2.8
1. Answer the following questions.
a. What type of numbers are said to be co-prime numbers?
b. What is the H.C.F. of two co-prime numbers?
c. Indentify whether 15 and 16 are co-prime numbers or not.
d. Indentify whether 35 and 36 are co-prime numbers or not.
2. Factorise 24 and 30 completely, then:
(i) make the set of factors of 24 and 30.
(ii) make the set of common factors.
(iii) find the greatest common factor.
(iv) what is this value called?
3. Find the H.C.F. of the following numbers by making the set of factors.
(a) 8, 12 (b) 15, 18 (c) 24, 36 (d) 8, 20 (e) 18, 24
4. Find the H.C.F. of the following numbers by prime factor method.
(a) 12, 36 (b) 6, 27 (c) 20, 48 (d) 6, 18
(e) 10, 15 (f) 24, 36 (g) 24, 40
5. (a) Find the prime factors of 36 and 64.
(b) Find the product of the common prime factors.
(c) Test whether the product divides the number exactly or not.
6. (a) Find the largest number which divides 48 and 144 exactly.
(b) Find the greatest number that divides 72 and 96 without leaving remainder.
Oasis School Mathematics – 6 41
7. (a) Rs. 24 and Rs. 60 are to be divided among some students equally. Find the
maximum number of students receiving the money.
(b) 60 apples and 140 oranges are to be divided equally among a number of boys.
Find the greatest number of boys receiving the fruits in this way.
Answers
1. Consult your teacher. 2. (i) {1, 2, 3, 4, 6, 8, 12, 24}, {1, 2, 3, 5, 6, 10, 15, 30}
(ii) {1, 2, 3, 6} (iii) 6 (iv) H.C.F. 3. (a) 4 (b) 3 (c) 12 (d) 4 (e) 6
4. (a) 12 (b) 3 (c) 4 (d) 6 (e) 5 (f) 12 (g) 8
5. (a) 2×2×3×3 and 2×2×2×2×2×2 (b) 4 (c) Yes.
6. (a) 48 (b) 24 7.(a) 12 (b) 20, each boy receives 3 apples and 7 oranges.
2.8 Lowest Common Multiple (L.C.M.)
Let us take numbers 4 and 6.
Now, Set of the multiples of 4 = {4, 8, 12, 16, 20, 24, ...........}
Set of the multiples of 6 = {6, 12, 18, 24, ................}
Set of common multiples = {12, 24, ....................}
Smallest common multiple = 12
∴L.C.M. = 12
Lowest common multiple of two or more than two numbers is the smallest number
which is exactly divisible by each of the given numbers.
Activity 1
Take many stripes of paper of length 2 units having blue colour.
Similarly take many stripes of paper of length 3 units.
First stripe Second stripe Third stripe
First stripe Second stripe
42 Oasis School Mathematics – 6
Keep stripes of length 2 units one after another in one row.
Similarly keep the stripes of length 3 units one after another in another row. Check at
which position two different colour stripes end. That point (6) is the L.C.M. of two
numbers (2 and 3).
Activity 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lets take any two numbers 3 and 4. Starting from 0, jump 3 steps four or five
times. Again, starting from '0' jump 4 steps four or five times. Check where
these two jumping meet at a common point. The common point is 12.
Hence, the LC.M. of 3 and 4 is 12.
Methods of finding L.C.M.
L.C.M. of two or more numbers can be obtained by the following methods.
(a) Set of multiples method (b) Prime factor method
Set of multiples method Steps:
Let's get the idea of getting L.C.M. • Write the set of multiples of given
with the help of given example: number.
To find the L.C.M. of 12 and 8 • Make the set of common multiples
from sets.
The set of multiples of 12
= {12, 24, 36, 48, ...............} • Select the smallest multiple which
The set of multiples of 8 is the L.C.M. of given numbers.
= {8, 16, 24, 32, 40, 48, .....}
The common multiples of 12 and 8 = {24, 48, ....................}
The smallest common multiple is 24. Steps:
∴ L.C.M. = 24 • Resolve each of the given numbers
into their prime factors.
Prime factor method
• Take out the common prime
Let's get the idea of getting L.C.M. by factors.
prime factor method with the help of
given example; • Take out the remaining prime
While finding the L.C.M. of 18, 24, factors which are not common.
and 30.
• Multiply all the factors from step
Here, (ii) and (iii) which is the required
L.C.M
Oasis School Mathematics – 6 43
2 24 2 30 L.C.M. of three Numbers
2 12 3 15
26 5 = Common factor from all × common factor
from two × remaining factors from all.
3
Now,
24 = 2 × 2 × 2 × 3
30 = 2 × 3 × 5
∴ L.C.M. = 2 × 3 × 2 × 2 × 5 = 120
Common Remaining Remaining
factor of 24. factor of 30.
Note:
L.C.M. can also be obtained by multiplying all the prime factors having highest
power.
Example:
24 = 2 × 2 × 2 × 3 = 23 × 3
30 = 2 × 3 × 5
Product of prime factors having highest power is
3 × 23 × 5 = 120.
∴ L.C.M. of 24 and 30 is 120.
Worked Out Examples
Example: 1
Using set of multiples method, find the L.C.M. of 6 and 9.
Solution:
The set of multiples of 6 = {6, 12, 18, 24, 30, 36, ................}
The set of multiples of 9 = {9, 18, 27, 36, ...................}
The set of common multiples = {18, 36, ..............}
The smallest common multiple = 18
∴ L.C.M. = 18.
Example: 2
Using prime factorisation method, find the L.C.M. of 18 and 24.
Solution:
Here, 2 18 2 24
39 2 12
26
3
3
44 Oasis School Mathematics – 6
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