Prime Mathematics 4

Prime

Mathematics
Series

4

Raj Kumar Mathema

Dirgha Raj Mishra Bhakta Bahadur Bholan

Uma Raj Acharya Yam Bahadur Poudel

Naryan Prasad Shrestha Bindu Kumar Shrestha

Prime

Mathematics
Series

4

Approved by
Government of Nepal, Ministry of Education Curriculum
Development Centre Sanothimi, Bhaktapur as and additional

learning materials.

Raj Kumar Mathema Authors
Bhakta Bahadur Bholan Dirgha Raj Mishra
Yam Bahadur Poudel Uma Raj Acharya
Bindu Kumar Shrestha Naryan Prasad Shrestha

Editors
Anil Kumar Jha
Dhurba Narayan Chaudhary
Hari Krishna Shrestha

Language Editor
Mrs. Tara Pradhan

Pragya Books & Distributors Pvt. Ltd.

Printing history
First Edition 2074 B.S.
Second Edition 2077 B.S.

Authors
Raj Kumar Mathema
Dirgha Raj Mishra
Bhakta Bahadur Bholan
Uma Raj Acharya
Yam Bahadur Poudel
Naryan Prasad Shrestha
Bindu Kumar Shrestha

Editors
Anil Kumar Jha
Dhurba Narayan Chaudhary
Hari Krishna Shrestha

Layout and design
Rabi Man Shrestha

© Publisher
All rights reserved. No part of this book, or designs and illustrations here within, may be
reproduced or transmitted in any form by any means without prior written permission.
ISBN : 978-9937-0-2220-0
Printed in Nepal

Published by
Pragya Books & Distributors Pvt. Ltd.

Lalitpur, Nepal
Tel : 5200575
email : [email protected]

Preface

Prime Mathematics Series is a distinctly outstanding mathematics series
designed in compliance with Curriculum Development Centre (CDC) to meet
international standards. The innovative, lucid and logical arrangement of the
content makes each book in the series coherent. The presentation of ideas
in each volume makes the series not only unique, but also a pioneer in the
evolution of mathematics teaching.

The subject matter is set in an easy and child-friendly structure so that
students will discover learning mathematics a fun thing to do.A lot of research,
experimentation and careful gradation have gone into the making of the series
to ensure that the selection and presentation is systematic, innovative and both
horizontally and vertically integrated.

Prime Mathematics Series is based on child-centered teaching
and learning methodologies, so the teachers will find teaching this series
equally enjoyable. We are optimistic that this series shall bridge the existing
inconsistencies between the cognitive capacity of children and the course matter.

We owe an immense debt of gratitude to the publishers for their creative,
thoughtful and inspirational support in bringing about the series. Similarly, we
would like to acknowledge the tremendous support of teachers, educationists
and well-wishers for their contribution, assistance and encouragement in making
this series a success.

We hope the series will be another milestone in the advancement of teaching
and learning mathematics in Nepal. We solicit feedback and suggestions from
teachers, students and guardians alike so that we can refine and improvise the
series in the future editions.

Our team would like to express our special thanks to Mr. Nara Bahadur
Gurung, Mr. Ram Narayan Shah, Mr.Tulsi Kharel, Mr. Mani Ram Khabas, Mr. Umesh
Acharya, Mr., J. Phuldel, Mr. Kamal Raj Tripathee, Mr. Rudra Prasad Pokharel, Mr.
Uttam Prasad Panta, Mr. L.N. Upadhyaya, Mr. Shakti Prasad Acharya, Mr. Upendra
Subedi, Mr. Kul Narayan Chaudhary, Mr. Bishonath Lamichhane, Mr. Harilal
Lamichhane, Mr. Govinda Paudel, Mr. Krishna Aryal, Mr. Nim Bhujel, Mr. Santosh
Simkhada, Mr. Pashupati Upadhyaya, Mr. Dipak Adhikari, Mr. Mukti Adhikari, Mr.
Dipendra Upreti, Mr. Dipak Khatiwada, Mr. Narayan Nepal, Mr. Raj Kumar Dahal,
Mr. Bhim Raj Kandel, Mr. Prem Giri,Mr. Iswor Khanal, Mr. Balram Ghimire, Mr. Om
Kumar Chhetri, Mr. Ram Hari Bhandar, Mr. Krishna Kandel, Mr. Madhav Atreya,
Deep Raj Nigam Unai, Sangita Thapa, Shiva Devkota, Harihar Adhikari, Chandra
Dev Tiwari, Chura Gurung, Sagar Dhakal, Baikuntha Marahatha, Subash Bidari
Raghu Kandel, Sudip Poudel, Roshan Sapkota Sujan Dhungana, Tara Bahadur
Bhandari and Jiwan K.C. for their Painstaking effort in peer reviewing of this
book.

Contents

Unit Topic Page

1. Geometry 1

2. Number System 23

3. Basic Operation in Mathematics 53

4. Measurements 89

5. Mensuration 119

6. Fraction, Decimal, Percentage and
Unitary Method 133

7. Bill and Budget 173

8. Statistics 179

9. Set 189

10. Algebra 195

11. Model Question 223

U1nit Geometry

Estimated periods − 17

Objectives

At the end of this unit, the students will be able to:
• measure and draw angles between 0o to 180o using protractor.
• classify the angles on the basis of right angle.
• classify the triangles according to sides and angles.
• recognize the quadrilaterals.
• draw circles of different radii using compass.
• identify solids, their vertices, edges and faces.

Teaching Materials
Geometric box, chart paper, geo board, models of solids etc.

Activities
It is better to:
• perform the activities of measuring and drawing angles using protractor.
• show the angles of concrete objects to clarify the concept of types of angles.
• demonstrate the models of triangles to clarify their types.
• show the figures and models of quadrilaterals, circles.
• show the models of solids, show the nets of solids and demonstrate the process of
making their paper models.
• play a game with students group using geo-board about making of different shapes or
figures.

0 Inches 1 2 3 4 5 6 Geometrical instruments
For the geometrical constructions, we use different tools. Which we find in a
single box known as geometric box or instrument box.

1. Ruler or straight edge: It is a straight strip of plastic, wood or metal. It
is scaled in millimeters, centimeters and inches. It is used to measure
the length of line segments and to draw straight line segments.

mm 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15

centimeters

2. Compass (or Compasses): It is a V shaped expandable device with
needle in one leg and pencil in other leg. It is used to draw circles,
angles and also to measure the lengths of line segments.
Historical fact

Euclid deviced a compass which used to collapse
when both the legs were lifted from the paper. It
was known as collapsing compass.

3. Divider: It is a V shaped expandable device
with needles in both legs. It is used to
measure the lines and angles.

4. Protractor : It is generally semi−circular
made of plastics and marked from 0° to
180°. It is used to draw angles of required
measure and to measure the angles.

5. Set squares : There are two set squares.
These are triangular pieces of plastics or
wood with angles (90º, 60º, 30º) and (90º,
45º, 45º). They are used to draw parallel
and perpendicular lines and also to draw
particular angles 90º, 60º, 45º, 30º, 75º,
105º, 120º, 135º, 150º etc.
Lines

Drawing a line segment:

We use a ruler or a straightedge to draw a line segment.
Process: For example to draw a line segment AB = 7.4cm
• Place the ruler on the paper at a suitable location.
• Keep the ruler fixed and mark the point A against 0 (zero) of the scale

and B against 4 small divisions after 7 corresponding to 7.4 cm.
• Now, run the pencil along the edge of the ruler joining the points A and B.

2 Prime Mathematics Book − 4

AB

mm 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15

centimeters

Note: The readings on scales must be observed from vertically above. 4 56

Drawing a line segment equal in length to the given line0 Inches 1 2 3 4 5 6

segment: 0 Inches 1 2 3 4 5 6 B C DX
• Draw a line CX. A
• Put the needle of compass to

one end A of the given line and
extend it till the tip of the pencil
is on the other end B.
• Now, put the needle of the
compass on one end C of the line
and mark the point D on the line
by the tip of the pencil.
Thus, AB = CD.

Measuring the length of given line segment:
(i) Using ruler: To measure the length of the line segment PQ.

• Adjust the edge of the ruler with required scale on the line so that
one end of the line segment is on the 0 (zero) of the ruler.

• Read the reading of the ruler shown by the other end.

P 7.5cm Q

mm 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15

centimeters

Here, PQ = 7.5cm

(ii) Using divider and ruler:
To measure the length of the line
segment AB.
• Place the tip of one leg of the
divider at one end point ‘A’ of
the line.
• Extend the divider till its tip of A B mm 1 cm 2 3 4 5 6

other leg reaches the point B. centimeters

Here, length of AB = 1.9cm.

Prime Mathematics Book − 4 3

• Now remove the divider from the line and apply over the ruler so
that the tip of one leg is against 0 (zero) of the scale. Read the
scale shown by the tip of the other leg.

Exercise 1.1

1. Draw the line segments of the following lengths using ruler:
(a) AB = 4.8cm (b) PQ = 5.5cm (c) MN = 8.3cm
(d) XY = 6.2cm (e) RS = 7.9cm (f) UV = 4.4cm
(g) KL = 7cm (h) ST = 2.7cm (i) YZ = 9.1cm.

2. Measure the following line segments using ruler: D
Q
(a) (b)
A BC N

(c) E F (d) P

(e) S M
R (f)

3. Draw the line segments equal in lengths to the following line segments:
S

(a) P Q (b) R B
M
(d)
(c) NA

(e) P (f) X
N
Y

4. Find the length of the sides of the following figures and find their
perimetear:
(a) A
(b) A D (c) P Q (d) A E

B CB CS BD
RC
4 Prime Mathematics Book − 4

Angles A
B
In the figure, two rays OA and OB meet at the point O. O
Thus, an angle is formed at O. The angle is named as
∠AOB or ∠BOA. The point O is called vertex and the rays
OA and OB are called arms. The symbol ∠ represents
angle. To name an angle, capital letters are used with
vertex in the middle.

Measuring angles: B

B

O BA OA OA
OA and OB coincide OB is rotated a OB is rotated a
each other. No angle little and angle little more and a
is formed. We say ∠AOB is formed. bigger angle ∠AOB
∠AOB = 0º (zero is formed.
degree)

Measuring angle is the measure of the amount of rotation of OB about OA. We
use a protractor to measure the angle.

Process: 110 100 90 80 70 60 B
• Put the center of the protractor at the 120 A
130
vertex O. 50 60 70 80 90 100110 120 50
• Adjust one of the arms, say OA, on the 140 40
150 130 40
zero line or base line of the protactor. 140 30
• Read the angle shown by other arm OB.
160 30 150 20
Here, in the figure, ∠AOB = 30° 20 160
170 10 170 10

180 0 O 180 0

If circumference of a circle is divided into 360 90
equal parts. Then angle subtended by each 120 60
part at the centre is 1º. 150 30

180 0

210 330
240 300
270

Prime Mathematics Book − 4 5

Historical fact

The division of the circumference of a circle into 360 equal parts owe to the ancient
Babylonians. Their mathematical texts written in clay tablets from 2100 BC to 600
BC are found.

There existed a distance unit, a sort of mile about seven of our mile. Time taken to travel
12 such miles was equal to a complete day. Which is one complete revolution of the
sky. The mile time was subdivided into 30 equal parts. Thus, the idea, circumference
= 12 × 30 = 360 parts originated.

To draw an angle of given measure: C130120110100 90 80 70 6050

Example: To draw an angle 50º. 140 40
Process: 150 30
• Draw a line segment, say AB. 160 20

• Adjust a protractor with its vertex at one 170 10 B
end A and its zero line on AB. 180
A0
• Mark a point C corresponding to 50º of C

the protractor.
• Remove the protractor and join AC using scale.
• Thus, ∠BAC = 50º is drawn. A B

Types of angles B A
90º
Angles are classified on the basis of their measures
comparing with 90º and 180º. O
1. Right angle: An angle whose measure is exactly 90º

is called a right angle.
Here, ∠AOB = 90º, so it is a right angle. Arms
OA and OB are said to be perpendicular to each
other.

2. Straight angle: An angle whose measure is exactly
180º (two right angles) is called a straight angle. B O A
Here, ∠AOB = 180º. It is a straight angle.

3. Acute angle: An angle whose measure is less than 90º is called an acute angle.
∠AOB, ∠PQR and ∠XYZ are all less than 90º. These are acute angles.
A PX

O 30º B Q 50º R Y 80º Z

6 Prime Mathematics Book − 4

4. Obtuse angle: An angle which is greater than 90º but less than 180º is
called an obtuse angle. A
Here, ∠AOB and ∠PQR P 150º R
are obtuse angles. 120º B
Q
O

5. Reflex angle: An angle which is greater than A Acute angle
180º is called a reflex angle. B

Here, the indicated angle 300º is greater than
180º. It is a reflex angle. [Our protractor can
draw or measure angles measuring up to 180º. O 300º
To draw or measure a reflex angle we draw Reflex angle
or measure corresponding acute or obtuse Reflex angle

angle and remaining angle is its reflex angle. OA
If ∠AOB is a reflex angle, its corresponding ∠AOBObtuse angle
acute/obtuse angle = 360º − reflex ∠AOB.]

Example 1: Draw angle 210º. B

Here, 210º is a reflex angle. B 150º
∴ Its corresponding obtuse angle is O
360º − 210º = 150º 210º A

So, draw angle ∠AOB = 150º, then reflex
∠AOB = 210º.

6. Complete angle (Point angle) : An angle A B
whose measure is exactly 360º is called
complete angle. 360º

O

Exercise 1.2
1 State the size of angle that are shown on the protractor:

a)

130 110 100 90 80 70 B
120 60
A ∠ AOB = ..........
140 50 60 70 80 90 100110 120 50
150 40 Prime Mathematics Book − 4 7
130 40
140 30

160 30 150 20
20 160
170 10 170 10

180 0 O 180 0

b) X c) W

Y X

∠ XOY = .......... ∠ XOW = ..........
d) M
e)

P

NR

∠ MON = .......... ∠ POR = ..........

2. Name the angles. Write its vertex and arms:

(a) A (b) x (c) P
R
BC Y

ZQ

................................. ................................. .................................

(d) J (e) (f) Y

KL M LK XW

................................. ................................. .................................

3. Name the following angles. Write its measure and type:

(a) C (b)

BA R QP
.................................
.................................

8 Prime Mathematics Book − 4

(d) (e)
L B

MN DC

................................. .................................
4. Draw the following angles using a protractor in your exercise book.

(a) 30º (b) 70º (c) 90º (d) 135º (e) 45º

(f) 80º (g) 100º (h) 120º (i) 150º (j) 165º

5. Classify the following angles as right angle, acute angle, obtuse
angle, straight angle or reflex angle:

(a) 120º (b) 90º (c) 180º (d) 270º

(e) 70º (f) 100º (g) 90º (h) 130º

(i) 200º (j) 89º (k) 150º (i) 225º

Right angle: ........................................ Straight angle: ....................................

Acute angle: ....................................... Reflex angle: .......................................

Obtuse angle: .....................................

6. Measure the following reflex angles:

(a) B A (b) C B (c) D C
E
D
C ∠CDE = …………………..
∠BCD = ………………….. ∴ Reflex ∠CDE
∠ABC = …………………..
= 360º−∠CDE
∴ Reflex ∠ABC ∴ Reflex ∠BCD = 360º − ……………..
= ……………..
= 360º−∠ABC = 360º−∠BCD

= 360º − ………….. = 360º − …………..

= …………….. = ……………..

Prime Mathematics Book − 4 9

(d) (e) Q (f) M
D P R

E F

∠DEF = ………………….. ∠PQR = ………………….. LN
∴ Reflex ∠DEF ∴ Reflex ∠PQR
= 360º−∠PQR ∠LMN = …………………..
= 360º−∠DEF = 360º − …………….
= 360º − ……………. = …………….. ∴ Reflex ∠LMN
= ……………..
= 360º−∠LMN

= 360º − ……………..

= ……………..

7. Measure the interior angles of the following triangles:

(a) A (b) P

C Q R
B
∠BAC = ………………… ∠PQR = …………………
∠ABC = ………………. ∠PRQ = ……………….
∠ACB = ..……………….. ∠QPR = ..………………..

(c) M X
(d)

Y Z
O

N ∠YXZ = …………………
∠MNO = …………………

∠NMO = ………………. ∠XYZ = ……………….

∠MON = ..……………….. ∠XZY = ..………………..

8. Define: (b) obtuse angle
(a) acute angle (d) straight angle
(c) right angle

(e) reflex angle

10 Prime Mathematics Book − 4

Plane figures

Any position of a plane surface bounded by one or more lines (straight or
curve) is called a plane figure. If a plane figure is bounded by straight lines,
it is called a rectilinear figures.

Triangle:
A triangle is a plane figure bounded by three straight lines.
• The given figure is a triangle. It has three corners or A

vertices which are denoted by capital letters A, B and
C. AB, BC and AC are three sides and ∠BAC, ∠ABC and
∠ACB are three angles. BC

• A triangle is named according to the letters representing the vertices.

AX

BC YZ The symbol delta
It is a triangle It is a triangle '∆' represents
ABC or ∆ABC XYZ or ∆XYZ triangle.

In a triangle ABC, the letters A, B, C denote the angles A

and the letters a, b, c denote the lengths of the sides c b
opposite to A, B and C respectively.
Sum of the angles of a triangle is 180º.
Ba C

Historical fact

Leonhard Euler (1707 – 1783, Switzerland) used a, b, c denoting

lengths of sides opposite to the vertices A, B and C respectively

and s for semi−perimeter of triangle
a + b + c
ABC. i.e. s = 2 , where perimeter of a triangle is a + b + c.

Types of triangles:
Parts of a triangle are three vertices, three angles and three sides. Triangles
are classified on the basis of sides and angles.
A
Types of triangles according to the sides:
1. Equilateral triangles: If the lengths of all the sides
are equal, it is called an equilateral triangle. In the
given triangle ABC, sides AB = BC = AC. B C

It is an equilateral triangle.

Note: An equilateral triangle is also equiangular. (which equals to 60o each)

Prime Mathematics Book − 4 11

2. Isosceles triangle : If any two sides of triangle are equal, A
it is called an isosceles triangle.
Here, in triangle ABC, AB = AC. So, it is an isosceles triangle.
∠ABC and ∠ACB are called base angles. Base angles of an
isosceles triangle are equal i.e. ∠ABC = ∠ACB B C

Note:

In an isosceles triangle, the term vertex is applied to the point at which the equal sides intersect. The
angle at the vertex is vertical angle and side opposite to it is called base and the angles opposite to the
equal sides are base angles.

3. Scalene triangle : If the sides of a triangle are A

all unequal, it is called a scalene triangle.
Here, in Δ ABC, AB ≠ BC ≠ AC. So, it is a scalene
triangle. C
B

Types of triangles according to the angles: A C
B
1. Right angled triangle : If one of the angles of a
triangle is a right angle (90º), it is called a right
angled triangle.
Here, in the figure, ∠ABC = 90º, so Δ ABC is a
right angled triangle.

2. Acute angled triangle : If all three angles of a A
triangle are acute, it is called an acute angled BC
triangle. Here, in the given figure, all the angles
are acute. It is an acute angled triangle.

3. Obtuse angled triangle : If one of the angles of A B C
a triangle is obtuse angle, it is called an obtuse
angled triangle.
Here, in the figure, ∠ABC is an obtuse angle, so it
is an obtuse angled triangle.

Note: Every triangle must have at least two acute angles.

12 Prime Mathematics Book − 4

Exercise 1.3

1. (a) Write the types of triangles according to the sides.
(b) Write the types of triangles according to the angles.

2. Write the types of the following triangles in which :
(a) one of the angles is 90º.
(b) all the sides are equal.
(c) two sides are equal.
(d) all the angles are less than 90º.
(e) one of the angles is greater than 90º but less than 180º.
(f) all the angles are equal.
(g) all the angles are unequal.

3. Write the types of the following triangles:

(a) A (b) B (c) C

D

B CC DE
E
(d) D (f) F
(e) 70º
45º 20º

90º 45º F F 60º 50º G G 120º 40º H

E

(g) G (h) H

60º 40º

H 60º 60º I I 70º 70º J

4. Estimate the types of the following triangles:
(a) (b) (c)

Prime Mathematics Book − 4 13

(d) (e) (f) 70O

50O 60O

5. Measure the lengths of sides of the following triangles and write
their types:

(a) A (b) M (c) W

BC NO X Y
AB = ...........
BC = ........... MN = ........... WX = ...........
CA = ...........
ΔABC is a ............... MO = ........... XY = ...........

NO = ........... WY = ...........

ΔMNO is a ............... ΔWXY is a ...............

6. Measure the angles of the following triangles and write their types:

(a) A (b) P (c) L

B C QR MN

∠A = .......... ∠P = .......... ∠L = ..........
∠M = ..........
∠B = .......... ∠Q = .......... ∠N = ..........
ΔLMN is an .........triangle
∠C = .......... ∠R = ..........

ΔABC is an .......... triangle ΔPQR is an .........triangle

7. What type of triangle is it ?

It is an triangle. A
BC
The vertex is

The vertical angle is

Base is

Base angles are and .

14 Prime Mathematics Book − 4

Quadrilaterals A D
Quadrilaterals are the closed figures bounded by four
straight lines. There are four sides, four vertices and B C
four angles in a quadrilateral. In the given quadrilateral A D
AB, BC, CD and AD are its four sides; A, B, C and D are
its four vertices and ∠BAD, ∠ABC, ∠BCD and ∠ADC are B C
its four angles. A B
Different types of quadrilaterals:
Square: D C
A square is a quadrilateral in which all the four sides are S
equal and all the four angles are right angles (90º). In
the given figure, ABCD is a square. R

Rectangle:
A rectangle is a quadrilateral having opposite sides equal
and all four angles 90º. In the given figure, ABCD is a
rectangle, where opposite sides AB = DC and AD = BC.
Angles ∠A = ∠B = ∠C = ∠D = 90º.

Rhombus: P
A rhombus is a quadrilateral having all four sides equal
and opposite angles equal. In the figure, PQRS is a Q
rhombus in which sides PQ = QR = RS = PS
Opposite angles ∠QPS = ∠QRS and ∠PQR = ∠PSR.

Parallelogram: K N
A parallelogram is a quadrilateral having opposite M
sides equal and opposite angles equal. The given
figure, KLMN is a parallelogram in which opposite
sides KL = MN and KN = LM and opposite angles L
∠LKN = ∠LMN and ∠KLM = ∠KNM

Circle

A circle is a closed figure bounded by a regular curved line. e radius
• The regular curved boundary is called circumference. cumferenc center
• A point inside the circle which is equidistant from
cir diameter
each point in the circumference is called the center.
• A straight line joining center and a point on the

circumference is called a radius. (Plural of radius is radii).
• A line drawn through the centre and joining any two

points on the circumference is called a diameter.
The given figure is a circle with center O. OA is a

Prime Mathematics Book − 4 15

radius and BC is a diameter. BA
The length of diameter is double the length of radius.
Here, BC = 2(OA) O
C

Constructing a circle of given radius :
Drawing outlines around circular objects, we get a circle but we use pencil
compasses to construct a circle of desired size.
Process:
Let’s draw a circle of radius 3cm.
• Draw a straight line segment OA = 3cm using a ruler.
• Extend your pencil compasses up to 3cm.
• Considering O as center, put the needle of the compass
at O and pencil at A. Rotate the compass so that the
needle is fixed at O and pencil runs around drawing O 3cm A

circumference.
Precaution: Use the compass carefully. Never play with it.

Exercise 1.4

1. Define the following with a figure:
(a) Quadrilateral (b) Square (c) Rectangle
(d) Parallelogram (e) Rhombus (d)

2. Name the following figures: (c)
(a) (b)

(e) (f) (g)

3. Measure the sides and angles of the following figures and determine
whether they are square, rectangle, rhombus or parallelogram:
(a) (b) (c) (d)

16 Prime Mathematics Book − 4

4. Draw circles of the following radii using pencil compasses:
(a) 2.6cm (b) 3.5cm (c) 4cm (d) 4.8cm
(e) 5cm (f) 5.2cm (g) 2.5cm (h) 2.9cm

5. Find the diameter of the circles having the following radii:
(a) 1.7cm (b) 2.2cm (c) 3.4cm (d) 4.3cm
(e) 5cm (f) 5.6cm (g) 2.5cm (h) 3.5cm

6. Find the radii of the circle whose diameters are:
(a) 3.2cm (b) 4cm (c) 4.6cm (d) 6cm
(e) 6.4cm (f) 7.6cm (g) 8.2cm (h) 9cm

7. Define the following parts of the circle:
(a) Center (b) Radius (c) Diameter (d) Circumference

Solid Figures
Plane figures like triangle, quadrilateral, circle etc are two dimensional
figures which have length and breadth but not height.
Solid figures are the figures of the solids having length, breadth and height.
Here we will discuss about some special geometrical solid shapes.

Cuboid: A cuboid consists 6 rectangular faces, 8
vertices (corners) and 12 edges, where the faces in-
tersect at right angles and opposite faces are exactly
same.

Top 6 7
Back
Side 5 8
Side
23

Front Bottom 1 4 12 edges
8 vertices
6 faces

Net of a cuboid: Suppose, we want to make a paper box 4cm × 3cm × 2cm

2
2

4 2 3
2 4

Prime Mathematics Book − 4 17

• Take a piece of chart paper or cardboard paper.
• Draw net of the cuboid of the desired dimensions on it.
• Cut outside the net along the doted lines and fold along the continuous

lines.
• Put glue along the shaded parts. Place the glued parts inside and join

the edges.
• Thus, a paper box 4cm × 3cm × 2cm is ready.
Cube: A cube is a solid having 6 plane square faces, 8 vertices and 12 edges
of equal length.

A cube 6 faces 8 corners 12 edge
top − bottom
front − back Circular base
left side − right side Lateral curved
surface
Cylinder: A cylinder is a solid having lateral
side of regularly curved surface and two circular Circular base
plane faces at two ends. It has no vertex. It
has two regularly curved edges.

Curved edge

Net of a cylinder:
To make a paper cylinder of diameter 7cm and height 10cm.
• Take a chart paper or cardboard paper.
• Draw the net of given
7cm

measures as in the figures. 22cm
• Cut along the lines
• Put glue on shaded part.
• Join the ends of rectangle. 10cm

• Adjust the circles on the base
and join with cello-tape.
Thus, a closed cylinder is
obtained.

18 Prime Mathematics Book − 4

Cone: Vertex

A cone is a solid having a vertex, a curved surface, a Curved surface

plane circular base and a curved edge. Plane circular
surface

Here in the figure, the sharpened part of the pencil Curved edge

is a cone.

Pyramid:
A pyramid is a solid having a polygonal base and plane triangular lateral face
with a common vertex.

Historical fact

The great pyramid of Gize (Egypt) was erected at around
2900 B.C. The structure is a tomb which was accomplished
by 100,000 labours in thirty years.

Exercise 1.5

1. Give at least two examples of the following solids:
(a) cuboid (b) cube (c) cylinder
(d) cone (e) sphere

2. Complete the following:

Name of solid figure No. of faces No. of edges No. of corners

Cuboid 6
Cube
Cylinder
Sphere
Cone
Square base pyramid

3. Write the shapes of the following objects:
A dice
A pipe
Ice−cream cone
A cupboard

Prime Mathematics Book − 4 19

Unit Revision Test

1. Draw line segments of the following lengths:
(a) 4.7cm (b) 4.3cm

2. Measure the given lines: N
(a) (b)
A BM

3. Write the name, measure and types of the following angles:

(a) A (b) P

BC QR

(c) L (d) BC D

MN

4. Draw the following angles:

(a) 70º (b) 110º

5. Write the types of triangles according to the sides:

6. Write the types of the following triangles:

(a) A (b) P

45º 70º

90º 45º C Q 55º 55º R

B

7. Define : (a) square (b) rectangle

8. Draw circles of radii: (i) 3.7cm (ii) 4cm

9. Give two examples of: (i) cuboid (ii) cylinder

10. Write the number of faces, edges and corners of
(a) cuboid (b) cone.

20 Prime Mathematics Book − 4

Answers:

Exercise: 1.1

1. Show to your teacher.
2. Show to your teacher.
3. Show to your teacher.
4. Show to your teacher.

Exercise:-1.2

1.a) 30° b) 70° c) 90° d) 120° e) 150°

2.a) <ABC, B, AB and BC b) <xyz, y, xy, yz c) <PQR, Q, PQ, and QR.

d) <JKL, K, JK, KL e) <MLK, L, ML, LK f) <YXW, X, YX, WX.

3. a) <CBA, 90° right angle b) <PQR, 180° Straight angle

c) <LMN, ... obtuse angle d) <BCD reflex angle.

4. Show to your teacher.

5. a) obtuse b) right c) straight d) reflex e) acute f) obtuse

g) right h) obtuse i) reflex j) acute k) obtuse l) reflex.

6. Show to your teachers.

7. Show to your teachers.

8. a) Angle lying between 0° to 90°. b) Angle lying between 900 to 180°.

c) Angle whose measurement is 90°. d) Angle whose measurement is 180°.

e) Angle lying between 180° to 360°.

Exercise: 1.3

1. a) Equilateral, Isosceles and Scalene. b) Acute, obtuse, right.

2. a) Right angled triangle. b) Equilateral triangle.

c) Isosceles triangle. d) Acute angled triangle.

e) Obtuse angled triangle. f) Equilateral triangle.

g) Scalene triangle.

3. a) Equilateral triangle. b) Scalars triangle.
c) Isosceles triangle. d) Isosceles or right angled triangle
e) Acute angled triangle. f) Obtuse angled triangle.
g) Equilateral triangle. h) Isosceles triangle.

Prime Mathematics Book − 4 21

4. a) Right angled triangle. b) Equilateral triangle
c) Isosceles triangle d) Obtuse angled triangle.
e) isosceles triangle triangle. f) Acute angled triangle.

5. Show to your teacher.
6. Show to your teacher.
7. Isosceles triangle, A, BC, B and C.

Exercise: 1.4

1. Show to your teacher.

2. a) Square b) Circle c) Parallelogram d) Rectangle e) Rhombus

f) Quadrilateral g) Equilateral triangle

3. a) Parallelogram b) Square c) Rectangle d) Rhombus.
4. Show to your teacher.

5. a) 3.4 cm b) 4.4 cm c) 6.8 cm d) 8.6 cm e) 10 cm f) 11.2 cm
g) 5 cm h) 7 cm

6. a) 1.6 cm b) 2 cm c) 2.3 cm d) 3 cm e) 3.2 cm f) 3.8cm
g) 4.1 cm h) 4.5 cm.

7. Show to your teacher.

Exercise: 1.5

1. Show to your teacher.
2. Show to your teacher.

22 Prime Mathematics Book − 4

U2nit Number System

Estimated periods − 34
Objectives

At the end of this unit, the students will be able to:
• read and count numbers upto 99 .
• place the given numbers in a place value table.
• write the numbers in words.
• write the number word in numeral.
• find the factors and multiples of a given number.
• distinguish between prime and composite number.
• round off of numbers to the nearest 10, to the nearest 100 and to the nearest 1000.
• convert Hindu-Arabic number to Roman numeral and vice versa.

Teaching Materials

Place value chart, abacus, number cards, Roman Numeral chart etc.

Activities
It is better to:
• ask the students to represent the given number on place value table and write in words.
• ask the students to represent the same given number on Hindu -Arabic system and International
system and ask them to put commas.
• while rounding off, ask the students to make a number line and see the given number in nearer
to which one.
• drill the students to convert Hindu- Arabic numeral to Roman numeral and vice versa.
• play a game about prime /even/composite/odd number
• use table tenis balls with numbers from 1-100
• make groups to play the game

Let’s recall the things that you have
learnt in previous classes.

Smallest number of one digit = 1
Smallest number of two digits = 10
Smallest number of three digits = 100
Smallest number of four digits = 1000

Students, you have learnt about the 5 digit
numbers in class III. In this class we are going

to study about the numbers which contain
more than 6 digits.

Let’s take 14,25,698 and represent it in a place value table.

Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones

14 2 5 6 98

Here,
8 is in ones place, so its place value is 8.
9 is in tens place, so its place value is 90.
6 is in hundreds place, so its place value is 600.
5 is in thousands place, so its place value is 5000.
2 is in ten thousands place, so its place value is 20,000.
4 is in lakh place, so its place value is 4,00,000.
1 is in ten lakh place, so its place value is 10,00,000.

Thus the number in words is fourteen lakh, twenty five thousand six hundred
and ninety eight.

Smallest number of five digit = 10,000. 9,99,999
Smallest number of six digit = 1,00,000. +1
Also, 10,00,000
The greatest number of one digit = 9
The greatest number of two digits = 99
The greatest number of three digits = 999
The greatest number of four digits = 9999
The greatest number of five digits = 99999
The greatest number of six digits = 999999
Let’s see what happens when 1 is added to 999999

It is ten lakh and it contains seven digits.

24 Prime Mathematics Book − 4

Similarly, let’s take 26,95,401
Represent it in a place value table.

Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones

26 9 5 4 01

Thus 26,95,401 in words is twenty six lakh ninety five thousand four hundred
and one.

Take, 99,99,999
Represent it in a place value table.

Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones

99 9 9 9 99

Ninety nine lakh ninety nine thousand nine hundred and ninety nine.

Can you add 1 to 99,99,999 ? Yes sir, I can.
It is one crore. 99,99,999

+1
1,00,00,000

Oh! It’s an eight digit number.
What is it called, sir?

Represent 9,21,04,538 in a place value table.

Crores Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones

9 21 0 4 5 38

Nine crore twenty one lakh four thousand five hundred and thirty eight.

Let’s see, how to write a given
number in expanded form.
Let’s expand 9,15,23,487

9,15,23,487 = 9,00,00,000 + 10,00,000 + 5,00,000 + 20,000 + 3,000 + 400 +
80 + 7

Similarly,
8,24,56,709 = 8,00,00,000 + 20,00,000 + 4,00,000 + 50,000 + 6,000 + 700 +

00 + 9

Prime Mathematics Book − 4 25

Can you sum the numbers 9,99,99,999 and 1?

It is a nine digit number. Yes sir, I can.
Let’s deal with some nine 9,99,99,999

digit numbers. +1
10,00,00,000

Oh! It's ten crores.

Represent 25,49,53,678 in a place value table.

Ten Crores Ten Lakhs Ten Thousands Hundreds Tens Ones
crores Lakhs Thousands

2549 5 3 6 78

Twenty five crore forty nine lakh thirty three thousand six hundred and seventy
eight.
Sir, how do we put comma?
Could you explain?

Thank you for the question. The system that we are studying
right now is Nepali system of place value. In this system first we
put comma after three digits from the right and after every
two digits. For example, 2,89,34,001.

Request to the teacher : Please give some more examples and involve the
students to do them in the board.

Exercise 2.1
1. Write the numerals by representing them in a place value table:

(a) Ten lakh fourteen thousand six hundred and fifty eight.

(b) Two lakh four thousand and fifty seven.
(c) Five crore seventy lakh forty nine thousand and one.

(d) Twenty four crore forty seven thousand eight hundred and ninety
seven.

(e) Seventy five crore twenty three lakh five thousand six hundred
and four.

26 Prime Mathematics Book − 4

2. Write in words by representing the given numbers in a place value table:
(a) 14,00,29,605 = ___________________________________________
(b) 5,72,46,893 = ___________________________________________
(c) 75,47,000 = ___________________________________________
(d) 9,68,457 = ___________________________________________
(e) 48,69,532 = ___________________________________________

3. Write in expanded form:

(a) 54,27,61,289 = ++

+ + + + ++

(b) 68,29,43,569 = ++

+ + + + ++

(c) 75,00,649 = + ++

+ ++

(d) 89,74,653 = + ++

+ ++

(e) 8,92,574 = + ++

++

4. Write in short form:

(a) 50,00,00,000 + 4,00,00,000 + 70,00,000 + 2,00,000 + 80,000 + 9,000

+ 100 + 60 + 3 =

(b) 70,00,00,000 + 8,00,00,000 + 50,00,000 + 9,00,000 + 10,000 + 2,000

+ 200 + 30 + 4 =

(c) 80,00,00,000 + 9,00,00,000 + 70,00,000 + 6,00,000 + 50,000 + 4,000

+ 300 + 20 + 1 =

(d) 100,00,00,000 + 90,00,000 + 2,00,000 + 40,000 + 7,000 + 600 + 30 + 8

=

(e) 8,00,00,000 + 20,00,000 + 40,000 + 9,000 + 700 + 30 + 1

=

Prime Mathematics Book − 4 27

5. Write the place value of the circled digits:

(a) 9,12,34,568 (b) 4,53,68,971 (c) 6,79,42,138

(d) 5,78,14,296 (e) 17,80,42,956 (f) 17,68,42,159

(g) 75,82,460 (h) 89,73,245 (i) 7,84,201

6. Put commas in the following numbers according to Nepali Place
Value system:

(a) 3145789 (b) 420569 (c) 124537693

(d) 52748961 (e) 162842179 (f) 40576893

(g) 981741382 (h) 124875396 (i) 482956107

Hindu- Arabic numerals and Devanagari numerals.

Digits of Hindu- Arabic numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Digits of Devanagari numerals ), !, @, #, $, %, ^, &, *, (

Place value table in Hindu Arabic numeral and devanagari numerals

Example : 1. Rewrite the Hindu Arabic number 79845 into devanagari numbers

with place value table and also write the number name in devanagri system.

Solution :

79845

bz xhf/ xhf/ ;o bz Ps

79 8 4 5

cIf/df M pgfgc;L xhf/ cf7 ;o kF}tfln; .

Example. 2 Rewrite the devanagari numeral 23465 place into Hindu Arabic

Numeral with value table and also write the number name in Hindu Arabic

system.
Solution :

@#$^% Thousand Hundred Tens Ones
3 4 6 5
Ten Thousands
2

In words: Twenty three thousands four hundred sixty five.

28 Prime Mathematics Book − 4

Exercise 2.2

1. Rewrite these numbers in devanagari numbers

(a) 17 (b) 58 (c) 328

(d) 4503 (e) 25735 (f) 87658

2. Rewrite these numbers in Hindu Arabic numbers

(a) 12 (b) 68 (c) 353

(d) 6214 (e) 26453 (f) 91837

3. Rewrite these numbers in devanagari digits and also write the

number names.
(a) Three hundred seventy five
(b) Two thousand sixty two
(c) Fifty one thousand one hundred seventy.
4. Write these numbers in Hindu Arabic digits and also write number
name

(a) ;ft ;o 5kGg
(b) Ps xhf/ ^ ;o rjGg
(c) rfln; xhf/ gf} ;o kRrL;

International Place Value system:

Do you know, the place value system, that
you are using is not used in my country, as well as in
different countries. My system is International Place

Value System.

Let’s compare the Nepali Place Value System and International Place Value System.
Nepali Place Value system.

Crores Lakhs Thousands Hundreds Tens Ones

Ten crores Crore Ten lakhs Lakh Ten Thou-
thou- sand
sands

10,00,00,000 1,00,00,000 10,00,000 1,00,000 10,000 1,000 100 10 1

Prime Mathematics Book − 4 29

International Place Value system.

Million Thousand Hundred Ten Ones

Hundred Ten Million Hundred Ten Thou-
Millions Millions thousand thousand sand

100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

From the above comparison it is understood that

Nepali Place International Nepali Place International
Value system Place Value Value system Place Value

1 one system 1 lakh system
1 ten 1 one 10 lakhs 100 thousands
1 hundred 1 ten 1 crore 1 Million
1 thousand 1 hundred 10 crores 10 Million
10 thousand 1 thousand 1 Arab 100 Million
10 thousands 1 Billion

Let's Learn :
Represent 122,145,698 in a place value table.

Hundred Ten Millions Hundred Ten Thousands Hundreds Tens Ones
Millions Millions Thousands Thousands 6 98

1 22 1 4 5

One hundred twenty two million one hundred forty five thousand six hundred
and ninety eight.

In the International Place Value
System, we put comma after the third

digit from the right.
For example, 246,125,940.

Represent 24,568,971 in a place value table.

Ten Millions Hundred Ten Thousand Hundred Tens Ones
Millions 4 Thousands Thousands 8 9 7 1

2 5 6

Twenty four million five hundred sixty eight thousand nine hundred and
seventy one.

30 Prime Mathematics Book − 4

Let’s learn the place value of the
digits of a number in the international place
value system. Let’s represent 146,257,983

in a place value table.

Hundred Ten Millions Hundred Ten Thousand Hundred Tens Ones
Millions Millions Thousands Thousands

1 46 2 5 7 9 83

Here,
3 is in ones place so its place value is 3.
8 is in tens place so its place value is 8 tens.
= 8 × 10 = 80
9 is in hundreds place so its place value is 9 hundreds.
= 9 × 100 = 900
7 is in thousands place so its place value is 7 thousands.
= 7 × 1000 = 7000
5 is in ten thousands place so its place value is 5 ten thousand.
= 5 × 10,000 = 50,000
2 is in hundred thousands place so its place value is 2 hundred thousands.
= 2 × 100,000 = 200,000
6 is in millions place so its place value is 6 millions.
= 6 × 1,000,000 = 6,000,000.
4 is in ten millions place so its place value is 4 ten millions.
= 4 × 10,000,000 = 40,000,000
And,
1 is in hundred millions place so its place value is
1 hundred million = 1 × 100,000,000 = 100,000,000.

Prime Mathematics Book − 4 31

Exercise 2.3

1. Write the place value of the circled digits in the international place
value system:

(a) 12,256,498 (b) 34,958,201

(c) 591,615,254 (d) 1,962,587

(e) 982,685 (f) 82,704

(g) 2,710 (h) 192,258

2. First represent the given numbers in place value table in international

place value system and write in words:

(a) 198, 253,670 (b) 25,943,798

(c) 5,958,602 (d) 569,700

(e) 240,810 (f) 6,529,846

(g) 78,453,269 (h) 397,538,275

3. How many thousands are there in :

(a) 3,600,000 (b) 40,000,000

(c) 218,000,000 (d) 700,000,000

(e) 4580,000 (f) 17,200,000

4. First represent the given number names in place value table in
international place value system and write in numerals:
(a) Two hundred thirty eight million three hundred forty two thousand
six hundred and five.
(b) Fifty two million six hundred forty thousand nine hundred and five.
(c) Nine millions thirty four thousand five hundred and seventy six.
(d) Three hundred forty five thousand two hundred and ninety eight.
(e) Four hundred ninety five thousand seven hundred and five.

32 Prime Mathematics Book − 4

5. Write the numbers in expanded form in international place value

system:

(a) 241,567,908 (b) 453,694,721

(c) 94,704,563 (d) 57,894,213

(e) 8,793,465

6. Write in short form:

(a) 200,000,000 + 30,000,000 + 5,000,000 + 700,000 + 50,000 + 1000

+ 900 + 80 + 7

(b) 300,000,000 + 50,000,000 + 4,000,000 + 800,000 + 40,000 + 6000

+ 900 + 70 + 1

(c) 40,000,000 + 5,000,000 + 900,000 + 70,000 + 8000 + 600 + 10 + 2

(d) 90,000,000 + 6,000,000 + 700,000 + 50,000 + 1000 + 300 + 20 + 8

(e) 8,000,000 + 700,000 + 50,000 + 2000 + 100 + 90 + 6

7. Write the short form of:
(a) 7x1000000 + 6x100000 + 4x10000 + 3x1000 +2x100 + 7x10+4
(b) 8x10000000 + 5x1000000 + 9x100000 +1x10000 + 4x1000 + 2x100+
4x10+8
(c) 9x1000 +8x100 + 7x10 +6
(d) 8x1000 + 5x 100 + 6x10 +2

8. Write the largest 4 digits number using the digits 6,2, 3 and 5 without
repeating the digits.

9. Find the largest and smallest 4 digits 2,0,8 and 7 number using the
digits

Prime Mathematics Book − 4 33

Factors: Factors and Multiples

Let’s observe the following examples

1 × 14 = 14 2 × 7 = 14
∴ 1 and 14 are factors of 14 ∴ 2 and 7 are factors of 14

So, the factors of 14 are 1, 2, 7 and 14.
Similarly,

1 × 12 = 12 ; 1 and 12 are factors of 12.
2 × 6 = 12 ; 2 and 6 are factors of 12.
3 × 4 = 12 ; 3 and 4 are factors of 12.
∴ 1, 2, 3, 4, 6 and 12 are factors of 12.

1×6 2×3 In the example, we write 6 as 1×6 and 2×3.

So, 1, 6, 2, 3 are the factors of 6. On the other-

6 hand, the numbers which
divides 6 exectly are the factors of 6.

1×24 4×6 In the example, we write 24 as 1×24, 2×12,
8×3 3×8 and 4×6.
24
So, 1, 2, 3, 4, 6, 8, 12, 24 are the factors of 24.

12×2 It means the numbers which are
exactly divisible to 24 are the factors of 24.

Here, 1, 2, 3, and 6 are all the possible factors of 6 . Similarly 1, 2, 3, 4, 6, 8, 12 and
24 are all the possible factors of 24.

So, the factors of any number are those numbers which divide the number
leaving remainder zero.
Factors of a number are always less than or equal to the number itself.

See, more examples.

2 × 1 = 2; 2 and 1 are only factors of 2
3 × 1 = 3; 3 and 1 are only factors of 3
5 × 1 = 5; 5 and 1 are only factors of 5
7 × 1 = 7; 7 and 1 are only factors of 7

Some numbers consists of at least two factors; 1 and the number itself.

34 Prime Mathematics Book − 4

Multiplies: 4×1=4
4×2=8
Observe the following examples: 4 × 3 = 12
4 × 4 = 16
∴ 4, 8, 12, 16, 20 and 24 are some multiplies of 4. 4 × 5 = 20
4 × 6 = 24

Let’s investigate the multiples of the numbers.
Multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 ........ etc.

Multiples of 3 are 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ........................ etc.

Multiples of 5 are 0, 5, 10, 15, 20, 25 ..................................... etc.

Above examples show that the multiples of a number those are exactly divisible by
the number without leaving remainder.
Can you investigate the multiples of 6, 7, 8, 9 & 10 and so on?

Thus, when we multiply the number by 1, 2, 3, we get the multiples of the
number. Multiples of any number are always greater than or equal to the
number itself.

2÷1=2 3÷1=3 5÷1=5 7÷1=7
2÷2=1 3÷3=1 5÷5=1 7÷7=1

2 is exactly divisible by 1 and itself 2.
3 is exactly divisible by 1 and itself 3.
5 is exactly divisible by 1 and itself 5.
7 is exactly divisible by 1 and itself 7.

Such numbers are prime numbers.

Therefore, 2, 3, 5 and 7 are some prime number.

So, the numbers which are exactly divisible by 1 and itself only are called
prime numbers.

Prime Mathematics Book − 4 35

Prime Factors:
Let’s find the possible factors of 12.

1×12 2×6

12

3×4

The possible factors of 12 are 1, 2, 3, 4, 6 and 12.

Among them 2 and 3 are prime numbers. So
they are called prime factors.

* Process of finding prime factors.

Let’s observe the following example.

2 20 Divide the number by Prime
2 10 Number until the quotient

5 becomes a prime.
20÷2=10, 10÷2=5
So,
20= 2×2×5

Similarly, Division is continued by prime
2 54 number till the quotient
3 27 becomes a prime number.
39
3 (It is also called prime factorisation)

Here, 54÷2=27
27÷3=9
9÷3=3

So, 54 = 2×3×3×3

36 Prime Mathematics Book − 4

Composite Numbers:
See the examples:

4÷1=4 6÷1=6 8÷1=8
4÷2=2 6÷2=3 8÷2=4
4÷4=1 6÷3=2 8÷4=2
6÷6=1 8÷8=1

4 is exactly divisible by 4, 2 and 1. So 4 is not a prime number.
It is a composite number.
6 is exactly divisible by 2, 3, 1 and 6. So, 6 is not a prime number.
It is a composite number.
8 is exactly divisible by 2, 4, 1 and 8. So, 8 is not a prime number.
It is a composite number.

So, the numbers which are exactly divisible by any other numbers
besides by 1 and itself are called composite numbers.
Therefore 4, 6, 8, 9, 10, 12,………… are some composite numbers.

Thus, prime numbers have only two factors and they are 1 and the
number itself. Composite numbers have more than two factors.

Let’s learn how to find out the
prime numbers and composite

numbers from 1 to 100.

Step 1 : 1 is not a prime number. Actually 1 is neither prime number nor
composite number. Although 1 is divisible by 1 and by itself. Cross it.

Step 2 : The next number is 2, cross all other even numbers.
Step 3 : The next number is 3, cross all the multiples of 3.
Step 4 : The next number is 5, cross all the multiples of 5.
Step 5 : The next number is 7, cross all the multiples of 7.

Prime Mathematics Book − 4 37

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

After crossing 1 and all the composite numbers between 1 and 100, 25 prime
numbers are left and they are :

2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61
67 71 73 79 83 89 97

The numbers besides the above prime numbers are the composite numbers
from 1 to 100.
Between 1 and 100, there are eight pairs of prime numbers, which are
separated by one composite number. They are 3 and 5, 5 and 7, 11 and 13,
17 and 19, 29 and 31, 41 and 43 , 59 and 61, 71 and 73.

2 3 5 7 11 13 17 19 23 29 31
37 41 43 47 53 59 61 67 71 73
79 83 89 97

Such prime numbers are called twin prime numbers.

38 Prime Mathematics Book − 4

Note: Seive of Eratosthenes (276 to 194 B.C.)

For finding all the primes below a given integer 'n' ,write down all the integers
from 2 to n in their natural order and then strike out all the multiples of the
primes ≤ n. The number left on the list (which do not fall through the sieve)
are primes.
To find the primes less than 100, list all the integers from 2 to 100 and strike
out all the multiples of primes ≤ 100 i.e. 10 which are 2, 3, 5, 7
Similarly, to find the primes less than 225, list all the primes from 2 to 225 and
as 225 =15, strike out all the multiples of primes ≤ 15 i.e. 2, 3, 5, 7, 11, 13

Exercise 2.4

1. Write the all the possible factors and also mention the prime factors:

(a) 50 (b) 150 (c) 200

(d) 300 (e) 400 (f) 575
(g) 680 (h) 940 (i) 1000

2. Find the first five multiples of the given numbers:

(a) 12 (b) 14 (c) 21
15
(d) 25 (e) 27 (f) 53

(g) 17 (h) 32 (i)

3. Write down all the prime numbers: (b) Between 5 and 18
(a) Less than 10 (d) Between 20 and 40
(c) Less than 25 (f) Less than 60
(e) Less than 40 (h) Between 80 and 100
(g) Between 40 and 60

4. Write all the composite numbers: (b) Between 5 and 17
(a) Less than 10 (d) Between 10 and 30
(c) Less than 30 (f) Between 70 and 100
(e) Between 30 and 60 (h) Less than 80
(g) Less than 50

Prime Mathematics Book − 4 39

5) Write down all the possible factors of the following numbers and circle the
prime factors.

a) 8 = ...................................
b) 7 = ...................................
c) 15 = .................................
d) 20 = ................................
e) 16 = .................................
f) 28 = .................................
g) 32 = ................................
h) 48 = ...............................
i) 54 = .................................
j) 72 = ...............................

6) Express the number in the product prime factors by prime factorisation.

(a) 20 (b) 15 (c) 28 (d) 45
(e) 60 (f) 36 (g) 90 (h) 80
(i) 120 (j) 144 (k) 175 (l) 100
(m) 180 (n) 75 (o) 128 (p) 250
(q) 375

Test of divisibility

Let's see the rules to investigate whether a number is exactly divisible

by 2,3,5,7 and 11 or not.

Divisible by Properties

2 If the number in ones place is even, it is

exactly divisible by 2. Eg.8, 20, 74, 870,

etc. are exactly divisible by 2.

3 If the sum of the digits of a number is

exactly divisible by 3, the number is also

exactly divisible by 3. Eg. 123, 456,801,

etc are exactly divisible by 3.

40 Prime Mathematics Book − 4

5 If a number consists of O or 5 in its ones
place, the number is exactly divisible by
5. Eg. 50,805, 715, 255, etc. exactly
divisible by 5.

Take the last digit of the number.
7 Double the last digits and subtract the

double of last digit from the number
formed by using the rest of the digits. If
the difference is exactly divisible by 7,
the numbers is divisible by 7 Eg. 343, 742,
1274, etc. are exactly divisible by 7.
11 In a number, if the difference between
ones digits and number with remainig
digits is either zero (0) or multiple of 11
then the number is also exactly divisible by
11. Eg. 132, 77, 187, 231, etc. are exactly
divisible by 11.

Exercise 2.5
1) Copy the numbers exactly divisible by 2.

50, 27, 82, 56, 43, 87, 129, 786

2) Copy the numbers exactly divisibe by 3.
45, 82, 343, 432, 816, 512

3) Copy the numbers exactly divisible 5.
88, 75, 820, 752, 315, 456

4) Copy the numbers exactly divisible by 7.
78, 812, 562, 56, 858, 343

5) Copy the numbers exactly divisible by 11.
705, 88, 759, 687, 825, 905

Prime Mathematics Book − 4 41

6) Write down among 2, 3, 5, 7 and 11 which number can exactly divide the
following numbers.

a) 12 = ........................... and ........................
b) 30 = .........................., ......................., and ......................
c) 22 = .......................... and .....................
d) 56 = ......................... and .......................
e) 70 = ........................., .......................... and ........................
f) 66 = ........................., .......................... and ........................

Highest Common Factors (H.C.F)

Let’s investigate the Highest Common Factors of two number. Let us consider the two
number 24 and 18.

Factors of 24 are 1, 2, 3, 4, 6, 12, and 24
Factors of 18 are 1, 2, 3, 6, 9 and 18.

Common factors are 1, 2, 3, 6. Among them 6 is the highest factor

So the HCF of 24 and 18 is 6.
ie; HCF = 6

This method is known as the finding HCF by definition method

Finding HCF by prime factorisation method

Finding the HCF 24 and 18

2 24 2 18

2 12 39
26 and 3

3
24 = 2 × 2 × 2 × 3
18 = 2 × 3 × 3

HCF = 2×3 = 6

42 Prime Mathematics Book − 4

Finding HCF by prime factorisation method

Finding the HCF 16, 24 and 32

2 16 2 24 2 32
12 2 16
28 2 6 and 2 8
3 24
24 2
2
2
16 = 2 × 2 × 2 × 2
24 = 2 × 2 × 2 × 3
32 = 2 × 2 × 2 × 2 × 2

HCF = 2×2 × 2 = 8

Exercise 2.6

1) Write all the possible factors of the following numbers and their HCF. by defi-

nition method.

a) 9, 12 b) 36, 48 c) 16, 20 d) 54, 72 e) 18, 24

f) 90, 64 g) 28, 42 h) 50, 60 i) 32, 24 j) 32, 40

2) Find the HCF of the following numbers by prime factorisation method.

a) 9, 15 b) 8, 20 c) 21, 28 d) 27, 36 e) 20, 30

f) 50, 75 g) 16, 32, 24 h) 8, 10, 12 i) 20, 40, 30

Lowest Common Multiples (L.C.M)

Let’s take two numbers 6 and 9.
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42 ........... etc.
Multiples of 9 are 9, 18, 27, 36, 45, 54, .................. etc.

Common multiples of 6 and 9 are 43
{18, 36, 54, .........}
Among these common multiples, the lowest Common multiple is 18.

the LCM of 6 and 9 is 18.

(It is known as finding the LCM by definition method)

Prime Mathematics Book − 4

* Process of finding LCM by division method)

i) To get the LCM of two or more numbers having common factor 1 then multiply
them to find LCM.
eg. LCM of 3 and 5 = 3 × 5 = 15
LCM of 2, 5 and 7 = 2 × 5 × 7 = 70
This means the product of prime numbers is the Lowest Common Multiple.

ii) To find the LCM of two or more than two numbers having common factors
among them, divide them by the common prime factors until the common factor
becomes 1.

Example 1
Find the LCM of 12 and 20

Solution :
2 12, 20
2 6, 10
3, 5

LCM of 12 and 20 = 2 × 2 × 3 × 5 = 60
Example 2

Find the LCM of 18, 30, 32
Solution 2 18, 30, 36

3 9, 15, 18
3 3, 5, 6

1, 5, 2
LCM of 18, 30 and 32 = 2 × 3 × 3 ×1 × 5 × 2 = 180

Exercise 2.7

1) Find the LCM of the following numbers by definition method :

a) 2 and 5 b) 5 and 6 c) 7 and 9 d) 2 and 3

2) Find the LCM of the given numbers by division method :

a) 6 and 10. b) 12 and 15 c) 16 and 20 d) 15 and 20

e) 18 and 21 f) 20 and 25 g) 24 and 32 h) 6, 8, and 12

i) 6, 9 and 12 j) 12, 15, 24 k) 10, 20, 30

44 Prime Mathematics Book − 4

Rounding off the numbers

My weight is But, Sir, You seem
70 kg. to have more weight.

Yes, you are correct. Actually my
weight is 73 kg but when I round it to off
to nearest ten, my weight is 70kg. It is
easy to remember.

Sir, could you explain
clearly how to round off
To round off the number 154 to the numbers ?
nearest ten, let’s take the help of
number line.

140 150 154 155 160

So, in the figure, the number 154 is nearer to 150 than to 160. So, while
rounding off 154 to nearest ten, it is 150.

Jay, I think you are Sir, It cost
wearing a new shirt. Rs. 700.

What is its cost?
But it doesn’t seem that

it costs that much.

Sir, actually its cost is Rs.
680. But, I rounded its off to
nearest hundred, so I said Rs.

700.
Jay, you are correct and
you have rounded off correctly.

Let me explain once again to round
off a number to nearest hundred. Let’s take 690.

Let, me represent it in a number line.

Prime Mathematics Book − 4 45