Vedanta Excel in Mathematics Book 5 Final (2077)

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

vedanta

Excel in

MATHEMATICS

5Book

Author
Hukum Pd. Dahal

Editor
Tara Bahadur Magar

Vedanta Publication (P) Ltd.

Vanasthali, Kathmandu, Nepal
+977-01-4382404, 01-4362082
[email protected]
www.vedantapublication.com.np

vedanta

MExcaeltihn ematics

5Book

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

Published by:
Vedanta Publication (P) Ltd.

Vanasthali, Kathmandu, Nepal
+977-01-4382404, 01-4362082
[email protected]
www.vedantapublication.com.np

Preface

The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical teaching
learning activities and methodologies extracted from Teachers' training, workshops, seminars and
symposia. It is an innovative and unique series in the sense that the contents of each textbooks of
the series are written and designed to fulfill the need of integrated teaching learning approaches.
Excel in Mathematics is an absolutely modified and revised edition of my three previous series:
'Elementary mathematics' (B.S. 2053), 'Maths In Action (B. S. 2059)' and 'Speedy Maths' (B. S. 2066).
Excel in Mathematics has incorporated applied constructivism. Every lesson of the whole series
is written and designed in such a manner, that makes the classes automatically constructive and
the learners actively participate in the learning process to construct knowledge themselves, rather
than just receiving ready made information from their instructors. Even the teachers will be able
to get enough opportunities to play the role of facilitators and guides shifting themselves from the
traditional methods of imposing instructions.
Each unit of Excel in Mathematics series is provided with many more worked out examples.
Worked out examples are arranged in the hierarchy of the learning objectives and they are reflective
to the corresponding exercises. Therefore, each textbook of the series itself is playing a role of a
‘Text Tutor’. There is a well balance between the verities of problems and their numbers in each
exercise of the textbooks in the series.
Clear and effective visualization of diagrammatic illustrations in the contents of each and every
unit in grades 1 to 5, and most of the units in the higher grades as per need, will be able to integrate
mathematics lab and activities with the regular processes of teaching learning mathematics
connecting to real life situations.
The learner friendly instructions given in each and every learning contents and activities during
regular learning processes will promote collaborative learning and help to develop learner-
centred classroom atmosphere.
In grades 6 to 10, the provision of ‘General section’, ‘Creative section - A’ and ‘Creative section - B’
fulfills the coverage of overall learning objectives. For example, the problems in ‘General section’
are based on the Knowledge, understanding and skill (as per the need of the respective unit)
whereas the ‘Creative sections’ include the Higher ability problems.
The provision of ‘Classwork’ from grades 1 to 5 promotes learners in constructing knowledge,
understanding and skill themselves with the help of the effective roles of teacher as a facilitator
and a guide. Besides, teacher will have enough opportunities to judge the learning progress and
learning difficulties of the learners immediately inside the classroom. These classworks prepare
learners to achieve higher abilities in problem solving. Of course, the commencement of every unit
with 'Classwork-Exercise' may play a significant role as a 'Textual-Instructor'.
The 'project works' given at the end of each unit in grades 1 to 5 and most of the units in higher
grades provide some ideas to connect the learning of mathematics to the real life situations.
The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities
to integrate mental maths and manual maths simultaneously. Moreover, the problems in ‘Section
A’ judge the level of achievement of knowledge and understanding and diagnose the learning
difficulties of the learners.
The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an important
role of ‘placement evaluation’ which is in fact used by a teacher to judge the level of prior
knowledge and understanding of every learner to make his/her teaching learning strategies.

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

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

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

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

Acknowledgments

In making effective modification and revision in the Excel in Mathematics series from my previous
series, I’m highly grateful to the Principals, HOD, Mathematics teachers and experts, PABSON,
NPABSAN, PETSAN, ISAN, EMBOCS, NISAN and independent clusters of many other Schools
of Nepal, for providing me with opportunities to participate in workshops, Seminars, Teachers’
training, Interaction programmes and symposia as the resource person. Such programmes helped
me a lot to investigate the teaching-learning problems and to research the possible remedies and
reflect to the series.
I’m proud of my wife Rita Rai Dahal who always encourages me to write the texts in a more
effective way so that the texts stand as useful and unique in all respects. I’m equally grateful to
my son Bishwant Dahal and my daughter Sunayana Dahal for their necessary supports during the
preparation of the series.

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

Grateful thanks are due to Mr. Tara Bahadur Magar for his painstakingly editing of the series.

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

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

Last but not the least, I’m heartily thankful to Mr. Pradeep Kandel, the Computer and Designing
Senior Officer of the publication house for his skill in designing the series in such an attractive
form.

Hukum Pd. Dahal

Contents Page No.

Unit Number System 5 - 18
1 1.1Countingandwritingnumbers-lookingback,1.2Hindu-Arabicnumbersystem,

1.3 Place, place value and face value, 1.4 Place and place value of numbers upto

arab, 1.5 Nepali and International place name system,1.6 b]jgfu/L ;V+ of, 1.7 Use of
commas in Nepali and International System, 1.8 Expanded forms of numbers,

1.9 The greatest and the least numbers, 1.10 The greatest and the least numbers

formed by any digits, 1.11 Rounding off numbers - Estimation

Unit Fundamental Operations 19 - 38
2 2.1 Addition and substraction - Looking back, 2.2 Multiplication and

division - Looking back, 2.3 Division fact, 2.4 Multiplication and Division of bigger

numbers, 2.5 Multiplication and division of 10, 100, 200, 3000, … and so on,

2.6 Simpli ication - A single answer of a mixed operation, 2.7 Use of brackets in

Simpli ication

Unit Properties of Whole Numbers 39 - 64
3 3.1 Various types of numbers - Looking back, 3.2 Test of divisibility,

3.3 Factors and multiples, 3.4 Prime factors and process of inding prime factors,

3.5 Process of inding multiples, 3.6 Common factors and common multiples,

3.7 Highest Common Factor (H.C.F), 3.8 Lowest Common Multiple (L.C.M.),

3.9 Process of inding H.C.F., 3.10 Process of inding L.C.M., 3. 11 Square and

square root, 3.12 Process of inding square and square root, 3.13 Cube and cube

root, 3.14 Process of inding cube and cube root

Unit Fractions 65 - 89
4 4.1 Equivalent Fractions - Looking back, 4.2 Process of inding equivalent

fractions, 4.3 Reducing fractions to their lowest terms, 4.4 Like and unlike

fractions, 4.5 Conversion of unlike fractions into like fractions, 4.6 Proper and

improper fractions, 4.7 Mixed number, 4.8 Addition and subtraction of like

fractions - Looking back, 4.9 Addition and subtraction of unlike fractions,

4. 10 Multiplication of fractions by whole numbers, 4.11 Multiplication of

fractions by fractions, 4.12 Finding the value of fraction of number in a collection,

4.13 Division of a whole number by a fraction, 4.14 Division of a fraction by a

whole number, 4.15 Division of a fraction by a fraction.

Unit Decimal 90 - 113
5 5.1 Tenths, hundredths and thousandths - looking back, 5.2 Place and place value of

decimal numbers, 5.3 Comparison of decimal numbers, 5.4 Conversion of a decimal

number into a fraction, 5.5 Conversion of a fraction into a decimal, 5.6 Addition

and subtraction of decimal numbers, 5.7 Multiplication of decimal numbers by

whole numbers, 5.8 Multiplication of decimal numbers by 10, 100 and 1000,

5.9 Multiplication of decimal numbers by decimal numbers, 5.10 Division of decimal

numbersbywholenumbers,5.11 Divisionofdecimalnumbersby 10,100and1000,

5.12 Division of whole numbers and decimal numbers by decimal numbers,

5.13 Rounding off decimal numbers, 5.14 Use of decimals

Unit Percent 114 - 120
6 6.1 Percent - How many out of 100 ? - Looking back, 6.2 Conversion of percent

into fraction, 6.3 Conversion of fraction into percent, 6.4 Conversion of percent

into decimal and decimal into percent, 6.5 Finding the value of the given percent

of a quantity,

Unit Buying and Selling 121 - 125
7 7.1 Cost price (C. P.) and selling price (S. P.) - Looking back, 7.2 Pro it and

loss - Looking back, 7.3 Pro it percent and loss percent

Unit Unitary Method and Simple Interest Page No.
8 8.1 Unitary method - Looking back, 8.2 Unit value, 8.3 Rate of cost, 8.4 Simple 126 - 136
Interest - Introduction, 8.5 Rate of interest 137 - 141
142 - 160
Unit Ratio
9 9.1 Ratio-Introduction, 9.2 Ways of writing a ratio, 9.3 Terms of a ratio 161 - 180

Unit Time, Money, Bill and Budget 181 - 192
10 10.1 Telling time - Looking back, 10.2 24 - hour clock system, 10.3 Conversion 193 - 208
209 - 226
of units of time, 10.4 Addition and subtraction of time - Looking back,
10.5 Multiplication and division of time, 10.6 Money - Looking back, 227 - 243
10.7 Conversion of money, 10.8 Addition and subtraction of money,
10.9 Multiplication and division of money, 10.10 Bill, 10.11 Budget 244 - 254
Unit Algebra - Algebraic Expressions 255-265
11 11.1 Constant and variable - Looking back, 11.2 Operation on constant and 266 - 271
variable, 11.3 Algebraic term and expression, 11.4 Types of algebraic expressions, 272 - 285
11.5Evaluationofalgebraicexpression,11.6Coef icient,base,powerandexponent 286 - 288
of algebraic term, 11.7 Like and unlike terms, 11.8 Addition and subtraction of
monomial expressions, 11.9 Addition and subtraction of polynomial expressions,
11.10 Multiplication of algebraic expressions, 11.11 Multiplication of binomial
expressions, 11.12 Division of monomial expressions, 11.13 Simpli ication by
removing brackets
Unit Algebra - Equation
12 12.1 Open mathematical sentence (or statement) - review, 12.2 Equation - review,
12.3 Solving equation, 12.4 Use of equation
Unit The Metric Measurement System
13 13.1 Measurement of length and distance - Looking back, 13.2 Conversion of
units of length, 13.3 Addition and subtraction of lengths, 13.4 Multiplication and
division of lengths, 13.5 Measurement of weight, 13.6 Measurement of capacity
Unit Perimeter, area and Volume
14 14.1 Perimeter of plane shapes - Looking back, 14.2 Perimeter of triangle
and quadrilateral, 14.3 Perimeter of rectangle and square, 14.4 Area of plane
shapes - Looking back, 14.5 Area of rectangle and square - Looking back,
14.6 Area of land, 14.7 Volume of solids - space occupied by solids (Review),
14.8 Volume of Cube and Cuboid
Unit Geometry - Line and Angle
15 15.1 Point, Line and line segment - Looking back, 15.2 Perpendicular line segment,
15.3 Parallel line segments, 15.4 Intersecting line segments, 15.5 Angle - Review,
15.6 Measurement of angles, 15.7 Construction of angles, 15.8 Types of angles
by their sizes, 15.9 Types of pairs of angles by their structures and properties.,
15.10 Transversal, 15.11 Pairs of angles made by transversal,
Unit Geometry - Plane Shapes
16 16.1 Plane shapes - Looking back, 16.2 Triangle, 16.3 Types of triangle by
sides, 16.4 Types of triangles by angles, 16.5 Sum of the angles of a triangle,
16.6 Quadrilaterals, 16.7 Sum of the angles of a quadrilateral, 16.8 Circle,
Unit Statics
17 17.1 Charts-Introduction, 17.2 Bar graph–Review, 17.3 Measurement of
temperature–Review, 17.4 Ordered pairs–Review, 17.5 Coordinates– Introduction,
Unit Set
18 18.1 Set – Looking back, 18.2 Set– A well –de ined collection, 18.3 Membership of
a set, 18.4 Methods of writing members of a set, 18.5 Types of sets
Answers
Evaluation Model

Unit Number System

1

1.1 Counting and writing numbers- looking back.

Classwork - Exercise

1. Let's count the blocks of thousands, hundreds, tens and ones. Then complete
the place value tables. Write the numerals and the number names.

(a) Th H T O
1 001
1001

One thousand one
(b) Th H T O

c) Th H T O
d) Th H T O
e) Th H T O
f) Th H T O

5 vedanta Excel in Mathematics - Book 5

Number System

2. Let's tell and write in numerals and in words, how many rupees altogether?
(a) Rs
rupees.
(b) Rs
rupees.
(c) Rs
rupees.
(d) Rs
rupees.
(e) There are 100 number of notes in each bundle of Rs 10, Rs 100, Rs 500
and Rs 1000 notes. Let's tell and write in numerals and in words, how
many rupees there are.
1 bundle of Rs 10 =

1 bundle of Rs 100 =

1 bundle of Rs 500 =

1 bundle of Rs 1000 =

3. Let's read the price of these items. Tell and write the price in words.

Rs 18,750.00 Rs 1,05,000.00 Rs 99,999.00 Rs 25,16,400.00

a) Refrigerator :

b) Washing machine :

c) Mobile :

d) Car :

e) Which one is the most expensive and which one is the cheapest item?

vedanta Excel in Mathematics - Book 5 6

Number System

4. Let's read these interesting facts. Rewrite the number names in numerals.

a) Thedistanceofmoonfromearthisabout threelakheighty-fourthousand
four hundred kilometres.

b) The earth moves round the sun at a speed of about one lakh seven
thousand kilometres per hour.

(c) The radius of earth is about sixty-three lakh seventy one thousand
metres.

(d) Russia is the largest country in the world in area. It's area is one crore
seventy-one square kilometres.

1.2 Hindu- Arabic number system
The Hindu- Arabic number system was developed by the Hindus and spread
by the Arabs all over the world. Therefore, it is well known as Hindu-Arabic
number system. In this number system we use ten symbols: 0, 1, 2, 3, 4, 5, 6, 7,
8 and 9. These symbols are called digits. We use these digits to write numerals
of any number of digits such as 7, 98, 572, 16304, and so on.

Number : A number is a count of objects or quantities. For example,
ive ingers, twenty- ive people, forty kg, and so on.

Numeral : A numeral is a symbol that represents a number. For
example, 5 ingers, 25 people, 40 kg, and so on.

Digit : A digit is a single symbol used to make numerals.

For example, in 307, the digits are 3, 0 and 7.

1.3 Place, place value and face value

Let's take a numeral 1235.

1235 Ones =5×1 =5

Tens = 3 × 10 = 30

Hundreds = 2 × 100 = 200

1 × 1000 2 × 100 3 × 10 5 × 1 Thousands = 1 × 1000 = 1000

Here, ones, tens, hundreds and thousands are the places of the digits of the
numeral 1235. A digit itself at any place of the numeral is called the face value.
The product of face value and its place is the place value.

7 vedanta Excel in Mathematics - Book 5

Number System

1.4Place and place value of numbers upto arab

Let's read these instructions carefully . Investigate the idea about how the
bigger numbers form.

Counting by 1 hundred, what comes after 9 hundred?

It is 10 hundred = 1000 = 1 thousand

Counting by 1 thousand, what comes after 9 thousand?

It is 10 thousand = 10000

Counting by 10 thousand, what comes after 90 thousand ?

It is 100 thousand = 100000 = 1 lakh

Counting by 1 lakh, what comes after 9 lakh ? It is 10 lakh = 1000000

Counting by 10 lakh , what comes after 90 lakh ?

It is 100 lakh = 10000000 = 1 crore

Counting by 1 crore, what comes after 9 crore ?

It is 10 crore = 100000000 = 10 crore 1 lakh = 100 thousand

Counting by 10 crore, what comes after 90 crore ? 1 crore = 100 lakh
It is 100 crore = 1000000000 = 1 arab 1 arab = 100 crore

1 arab = 1000000000 is a 10- digit numeral.

The place value table given below shows 1000000000 (One arab).

Periods Arab Crores Lakhs Thousands Units

Places Arab Ten- Crores Ten- Lakhs Ten- Thousands Hundreds Tens Ones
Numeral 1 Crores Lakhs Thousands 0 0 00

0 0 00 0

Now, let's take a 10-digit numeral 2351872496 and ind the place and place value
of each digit.

Numeral Places Face Value Face Value × Place = Place Value

23 5 1 8 7 2 4 9 6 Ones 6 6×1 =6
9 × 10 = 90
Tens 9 4 × 100 = 400
2 × 1000 = 2000
Hundreds 4 7 × 10000 = 70000
8 × 100000 = 800000
Thousands 2 1 × 1000000 = 1000000
5 × 10000000 = 50000000
Ten-thousands 7 3 × 100000000 = 300000000
2 × 1000000000 = 2000000000
Lakhs 8

Ten-lakhs 1

Crores 5

Ten-crores 3

Arabs 2

Now, the number name of 2351872496 is: Two arab thirty- ive crore eighteen
lakh seventy-two thousand four hundred ninety-six.

vedanta Excel in Mathematics - Book 5 8

Number System

1.5 Nepali and International place name system

Let's compare the places of digits of numerals between Nepali and International
system.

Nepali place International place Place values Number of digits
names names
Ones Ones 1 One
Tens Tens 10 Two
Hundreds Hundreds 100 Three
Thousands Thousands 1000 Four
Ten-thousands Ten-thousands 10000 Five
Lakhs Hundred-thousands 100000 Six
Ten-lakhs Millions 1000000 Seven
Crores Ten-millions 10000000 Eight
Ten-crores Hundred-millions 100000000 Nine
Arabs Billions 1000000000 Ten

The table given above shows that only after ten-thousand the place names of digits
are different in Nepali and International system.

Now, Let's compare the places of the digits of a numeral 2530498167 in Nepali
and International systems.

Billions Arabs Ten Crores Crores Millions Ten Lakhs Lakhs Ten Ten Thousands Thousands Hundreds Hundreds Tens Ones
Thousands Thousands
Hundred Ten Hundred Tens Ones
Millions Millions thousands

2530498167

In Nepali system, the number name is:
Two arab ifty-three crore four lakh ninety-eight thousand one hundred sixty-seven.
In International system, the number name is:
Two billion ive hundred thirty million four hundred ninety-eight thousand one
hundred sixty-seven.

1.6 b]jgfu/L ;V+ of
b]jgfu/L ;+Vof k0| ffnL klg lxGb–' c/ljs ;+Vof k|0ffnLdf g} cfwfl/t 5 . b]jgfu/L
cs+ x?nfO{ hgfpg] ;s+ t] x? dfq lxGb'–c/las c+s hgfpg] ;s+ ]tx? eGbf km/s 5g\ .

9 vedanta Excel in Mathematics - Book 5

Number System

lxGb–' c/ljs 0 1 2 3 4 5 6 7 8 9
c+s Zero One Two Three Four Five Six Seven Eight Nine

bj] gfu/L ) ! @ # $ % ^ & * (
cs+ ;'Go Ps bO' { tLg rf/ kfFr 5 ;ft cf7 gf+}

b]jgfu/L ;V+ of k|0ffnLdf klg ) b]lv ( ;Ddsf cs+ x?sf] ko| fu] u/L s'g} klg 7"nf] cyjf ;fgf]

;+Vof nV] g ;lsG5 . h:t} M Ps xhf/ ;ft ;o krf;L – !,&*% (One thousand seven hundred

eighty- ive) k}tfln; xhf/ gf} ;o tL;– $%,(#) (Forty- ive thousand nine hundred thirty)

b]jgfu/L ;V+ of k|0ffnLdf :yfg–dfgsf] gfd lgDg cg';f/ xG' 5 .

Arabs Ten- Crores Ten-lakhs Lakhs Ten-thousands Thousands Hundreds Tens Ones
Crore

c/a bz s/f]8 bz nfv nfv bz xhf/ xhf/ ;o bz Ps

s/f8]

@% # )$ ( * ! ^&

o; cg;' f/ @%#)$(*!^& ;+Vofsf] gfd b'O{ c/a lqkGg s/f8] rf/ nfv cG7fgAa]

xhf/ Ps ;o ;t\;¶L xG' 5 .

1.7 Use of commas in Nepali and International system

We use commas to separate the periods of digits of numerals. It makes easier
to read and write number names of bigger numbers.

We start to use commas only in 4-digit and greater than 4-digit numerals both
in Nepali and International system.

Let's compare the way of using commas in Nepali and in International system.

In Nepali system International system
72,58,36,194 725,836,194

using comma to separate the using comma to separate
digits at units period the digits at units period

using comma to separate the using comma to separate the
digits at thousands period digits at thousands period

using comma to separate Number name is:
the digits at lakhs period Seven hundred twenty- ive million
eight hundred thirty-six thousand
Number name is: one hundred ninety-four.
Seventy-two crore ifty-eight lakh thirty-six
thousand one hundred ninety-four.

In this way, in Nepali system, each period of two digits is separated by commas.
In International system, each period of three digits is separated by commas.

vedanta Excel in Mathematics - Book 5 10

Number System

1.8 Expanded forms of numbers

Let's take a number 2314 and study the illustrations given below, how it is
expanded.

2314 = 3 × 100 1×10 4×1 =2 × 1000 + 3 × 100 + 1 × 10 + 4 ×1

2 × 1000

Similarly ,

3005 = 3 × 1000 + 0 × 100 + 0 × 10 + 5 × 1 = 3 × 1000 + 5 × 1

6040 = 6 × 1000 + 0 × 100 + 4 × 10 + 0×1 = 6 × 1000 + 4 × 10

59183 = 5 × 10000 + 9 × 1000 + 1 × 100 + 8 × 10 + 3 × 1

8670504 = 8 × 1000000 + 6 × 100000 + 7 × 10000 + 5 × 100 + 4 × 1

In 2078,
After 2, there are three digits. So, 2 × 1000
After 7, there is one digit. So, 7 × 10
The last digit 8 is at ones place. So, 8 × 1
2078 = 2 × 1000 + 7 × 10 + 8 × 1

Let's Remember!
500103 = 5 × 100000 + 0 × 10000 + 0 × 1000 + 1 × 100 + 0 × 10 + 3 × 1
Here, 0 × 10000 + 0 × 1000 + 0 × 10 = 0 and it is not necessary to write in
the expanded form.
So, 500103 = 5 × 100000 + 1 × 100 + 3 × 1
In this way, we can expand a number as the sum of its digits multiplied by
their places.

1.9 The greatest and the least numbers
Among 0 to 9 , the greatest digit is 9 . So, 9 is the greatest 1 -digit number. 99
is the greatest 2 - digit number, 999 is the greatest 3 - digit number and so on.
1 is the least counting number. So, 1 is the least 1 -digit number. 10 is the least
2 -digit number, 100 is the least 3 - digit number and so on.
11 vedanta Excel in Mathematics - Book 5

Number System

1.10 The greatest and the least numbers formed by any digits
Let's take any six digits 7, 3, 0, 9, 2 and 5 .
Arranging the digits in descending order o 975320
It is the greatest 6 -digit number.
Arranging the digits in ascending order o 203579
It is the least 6 -digit number.
Remember, 023579 is not a 6 -digit number. 023579 is same as 23579
which is a 5 -digit number.

EXERCISE 1.1

Section A - Classwork
1. Let's tell and write the answer in the blank spaces.

a) The digits of the numeral 380642 are
b) Howmanydigitsarethereinthenumeral40509817?
c) What is the face value of 6 in 1632409 ?
d) What is the place value of 9 in 20497350 ?
e) The digit at ten-lakhs place of the numeral 142068537 is
f) The digit at hundred million place of the numeral 501652084 is
2. Let's tell and write the correct answer of these questions.

a) How many hundreds make 1 thousand ?

b) How many thousands make 1 lakh ?

c) How many lakhs make 1 crore ?

d) How many crores make 1 arab ?

e) How many lakhs are there in 1 million?

f) How many millions are there in 1 crore?

g) How many crores are there in 100 million?

h) How many billions are there in 1 arab?

vedanta Excel in Mathematics - Book 5 12

Number System

3. Let's rewrite these numerals using commas in Nepali and in
International system.

Numerals Use of Commas in Nepali Use of Commas in
192750 System International System

4308679

20509180

370084300

1201610240

4. a) Let's write the number name in Nepali system.
(i) 10001
(ii) 1000100
b) Let's write the number name in International system.
(i) 300005
(ii) 300500100

5. a) ;G' o bl] v gf} ;Ddsf b]jgfu/L c+sx? nV] gx' f];\ .

b) krxQ/sf] c+u|h] L ;V+ of gfd s] x'G5 <

c) 49 sf] bj] gfu/L ;+Vof / ;V+ of gfd s] x'G5 < ,

6. a) The least and the greatest number of 6-digit are

b) The least and the greatest number of 9-digit are ,

c) The least number formed by the digits 2, 0, 5, 7, 4 is

d) The greatest number formed by the digits 4, 9, 2, 0, 7, 3, 1 is
7. a) The expanded form of 30402 is

b) The short form of 2 × 1000000 + 6 × 1000 + 1 × 10 is

13 vedanta Excel in Mathematics - Book 5

Number System

Section B

8. Let's write the face value, place name and place value of the red coloured
digits of the numerals in Nepali and International system.

a) 95731 b) 62915 c) 2753460 d) 5816209

e) 82413596 f) 314702600 g) 770038002 h) 4162030575

9. Let's write the numerals using commas. Then write the number names
from the place value tables.

a) Ten-lakhs Lakhs Ten-thousands Thousands Hundreds Tens Ones

17 0 5 3 94

b) Ten-crores Crores Ten-lakhs Lakhs Ten-thousands Thousands Hundreds Tens Ones

3 6 05 1 8 0 50

c)

Arabs Ten-crores Crores Ten-lakhs Lakhs Ten-thousands Thousands Hundreds Tens Ones

40 7 62 3 3 9 17

d) Millions Hundred-thousands Ten-thousands Thousands Hundreds Tens Ones

21 5 0 7 12

e) Ten- Millions Hundred- Ten-thousands Thousands Hundreds Tens Ones
millions thousands 5 8 2 07

50 9

10. Let's write these numerals in the expanded forms.

a) 5263487 b) 20470155 c) 640035209 d) 305020700

11. Let's write the numerals in the short forms.

a) 9 × 100000 + 2 × 10000 + 5 × 1000 + 7 × 100 + 4 × 10

b) 6 × 1000000 + 3 × 10000 + 2 × 1000 + 1 × 100 + 8 × 1

c) 2 × 10000000 + 1 × 1000000 + 4 × 10000 + 9 × 1000 + 6 × 100

d) 3 × 100000000 + 3 × 10000000 + 5 × 100000 + 2 × 1000 + 5 × 1

12. Let's ϐind how many millions in - 10 lakh = 1000000 = 1 million
a)10 lakh b) 30 lakh c) 70 lakh 1 crore = 10000000 = 10 million

d) 1 crore e) 4 crore f) 9 crore 1 million = 1000000 = 10 lakh
13. Let's ϐind how many lakh or crores in - 10 million = 10000000 = 1 crore

a)1 million b) 4 million c) 8 million

d) 10 million e) 50 million f) 70 million

vedanta Excel in Mathematics - Book 5 14

Number System

14. Let's read the information carefully then rewrite the numerals in number
names in Nepali system.

a) Brunei is the least populated country in Asia. It's population is about
4,39,336 in 2019.

b) The estimated population of Nepal is about 3,02,60,244 in 2077 B.S.

c) China is the country with the largest population in the world. It's population
is about 1,42,45,48,266 in 2020.

d) Australia is the continent with the least population in the world. It's
population is 2,53,98,177 in 2020.

15. Let's write the numerals of the number names using commas in Nepali
system.
a) The area of Nepal is one lakh forty-seven thousand one hundred
eighty-one square kilometres.
b) The cost of a car is thirty-two lakh four thousand nine hundred rupees.
c) 1 lakh U.S. dollar is nearly equal to one crore twelve lakh six thousand
Nepali rupees.
d) The estimated budget of a drinking water supply project is sixteen crore
nine lakh ϐive thousand seven hundred Nepali rupees.

16. Let's write the numerals of the number names. Then rewrite the number
names in Nepali System.

a) The cost of a motorbike in Nepal is two hundred ϐifty-ϐive thousand rupees.

b) The population of a province is three million forty-eight thousand nine
hundred.

c) The speed of light in air is about seventeen million nine hundred
eighty-seven thousand ϐive hundred twenty kilometres in 1 minute.

d) The distance of the sun from the earth is about one hundred forty-nine
million six hundred thousand kilometres.

e) Nepal Government has allocated a budget of two billion ϐifty million four
hundred thousand rupees to develop a hydropower in far western region.

17. s_ Ps ;o krf; xhf/nfO{ bj] gfu/L ;+Vofdf nV] gx' f];\ / gk] fnL k4lt cg;' f/ ;+Vofsf] gfd
n]Vg'xf];\ .

v_ 2547698 nfO{ bj] gfu/L ;V+ ofdf n]vL g]kfnL k4lt cg';f/ ;+Vofsf] gfd n]Vg'xf];\ .
u_ k}tL; ldlnognfO{ bj] gfu/L ;+Vofdf nV] g'xf];\ . of] ;+Vofdf slt s/f8] / nfv xG' 5 <

18. a) Find the difference of the greatest and the least numbers of 5 -digits.
b) Find the sum of the greatest and the least numbers of 6- digits.

15 vedanta Excel in Mathematics - Book 5

Number System

c) Find the difference of the greatest and the least numbers of ive digits
formed by 3,7,0,1 and 9.

d) Find the sum of the greatest and the least numbers of seven digits formed
by 8,0,2,4,1,3 and 6.

19. It's your time - Project Work !

a) Let's write a numbers for each of 7 -digit, 8- digit, 9 - digit and 10 -digit.
(i) Show each numbers in place value table.

(ii) Write the number name of each numeral in Nepali system and International
system.

b) Let's write a 7 -digit number. Increase this number by 10 lakh in each case and
write next three numbers.

c) Let's write a 9 -digit number. Increase this number by 10 crore in each case
and write next three numbers.

20. a) Let's visit to the available website in your school computer or your own
computer or your family member's mobile. Search and write the present
population of Nepal, Bhutan, Bangaladesh and India.

(i) Write the population in words in Nepali and in International systems.

(ii) Compare the population of these countries.

(iii) Can you search the population of your district ? If so write the present
population of your district.

b) Let's make 10 lash-cards of equal size by cutting a chart paper.

(i) Write the number 0 to 9 in each lash-card and make two sets of number
cards.

012 345678 9

(ii) Play with a partner to make the greatest and the least numbers of 2 -digit,
3 -digit, .... 9 - digit by drawing the different lash-cards.

1.11 Rounding off numbers - Estimation

Let's investigate the rules for rounding off numbers to the nearest tens,
hundreds, thousands, etc.

a)

300 310 320 330 340 350 360 370 380 390 400
375

Rounding of 330 to the nearest hundreds is 300.

Rounding of 375 to the nearest hundreds is 400.

vedanta Excel in Mathematics - Book 5 16

Number System

b)

2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
2480

Rounding off 2500 to the nearest thousands is 3000.

Rounding off 2480 to the nearest thousands is 2000.

Similarly,

c) Rounding of 674 to the nearest tens is 670 and to the nearest hundreds is
700.

d) Rounding of 1280 to the nearest hundreds is 1300 and to the nearest
thousands is 1000.

EXERCISE 1.2
Section A - Classwork
1. Let's use the given number lines and round off the numbers to the

nearest tens, hundreds or thousands.
a)

400 410 420 430 440 450 460 470 480 490 500

(i) Rounding off 423 to the nearest tens is
(ii) Rounding off 464 to the nearest hundreds is

b)

900 910 920 930 940 950 960 970 980 990 1000

(i) Rounding off 955 to the nearest tens is
(ii) Rounding off 972 to the nearest thousands is

c)

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

(i) Rounding off 1236 to the nearest tens is
(ii) Rounding off 1236 to the nearest hundreds is
(iii) Rounding off 1236 to the nearest thousands is

d)

30000 31000 32000 33000 34000 35000 36000 37000 38000 39000 40000

(i) Rounding off 37490 to the nearest hundreds is
(ii) Rounding off 37490 to the nearest thousands is
(iii)Rounding off 37490 to the nearest ten-thousands is

17 vedanta Excel in Mathematics - Book 5

Number System

Section B
2. Let's express the numbers to the round ϐigures.

a) There are 482 students in a school. Write the number in round igure to the
nearest (i) tens (ii) hundreds

b) The distance between Damak to Butwal is 539 km. Write the distance in
round igure to the nearest (i) tens (ii) hundreds

c) The cost of a mobile is Rs. 1,750. Write the cost in round igure to the nearest
(i) hundreds (ii) thousands.

d) The population of a village is 8,360. Write the population in round igure to
the nearest (i) hundreds (ii) thousands.

e) The monthly income of a family is Rs. 27,820. Write the income in round
igure to the nearest (i) hundreds (ii) thousands (iii) ten-thousands

f) The price of a motorcycle is Rs. 2,45,250. Write the price in round igure to
the nearest (i) hundreds (ii)thousands (iii)ten-thousands.

3. a) Round off 493 to the nearest tens then to the nearest hundreds.

b) Round off 2,775 to the nearest hundreds then to the nearest thousands.

c) Round off 8,980 to the nearest thousands then to the nearest ten- thousands.

It's your time - Project Work !
4. a) How many students are there in your class? Round off this number to the

nearest tens.

b) How many students are there in your school? Round off this number to the
nearest (i) tens (ii) hundreds (if possible).

5. a) Write a 4-digit number with non-repeated digits. Then, round off your number
to the nearest (i) tens (ii) hundreds (iii) thousands.

b) Write a 6-digit number with non-repeated digits. Then round off your number
to the nearest (i) tens (ii) hundreds (iii) thousands (iv) ten-thousands.

vedanta Excel in Mathematics - Book 5 18

Unit Fundamental Operations

2

2.1 Addition and substraction - Looking back

Classwork - Exercise

1. At first, let's add the numbers that make 10, 20, ...100, ... Then complete
the addition.

3+9+7 = 10 + 9 = 19 65 + 85 + 15 = 100 + 65 = 165

a) 4 + 7 + 6 = b) 8 + 12 + 6 =

c) 5 + 4 + 45 = d) 17 + 60 + 40 =

e) 75 + 25 + 28 = f) 180 + 100 + 20 =

g) 240 + 200 + 60 = h) 350 + 395 + 5 =

2. Let's write the missing numbers then add quickly and find the sums.

13 + 14 = (10 + 10) + (3 + 4) = 20 + 7 = 27

25 + 33 = (20 + 30) + (5 + 3) = 50 + 8 = 58

a) 12 + 16 = (10 + 10) + ( +6) =

b) 23 + 13 = ( + 10) + (3 + ) =

c) 24 + 21 = ( + ) + (4 + 1) =

d) 32 + 31 = ( + ) + (2 + ) =

e) 44 + 35 = ( + 30) + ( + ) =

f) 57 + 23 = (50+ 20) + (7+ 3) =

g) 48 + 34 = ( + 30) + (8+ ) =
3. Let's learn the trick then subtract quickly.

27 – 14 = (27 – 10) – 4 = 13 46 – 24 = (46 – 20) – 4 = 22

a) 24–11 = (24–10) – = b) 28 – 13 = (28 – 10) – =

19 vedanta Excel in Mathematics - Book 5

Fundamental Operations

c) 36 –15 = (36 – ) – 5 = d) 39 – 21 = (39 – )– 1 =
)– =
e) 55 –23 = (50 – ) – 3 = f) 67 – 35 = ( – )– =

g) 74 –42 = ( – ) – = h) 90 – 51 = ( –

4. Let's tell and write the answer as quickly as possible.

a) 5 + 8 = Then 13 – 5 = and 13 – 8 =

b) 7 + 9 = Then 16 = 9 and –9 = 7

c) 12 + 6 = Then –12 = 6 and 18 – = 12

d) 20 + 30 = Then 50– = 20 and 50 – = 30

e) 25 + 15 = Then 25 = 15 and – 15 = 25

5. Quiz time !

a) The sum of two numbers is 14 and the smaller number is 5. The bigger
number is

b) The difference of two numbers is 7 and the bigger number is 18. The smaller

number is Sum = 12

c) The sum of two numbers is 15 and the difference difference = 2, let's think ...

is 5. The numbers are and 11 + 1, 11 – 1 10 + 2, 10 – 2
9 + 3, 9 – 3 8 + 4, 8 – 4

d) The sum of two numbers is 24 and the difference 7 + 5, 7 – 5 ‘7 + 5 = 12 and 7–5 = 2!

is 4. The numbers are and

e) If a + b + c = 20 and a + b = 14 , then c =

f) If p + q + r = 27 and q + r = 18, then p =

6. Puzzle time !

a) Let's ill in the missing numbers to complete the sums.

i) + 13 = 25 ii) – = iii) – 6 =

+ + +– – – +– +

+= – 5=1 + =7

= = == = = == =

22 + = 40 30 – 11 = – 4 = 16

vedanta Excel in Mathematics - Book 5 20

Fundamental Operations

b) The sum of the numbers in each row, column and diagonal is the same. Let's
complete the magic squares.

i) ii) iii) 1
11 13 17

14 11 14

7 13 3 10

Sum is 30 Sum is 42 594

Sum is 34
c) Each hexagon is made by adding up the numbers in the two hexagons

below it. Let's tell and write the missing numbers.

a) b)

21 24 36 13
9 12 10 16

5

EXERCISE 2.1
Section A - Class work
1. Let's tell and write the answer as quickly as possible.

a) 75 girls and 50 boys, how many students altogether ?
b) Among 480 people, 230 are women. Number of men are
c) By how much is Rs 5010 more than Rs. 4999 ?
d) By how much is Rs. 9990 less than Rs. 10001 ?
e) What should be added to 7777 to get 9999 ?
f) What should be subtracted from 10000 to get 9999 ?

21 vedanta Excel in Mathematics - Book 5

Fundamental Operations

2. Let's tell and write the missing digits in the following sums.

a) 3 6 b) 2 5 c) 9 5 d) 5 2

+4 +6 8 – –6

7 87 5 8 147

3. a) The numbers in the circles have been added in pairs and 18
the sum of each pair is in the square between the circles. 25 28
Complete these puzzles. 7 17 10

(i) 15 (ii) (iii) 36

35 25 48
9 17 30 52

b) Let's add as shown and complete these addition puzzles.

(i) (ii) + 5 (iii) + 2

+ 9 38

6 15 4 12 7 17

4 12 14 13 25

7 10 6 13 11

Section B
4. Let's rewrite these problems and solve them.

a) 694 girls b) 4731 men c) Rs 7350 d) 9420 people

+ 586 boys + 4699 women – Rs 4580 – 5875 women

students people kg men

e) selling price = Rs 3245 f)buying price = Rs 5710
buying price = Rs 2965 selling price = Rs 5360
Pro it = Rs loss = Rs

vedanta Excel in Mathematics - Book 5 22

Fundamental Operations

5. Let's add or subtract and check your answer by subtraction or addition.

a) 4658 b) 3895 c) 8012
+2457 – 4637
–2457 + 5325 –3895 + 4637

6. Let's read these problems carefully and solve them.

a) A fruit seller bought some apples for Rs. 3,750 and orange for
Rs 2,875. She/he sold all fruits for Rs 7,550.

(i) Find the total cost of fruits for the fruit seller.
(ii) How much money did she/he gain or loss ?

Solution (ii) selling price =
(i) cost of apples = buying price =
gain =
cost of oranges =
total cost =

b) The male population of a town is twelve thousand six hundred
ϔifty-four and the female population is thirteen thousand seventy.

(i) Find the total population of the town.
(ii) How many more females are there than males ?

Solution (ii) female population =
(i) male population = 12,654 male population =
difference =
female population = 13,070
total population =

c) A farmer had 1,736 chickens in his poultry farm. 258 of them died during
the epidemic of 'Bird Flu'. How many more chickens does he need to have
2,500 ?

d) Mrs. Pariyar bought a second hand scooty for Rs 95,600 and she spent
Rs 7,850 for it's repairment. Then she sold it for Rs 1,10,500.

(i) Find the total cost of the scooty with it's repairment.
(ii) How much money did she gain or loss ?

23 vedanta Excel in Mathematics - Book 5

Fundamental Operations

e) Last week, Dakshesh visited a Population 18450
municipality of ice and study the 17450
population of the municipality in 16450 Male Female Children
a bar graph. 15450
14450
(i) Find the total population of the 13450
municipality. 12450
11450
(ii) How many less number of 10450
children are there than the adult
population? O

It's your time - Project work !

7. Let's make groups of 5/5 students and conduct a survey to ind the number of
girls and boys in your school in primary level. Write the numbers in the table
and answer the questions.

Classes Number of girls Number of boys Total
1
2
3
4
5

Total

a) Find the total number of students in each class.

b) Find the total number of girls in the primary level.

c) Find the total number of boys in the primary level.

d) Find the total number of students in the primary level.

e) Which number is grater in each class, girls or boys and by how many ?

f) How many more girls or boys are there in the primary level of your school?

8. Let's visit the available website in your school computer or in your computer
or in your family member's mobile.

a) Search the live male and female population of Nepal. Write the population
in words and in numerals.

b) Calculate today's total population of Nepal.

vedanta Excel in Mathematics - Book 5 24

Fundamental Operations

2.2 Multiplication and division - Looking back

Classwork - Exercise

1. Let's tell and write the correct numbers in the blank spaces.

a) + + + +++
=×=
dots

b) + + + + ++++

=×= dots

c) + + ++
=×=
dots

Did you understand the relation between addition and multiplication ?
Multiplication is repeated addition.

2. Let's tell and write the products as quickly as possible

a) 4 × 1 = , 9 × 1 = , 10 × 1 = , 15 × 1 =

b) 5 × 0 = , 7 × 0 = , 12 × 0 = , 18 × 0 =

c) 3 × 7 =7 × 3 = , 5 × 8 = 8 × 5 = , 10 × 7 = 7 × 10

Did you understand some interesting facts about multiplication ?
Fact I: The product of any number and 1 is the number itself.
Fact II: The product of any number and 0 is always 0.
Fact III: The product remains the same even if the order of multiplier and

multiplicand is changed.

3. Let's tell and write the correct numbers in the blank spaces.

a) How much is 8 apples shared between 2 pupils ?

Each gets ÷ = apples.
How many twos are there in 8 ?

÷=

25 vedanta Excel in Mathematics - Book 5

Fundamental Operations

b) How much is 12 marbles shared between 3 girls ?
Each gets ÷ = marbles.

How many threes are there in 12 ?
÷=

4. Let's tell and write the correct numbers in the blank spaces.

a) 4 × 2 = 8 and 8 ÷ 4 = 2

b) × = and 15 ÷ 5 =

c) × = and ÷ =

d) × = and ÷ =

5. Quiz time !

a) The product of two numbers is 42 and one of them is 7.

The other number is

b) Multiplicand is 9 and the product is 72. Multiplier is

c) Multiplier is 10 and the product is 100. Multiplicand is

d) The quotient of 54 divided by a number is 6. The number is

e) Dividend is 42, divisor is 8, quotient = remainder =

6. Puzzle time !
a) Let's ill in the missing numbers to complete the sums.

× 5 = 20 72 ÷ =8 ÷ 3=0
× ××
×× × ÷ ÷÷
÷ =2
×= ÷ 3=2 = ==

== = = == ÷ 9 = 10

12 × = 120 ÷=

vedanta Excel in Mathematics - Book 5 26

Fundamental Operations

b) Let's multiply as shown and complete these multiplication puzzles .

(i) × 6 4 8 (ii) × 9 (iii) ×5
48
3 18 5 35 60 50

7 28 36 7 63

9 72 3 18

2.3 Division fact

Let's learn the following facts about division.

Fact I : When a number is divided by 1, the quotient is the number itself.

5 ÷ 1 = 5, 8 ÷ 1 =8, 12 ÷ 1 = 12 and so on.

Fact II : When a number is divided by itself, the quotient is always 1 .

4÷4=1, 6 ÷ 6 = 1, 9 ÷ 9 = 1 and so on .

Fact III : When 0 is divided by any nonzero number, the quotient is always 0.

0 ÷ 3 = 0, 0 ÷ 8 = 0, 0 ÷10 = 0 and so on.

Fact IV : When any non-zero number is divided by 0, the quotient is in inite.
2 ÷ 0 = in inite, 7 ÷ 0 = in inite and so on.

Fact V : Dividend = Divisor × Quotient + Remainder.

In 9 ÷ 4 = 2 is the quotient and 1 is remainder.

So, 9 = 4 × 2 + 1 = 8 + 1 = 9 = Divisor × Quotient + Remainder

2.4 Multiplication and Division of bigger numbers

Let's study the following examples and learn about the multiplication and
division of bigger numbers.

Example 1 : Multiply a) 4328 by 27 b) 3615 by 236

Solution

a) 4 3 2 8 b) 3 6 1 5
× 2 7 o 20 + 7 × 2 3 6 o 200 + 30 + 6
30296 m 7 × 4328 21690 m 6 × 3615
108450 m 30 × 3615
+ 86560 m 20 × 4328
116856 + 723000 m 200 × 3615
853140

27 vedanta Excel in Mathematics - Book 5

Fundamental Operations

Example 2 : Divide a) 510 ÷ 36 b) 2880 ÷ 125
Solution

a) 36 ) 510 )14 Here, divisor 36 has two digits. So, at irst try to divide
–36 two digits 51 of the dividend 510.

150 51 ÷ 36 = 1 time and 15 remainder. Then, bring down 0
–144 to the remainder and continue the process.

6
Q = 14 and R = 6

b) 125) 2880)23 Divisor 125 has three digits. So, try to divide three digits
–250 288 of the dividend 2880 .

380 Then continue the process as above
– 375

5
Q = 23 and R = 5

2.5 Multiplication and division of 10, 100, 200, 3000,… and so on

Let's learn some tricky ways about the multiplication and division of 10, 100,
200, 3000, 50000, … and so on.

Example 3 : Multiply a) 600 × 300 b) 320 × 40

Solution Multiply the non-zero numbers: 6 × 3 = 18. Write as
a) 600 × 300 = 180000 many zeros at the end of the product as the multiplier
and multiplicand have.

b) 320 × 40 = 12800 32 × 4 = 128, then 320 × 40 = 12800

Example 4 : Divide a) 12000 ÷ 400 b) 12500 ÷ 50
Solution

a) 12000 ÷ 400 = 120 ÷ 4 Equal number of zeros from 12000 and
= 30 400 are cancelled. Then 120÷ 4 = 30 !

b) 12500 ÷ 50 = 1250 ÷ 5 It's easy! Equal number of zeros
= 250 from 12500 and 50 are cancelled.
Then 1250 ÷ 5 = 250 !!
vedanta Excel in Mathematics - Book 5
28

Fundamental Operations

EXERCISE 2.2

Section A - Class work
1. Let's multiply row and column, then ϐind the total number of dots.

a) b) c)

6 ×7= ×= ×=

2. Let's divide the total number of dots by the number of rows or columns.
a) b)

÷ = dots in each row ÷ = dots in each column

3. Let's tell and write the missing numbers as quickly as possible.

a) 6 × = 30 b) ÷ 5 = 6 c) × 4 = 32

d)32 ÷ =4 e) × 7 = 63 f) ÷ 9 = 7

g) 10 × = 60 h) 60÷ = 10 i) × 8 = 72

j) ÷ 8 = 9 k) 49 ÷ =7 l) × 7 = 49

4. Let's tell and write the products or quotients quickly.

a) 6 × 70 = , 6 × 70 = , 600 × 70 =

b) 400 ÷ 4 = , 400 ÷ 40 = , 4000÷400 =

c) 50 × 90 = , 50 × 900 = , 500 × 900 =

d) 560 ÷ 8 = , 560 ÷ 80 = , 5600 ÷ 80 =
29
vedanta Excel in Mathematics - Book 5

Fundamental Operations

Let's tell and write the answer as quickly as possible.

5. a) 80 number of Rs 5 notes = Rs

b) 100 number of Rs 20 notes = Rs 50 number of Rs 50 notes
c) 50 number of Rs 50 notes = Rs = 50 × Rs 50 = Rs 2,500

d) 70 number of Rs 100 notes = Rs

e) 100 number of Rs 500 notes = Rs

f) 100 number of Rs 1000 notes = Rs

6. a) Number of Rs 5 notes in Rs 100 = Number of Rs 10 notes in
b) Number of Rs 10 notes in Rs 1000 = Rs 1000 = Rs 1000 ÷ Rs 10
c) Number of Rs 50 notes in Rs 5000 =
d) Number of Rs 500 notes in Rs 35000 = = 100 notes

7. Let's tell and write the correct answer as quickly as possible.

a) 1cm = 10 mm, then 6 cm =

b) 1 m = 100 cm, then 5 m =

c) 1 km = 1000 m, then 7 km =

d) 1 kg = 1000 g, then 4 kg =
e) 1 l = 1000 ml, then 9 l =

f) 1 hour = 60 minutes, then 4 hours =

g) 1 minute = 60 seconds, then 5 minutes =

h) 1 quintal = 100 kg, then 100 quintals =

i) 1 metricton = 1000 kg, then 10 metricton =

8. Let's tell and write the correct answer as quickly as possible.

a) 10 mm = 1 cm, then 50 mm =

b) 100 cm = 1 m, then 300 cm =

c) 1000 m = 1 km, then 6000 m =

d) 1000 g = 1 kg, then 7000 g =
e) 1000 ml = 1 l , then 4000 ml =

f) 60 minutes = 1 hour, then 180 minutes =

vedanta Excel in Mathematics - Book 5 30

Fundamental Operations

Section B
9. Let's answer the following questions.

a) Is the multiplication of natural numbers a repeated addition? Justify your
answer with two examples.

b) Is the division of whole numbers a repeated subtraction ? Justify your
answer with two examples.

c) What is the difference between the meaning of 2 × 3 = 6 and 3 × 2 = 6 ?

d) Dividend = Divisor × Quotient + Remainder, justify it with two examples.
10. Let's multiply.

a) 85 × 56 b) 398 × 74 c) 5260 × 290 d) 7413 × 325
11. Let's divide.

a) (i) 784 ÷ 7 (ii) 9640 ÷ 7 (iii) 2587 ÷ 8 (iv) 63729 ÷ 9

b) (i) 1625 ÷ 12 (ii) 1280 ÷ 25 (iii) 2856 ÷ 136 (iv) 78760 ÷ 254
12. a) The rate of cost of apples is Rs 135 per kg. Find the cost of 6 kg of apples.

b) The cost of 9 kg of rice is Rs 765, ind the rate of cost of rice.

c) Each of 35 students of class V donated Rs 160 to a charity. How much was
the total amount of donation ?

d) Each of 125 families in a locality donated equal amount of money to make a
fund of Rs 6,37,500 for building a library. How much money did each family
donate ?

e) A packet of 500 ml of milk gives 17 g of protein. How much protein do we
get from 48 packets of milk ?

f) We get roughly 2592 calories from 36 boiled eggs. Estimate the amount of
calories found in 1 boiled egg.

13. a) The distance between place A and place B is 45 km. A local bus
carries passengers from A to B and B to A 6/6 times everyday. How many
kilometres does the bus travel in a day ?

b) In a hall, chairs are arranged in 24 rows with 24 chairs in each row. How
many chairs are there in the hall ?

c) In a school assembly, students are arranged in 18 columns with 27 students
in each column. How many students are there in the assembly ?

31 vedanta Excel in Mathematics - Book 5

Fundamental Operations

14. a) When some sweets are divided between 9 children, each gets 7 sweets and
2 sweets are left to divide. How many sweets are there altogether ?

b) In a school assembly, students are arranged in 15 rows with 18 students in
each row. If 7 students are left to arrange in this way, how many students
are there in the assembly ?

It's your time - Project work !

15. a) Estimate how many kilometres (or metres) do you travel everyday while
coming to school and going to your home ? Calculate the distance travelled
by you in a week (except weekend).

b) Estimate how many grams (or kilograms)of rice do your family consume in
1 day ? Discuss with your family members and estimate the quantity of rice
consumed in (i) 1 week (ii) 1 month (iii) 1 year.

c) A 8 - 10 year old child needs to drink roughly 1500 ml of water each day.
Now, estimate how many millilitres of water do you drink in (i) 1 day
(ii) 1 week (iii) 1 month (iv) 1 year ?

d) (i) How many class periods do you have in a week ?

(ii) How long is your each class period ?

(iii) How many hours and minutes are there in one week class periods ?

16. a) Let's Write any three 2 -digit numbers. Divide them separately by any three
1 -digit divisor. Then show that :

Dividend = Divisor × Quotient + Remainder

Let's stick your indings on the school's wall - magazine !

b) Let's draw 20 circles in 1 row, 2 rows, 4 rows, 5 rows, 10 rows and 20 rows
with equal number of circles in each row in a chart paper. Find the number
of circles in each row by using division process. Stick your indings on the
school's wall-magazine.

c) Let's draw 30 circles equally in 2, 3, 5, 6, 10 and in 15 rectangular boxes
separately. Then ind how many twos, threes, ives, sixes, tens and ifteens
are there in 30 by using division process. Stick your indings on the school's
wall- magazine.

vedanta Excel in Mathematics - Book 5 32

Fundamental Operations

2.6 Simplification - A single answer of a mixed operation

Classwork - Exercise

1. Let's perform these operations. Then tell and write the answer.

a) Add 7 and 9 and subtracts 6 .

7 + 9 – 6 = 16 – 6 =

b) Subtract 5 from 18, then add 8 .

– + =+ =

c) Subtract 4 from 20, then again subtract 12.

–– =– =

d) Multiply 6 and 8, then add 5.

× + =+ =

e) Divide 63 by 7, then subtract 4 .

÷ – =– =

The problems given above have more than one operations. Such problems are
called the mixed operations. In mixed operation, we should perform division,

multiplication, addition and substraction in order and get a single and simple
answer. The process is called Simpliϐication.

Now, let's learn more about the order of operations from the following
examples.

Order of addition and substraction

Example 1 : Simplify a) 17 + 13 – 9 b) 24 – 15 + 10 c) 27 – 12 – 8

Solution

a) 17 + 13 – 9 = 30 – 9 b) 24 – 15 + 10 = 9 + 10 c) 27 – 12 – 8 = 15 – 8

= 21 = 19 =7

Order of multiplication, addition and substraction

Example 2 : Simplify a) 6 × 9 + 7 b) 8 + 5 × 8 c) 36 - 4 × 7

Solution

a) 6 × 9 + 7 = 54 + 7 6 × 9 + 7 = 6 × 16 = 96,
= 61 Which is the wrong order !

33 vedanta Excel in Mathematics - Book 5

Fundamental Operations

b) 8 + 5 × 8 = 8 + 40 8 + 5 × 8 = 13 × 8 = 104,
= 48 Which is the wrong order!

c) 36 - 4 × 7 = 36 - 28 36 - 4 × 7 = 32 × 7 = 224,
=8 Which is the wrong order!

Order of division and multiplication b) 5 × 30 ÷ 6
Example 3 : Simplify a) 30 ÷ 6 × 5
Solution 30 ÷ 6 × 5 = 30 ÷ 30 = 1,
Which is the wrong order!
a) 30 ÷ 6 × 5 = 5 × 5
= 25

b) 5 × 30 ÷ 6 = 5 × 5 Alternative process
= 25 5 × 30 ÷ 6 = 150 ÷ 6

= 25

Example 4 : Simplify 4 × 15 ÷ 5 + 20 - 14

Solution Another process Shorter process

4 × 15 ÷ 5 + 20 – 14 4 × 15 ÷ 5 + 20 – 14 4 × 15 ÷ 5 + 20 – 14

= 4 × 3 + 20 – 14 = 60 ÷ 5 + 20 – 14 = 4×3+6

= 12 + 20 – 14 = 12 + 20 – 14 = 12 + 6

= 32 – 14 = 32 – 14 = 18

= 18 = 18

In this way, while simplifying mixed operations, at irst we should perform
division (D), then multiplication (M), addition (A) and Subtraction (S) in
order. We can remember this order as DMAS rule.

2.7 Use of brackets in Simplification

Let's study the given examples carefully and learn to use brackets in
simpli ications.
Example 5 : Find the product of 4 and the sum of 7 and 8 .
Solution

vedanta Excel in Mathematics - Book 5 34

Fundamental Operations

Here, the mathematical expression is 4 × ( 7 + 8 ) but not 4 × 7 + 8.

So, 4 × ( 7 + 8 ) = 4 × 15 But, 4 × 7 + 8 = 28 + 8 = 36 and it is
= 60 the wrong answer for the given problem.

In the above problem, at irst we need to ind the sum of 7 and 8. Then the
sum is multiplied by 4. So, to ind the sum at irst we enclose 7 + 8 in the
brackets ( ).

Example 6: Find 5 times the difference between 18 and 12 is divided
by 10.

Solution

Here, the mathematical expression is {5 × (18 – 12)} ÷ 10

{ 5 × (18 – 12)} ÷ 10 = {5 × 6} ÷ 10

= 30 ÷ 10 = 3

In this case, we write the difference between 18 and 12 inside the small
brackets ( ). Then we write 5 times the difference between 18 and 12 inside
the middle or curly brackets { }.

Example 7: Simplify 17 + 15 × {(7 – 4) + 5} ÷ 12
Solution

17 + 15 × {(7 – 4) + 5} ÷ 12

= 17 + 15 × {3 + 5} ÷ 12 At first, perform the operation inside ( ).

= 17 + 15 × 8 ÷ 12 Then, perform the operation inside { }.

= 17 + 5 × 82 Then perform the division, multiplication and
123 addition in order.
15

= 17 + 5 × 2 = 17 + 10 = 27

Example 8: On a day, there were 25 students present in class ϔive. 15 of
them were girls and the rest were boys. If only 3 boys were
absent on that day, ϔind the number of boys in class ϔive.

Solution

Here the mathematical expression is (25 – 15) + 3

(25 – 15) + 3 = 10 + 3

= 13

Hence, there are 13 boys in class ive.

35 vedanta Excel in Mathematics - Book 5

Fundamental Operations

EXERCISE 2.3
Section A - Class work
1. Let's simplify mentally. Tell and write the answer quickly.

a) 5 is subtracted from the sum of 4 and 9.

b) 7 is added to the product of 3 and 6 .

c) The quotient of 20 divided by 5 is subtracted from 16.

d) The product of 9 and 2 is divided by 6 .

e) The product of 10 and the difference of 8 and 5.
2. Let's simplify mentally. Tell and write the answer quickly.

a)12 – 4 + 7 = b) 15 – 5 – 4 =

c) 6 × 8 + 2 = d) 9 + 3 × 7 =

e) 45 ÷ 5 × 4 = f) 8 × 56 ÷ 7 =

g) 10 × (9 – 4)= h) 72 ÷ (5 + 3) =

3. Let's ϐill in the blanks of each crossword puzzle with the correct numbers.

48 ÷ =6 – – 50 = 110
×2 × + = 90

= 60

==

+ 60 =

4. Let's insert the appropriate sign ( +, – , × or ÷) in the blank spaces to get
the given answer.

a) 7 4 2 = 30 b) 10 5 9 = 18

c) 21 6 4 = 11 d) 8 24 4 = 14

e) 12 40 8= 7 f) 18 3 5 = 3
36
vedanta Excel in Mathematics - Book 5

Fundamental Operations

5. Let's enclose the operation which is to be performed at ϐirst using small
brackets to get the given answer.

a) 3 × 6 + 2 = 24 b) 17 – 5 ÷ 4 = 3

c) 30 ÷ 3 × 2 = 5 d) 16 – 7 × 5 = 45

Section B

6. Let's simplify these mixed operations.

a) 7 + 9 – 3 × 5 b) 6 × 4 – 4 × 5

c) 15 – 4 + 7 × 18 ÷ 9 d) 32 ÷ 4 × 3 + 10 – 8

e) 24 + 3 × 15 ÷5 × 2 – 10 f) 6 × 27 ÷ 9 – 7 × 2 + 2

g) 20 – (9 + 4) h) (20 – 9) + 4

i) 7 × (5 + 3) j) 30 ÷ (4 × 4 – 6)

k) (6 + 3) × (6 – 3) l) (50 – 5 × 4) ÷ (2 + 36 ÷ 9)

7. Let's simplify these mixed operations.

a) 27 – { 9 + (12 – 5)} ÷ 4 b) 63 ÷ {(5 + 3) × 4 – 23}

c) 5 × {14 – (17 + 7) ÷ 6} d) 8 + 12 × {(9 – 7) + 3} ÷ 15

e) 18 – 10 × {4 + (12 – 7)} ÷ 18 f) 25 – 11 +{18 – (45 ÷ 9 × 2} ÷ 4

g) 24 ÷ 3 × { 6 + 2 × (14 – 4) ÷ 5} – 10

h) (4 × 9) ÷ {10 + 22 ÷ (5 + 2 × 3)} – 3

8. Let's rewrite these operations using ( ) and { } brackets at the appropriate
places. Then simplify and get the given answer.

a) 4 × 8 + 7 ÷ 10 = 6 {4 × (8 + 7) } ÷ 10

b) 5 × 6 + 3 ÷ 5 = 9 c) 4 + 12 – 5 × 3 = 25

d) 18 ÷ 2 × 4 + 5 = 1 e) 8 × 27 – 6 ÷ 7 = 24

9. Let's make mathematical expressions and simplify.

a) The product of 5 and the sum of 4 and 6.

b) 6 times the difference of 12 and 5.

c) The product of 9 and 4 is divided by 6 .

d) 3 times the sum of 3 and 5 is divided by 4 .

e) The sum of 4 and 5 is subtracted form one-third of the difference of 40 and 4.

f) The difference of 8 and 3 is multiplied by one-quarter of the sum of 7 and 5.
37 vedanta Excel in Mathematics - Book 5

Fundamental Operations

Let's simplify and solve these problems by making mathematical
expressions.

10. a) Sahayata had 10 colour pencils. She bought a few more colour pencils for
Rs 50 at the rate of Rs 10 each. How many colour pencils does she have
now ? (Hint: 10 + 50 ÷ 10)

b) After buying 6 sweets at Rs 12 each, Dakshesh had Rs 28 left. How much
money did he have at irst ?

c) On Sunday, there were 28 students present in class ive. 18 of them were
boys and the rest were girls. If only 4 girls were absent on that day, ind
the number of girls in class ive.

d) The cost of 1 kg or rice is Rs 90 and 1 kg of sugar is Rs 75. Find the total
11. a) cost of 3 kg of rice and 4 kg of sugar.

A sick person takes 10 ml of medicine three times a day. How much
medicine does she/he take in a week ?

(Hint : 7 × (10 ml + 10 ml + 10 ml)

b) The distance between Sunayans's house and her school is 6 km. How many
kilometres does she travel in 6 days ?

c) Mr. Lama earns Rs 5,600 in a week. He spends Rs 300 for food and Rs 50
for transportation everyday. How much money does he save in a week ?

d) Teacher divided 60 fruits equally between 16 girls and 14 boys of class
ive. How many fruits will each student get ?

It's your time - Project work !

12. a) Let's make any four your own mixed expressions using all four signs
( +, – , ×, ÷ ) in each expression . Simplify them and get the correct answer.

b) Let's rewrite these simpli ications and ind the mistakes. Then complete
the simpli ication in the correct way. You can display your work in your
school's wall-magazine.

24 – 8 – 7 6+2×7 15 - 4 × 3 48 ÷ 4 × 2
= 24 – 1 = 8×7 = 11 × 3 = 48 ÷ 8
= 23 = 56 = 33 =6

vedanta Excel in Mathematics - Book 5 38

Unit Properties of Whole Numbers

3

3.1 Various types of numbers - Looking back

Classwork - Exercise

Let's tell and write the correct answer of these questions.
1. a) Natural numbers less than 10

b) Whole numbers less than 10
c) Odd numbers less than 10
d) Even numbers less than 10
e) Prime numbers less than 10
f) Composite numbers less than 10
2. a) What is the least natural number ?
b) What is the greatest natural number ?
c) What is the least whole number ?
d) What is the greatest whole number ?

Natural numbers
We count the number of objects by 1, 2, 3, 4, 5,… Therefore, these are the counting
numbers. The counting numbers are also called the natural numbers.
1 is the least natural number and the greatest natural number is in inite.

Whole numbers
Suppose, you have 2 sweets. You eat 1 sweet and you give 1 sweet to your friend.
Now, how many sweets are left with you?
How much is left when 3 is subtracted from 3 ?
The answer of each of these questions is 'None'.
In counting, none means zero (0). Therefore, zero also counts the number of
objects. However, it counts 'there is no any number of object.'
In this way, counting numbers include zero (0) also. The set of natural numbers
including zero (0) are call the whole numbers. 0, 1, 2, 3, 4, 5,… are the whole
numbers. 0 is the least whole number and the greatest whole number is in inite.

39 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

Odd and even numbers 1 pencil (unpaired)
Can you make a pair with 1 pencil ?

Can you make a pair with 2 pencils ? 2 pencils (paired)

1, 3, 5, 7, 9, … are unpaired numbers. They are called Odd numbers.

2,4,6,8,10, … are paired numbers. They are called even numbers.

Again, let's divide some of these odd and even numbers by 2.

2 ÷ 2 = 1 quotient and 0 remainder. 2 is an even number. When an even number
3 ÷ 2 = 1 quotient and 1 remainder. 3 is an odd number. is divided by 2, the
4 ÷ 2 = 2 quotient and 0 remainder. 4 is an even number. remainder is always 0.
5 ÷ 2 = 2 quotient and 1 remainder. 5 is an odd number.
When an odd number
is divided by 2, the
remainder is always 1.

Can you investigate the idea to identify the given natural number is an odd or an
even number ? Discuss with your friends.

Now, let's take some bigger numbers and see the digits at ones place of these
numbers.

In 81, 93, 125, 4687, …the digits at ones place are odd numbers.

Therefore, 81, 93, 125, 4687, … are odd numbers.
In 90, 132, 576, 3714, … the digits at ones place are zero or even numbers.

Therefore, 90, 132, 576, 3714, … are even numbers.

Prime and composite numbers

Let's study the given illustrations carefully and investigate the idea about prime
and composite numbers.

2 ÷ 1 = 2 2 is exactly divisible by 1 or by the number itself.

2 ÷ 2 = 1 Therefore, 2 is a prime number.

3 ÷ 1 = 3 and 3 ÷ 3 = 1 o 3 is exactly divisible by 1 or by the number itself.
Therefore, 3 is a prime number. Similarly, 5, 7, 11, 13, … are also the prime numbers.

On the other hand,
4 ÷ 1 = 4, 4 ÷ 2 = 2 and 4 ÷ 4 = 1 o 4 is exactly divisible by not only 1 and by itself.
It is exactly divisible by 2 also. So, 4 is a composite number.

vedanta Excel in Mathematics - Book 5 40

Properties of Whole Numbers

6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2 and 6 ÷ 6 = 1 o 6 is exactly divisible by not only
1 and by itself. It is exactly divisible by 2 and 3 also. Therefore, 6 is a composite
number.
Similarly, 8, 9, 10, 12, 14, 15 ,… are also the composite numbers.
Remember, 1 is neither a prime number nor a composite number.

EXERCISE 3.1
Section A - Class work
1. Let's and tell and write the correct answer in the blank spaces.

a) What are the least natural and whole numbers ?

b) What are the greatest natural and whole numbers ?

c) Are all natural numbers whole numbers ?

d) Are all whole numbers natural numbers ?

e) Is the difference of 7 and 7 a natural number ?

f) Is the difference of 7 and 7 a whole number ?

g) Is 1 a prime, composite or none of these types of number?

h) Odd numbers between 20 and 30 are

i) Even numbers between 30 and 40 are

j) Prime numbers less than 20 are

k) Composite numbers between 10 and 20 are
Section B
2. Answer the following questions.

a) In what way a set of natural numbers is different from the set of whole
numbers ?

b) Can you make complete pairs of marbles form 15 marbles? What type of
number is 15, an odd or an even ?

c) Can you make complete pairs of pencils from 18 pencils ? What type of
number is 18, an odd or an even ?

d) How do you say 7 is a prime number ?
e) How do you say 9 is a composite number ?
f) Why is 1 called neither a prime nor a composite number ? Discuss with your

teacher .
41 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

3. Let's copy and complete the pattern of odd and even numbers.
a) 151, 153, , , , , 163.
b) 180, , 184 , , , 192
c) 319, , 323, , , , 331

d) 596, , , , , 606

4. It's your investigation !

a) Is the sum of any two whole numbers always a whole number ? Answer it
with at least 3 examples.

b) Is the sum of any two natural numbers always a natural number? Answer it
with at least 3 examples.

c) Is the sum of any two odd numbers always an odd or an even number ?
Justify your answer with at least 5 examples.

d) Is the sum of an even and an odd numbers always an odd or an even number?
Justify your answer with at least 5 examples.

e) Is the product of any two odd numbers always an odd or an even number ?
Justify your answer with 5 examples.

f) Is the product of an even and an odd numbers always an odd or an even
number ? Justify your answer with 5 examples.

5. It's your time - Project work !

a) Let's draw (i) 9 (ii) 11 (iii) 12 (iv) 15 (v) 20 circles in a chart paper. Colour
each pair of circles and identify odd and even numbers.

b) Let's write the natural numbers 2 3 4 5 6 7 8 9 10
from 2 to 100 as shown in the 11 12 13 14 15 16 17 18 19 20
given table. 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
(i) Circle the number 2 and cross 41 42 43 44 45 46 47 48 49 50
out all the multiples of 2 . 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
(ii) Circle 3 and cross out all the 71 72 73 74 75 76 77 78 79 80
multiples of 3. 81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
(iii) Circle 5, 7 and other remaining
numbers which are not crossed
and continue the process.

(iv) Now, list the numbers which are
circled. What type of numbers
are in the circles?

vedanta Excel in Mathematics - Book 5 42

Properties of Whole Numbers

(v) List the numbers which are crossed out. What type of numbers are
crossed out ?

In mathematics, it is a simple ancient rule for inding all prime numbers up
to any given limit. It was invented by a Greek Mathematician 'Eratosthenes'
and it is well known as 'Sieve of Eratosthenes.'

3.2 Test of divisibility

When a dividend is divisible by a divisor with no remainder, the dividend is
called exactly divisible by the divisor. For example:

14÷2 = 7 quotient with 0 remainder. 14 is exactly divisible by 2.

15÷2 = 7 quotient with 1 remainder. 15 is not exactly divisible by 2.

Now, let's learn a few rules about the test of divisibility.

Exactly Rules of divisibility test
divisible by

2 The digit at ones place of any number is 0 or even number. 70,
152, 690, 834, 1996, 4758, … are exactly divisible by 2.

The sum of the digits of any number is exactly divisible by 3.

3 In 372, 3+7+2 = 12 and 12 is exactly divisible by 3. So, 372 is

exactly divisible by 3.

The number formed by last two digits of any even number

4 is exactly divisible by 4. So, 96, 208, 512, 1372, … are exactly

divisible by 4.

5 The digit at ones place is 0 or 5. So, 90, 140, 365, 725, 4135,
9800, … are exactly divisible by 5.

6 Any even number exactly divisible by 3 are also exactly divisible
by 6. So, 96, 210, 924, 5328, … are exactly divisible by 6.

The number formed by last three digits of any even number is

8 exactly divisible by 8. So, 152, 640, 2344, … are exactly divisible

by 8.

The sum of the digits of any number is exactly divisible by 9.

9 In 594, 5+9+4 = 18 and 18 is exactly divisible by 9. So, 594 is

exactly divisible by 9.

10 The digit at ones place is 0. So, 80, 430, 1650, 7960, … are
exactly divisible by 10.

43 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

3.3 Factors and multiples
Let's study the given examples and investigate the ideas of factors and multiples.

In how many ways can you In how many ways can you
make 18 by multiplication ? make 24 by multiplication ?

1 × 18 2×9 1 × 24 2 × 12

18 24

3×6 3×8 4×6

Therefore, 1, 2, 3, 6, 9 and 18 are Therefore, 1, 2, 3, 4, 6, 8, 12 and
the factors of 18. 24 are the factor of 24.

18 is the multiple of 1, 2, 3, 6, 9 24 is the multiple of 1, 2, 3, 4, 6,
and 18. 8, 12 and 24

The factors of a number can exactly divide the number.

18÷1 = 18, 18÷2 = 9, 18÷3 = 6, 18÷6 = 3, 18÷9 = 2 and 18÷18 = 1

24÷1 = 24, 24÷2 = 12, 24÷3 = 8, 24÷4 = 6, 24÷6 = 4, 24÷8 = 3
24÷12 = 2 and 24÷24 = 1

3.4 Prime factors and process of finding prime factors
Let's take a number 30. All possible factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
Among these factors, 2, 3 and 5 are the prime factors because 2, 3 and 5 are

the prime numbers.

Now, let's investigate the rule of inding prime factors of the given numbers
from the following examples.

24 ÷ 2 = 12, 12 ÷ 2 = 6 and 6 ÷ 2 = 3 Factor Tree
24
We can show this successive division
in the following way.

2 24 2 × 12

2 12 2× 2× 6
26 2×2×2×3
24 = 2 × 2 × 2 × 3
3
So, 24 = 2 × 2 × 2 × 3 44

vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

Thus, to ind the prime factors of a given number, we should start to divide
the number by the lowest prime number. We should continue division till the
quotient becomes a prime number.

3.5 Process of finding multiples

Classwork - Exercise

1. Let's multiply and write the products in the blank spaces.

a) 3 × 1 = , 3×2= , 3×3= 3, 6, 9, 12, 15 are the
3×4= , 3×5= , ϔirst ϔive multiples of 3.

b) 4 × 1 = , 4× 2= , 4×3= 4, 8, 12, 16, 20, 24 are the
ϔirst six multiples of 4
4×4= , 4×5= , 4×6=

2. a) What are the irst ive multiples of 2 ?

b) What are the irst ive multiples of 5 ?

In this way, to ind the multiples of a given number, we should multiply the
number by any natural number.

EXERCISE 3.2
Section A - Class work

1. Let's circle the numbers which are exactly divisible by the given numbers.

a) by 2 o 70 94 123 432 900 1755 2316 3978

b) by 3 o 85 90 113 237 801 1680 2003 6516

c) by 4 o 64 94 128 350 716 1538 3940 9100

d) by 5 o 75 57 156 205 670 2700 4508 6015

e) by 6 o 86 90 144 352 594 3036 5600 8100

f) by 8 o 88 98 112 248 348 2400 3124 7640

g) by 9 o 79 99 163 351 693 2990 4950 9990

h) by 10 o 70 95 130 345 900 1050 3005 5100
45 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

2. Let's use the rule of divisibility test then tell and write 'True' or 'False'.

a) Is 183 exactly divisible by 3 ? ++ = True

b) Is 267 exactly divisible by 9 ? ++ =

c) Is 359 exactly divisible by 3 ? + +=

d) Is 684 exactly divisible by 9 ? ++ =

3. First ϐind the multiples. Then, tell and write the all possible factors of the
multiple. Circle and list the prime factors.

a) 1 × 6 = , 2 × 3 =

All possible factors of 6 are , , and

The prime factors of 6 are and

b) 1 × 10 = , 2 × 5 =

All possible factors of 10 are , , and

The prime factors of 10 are and

c) 1 × 12 = , 2 × 6 = , 3 × 4 =

All possible factors of 12 are ,,, , and

The prime factors of 12 are and

d) 1 × 15 = , 3 × 5 =

All possible factors of 15 are , , and

The prime factors of 15 are and

4 Let's divide the given numbers by the prime numbers till the quotient
becomes a prime number.

a) 2 20 b) 2 24 c) 2 28
2 2 2

2

20 = × × 24 = ×× × 28 = × ×
46
vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

5 Let's tell and write the correct numbers in the empty circles. Then complete
the 'Factor Tree'.

a) 16 b) 18 c) 30

2× 2× 2×

×2× × ×3 ×3×

×××

16 = × × × 18 = × × 30 = × ×

6. Let's tell and write these numbers as the product of their prime factors.

a) 4 = × b) 6 = × c) 8 = × ×

d) 9 = × e) 10 = × f) 12 = × ×

Section B
7. Let's write these numbers.

a) Three 3 -digit numbers exactly divisible by 3 and 9.
b) Three 3 -digit numbers exactly divisible by 4.
c) Three 3 -digit numbers exactly divisible by 6.
d) Three 4 -digit numbers exactly divisible by 8.

8. Let's write the answer of the questions.
a) Why is 5 called a factor of 10 ?
b) Is 6 a factor of 20 ? Why ?
c) Why is 12 called a multiple of 4 ?
d) Is 30 a multiple of 8 ? Why ?
e) Why are 2 and 7 called prime factors of 14.
f) Is 9 a prime factor of 18 ? Why ?

9. Let's ϐind the ϐirst 5 multiples of the following numbers.
a) 5 b) 7 c) 8 d) 9 e) 10 f) 12
47 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

10. Let's write all possible factors of these numbers. Then list out the prime
factors.

a) 4 b) 6 c) 8 d) 9 e) 10 f) 12

g) 14 h) 15 i) 16 j) 18 k) 20 l) 36

11. Let's ϐind the prime factors of these numbers and write them as the
product of their prime factors.

a) 2 18 b) 10 c) 12 d) 14 e) 15 f) 16
3 g) 20 h) 24 i) 25 j) 27 k) 28
9 l) 30 m) 32 n) 36 o) 40 p) 42
3

18 = 2 × 3 × 3 q) 44 r) 45 s) 48 t) 50 u) 54

12. Let's draw 'factor-tree' and show these numbers as the product of their
prime factors.

a) 8 b) 12 c) 16 d) 18 e)20 f) 24

It's your time- Project work !

13. a) Let's write prime numbers less than 10. List all possible factors of each
prime number.

b) Again, write prime numbers between 10 and 20. List all possible factors
of each prime number.

c) What conclusion can you make from the above activities ?

d) Let's write composite numbers less than 10. List all possible factors of
each composite number. What is the minimum number of factors of these
composite numbers.

e) What are the possible factors of 1 ? Discuss, why 1 is neither a prime nor
a composite number.

3.6 Common factors and common multiples

Classwork - Exercise

1. Let's tell and write the possible factors of each pair of numbers. Then
circle the common factors.

a) Possible factors of 4 48
Possible factors of 6

b) Possible factors of 6
Possible factors of 9

vedanta Excel in Mathematics - Book 5