Imagine_Maths_CB_Grade6_Sample_Part1

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MATHEMATICS

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NEP 2020 based

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NCF compliant

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CBSE aligned

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MATHEMATICS Master Mathematical Thinking

Grade 6

Fo re wo rd

Mathematics is not just another subject. It is an integral part of our lives. It shapes the very foundation of our understanding, personality and interaction with the world around us.

However, due to the subject’s abstract nature, the stress of achieving high academic scores and complex teaching methods, most children develop a fear of mathematics from an early age. This fear not only hinders their mathematical thinking, logical reasoning and general problem solving abilities, but also negatively impacts their performance in other academic subjects. This creates a learning gap which widens over the years.

The NEP 2020 has distinctly recognised the value of mathematical thinking among young learners and the significance of fostering love for this subject by making its learning engaging and entertaining. Approaching maths with patience and relatable real-world examples can help nurture an inspiring relationship with the subject. It is in this spirit that Uolo has introduced the Imagine Mathematics product for elementary grades (1 to 8). This product’s key objective is to eliminate the fear of mathematics by making learning exciting, relatable and meaningful for children. This is achieved by making a clear connection between mathematical concepts and examples from daily life. This opens avenues for children to connect with and explore maths in pleasant, relatable, creative and fun ways.

This product, as recommended by the NEP 2020 and the recent NCF draft, gives paramount importance to the development of computational and mathematical thinking, logical reasoning, problem solving and mathematical communication, with the help of carefully curated content and learning activities.

Imagine Mathematics strongly positions itself on the curricular and pedagogical approach of the Gradual Release of Responsibility (GRR), which has been highly recommended by the NEP 2020, the latest NCF Draft and other international educational policies. In this approach, while learning any new mathematical concept, learners first receive sufficient modelling, and then are supported to solve problems in a guided manner before eventually taking complete control of the learning and application of the concept on their own. In addition, the book is technologically empowered and works in sync with a parallel digital world which contains immersive gamified experiences, video solutions and practice exercises among other things. Interactive exercises on the digital platform make learning experiential and help in concrete visualisation of abstract mathematical concepts.

In Imagine Mathematics, we are striving to make high quality maths learning available for all children across the country. The product maximizes the opportunities for self-learning while minimising the need for paid external interventions, like after-school or private tutorial classes.

The book adapts some of the most-acclaimed, learner-friendly pedagogical strategies. Each concept in every chapter is introduced with the help of real-life situations and integrated with children’s experiences, making learning flow seamlessly from abstract to concrete. Clear explanations and simple steps are provided to solve problems in each concept. Interesting facts, error alerts and enjoyable activities are smartly sprinkled throughout the content to break the monotony and make learning holistic. Most importantly, concepts are not presented in a disconnected fashion, but are interlinked and interwoven in a sophisticated manner across strands and grades to make learning scaffolded, comprehensive and meaningful.

As we know, no single content book can resolve all learning challenges, and human intervention and support tools are required to ensure its success. Thus, Imagine Mathematics not only offers the content books, but also comes with teacher manuals that guide the pedagogical transactions that happen in the classroom; and a vast parallel digital world with lots of exciting materials for learning, practice and assessment. In a nutshell, Imagine Mathematics is a comprehensive and unique learning experience for children.

On this note, we welcome you to the wonderful world of Imagine Mathematics. In the pages that follow, we will embark on a thrilling journey to discover wonderful secrets of mathematics—numbers, operations, geometry and measurements, data and probability, patterns and symmetry, algebra and so on and so forth. Wishing all the learners, teachers and parents lots of fun-filled learning as you embark upon this exciting journey with Uolo. ii

We know that numbers are basic units of mathematics and are used for counting, measuring and comparing quantities. We have also previously learned that each digit in a number has a value, which we call the place value.

Let us take a 2-digit number as an example! The digit on the left is at ten’s place while the digit on the right is at one’s place.

K ey El ements o f a C h apt e r— a Q u i c k G lanc e

Similarly, in the number 2548, 5 represents 5 hundreds, or 500. However, in the number 56, 5 represents 5 tens, or 50. Therefore, even if a digit is same, its value always depends on where it is in the number.

Th

H

T

O

2

5

4

8

4 Tens

8 Ones

2 Thousands 5 Hundreds

Let's Warm-up

Introductory

Concept

Write the correct place value of the coloured numbers.

Understanding Multiplication

page with a 5 multiples introduction Find the first of 6. Check__________ by dividing if 92 and 96 are multiples of 6.

1 32

Do It Together

ays is always fun!quick You first decide on a place to visit, warm-up 2 548 you want__________ 1 2 what you can do there, and then finally make arrangements.

3 876 __________

4 4563 o Ooty. The train departs every second day.

5

Sanju struggles to find how many words will he learn in whole January.

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 +2 + 2 + 2 + 2………

This is too much! Sanju gets confused and asks his father to help.

__________

4 5 6 7Sunday 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Saturday

30

31

1

2

6

7

8

9

13

14

15

16

20

21 question 22

23

3

6 × 2 = ____

6 × ____ = ____

6 × ____ = 24

29

Multiplication by 1-digit Number We know that the number obtained from multiplication is the product.

TTh

Th

H

T

O

1

2

3

4

4

+

Thinkofand Tell I scored out12,344 of 5. + 1115 = _____________.

The first five multiples 6 are 6, ____, ____, 24, ____._________ So, 17 What do you think do

31

even number?

1

1

1

6

Find the factors of 36 using the division method.

Divide numbers by 36

What do we get?

Check the remainder

Are the numbers factors of 36?

36 ÷ 3

12

____

____

0

____

=31 ×

5

multiplicand

TTh

2

multiplier

= 62

Th

H

On multiplying a number by 1, the product is always the number itself.

T

O

On multiplying a number by 0, the product is always zero.

3

5

product

We can find the product of two numbers by placing them horizontally next to each other.

Let us multiply 123 and 3.

What do we know?

96

O

We can multiply two numbers in any order.

The product always remains the same.

Horizontal Method

Step 1

Step 2

Step 3

Multiply by ones.

Multiply by tens.

Multiply by hundreds.

Multiply 3 and 3 ones.

Multiply 3 and 2 tens.

123 × 3 = 369

Did You Know?

123 × 3 = __ __9

= __69 Rinne Tsujikubo123 of× 3Japan broke The productthe of 123 and 3 is 369.world record for Guinness fastest mental arithmetic on Chapter 3 • Multiplication January 17, 2023 by correctly adding 15 sets of three-digit numbers in 1.62 seconds.

Let us start finding the total number of books step by step.

n the following dates - 2, 4, 6, 8, 10, 12, 14, 16 and so on.

6 92

ultiples

T

In the school library, there are 1219 fiction 9/11/2023 4:24:58 PM books and 1567 non-fiction books. How many books are there in total?

Are the multiples of an even number always an

Do It Together

H

to the concept

Think and Tellhave in common?

30

Th

Fun fact, related Story Sums

all the circled numbers

24

Multiplication Rules

The number to be multiplied is multiplicand and the number by which we multiply is multiplier.

Do It Together

UM24CB_G4.indb 1

28

Sanju’s father helps him find the number of words using multiplication.

Add 12,344 and 1115.

6 × ____ = ____

10

A quick-thinking

He has learnt 14 words already! Hurray!

example

0 1 Friday 2 3 Thursday

6×1=6

2 + 2 + 2 + 2 + 2 + 2 + 2 = 14

__________

9958

Sanju and his father play a newspaper game where he learns 2 new words each day. They have been playing this game for a week. Sanju is trying to find all the words he has learnt.

He starts adding quickly.

with a real-life

Wednesday

27

Real Life Connect

49

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9/11/2023 4:25:30 PM

The total number of fiction books = 1219 The total number of non-fiction books = 1567 What do we need to find?

dividing 92 by weby get 2 remainder.

On dividing 96 by 6, we get ____ remainder.

36 ÷ 1 36 the number 0 of a number areOn the products we6,get multiplying by 1,Yes2, 1 2 1 9 Face Value Numbers Thinkinand Telltotal number =6-digit The of fiction books 36 ÷ 2 ____ 0 ____ 1 5 6 7 + The total number of non-fiction books Do we need+of to go6.

So, 92 _______________________________ of 6.

So, 96 ___________________________ We learnt about the concept of face value in the previous section. It is defined as the

an be found by using multiplication tables36as follows: ÷4 ____

The total number of books in the library

beyond 6=to1219 find + 1567

Th

H

T

O

2

7

8

6

numericalmore value ofofthe factors 36? digit on a particular place in a number.

Solve to find the answer.

out 36 ÷ 5 7 1 ____ For example, the Pincode on Ajay’s 3 × 6 = An 18important So,Pointing the total number of books in the library is 2786. ____ 0 Yes letter was 781005.

The face value of It Yourself 5A 36 ÷ 6 keep in commonly made 3 × 7 = point 21 toDo Error Alert!

The city NGO organised a two-day donation drive. the Lakhs placeOnisthe simply 7.of the drive, 1366 clothes were Remember!

Remember! first day So, the factors of 36 are ___________________________________________________________________.

3 × 8 = mind 24 mistakes and When we add 1000 to a collected. On the second day of the drive, 1000confuse Face Value Never with Place Value! For A number is a multiple of itself too.

Similarly, we say that the face value of 4-digit number, only the clothes were collected. How many clothes were 3 × 9 = 1 27Colour the balloons that are multiples of 2. howinto avoid example, in 781005, the Value of the Lakhs digit Face in the thousands For example, multiples of 5 are 5, total? the digit on thecollected Ten Thousands place place changes. place is 7 and the Place Value is 7 × 100000 = 700000.

Do15, It Yourself 3 × 10 = 30 10, 20, 25,5C 30 and so on! them 4

Round off each dividend to the nearest 10s and 100s and then divide.

To the nearest 10s

Example 4

To the nearest 100s

a 1147 ÷ 2

b 4589 ÷ 3 c 6478 ÷ 6

1

d 8974 ÷ 7

is 8.

2

3

e 5555 ÷ 5

What do we know?

1

4 6 5 8 7 Show 10 in different arrangements. Then, list the factors of 10.

2

Show 18 in different arrangements. Then, list the factors of 18.

4

1 238 children went to a school camp. If one tent can be shared by 4 children, number of clothes collected = 1366 + 1000 Find the factors of the following numbers using multiplication. write the placeTotal value chart for 348673:

9

10

860 people have been invited to a banquet. The caterer is arranging tables. Each table can seat 10 Number of clothes collected during the 5first day of the drive = 1366 people. About how many tables are needed?

[Round off the dividend to the nearest hundred].

Number of clothes collected duringin thethe second day ofplace the drivein = 1000 What is the face value of the number lakhs 348673?

eck if a number is a multiple of a number using division. If the remainder Word Problems What do we need to find?

Show 20 in different arrangements. Then, list the factors of 20.

We know that face value is the numerical value of the digit in a particular place. Let us gger number is a multiple of the other3number.

For example: 5

ainder

15 – 15 00

3 11

Example 6

5 d 3916 e 340 18 – 17 15 Find the factors of the following numbers using division.

5 a b c d e 9 11 12 13 15 leaves remainder 1 01

12

a 14

b 21

14

13

c 36

42

f

16

15

18

f

19 L

g 48

3

h 50

20 30 TTh

i

77

4

approximately how many tents will be needed for the camp?

Th

H

8

6

2

₹5734 is distributed between 2 groups of students. Each group has 11 students.

T

O

7

3

Estimate the amount of money that each student gets.

Points to Remember HOTS: end Is 18 a factor of 126? Explain your answer.

6 The face value ofChapter the number in the Lakhs place is 3. by 5, we get 0 remainder.

On dividing 16 by 5, we get remainder 1.

Is of 6 a the factorgiven of 64? Explain your answer.

7 Find the first five multiples numbers.

2 • The number being divided is called the dividend. The number by which we divide is Applicative summary called the divisor. The result of the division is called the quotient. The number left over Do It iple of 5.

So, 16 8is not multiple 5.10 have exactly TWO factors?

Which a numbers betweenof 1 and Thousands division is called the place remainder. in 800234? a 7 b 8 c 9 dTogether e 11value of the number in the after 10 What is the face and analytical • To check if our answer after division is correct, we can use: Dividend = (Quotient × UM24CB_G4.indb 30

18

Which number has the greatest number of factors between 5 and 15?

What is the smallest number that has exactly three factors?

s and Factors

h 14

Points to Remember m

Word Problems

19

i

place value chart for 800234: j 16 15 Let us write the

n 20

81

•

The place value table is divided into groups called periods.

•

5-digit numbers have 2 periods - Thousands Period and Ones Period.

•

Face value is the numerical value of the digit in a particular place in a number.

L

o TTh 25

Th

H

________

________

________

________

Tina bought 16 eggs. She wants to arrange them into a tray. In how many Chapter 5 • Multiples and Factors • 16-digit numbers have 3 periods - Lakhs Period, Thousands Period and Ones Period.

• • •

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When a number is divided by itself, then the quotient is 1.

•

When 0 is divided by any number, then the quotient is always 0.

•

When a number is divided by 10, the digit at the ones place forms the remainder and the remaining digits form the quotient.

•

When a number is divided by 100, the digit in the ones place and tens place forms the remainder and the remaining digits form the quotient.

T

O

________

________

The face value of the number in the thousands place is _________.

• When a number is divided by 1000, the digits in the ones place, tens place and

ways can she arrange the eggs?

Raman, a baker, baked biscuits.

He wants to place arranged from thehas smallest to the72 largest are said to be in an ascending order. the same number 2Numbers

a

thousands place form the remainder and the remaining digits form the quotient.

of biscuits each packet. are possible?

Numbers arranged in from the largest to theWhat smallestdifferent are said to bearrangements in a descending order.

Representing 6-digit Numbers

Rounding numbers is helpful when we need an estimate and when we want to convey 9/11/2023 4:26:02 PM numbers in an easier way.

77

Chapter 4 • Division

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Place value Scavenger Hunt:

to interactive

Materials Required: Newspapers, Magazines, or the Internet

For 6-digit numbers, the Lakhs place falls in the Lakhs Period.

Setting: Groups of 4

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digital resources 9/11/2023 4:26:15 PM

1

Divide the entire class into groups of 4.

classroom

2

Each group can be dedicated a particular category like City population, State population,

Lakhs Period

activity

3

Each group collects the data on their respective category from newspapers, magazines, or

4

Each groups’ data must include at least 5 numbers in their category.

Lakhs

5

Each group discusses and deduces the following for each number in their data:

Followers of celebrities, Number of speakers of a language, and Car and Bike prices. the internet.

Place value and face value of each digit.

Thousands Period Ten Thousands (TTh)

Correctly written number names.

Lakhs Period

Correctly order the numbers in ascending and descending order Round off the numbers to the nearest 10s, 100s and 1000s.

UM24CB_G4.indb 22

Ones Period

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

We can therefore represent our 6-digit number 781005 as:

Correct number representation.

22

9/11/2023 4:25:58 PM

We learnt that the Thousands period includes Ten Thousands and Thousands places. The QR Code: Access Ones period includes the Hundreds, Tens and Ones places.

Math Lab

multidisciplinary and fun

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

•

Let us continue to learn about the4:26:03 “periods” convention for large numbers.

9/11/2023 PM

92

A

83

Divisor) + Remainder.

•

Chapter 1 • Numbers up to 6-digits

9/11/2023 4:25:13 PM

7,81,005   

l

9

10

  

k 17

g 13

  

f 12 questions

9/11/2023 4:25:17 PM

Thousands Period

Ones Period

iii 9

G rad ual Rel ease of Re spon si bi li t y

The Gradual Release of Responsibility (GRR) is a highly effective pedagogical approach that empowers students to learn progressively by transitioning the responsibility from the teacher to the students. This method involves comprehensive scaffolding—including modelling, guided practice, and ultimately fostering independent application of concepts.

GRR, endorsed and promoted by both the NEP 2020 and NCF, plays a pivotal role in equipping teachers to facilitate age-appropriate learning outcomes and enabling learners to thrive.

The GRR methodology forms the foundation of the IMAGINE Mathematics product. Within each chapter, every unit follows a consistent framework: 1. I Do (entirely teacher-led)

2. We Do (guided practice for learners supported by the teacher) 3. You Do (independent practice for learners) GRR Steps

Unit Component

Snapshot

Numbers Beyond 9999 Ajay: Hello daddy, I found this letter, it is for you.

Real Life Connect

Father: Thank you.

Ajay: The letter has your name and the address of our home.

Father: Yes.

Real Life Connect

Ajay: But, what is this big number 781005?

From: Ajay Shukla, 12, Hathipol,

Guwahati - 781005

Father: It is a special code, also called the postal code. This code helps in finding the exact location in a city.

Theoretical explanation

Ajay: Okay daddy. But it has 6-digits, and I find it hard to read!

Facts about Multiples

• Every number is a multiple of 1 and the number itself.

ForAbout example, 55-digit × 1 = 5. Here, 5 is a multiple of 1 and 5.

All Numbers!

• Every multiple is either greater than or equal to the number itself.

ToFor help Ajay understand 6-digit numbers, let's first learn about 5-digit numbers. example, the multiples of 8 are 8, 16, 24, 32, 40, … and so on. Here, each multiple is

to that or greater 8. thousand nine hundred ninety-nine is the greatest 4-digit Weequal know 9999than – nine • Every number has an unlimited number of multiples. number.

I do

For example, the multiples of 7 are 7, 14, 21, 28, 35, …, 70, 77, …, 7000, …, 70000, …,

Now, we add 1 to this, we unlimited. get 10000. and when so on. Here, multiples of 7 are Example 1

Find the 5 multiples of 4.

9999 + 1first = 10000

Remember!

We can find the multiples of 4 by using the number line showing jumps of 10000 4. is the smallest 5-digit

10000 is read as “Ten Thousand”.

1

2

3

4

Let us learn more about 5-digit numbers!

0

1

2

3

4

5

6

7

8

5

number. 99999 is the greatest 5-digit number.

9 10 11 12 13 14 15 16 17 18 19 20

Place Values and Expanded Form in 5-digit Numbers We can find the multiples of 4 by using multiplication tables as follows:

We4 know that a 4-digit number has 4 places on the place value chart - ones, tens, ×1= 4 hundreds and thousands. The place on the left to the Thousands place is called the Ten Did You Know?

4×2= 8 Leap years are always Thousands place.

4 × 3 = 12

Examples

multiples of 4. For example

× 4 take = 16a 5-digit number 13435. The place value chart Let4 us for this can be written the years 2016, number 2020, 2024, ... as:4 × 5 = 20 are all leap years.

The five multiples TThfirstTh H T of 4Oare 4, 8, 12, 16 and 20.

Example 2

Did You Know?

1 3 4 3 5 Find the first 5 multiples of 5. Check by dividing whether 95 is a multiple of 5.

The number 4 is the 1 2 3 4 5 only number with the same number of Always remember that place value is the letters as its value in value of the digit in a number based on the English language.

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 its0position in the given number.

The place 100 5= ×400.

5 ×value 1 = 5 of “4” 5 × in 2 =13435 10 5is × 34 =×15 4 = 20

iv

2

The first five multiples of 5 are 5, 10, 15, 20 and 25.

5

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5 × 5 = 25

leaves remainder 0

95 – 95 00

19 On dividing 95 by 5, we get 0 remainder. So, 95 is a multiple of 5.

9/11/2023 4:24:58 PM

Find the factors of 36 using the division method.

Do It Together

GRR Steps

Divide numbers by 36

Unit Component

What do we get?

Check the remainder

36 ÷ 1

36

0

36 ÷ 2

____

0

____

Yes

36 ÷ 4

____

0

____

Think and Tell Do we need to go

Find 20 and 30. Also, common factor of both 36 ÷ 3the common 12 factors of ____ ____ find the highest beyond 6 to find the numbers. more factors of 36?

Do It Together

Find the method.

Step 1 factors7of 36 using1 the division 36 ÷ 5 ____

Do It Together

Factors of 20 36 ÷ 6 Divide numbers by 36

We do

Are the numbers

factors of 36?

Snapshot

1 2 0 What do we get?

____

4 5 10 20 Yes Check the Are the numbers remainder factors of 36?

Step 2 So, the factors of 36 are ___________________________________________________________________.

Do It Together

36 ÷of130 Factors

36

0

Yes

Step363÷ 2

____

0

____

Common 36 ÷ 3Factors

12

____

____

Do It Yourself 5C

1

of 20 and 30 3610 ÷ in 4 different arrangements. ____ Show Then, list 0 the factors of 10. ____

2

Show Then, list 1 the factors of 18. ____ 3618 ÷ in 5 different arrangements.

7

Do we need to go beyond 6 to find more factors of 36?

The common factors of 20 and 30 are ____________________________________________________.

Show 20 in different arrangements. Then, list the factors of 20.

3 4

Think and Tell

36 ÷ 6

____

0

Yes

Math Lab of the following Find the factorscommon numbers multiplication.

The highest factor of 20 using and 30 is __________.

So,a the b 21 of 36c are d 39 e 40 f 42 g 48 h 50 i 77 14 factors 36 ___________________________________________________________________.

Board Game of Multiples

Find the factors of the following numbers using division.

Do It Yourself 5D

5

Setting:bIn groupsc of 4

a 9

11

12

d 13

e 15

f

18

Materials grid as shown below, dice, crayons Do ItRequired: Yourself Number 5C

Find the common factorsyour of the following numbers.

1 18 Is a factor of 126? Explain answer.

6

8, 10 of 64? Explainbyour 12, answer.

15 IsMethod: 6a a factor

7

c 13, 16

d 14, 20

e 16, 18

Show 10 in different arrangements. Then, list the factors of 10. 45 j 72, i 54, 64 12 2681

g h 35, Each chooses their 20, 30 player 33, 10 44 50 factors?

1 f numbers between 1 and have colour. exactly TWO 8 1 Which

31

number.

4 player chooses multiple that Also, find the lowest and the highest 3 FindThe the factors ofathe pairsnumbers of of numbers.

3 Find thecommon factors of the following using multiplication.

18 20 22 48 Word common Problems number on Show the board and shades it with factors. it with a diagram. a 14 b 21 c 36 d 39 e 40 21 f 42 30 17g 4822 their colour.

Do It Yourself

a is6the b arrangements. c Then, d 3 40 8that has exactly 12 factors? number three Show 20smallest in different list the factors of 20. 50 e 436 10 3 What

Word Problems

1

a 16 and 24

1

16

a common factor of allyour the numbers.

Isc 6 0aisfactor of 64?

Explain answer. _______

e 6 is number a common factor 18, 30 and 66. _______ Which has theof greatest number of factors between 5 and 15?

f The lowest common factor of 20, 34, 39 and 42 is 1. _______ Chapter Checkup 10 What is the smallest number that has exactly three factors?

5

9/11/2023 4:26:15 PM

Radhe says, “The number 14 has a greater number of factors than 45.” Is he correct? Verify your answer.

Word Find theProblems factors.

c 72 d 88 eggs. She wants to arrange them into a tray. iIn how many g 16 1201 Tina bought 156 200 h 180

ways can she arrange the eggs?

e 98 95 j

222

Find the common factors of the given pairs of numbers.

2

Raman, a baker, has baked 72 biscuits. He wants to place the same number 9/11/2023 4:26:19 PM

Which of these pairs of numbers have the common factor of 4?

92

a 5 and 20

b 20 and 100

c 12 and 36

d 60 and 200

c 15

d 23

Write the first 5 multiples. a 7

b 11

e 30

Write the smallest number which is a common multiple of the given number pairs.

UM24CB_G4.indb 92

a 2 and 5

e 10 and 15

6

77

d 15 and 25 havebetween a total of 1 3 and common factors. _______ Which numbers 10 have exactly TWO factors?

a 7 and 14 c 9arrangements of biscuits b in each packet. are possible?

24 and 30 What different and 12 d 20 and 25

5

i

42

7

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4

9

of biscuits in each packet. What different arrangements are possible?

f

3

6 10

aChapter 45 5 • Multiples andbFactors 66 2

14

13 have no common _______ aand factor ofwho 126?

Explain factors. yourmost answer.

The player colours the number of multiples on the board is the winner.

5Isb 1811

9

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79

f 818

6

92 8

Chapter Checkup

33

h 50 11 60

b 21 and 42 c 63 and 18 d 55 and 100 e 48 and 84 Tina bought 16 eggs. She wants to arrange them into a tray. In how many

Inthe case a player gets 1 on the dice, they 67 factors of the following numbers using division. 7 5 4 4Find ways can she arrange the eggs?

Write if True or False. can choose any number on the board. b 11 c 72 dHe13 e is 15 9 12biscuits.

18 a baker, has baked wants to36 place samefnumber The biggest common factor of numbers 24 and 3.the _______ 2aa Raman, (Do you know why?)

You do

28

18 inhas different arrangements.

Then, list the of number therolls greatest of factors between 5factors and 15?the One player the number diceare and sees Which of the following numbers factors ofthe 78 and 96?

Circle correct 9 2 Which 5 18.

24 option.

15 Verify 42your answer.

25 35 2 2Show

b 3 and 7 f

10 and 25

c 5 and 8

g 11 and 22

9/11/2023 4:26:15 PM

d 4 and 14

h 20 and 24

Find the following. a Multiples of 4 that are smaller than 30.

Pearson, P. D., & Gallagher, G. (1983). Contemporary Educational Psychology.

b Multiples of 6 that are smaller than 50.

Fisher, D., & Frey, N. (2021). Better learning through structured teaching: A framework for the gradual release of responsibility. c Multiples of 8 that are greater than 30 but smaller than 80.

Fisher, D., & Frey, N. (2014). Checking for understanding: Formative assessment techniques for your classroom.

97

Chapter 5 • Multiples and Factors

v

Gradual Release of Responsibility UM24CB_G4.indb 97

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C o nt e nt s

1 2 3 4 5 6

vi

Large Numbers ....................................................................................................................1 Estimation and Operations on Large Numbers.............................................................16 Whole numbers .................................................................................................................33 Playing with Numbers: Factors and Multiples...............................................................53 Line Segments and Angles ...............................................................................................75 Triangles, Quadrilaterals and Polygons .........................................................................91

1

Large Numbers

Let's Recall The numbers used in our daily lives are written using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We can form different numbers ranging from 1-digit up to 7-digit numbers. Such numbers are written using commas after every period, starting from the ones period.

For example, let us say the population of a country is 2347709. This number can be written using commas by representing it in a place value chart.

Lakhs

Thousands

Ones

Number

Ten Lakhs (TL)

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

23,47,709

2

3

4

7

7

0

9

Each digit of a number has a place value and a face value. Place values of each digit are used to write its expanded form and number name. For example, the population of a country can be represented as shown below.

Face Value Place Value 2347709 9 ones

=9×

7 hundreds

=7×

0 tens

7 thousands

4 ten thousands 3 lakhs

2 ten lakhs

=0× =7×

=4×

1

=

9

100

=

700

10

1000

10000

= 3 × 100000

= 2 × 1000000

= =

=

=

0

7000

40000

300000

= 2000000

Expanded form: 2000000 + 300000 + 40000 + 7000 + 700 + 9 Number Name: Twenty-three lakh forty-seven thousand seven hundred nine

Let's Warm-up Write True or False.

1 The number 5865952 can be rewritten using commas as 5,865,952.

_________

2 The place value and face value of 0 in 9,80,564 is 0 in each case.

_________

3 The expanded form of 3,83,452 is 300000 + 80000 + 3000 + 400 + 50 + 2.

_________

4 The number name of 7,03,478 is seven lakh three hundred four thousand seventy-eight.

_________

5 Between 40,73,548 and 40,72,187, 40,73,548 is greater.

_________ I scored _________ out of 5.

Understanding Roman Numerals Real Life Connect

Krish went to the market to buy a clock. At a shop, there were some clocks with numbers 1 to 12 and some with unusual numbers. He was confused and could not read these unusual numbers.

Roman Numbers The clock with unusual numbers is another way of writing digits up to 9 except 0. Such numbers are called Roman numbers.

This is the oldest system of writing numbers and is called the Roman number system. It is still in use in some cases like numbered lists, clock faces, etc.

In this system, we use seven basic symbols to write different numbers. It does not have the concept of place value, so there is no representation for 0.

Roman Numbers

I

V

X

L

C

D

M

Numbers in the Indian System

1

5

10

50

100

500

1000

There are several rules to writing Roman numbers.

Rule 1: A Roman number symbol cannot be repeated more than three times. Repeating it means adding its value as many times as it is repeated.

Remember!

XXX = 10 + 10 + 10 = 30

The symbols V,

CC = 100 + 100 = 200

L, D can never be repeated or subtracted, whereas

Rule 2: If a symbol of smaller value is written to the right of a symbol of greater value, the resulting number is equal to the sum of the symbols.

VI = 5 + 1 = 6

CI = 100 + 1 = 101

DI = 500 + 1 = 501

MCXXXV = 1000 + 100 + 10 + 10 + 10 + 5 = 1135

2

the symbols I, X, C, and M can be repeated up to three times.

So, the numbers on the clock that Krish saw are read in the way given.

XII= 10 + 1 + 1 = 12 I=1

XI = 10 + 1 = 11

II = 1 + 1 = 2

X = 10

IX = 10 – 1 = 9

III = 1 + 1 + 1 = 3

VIII = 5 + 1 + 1 + 1 = 8 VII = 5 + 1 + 1 = 7

IV = 5 – 1 = 4

VI = 5 + 1 = 6

V=5

Write the number form of the Roman numbers.

Example 1

1 CDXXXIX = CD + X + X + X + IX

2 MCMLXXXIV = M + CM + L + X + X + X + IV

= (500 – 100) + 10 + 10 + 10 + (10 – 1)

= 1000 + (1000 – 100) + 50 + 10 + 10 + 10 + (5 – 1)

= 400 + 10 + 10 + 10 + 9 = 439

= 1000 + 900 + 50 + 10 + 10 + 10 + 4 = 1984

Write the Roman numbers for the numbers.

Example 2

1 97 = 90 + 7

2 1567 = 1000 + 500 + (50 + 10) + 7

= (100 – 10) + 7 = XC + VII = XCVII

= M + D + LX + VII = MDLXVII

Error Alert!

Remember!

The symbol I can be subtracted from V and X

215 = 100 + 100 + 5 + 5 + 5 = CCVVV

only, whereas the symbol X can be subtracted from L and C only. Similarly, the symbol C

215 = 100 + 100 + 10 + 5 = CCXV

Do It Together

can be subtracted from D and M only.

Match the numbers with their Roman numbers.

1 45

MCCCXC

2 98

CX

3 110

XLV

4 185

XCVIII

5 1390

CLXXXV

Chapter 1 • Large Numbers

3

Do It Yourself 1A 1

Write Roman numbers for the numbers. a 99

2

b 184

4

5

d 438

e 1564

f 1287

d CCCLXV

e MDCCCLXIV

f CMXCIX

Write number form for the given Roman numbers. a CDLXVI

3

c 285

b

XCVI

c CXV

Fill in the blanks using < or >. a CLXXXVII _______ CXLVII

b XC _______ CX

c CXCII _______ CC

d LXIX _______ LXXII

e CXCVII _______ CLXXV

f MCXLVI _______ MCMXXIV

Write the next number just before and after the Roman numbers given. a _______ CDLXV _______

b _______ CMLXVII _______

c _______ MCCCXLV _______

d

_______ MXV _______

Solve and express the final answer in Roman numbers. a CDXL – CCCXIV

b DXVIII + XCII

c MDXL – MCCLXX

d MCX + XC

Word Problem 1

There are CM people in one village of a town and CCCL in another village. How many people are there in total?

Understanding Large Numbers Real Life Connect

The data on the current population of a few countries were given.

While reading the article, Ritu was confused and could not read the numbers given in the data.

Indonesia

277534122

Russia

144444359

United States 339996563

Japan

123294513

Indian Number System The population of each of the different countries that was given in the data is 9-digit numbers.

Indonesia

277534122

Russia

144444359

United States 339996563

9-digit Numbers

Let us see how to read and write large numbers using place value.

4

Japan

123294513

Face Value and Place Value

We know that the greatest 8-digit number is 99999999. If we add 1 to this number, we get 100000000. This is the smallest 9-digit number with a new place called ‘Ten Crores’ place.

Crores

Lakhs

Thousands

Ones

Ten Crores Crores Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones (TC) (C) (TL) (L) (TTh) (Th) (H) (T) (O) 9

9

9

9

9

9

9

+

9 1

1

0

0

0

0

0

0

0

0

periods place Smallest 9-digit number

new place Ten Crores (C) place

We saw in the article that the population of Indonesia is 277534122. We can write it as: Crores

Lakhs

Thousands

Ones

Ten Crores Crores Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones (TC) (C) (TL) (L) (TTh) (Th) (H) (T) (O) 2

7

7

5

3

4

1

2

2

periods place

Every digit in a number has a fixed position according to which the digit has its value. The fixed position is called the place of a digit and the value of the digit is called the place value of the digit. The face value is the digit itself.

To find the place value of any digit, we multiply the digit by its value in a place of the number, whereas the face value does not depend upon the position of the digit in the number. For example, 277534122 can be represented as 277534122

Face Value 2 ones

2 tens

1 hundreds

4 thousands

3 ten thousands 5 lakhs

7 ten lakhs 7 crores

2 ten crores

Example 3

Place Value =2× =2× =1× =4× =3× =5× =7×

1 =

10 =

100 =

1000 =

10000 =

100000 =

1000000 =

2

20

100

4000

30000

500000

7000000

Remember!

The place value of zero is always 0. It may hold any place in a number, its value is always 0.

= 7 × 10000000 = 70000000

= 2 × 100000000 = 200000000

Write the face value and place value of the coloured digits in the given numbers.

1 43,07,64,936 — 0 is in the ten lakhs place. So, both place value and face value of 0 is 0.

2 85,50,99,225 — 8 is in the ten crores place. So, the place value of 8 is 80,00,00,000 and the face value of 8 is 8.

Chapter 1 • Large Numbers

5

Example 4

Find the sum of the place value and face value of 4 in 23,43,08,970.

Place value of 4 = 40,00,000 Face value of 4 = 4 Sum = 40,00,000 + 4 = 40,00,004

Do It Together

Write the place value and face value of 3 in 13,98,70,465.

3 is in the _____________ place.

The place value of 3 is _____________.

The face value of 3 is _____________.

Expanded Form and Number Names To form a number when the place values of all the digits are added, it is called the expanded form of the number. For example, the expanded form of 277534122 is 200000000 + 70000000 + 7000000 + 500000 + 30000 + 4000 + 100 + 20 + 2 = 277534122 Expanded Form

Standard form

Number Names To read and write 9-digit numbers in figures and words, we group the digits according to the periods and put commas after each period starting from ones. This helps us read the numbers easily. For example, 277534122 can be read in the following way.

27,75,34,122 Crores period

Lakhs period

Thousands period

Ones period

thirty-four thousand

Write the number for the number name.

Ninety crore twenty-one lakh eleven thousand two hundred two We can write it as 90,21,11,202 using commas.

Do It Together

  

Example 5

seventy-five lakh

  

twenty-seven crore

  

  

200000000 + 70000000 + 7000000 + 500000 + 30000 + 4000 + 100 + 20 + 2

one hundred twenty-two

Think and Tell

Can you now read and write the population of the United States, Russia, and Japan in the Indian number system?

Write the number in its expanded form and as a number name.

56,70,34,219 = _________________________________________________________________________________________________ Number name: ________________________________________________________________________________________________ ________________________________________________________________________________________________

6

Do It Yourself 1B 1

2

3

4

Write the face value and place value of the coloured digits in the numbers. a 58,76,03,653

b 18,90,37,890

c 20,56,78,450

d 27,89,73,653

e 38,57,64,999

f 81,87,50,658

Write the expanded form of the numbers. a 93,27,59,191

b 11,07,64,787

c 56,42,86,604

d 70,21,35,436

e 89,13,45,468

f 24,85,67,835

Write the number names for the numbers. a 23,34,26,578

b 74,36,54,679

c 97,87,64,224

d 11,98,03,456

e 46,57,68,979

f 57,52,65,389

Write numbers for the number names. a eighty crore ninety-nine lakh forty thousand two hundred one b sixteen crore twenty-six lakh ninety-two thousand one hundred forty-three c sixty-five crore twenty-two lakh forty-four thousand nine hundred

5

In a number 76,47,56,229, the digits 5 and 4 are interchanged to form a new number. Answer the questions. a Will there be a change in the face values of 5 and 4? b What will be the new place values of 5 and 4? c What is the difference in the place value of 5 in the new and the original number?

6

In 15,87,12,528; how many times the value of 5 in the Hundred place is the value of 5 in the Crores place?

Word Problem 1

A lottery ticket bearing the 9-digit number 272545619 was won by Mary in a lucky draw.

Rewrite the number using commas. Write its number name and expanded form.

International Number System Like the Indian number system, there is a globally recognised system of numbers called the international number system. For 9-digit numbers, there are ones period, thousands period, and the millions period.

Each period has three places.

Millions Hundred Ten Million Million (HM) (TM)

Thousands Million (M)

100,000,000 10,000,000 1,000,000 3 places Chapter 1 • Large Numbers

Ones

Hundred Ten Thousands Thousands Hundreds Tens Ones Thousands (HTh) (TTh) (Th) (H) (T) (O) 100,000

10,000 3 places

1,000

100

10

1

periods place

3 places

7

Face Value and Place Value Like the Indian number system, each digit of a number has a place value and a face value. We find the place value and face value in the same way as in the Indian number system.

Place value of a digit = digit or face value × value of the place

For example, let us take the population of Indonesia (277534122). Its face value and place value can be shown in this way.

Face Value Place Value 277534122 2 ones

=2×

1 hundred

=1×

2 tens

=2×

4 thousands

5 hundred thousands

=5×

100

=

1000

10000

100000

7 millions

=7×

2 hundred millions

= 2 × 100000000

7 ten millions

Example 6

=3×

=

10

=4×

3 ten thousands

1

1000000

= 7 × 10000000

2

=

20

100

=

4000

=

30000

=

500000

=

7000000

= 70000000

= 200000000

Write the face value and place value of the coloured digits in the numbers.

1 203,565,768 — 5 is in the hundred thousands place. The place value of 5 is 500,000. Its face value = 5.

2 909,358,767 — 9 is in the hundred millions place. The place value of 9 is 900,000,000. Its face value = 9.

Example 7

Find the difference between the place value and the face value of 8 in 682,344,590.

Do It Together

Write the place value and the face value of 4 in 134,870,965.

Place value of 8 = 80000000;

4 is in the millions place.

Face value of 8 = 8

So, the place value of 4 is _____________.

Difference = 80,000,000 − 8 = 79,999,992

The face value of 4 is _____________.

Expanded Form and Number Names Like the Indian number system, we put commas after each period starting with the ones, but read and written differently. For example, let us take the population of Indonesia (277534122).

277,534,122 Millions period

Thousands period

  

8

100 + 20 + 2

  

Number form:

+

  

Expanded form: 200000000 + 70000000 + 7000000 + 500000 + 30000 + 4000

Ones period

two hundred seventy-seven million

five hundred thirty-four thousand

one hundred twenty-two

Remember!

In every number in each system, the value of each digit is 10 times the value of the digit on its right.

What is the expanded form of 112,086,785? Also write its number name.

Example 8

Expanded form: 112,086,785 = 100,000,000 + 10,000,000 + 2,000,000 + 80,000 + 6000 + 700 + 80 + 5 Number name: one hundred twelve million eighty-six thousand seven hundred eighty-five Do It Together

Write the expanded form and number name for the given number.

902,564,875 = _________________________________________________________________________________________________ Number name: ________________________________________________________________________________________________ ________________________________________________________________________________________________

Do It Yourself 1C 1

2

3

4

Write the face value and place value of the coloured digits in the numbers. a 580,398,653

b 179,043,456

c 205,145,450

d 807,973,436

e 385,876,243

f 754,750,213

Write the expanded form of the numbers. a 932,542,191

b 110,764,214

c 724,286,604

d 787,135,436

e 891,096,468

f 248,567,465

Write the number names of the numbers. a 233,426,578

b 743,654,679

c 978,764,224

d 119,803,456

e 345,768,979

f 575,123,389

Write numbers for the number names. a Eight hundred twenty-six million one hundred twenty thousand sixty-six b One hundred ninety-two million five hundred four thousand sixteen c Seven hundred eighteen million seven hundred ten thousand one hundred fifty-six

5

In a number 764,156,229, the digits 7 and 1 are interchanged to form a new number. Answer the given questions. a Will there be a change in the place values of 7 and 1? b What will be the new place values of 7 and 1? c What is the difference in place value of 7 in the new and original number?

Chapter 1 • Large Numbers

9

Comparing and Ordering Numbers Recall that Ritu read and wrote the population of the countries in both Indian and international number systems.

Indonesia

Russia

United States

Japan

277534122

144444359

339996563

123294513

Let us now compare the population of different countries and arrange them.

Comparing Numbers To compare two numbers, let’s say 21,04,76,827 and 21,98,70,539, we follow the steps.

Step 1 Arrange the numbers in the place value chart and check the number of digits.

Both the numbers have 9 digits.

TC 2 2

C 1 1

TL 0 9

TC

C

TL

2

1

9

L 4 8

TTh 7 7

Th 6 0

H 8 5

T 2 3

O 7 9

L

TTh

Th

H

T

O

8

7

0

5

3

9

Step 2 Start comparing the digits from the left. The number with the greater digit is greater.

Here, the digits are the same up to the crores place. Compare the digits in the ten lakhs place.

2

2=2

We see that 0 < 9.

So, 21,04,76,827 < 21,98,70,539.

1

1=1

0

4

7

8

2

0<9

Think and Tell

Remember!

Can you compare the population of Russia and Indonesia

A number with more digits is always

and tell which country’s population is smaller?

greater.

Ordering Numbers

r

A

1

ng

di

n

e sc

2

3

de or

4

smallest to greatest

10

6

5

De 5

sc

4

en

3

di

ng

2

or de

r

1

greatest to smallest

7

To arrange the population of the countries in ascending order, we follow the steps.

Indonesia

Russia

United States

Japan

277534122

144444359

339996563

123294513

Step 1 Write the smallest number.

On comparing, we notice that all the numbers

TC

C

TL

1

4

4

2

have 9 digits and 1 < 2 < 3.

1<2<3

So, let us compare numbers starting with digit 1.

3

1

On comparing we get, 4 > 2.

1=1

So, 12,32,94,513 is the smallest of all.

7

3

2

L

TTh

Th

H

T

O

4

4

4

3

5

9

7

5

9

9

3

2

4

9

9

3

6

4

1

5

5

2

6

1

2

3

3

4>2

Step 2

Think and Tell

Compare the remaining numbers. Write the next smallest number.

Can you now arrange the population

Since 1 < 2 < 3, the next smallest number is 14,44,44,359.

of the countries in descending order?

Step 3 In the same way, compare the remaining numbers if any, and write the next smallest number or write the greatest number.

Since 2 < 3, the next smaller number is 27,75,43,122.

Therefore, 33,99,96,563 is the greatest number of all.

Thus, 12,32,94,513 < 14,44,44,359 < 27,75,43,122 < 33,99,96,563 is the required ascending order.

Forming Numbers Let us understand this using an example. Let us say we are given the digits 9, 5, 1, 0, 6, 7, 2, 8, and 3.

Let us try forming numbers so that each digit appears exactly once.

510286739

710653928

610287539

501376928

701328569

601357928

513728609

712835609

613570289

Think and Tell Can you form more such numbers?

Now what if we wanted to form the greatest and smallest 9-digit numbers using these digits exactly once each?

To form the greatest number, we write the digits in descending order, 987653210.

This becomes the greatest 9-digit number that can be formed using each of the digits exactly once!

To form the smallest 9-digit number, we write the digits in the ascending order, 102356789.

Remember!

0 cannot appear in the left-most place because then it will be a 8-digit number.

This becomes the smallest 9-digit number that can be formed using each of the digits exactly once!

Think and Tell

What if the repetition of 1 digit is allowed and we want to form 9-digit numbers using 9, 5, 1, 0, 6, 7, 2, and 3?

repeat the greatest digit?

Chapter 1 • Large Numbers

Why did we choose to

11

To write the greatest number, we will choose to repeat the greatest digit.

9

9

7

6

5

3

2

1

0

This becomes the greatest 9-digit number that can be formed, while repeating only 1 digit once!

To write the smallest number, we repeat the smallest digit once. Remember not to put a 0 in the highest place!

1

0

0

2

3

5

6

7

9

This becomes the smallest 9-digit number that can be formed, while repeating only 1 digit once!

Example 9

Arrange 100,000,000; 318,675,080; 702,354,758 and 450,500,809 in descending order.

Compare the numbers, and write the next greatest number. Or, write the smallest number.

Therefore, 702,354,758 > 450,500,809 > 318,675,080 > 100,000,000 is the descending order.

Example 10

Form the greatest and the smallest 9-digit number using the digits 1, 8, 6, 7, 0, 9, 2, 5 and 4. Repetition of digits is not allowed.

The greatest number is 987654210 and the smallest number is 102456789.

Do It Together

Arrange the following numbers in both ascending and descending order.

126,856,384; 165,121,349; 860,778,653; 400,897,234 Ascending Order: _____________________________________________________________________________________________ Descending Order: ____________________________________________________________________________________________

Do It Yourself 1D 1

Arrange the numbers in ascending order. a 100,356,782; 500,040,367; 887,210,460; 931,124,820 b 927,516,890; 360,841,910; 692,180,350; 826,020,031 c 500,216,138; 604,503,821; 650,241,567; 945,241,823

2

Arrange the following in descending order. a 826,374,510; 871,926,345; 670,814,256; 450,070,921 b 423,516,789; 801,210,450; 962,115,108; 678,203,001 c 543,343,867; 967,208,891; 788,216,134; 578,206,010

3

Write the greatest 9-digit number and the smallest 9-digit number using all the digits only once. a 5, 3, 4, 0, 8, 9, 1, 7, 2

4

b 8, 3, 9, 4, 7, 1, 6, 5

c 7, 5, 2, 0, 4, 9, 3, 1

Write the greatest 9-digit number and the smallest 9-digit number using: a two different digits

12

c 1, 0, 3, 5, 6, 2, 4, 8, 4

Write the greatest 9-digit number and the smallest 9-digit number using all the digits but not repeating 1 digit. a 2, 7, 1, 0, 8, 6, 4, 5

5

b 5, 7, 6, 2, 1, 0, 4, 3, 8

b five different digits

c four different digits

Word Problem 1

The teacher pasted flashcards of 10 digits 0 to 9 on the board. The students were asked to form different numbers using the digits given. Circle the numbers that can be made.

547386291, 764238105, 547689123, 632478591, 836576124, 421358967

Points to Remember • • • • • • • •

In the Roman number system, we use seven basic symbols to write different numbers, as shown in the table.

In the Indian number system, 9-digit numbers begin with the ten crores place in the place value chart (from the left).

In the international number system, for 9-digit numbers, we have the ones period, thousands period, and the millions period.

The place value of a digit is the position of a digit in the number.

The face value of a digit for any place in the given number is the digit itself.

When place values of all its digits are added to form a number, it is called the expanded form of the number.

A number with more digits is always greater.

To form numbers using the digits given, we arrange the digits in either ascending or descending order.

Math Lab Setting: In groups of 5

Number Extravaganza!

Materials Required: Index cards with random digits (0 – 9), coloured pencils, A4-size sheet of paper Method: All 5 members of each group must follow these steps.

1

In groups, collect the index cards from the teacher.

2

Construct a 9-digit number using the digits on the index cards.

3

Rearrange their digits to form the greatest and the smallest 9-digit number.

4

One of the members of the team comes up to the blackboard and writes the number formed and

5

Continue until all teams have written the numbers in words.

6

The team that writes the number correctly in the least time wins a small reward.

its number names.

Chapter 1 • Large Numbers

13

Chapter Checkup 1

Write Roman numbers for the numbers given. a 427

2

b 2087

b MXLXXV

e 215

f

5750

c MCDIX

d DCCXCVII

e CCXVIII

Rewrite the numbers in figures and words using both the Indian and the International number system. Also, write their expanded form. a 350427681

4

d 9015

Write the number form of the Roman numbers. a CCDXVII

3

c 635

b 420879502

c 635658421

d 901500084

Write the numbers for the number names. a Four hundred sixty million seven hundred twenty-two thousand two hundred thirty-nine b Eighty crore nine lakh fifty thousand sixty-two c One hundred million one hundred thousand thirty-nine d Sixty-three crore twelve lakhs fifty-eight thousand one hundred forty-three

5

6

Fill in the blanks using <, > or =. a 656,502,567 ______ 648,900,650

b 314,572,879 ______ 314,527,879

c 900,760,518 ______ 900,768,757

d 113,005,885 ______ 113,005,885

Arrange the following numbers in ascending order. a 67,23,56,475; 19,08,04,365; 68,91,63,896; 76,90,87,687 b 676,162,895; 676,817,980; 435,406,576; 324,335,678 c 87,12,63,256; 65,45,12,845; 97,12,36,125; 65,78,15,325

7

Arrange the following numbers in descending order. a 14,56,45,768; 25,36,45,787; 12,40,85,167; 14,43,56,787 b 810,868,428; 949,076,837; 909,087,897; 810,638,964 c 456,125,789; 654,785,125, 745,125,364, 745,124,698

8

Form the greatest and the smallest 9-digit number using the digits 1, 3, 6, 8, 4, 9, 0, 2, 5 only once. Also, form the

greatest and the smallest 9-digit number using these digits except 0. Arrange the numbers in both ascending and descending order.

9

A certain 9-digit number has only eights in the ones period, only sevens in the thousands period, and only fours in the millions period. Write the number in figures and words using both number systems.

10 Solve and express your answer in Roman numbers. a XLXX + CDXV

14

b MXCVII − DCXVI

Word Problem 1 Junaid opened a bank account. He was given a customer Id: 750635389.

Rewrite this number in figures, words and number names using both number systems.

Chapter 1 • Large Numbers

15

32

Estimation and Operations on Large Numbers

Let's Recall On day 1, a restaurant earned ₹23,084 by selling food and beverages. On the second day, the restaurant made ₹45,987. How much money did the restaurant make in two days?

To find the total amount of money made in two days, we will need to add.

1

1

1

2

3

0

8

4

6

9

0

7

1

+ 4 So, the restaurant made ₹69,071 in two days.

5

9

8

7

On day 3, the restaurant set a goal to make ₹90,000. By lunchtime, it had made ₹89,646. How much more money is needed to reach the goal by the end of day 3?

To find the answer, we will need to subtract.

8

9

9

9

10

9

0

0

0

0

0

0

3

5

4

+ 8 So, the restaurant needs to make ₹354 more to reach their goal.

9

6

4

6

Let's Warm-up Fill in the blanks.

1 65,700 + 4532 = __________

2 51,000 + 4000 = __________

3 13,000 – 5678 = __________

4 23500 – 12000 = __________

I scored _________ out of 4

Mean, Median and Mode Estimation in Large Numbers Real Life Connect

Disha was reading a newspaper. A news clipping in the newspaper read: The concert raised ₹52,00,67,890 as a donation for the frontline workers of the covid 19 battle in 2020.

Rounding Off Large Numbers In the news article, the number 52,00,67,890 is very inconvenient to read and say out loud.

We can also just read it as: More than ₹52,00,00,000 has been raised for the frontline workers of the COVID-19 battle through a concert.

This gives a fair idea of about how much money has been donated for the frontline workers. This is exactly what rounding off a number is!

To show rounded off numbers, terms like “about”, “over”, “more than”, and “approximately”, etc. are added to convey that the number is close to exact.

We can round off numbers to different places such as tens, hundreds, and thousands. Let us understand how we can do this.

Remember!

Rounding off a number to the nearest ten

Rounded off numbers are the nearest values of the exact numbers.

Look for the digit at the ones place of the number.

If the digit is less than 5, the digit is replaced by 0.

If the digit is 5 or greater than 5, the digit at the tens place is increased by 1 and the digit at ones place is replaced by 0. For example, look at the numbers given below.

213786992

rounded off to

2<5

213786990

706974268

rounded off to

8>5

706974270

Rounding off a number to the nearest hundred Look for the digit at the tens place.

If the digit is less than 5, the digits at the ones and tens places are replaced by 0.

If the digit is 5 or greater than 5, the digit at the hundreds place is increased by 1 and the digit at the tens and ones places are replaced by 0. For example, look at the numbers given below.

456782901

0<5

rounded off to

456782900

Rounding off a number to the nearest thousand

612438950

rounded off to

5=5

612439000

Look for the digit at the hundreds place.

Think and Tell

If the digit is less than 5, the digits at the ones, tens and hundreds place are replaced by 0.

to the largest place of a number?

Chapter 2 • Estimation and Operations on Large Numbers

How do we round off an 8-digit number

17

If the digit is 5 or greater than 5, the digit at the thousands place is increased by 1 and the digits at hundreds, tens and ones places are replaced by 0. For example, look at the numbers given below.

154376414

rounded off to

154376000

4<5

Example 1

356873660

rounded off to

6>5

356874000

Round off 86,46,00,679 to the nearest ten.

86,45,00,679 has 9 in the ones place. 9 > 5. So add 1 to the digit in the 10s place.

So, 86,45,00,679 rounded off to the nearest ten is 86,45,00,680.

Example 2

Which of the following is the rounded off number for 32,75,98,065, if it is rounded off to the nearest thousand? a 32,75,98,060

b 32,75,98,006

c

32,75,98,000

On rounding off 32,75,98,065 to the nearest thousand, we get 32,75,98,000. So, part c is the correct answer.

Do It Together

Round off 73,26,63,547 to the nearest ten, hundred and thousand.

So, the number 73,26,63,547 rounded off to the nearest 10 is ____________________.

Did You Know?

So, the number 73,26,63,547 rounded off to the nearest 100 is ____________________.

The population of India was 1,43,16,62,681 as of September 24, 2023.

So, the number 73,26,63,547 rounded off to the nearest 1000 is ____________________.

Do It Yourself 2A 1

Round off the numbers to the nearest ten. a 478

b 7,675

c 6352

d 5650

e 94,267

f 40,536

g 76,567

h 2,54,765

i

2

20,45,546

a 545

b 3098

c 4576

d 47,23

e 86,850

f 43,607

g 8,09,757

h 7,97,005

k 3,80,08,023

l

23,98,790

j

49,46,826

89,74,53,542

Round off the numbers to the nearest thousand. a 7747

b 4677

c 13,325

d 68,723

e 48,890

f 3,42,098

g 4,89,779

h 65,43,567

k 1,03,09,095

l

i

18

j

Round off the numbers to the nearest hundred.

i

3

8,76,746

31,41,457

j

4,65,45,758

30,47,57,698

4

Write all the possible numbers that can be rounded off to the nearest hundred as 4300.

5

What is the greatest and the smallest numbers that can be rounded off to the nearest thousand to give 7000 as

6

Using the digits 5, 1, 9 and 7, form numbers that when rounded off to the nearest hundred and thousand give

answer?

7500 and 8000 respectively.

Word Problem 1

Srikant estimated a number to the nearest ten and got the answer of 567790. What are the smallest and the greatest possible numbers that can make Srikant’s answer correct?

Estimation in Operations In the news article, the concert raised ₹52,00,67,890 in 2020. If another concert raised ₹50,00,000 more in 2021, then about how much did it raise in 2021? Let us see.

To solve such questions or any other, we round off the numbers and then perform the required operation on the two rounded off numbers.

So, to find the amount of money raised in 2021 through the concert, let us round off 52,00,67,890 to the nearest 10,00,000 and then add the two numbers.

Amount raised through the concert in 2020 = ₹52,00,67,890 Rounding off ₹52,00,67,890 to the nearest thousand = ₹52,00,00,000

+

Amount raised through concert in 2021 = ₹52,00,00,000 + ₹50,00,000

5

2

0

0

0

0

0

0

0

0

0

5

0

0

0

0

0

0

5

2

5

0

0

0

0

0

0

So, ₹52,50,00,000 was raised through the concert in 2021.

Estimating the Sum and Difference To find the estimated sum and difference, we round off each of the numbers and then add or subtract.

For example, let us say we have to find the estimated sum and difference of 6584 and 2431 to the nearest hundred.

6584 rounded off to the nearest hundred = 6600 2431 rounded off to the nearest hundred = 2400 1

+

6

6

0

0

2

4

0

0

9

0

0

0

–

6

6

0

0

2

4

0

0

4

2

0

0

So, the estimated sum of 6584 + 2431 = 9000.

The estimated difference of 6584 – 2431 = 4200.

Chapter 2 • Estimation and Operations on Large Numbers

19

Example 3

Find the estimated sum of 79,643 and 10,056 by rounding it off to the nearest thousand.

79,643 rounded off to the nearest thousand = 80,000

8

0

0

0

0

1

0

0

0

0

9 0 10,056 rounded off to the nearest thousand = 10,000

0

0

0

+

Example 4

42,431 rounded off to the nearest ten = 42,430 20,564 rounded off to the nearest ten = 20,560

Thus, the estimated sum of 79,643 and 10,056 when rounded off to the nearest thousand is 90,000.

Do It Together

What is the estimated difference of 42,431 and 20,564 if they are rounded off to the nearest ten?

–

1

13

13

4

2

4

3

0

2

1

8

7

0

2

0

5

6

Thus, the estimated difference of 42,431 and 20,564 when they are rounded off to the nearest ten are 21,870.

Solve and give the approximate answer.

1 765 + 213 (to the nearest hundred)

2 984 – 356 (to the nearest ten)

765

rounded off to

__________

984

rounded off to

__________

213

rounded off to

__________

356

rounded off to

__________

So, estimated sum of 765 and 213 = ______

So, estimated difference of 984 and 356 = ______

Estimating the Product Like estimated sum and difference, we can also find the estimated product of two numbers.

For example, let us say we have to find the estimated product of 84 and 28 to the nearest ten.

84

rounded off to nearest ten

80

28

rounded off to nearest ten

30

So, the estimated product of 80 × 30 is 2400.

Now, what if we have to find the estimated product of 568 × 45?

To find the estimated product in such cases, we round off each number to its greatest place and then multiply the rounded off numbers.

568

rounded off to hundred

600

45

rounded off to ten

50

Hence, the estimated product is 600 × 50 = 30,000.

Example 5

Find the estimated product of 834 and 25 by rounding off each number to its greatest place.

834 rounded off to hundred is 800.

25 rounded off to ten is 30.

800 × 30 = 24000 Thus, the estimated product of 834 and 25 is 24,000.

Do It Together

Muskan found the estimated product of 758 and 510 as 3,50,000 by rounding off each number to the nearest hundred. Is she correct? Give reasons.

758

rounded off to

__________

510

rounded off to

The estimated product of 758 and 510 is ______________.

So, Muskan’s estimated product is ______________ (correct/incorrect).

20

0

__________

Estimating the Quotient Like the estimated product, we can also find the estimated quotient. For example, let us say we have to find the estimated quotient of 96 and 14 to the nearest ten.

96 rounded off to the nearest ten = 100 14 rounded off to the nearest ten = 10 100 100 ÷ 10 = = 10 10 Thus, the estimated quotient of 96 and 14 is 10.

Now, what if we have to find the estimated quotient of 256 ÷ 21?

To find the estimated quotient in such cases, we round off each number to the greatest place of the smaller number and then divide the rounded off numbers.

Here, the smaller number is 21. So, we round off each number to the nearest ten.

256 rounded off to the nearest ten = 260 21 rounded off to the nearest ten = 20 260 ÷ 20 =

260 = 13 20

13 20 260 –20 60 –60 0

Find the estimated quotient for 5634 ÷ 231 by rounding off each number to the nearest hundred.

Example 6

5634 rounded off to the nearest hundred = 5600

28 200 5600 –400 1600 –1600 0

231 rounded off to the nearest hundred = 200 5600 = 28 5600 ÷ 200 = 200 Thus, the estimated quotient for 5634 ÷ 231 is 28.

Do It Together

Estimate the quotient of 332 ÷ 25.

332

rounded off to the nearest hundred

__________

25

rounded off to the nearest ten

__________

__________ ÷ __________= __________ Thus, the estimated quotient of 332 ÷ 25 is __________.

Do It Yourself 2B 1

2

Estimate the sum to the nearest ten, hundred, or thousand. Compare your answers with the actual answer. a 56 + 92

b 4356 + 9120

c 89,174 + 23,589

d 21,54,759 + 9,78,422

e 89,73,215 + 97,056

f 21,46,689 + 54,202

Estimate the difference to the nearest ten, hundred or thousand. Compare your answers with the actual answer. a 89 − 52 d

21,87,759 − 9,78,524

b 9356 − 4120

c 79,174 − 23,543

e 89,56,285 − 89,656

f 34,46,667 − 32,095

Chapter 2 • Estimation and Operations on Large Numbers

21

3

Multiply and give the approximate answer by rounding off each number to the nearest ten, hundred or thousand. a 78 × 23

4

e 215 × 956

f 406 × 202

b 4784 × 120

c 5437 × 235

d 2154 × 78

e 8973 × 705

f 7146 × 54

b 4356 ÷ 1025

c 9174 ÷ 2008

d 1256 ÷ 1032

e 715 ÷ 115

f 668 ÷ 122

e 715 ÷ 56

f 468 ÷ 42

Estimate the quotient. Also, find the actual answer and compare. a 87 ÷ 9

7

d 759 × 422

Find the estimated quotient. a 67 ÷ 13

6

c 8174 × 3,589

Find the estimated product. Also, find the actual answer and compare. a 12 × 9

5

b 6786 × 9123

b 4356 ÷ 920

c 8914 ÷ 358

d 1547 ÷ 97

A number is formed by interchanging the digits 6 and 1 in 465271. On rounding it off to the nearest ten, we get 415280. If the original number is rounded off to the nearest hundred, then what is the difference between the original rounded off number and the new rounded off number?

Word Problems 1

Reeti bought a suit for ₹3659, a saree for ₹6342, and a bedsheet. If she paid ₹12,160 to the

2

Chirag purchased 45 calculators for his stationery shop. If each calculator costs ₹560, then estimate the total cost of the calculators.

3

There are 7869 apple trees in an orchard. If there are 345 rows of apple trees, then find the estimated number trees in each row.

shopkeeper, about how much did she pay for the bedsheet?

Mean, Median Mode Operations on and Large Numbers Real Life Connect

A factory can produce 15,000 garments every 3 days.

First day: 4876 garments

Second day: 5823 garments

It produced the rest of the garments on the third day. How can we find the number of garments the factory produced on the third day? Let us see!

Addition and Subtraction Let us understand how we can find the number of garments produced by the factory on the third day.

Total number of garments produced by the factory in 3 days = 15,000 Number of garments produced on the first day = 4876 Number of garments produced on the second day = 5823 22

To find the number of garments produced on the third day, we will first add the production of garments in two days and then subtract it from the total production.

1

+

1

1

4

8

7

6

0

6

9

9

5

8

2

–

3

4

9

9

10

1

5

0

0

0

0

4

3

0

1

1

0

6

9

9

Thus, the number of garments produced on the third day is 4301.

Example 7

A printer prints 47,489 copies of a newspaper on Monday. The same printer prints 45,784 copies of a newspaper on Tuesday. How many copies does it print in two days?

Number of copies of newspaper printed on Monday = 47,489

1

Number of copies of newspaper printed on Tuesday = 45,784 So, the total number of copies of newspapers printed in two days is 93,273.

1

The weight of a cylinder filled with gas is 30 kg 100 g and the weight of the empty cylinder is 14 kg 500 g. What is the weight of the gas in the cylinder?

Weight of the cylinder filled with gas = 30 kg 100 g = 30.1 kg

2

Weight of the empty cylinder = 14 kg 500 g = 14.5 kg

9

3 0 – 1 4 1 5

Weight of the gas in the cylinder = 30.100 kg – 14.5 kg So, weight of the gas in the cylinder is 15.6 kg or 15 kg 600 g.

Do It Together

1

4 7 4 8 9 + 4 5 7 8 4 9 3 2 7 3

Total number of copies printed in two days = 47,489 + 45,784

Example 8

1

. . .

11

1 5 6

After a survey, Misha found that the total population of a city in India was 37,65,392, of which, 18,90,898 were males, 16,54,976 are females and the rest were children. How many children were there in the city?

Think and Tell

Total population of the city in India = 37,65,392

kg 100 g and 14 kg

How do you think 30 500 g is written as

Number of males in the city = 18,90,898

30.100 kg and 14.500

Number of females in the city = 16,54,976

kg, respectively?

Number of children in the city = ______________ – (______________ + 16,54,976) So, the number of children in the city was _________.

Do It Yourself 2C 1

2

Solve the problems. a 4,37,548 + 2345 + 7,65,675

b 76,98,608 + 23,13,542 + 2313 + 498

c 8,79,436 – 3,24,255

d 56,47,658 – 21,432

e 6,58,769 – 23,143 + 34,546

f 9,80,76,576 + 32,435 – 7,65,878

Find the missing digits. a

2 +

2 8

7 5

0

7

1 9

5

0

2

2

8

0

8

Chapter 2 • Estimation and Operations on Large Numbers

b

6

7 9

– 3

5

8

0

8

1 4

1

2

6

7

23

3

Which number when added to 46,98,799 gives 95,34,657?

4

Which number when subtracted from 62,54,65,790 gives 24,65,47,583?

5

By how much is 59,87,700 greater than 36,57,699?

6

The sum of two numbers is 1,09,80,536. If one of the addends is 58,76,989, then find the other.

Word Problems 1

There are 6470 bags of wheat, 23,890 bags of pulses, and 60,000 bags of rice in a store for sale.

2

A school library has 6758 books in English, 9875 books in French, 4215 books in Sanskrit and 3659

3

In a football match, the number of spectators on Monday was 76,986 and that on Tuesday was

4

A car has a total capacity of 87 L petrol. If 800 mL of it was used, how much petrol is left in the car?

5

The distance between Mercury and Earth is 9,63,77,000 km. The distance between Earth and

Find the total number of bags of food grains in storage.

books in other languages. How many books are there in the library?

23,465. On which day was there a greater number of spectators? By how much?

Jupiter is 6,56,96,000 km. Assuming that Mercury, Earth and Jupiter form a straight line, find the total distance between Mercury and Jupiter.

6

The income of a company during a year is ₹27,83,67,095. This is 56,90,482 more than the previous

7

The total budget allocated by the government for a construction project is ₹35,87,925. During

year. How much did the company earn in the previous year?

the project, Shyamali keeps track of expenses and payments made during the project. During the

project, she finds that the following expenses have been incurred and payments have been made. a ₹12,46,378 were spent on materials and labour. b Payment of ₹7,23,456 was received from the client. c An unexpected expense of ₹89,723 was incurred for some additional work.

Now, if more materials are to be purchased for ₹5,78,945, what is the current budget balance for the project? Is it sufficient to complete the project?

Multiplication and Division When the factory produces 15,000 garments in every 3 days, it is clear that a large number of garments are produced in a month. Now, let us see how we can find the number of garments produced in a month.

1 month has 30 days

Number of garments produced in every 3 days = 15,000

Number of garments produced in 1 day = 15,000 ÷ 3 = 5000 24

Number of garments produced in 30 days = 5000 × 30 = 150000 So, the number of garments produced in a month is 1,50,000.

Example 9

A car can cover 85 km in 1 hour. How much distance will it cover in 32 hours?

Distance covered by a car in 1 hour = 85 km

+

Distance covered by the car in 32 hours = 32 × 85 km

+

So, distance covered by the car in 32 hours is 2720 km.

Example 10

Do It Together

2 2

1 5 7

8

5

7

0

3 5 2

2 0 0

To stitch a pair of trousers 2 m 35 cm cloth is needed. Out of 52 m 25 cm, how many trousers can be stitched and how much cloth will be left?

22 Length of cloth available = 52 m 25 cm = 5225 cm 235 5225 –470 Length of cloth required for 1 pair of trousers = 2 m 35 cm = 235 cm 525 Number of trousers that can be made from the given cloth = 5225 ÷ 235 –470 Thus, 22 trousers can be made and 55 cm cloth will be left.

55 A school in a city has 5678 students. If the annual fee per student is ₹16,000, what is the total fee collected by the school?

1 Total number of students in a school = 5678 Annual fee per student = ₹16,000 Total fee collected by the school = _______ × _______ So, the total fee collected by the school is _________.

Do It Yourself 2D 1

2

Solve. a 54,76,587 × 40

b 9,87,98,709 ÷ 9

c (3,24,45,670 ÷ 5) × 13

d 65,78,599 ÷ 3

e 3,24,543 × 120

f (79,69,889 × 6) ÷ 2

Find the missing digits. a 564_756 × 21 = 11_623_76

b 22_36_7 × 56 = 12_96_792

c 45376575 ÷ 15 = 30_5_05

d 763658890 ÷ 2 = 381_2944_

3

The product of two numbers is 5,87,69,880. If one of the numbers is 30, then find the other.

4

What should be multiplied by 7 to get the product of 5,90,41,745?

5

What should be divided by 7,28,778 to get the quotient 46?

6

The quotient of two numbers is 9,04,35,460. If one of the numbers is 20, what is the other number?

Chapter 2 • Estimation and Operations on Large Numbers

25

Word Problems 1

A bag of rice weighs 5 kg 750 g. What will be the weight of 25 such bags?

2

A vessel contains 8 L 650 mL of water. How many glasses of capacity 500 mL each will be filled with the total quantity?

3

In a company, the HR department sent 2150 emails to their customers regarding an event to be organised. If there are 8 people in the HR department, how many emails were sent in total?

4

Sara gets a scholarship of ₹3408 each year. How much money will she receive in 5 years?

5

In five days, a receptionist makes 385 phone calls. How many calls does the receptionist make in a day?

6

A nurse orders 78 sets of protective gloves. If there are 546 gloves in total, how many gloves are there in one set?

7

A car‐racing event attracted 15,690 race‐car fans. After the race, the spectators were leaving the parking lot at a rate of 52 cars per minute. There were 3 people in every car on average. How many minutes would it take for all the cars to leave the parking lot?

Parentheses and Order of Operations We know that in the first month, the number of garments produced by the factory were 1,50,000. What if the production of garments in the factory increases by 3500 each month for the subsequent 6 months? Let us find the number of garments produced in 7 months.

Number of garments produced in the first month = 150000 For the next 6 months, production of garments is increased by 3500 each month.

Number of garments produced from the second month to the seventh month each month = 150000 + 3500 Number of garments produced in the 7 months = 150000 + [(150000 + 3500) × 6] Now, we can see that the above expression has more than 1 operation and more than 1 set of brackets.

2 operations

Addition and Multiplication

150000 + [(150000 + 3500) × 6]

2 brackets

small brackets (parentheses) and square brackets

Each operation and each bracket has an order for solving them. Let us see how the order is followed.

26

Small brackets Curly brackets Square brackets

So, the expression 150000 + [(150000 + 3500) × 6] as per the given order can be solved in this way.

Step 1: Solve the brackets in the order of (), {}, []. There is no curly bracket, so we only solve inside small and square brackets.

150000 + [(150000 + 3500) × 6]

Step 2: We do not have ‘of’. So, skip the step. Next, we have to solve the operations in the order of ÷, ×, +, −. We do not have the ‘÷’ and ‘×’ sign. So, we skip these operations. Next, we solve for ‘+’ sign.

= 150000 + 921000

= 150000 + [153500 × 6] = 10,71,000

Since we do not have any other operation in the expression, we stop here.

Thus, 150000 + [(150000 + 3500) × 6] = 10,71,000 Using Brackets in Solving Problems

When we use brackets, it is easy to interpret the problem and simplify the process of solving it without any confusion.

For example, Sanya bought 3 bouquets of 8 roses each, and Aakash bought 5 bouquets of 8 roses each.

Let us see how we can find the total number of roses they have altogether. There are two ways of solving this problem.

Method 1 (without the use of brackets):

Number of roses purchased by Sanya = 3 × 8 = 24 Number of roses purchased by Aakash = 5 × 8 = 40 Together, the number of roses they purchased = 24 + 40 = 64

Think and Tell

Which method is easier to use and find the answer?

Method 2 (with use of brackets):

The number of roses Sanya and Aakash purchased = 8 × (3 + 5) = 8 × 8 = 64 In each case, the answer is the same but methods are different. Method 2 was a much faster and easier way than method 1 because we used a pair of brackets to put the number of bouquets together.

So, using brackets appropriately is very important to make the mathematical calculations easier and faster.

Expanding Brackets

We can also use brackets to expand a number.

For example, to solve 5 × 102, we use brackets to expand the second number and then use the expansion to solve it. Let us see how.

5 × 102 = 5 × (100 + 2) = 5 × 100 + 5 × 2 = 500 + 10 = 510

Chapter 2 • Estimation and Operations on Large Numbers

27

Example 11

Simplify [10 + (15 ÷ 5) × 4 – (2 × 2)] [10 + (15 ÷ 5) × 4 – (2 × 2)] = [10 + 3 × 4 – 4]

(solve small brackets)

= [10 + 12 – 4] = 18

Example 12

(solve square brackets through DMAS)

John ate 6 chocolates from a packet of 26 chocolates. Carl ate 5 chocolates from the rest. How many chocolates are left? Use brackets to find your answer.

Number of chocolates eaten by John = 6 Number of chocolates eaten by Carl = 5 Total number of chocolates = 26 Total number of chocolates left = 26 – (6 + 5) = 26 – 11 = 15

Do It Together

Solve: 8 × 105

Error Alert!

8 × 105 = 8 × (______ + 5) = 8 × ______ + 8 × ______

44 – 8 + 5 × 3 = 36 + 5 × 3 = 41 × 3 = 123

= ______ + ______ = ______

44 – 8 + 5 × 3 = 44 – 8 + 15 = 36 + 15 = 51

Do It Yourself 2E 1

2

Find the value of the number statements. a 402 – 118 + 180 ÷ 45 × 162

b 168 ÷ 14 × 22 – 210 + 185

c 15 × 25 ÷

d 35 ÷ 5 × 9 + 7 × 30 ÷ 3 of 6

3 of 21 + 20 7

Expand using brackets to find the product. a 56 × 78

3

4

b (70 − 42) ÷ 14 + 15 of 3 – 7 × 5

c [82 – 18 ÷ 3 of 2] + (18 – 6) ÷ 4

d 2400 ÷ 10 × { (18 – 6 ) + ( 24 − 12)}

Insert brackets to make the calculations true. b 3 + 4 × 8 – 4 = 28

c 10 + 8 – 4 + 1 + 5 = 8

Insert signs (+, −, ×, ÷, =) to make the statements true. b 15 ____ 5 ____ 72 ____ 8 ____ 27 ____ 18

Using brackets in different places, how many different sums and answers can you find for these two statements? a 4+6×5–2

28

d 34 × 7

a 250 + [24 – {4 of 3 + (8 – 5)}]

a 8 ____ 3 ____ 4 ____ 5 ____ 44 ____ 12

6

c 63 × 142

Simplify.

a 6 + 5 × 3 – 1 = 20

5

b 89 × 112

b 8+8×1+3–8

Word Problems 1

Damini has 3 pizzas. Each pizza is divided into 9 slices. She wants to share the pizza equally among herself and five friends.

a Circle the number sentence which shows how many slices each one of them will

get. i

(3 × 8) + (1 + 5)

ii

3 × ( 8 + 1) + 5

(3 + 8) × (1 + 5)

iii

b How many slices of pizza does Damini get?

2

A florist buys 8 sets of carnations with 5 carnations each on the first day of opening the shop. The next day he brings 12 sets of carnations with 5 carnations each. How many carnations does he buy in total?

Multi-step Word Problems We know that in the first month, the factory produces 1,50,000 garments and in the next 6 months, the factory produces 9,21,000 garments. Now, let us say that for the next subsequent months in the year, the factory produces half the number of garments in the first month. Let us see how we can find the number of garments produced in the year.

Number of garments produced in the first month = 1,50,000

Number of garments produced in the next 6 months = 9,21,000 1 Number of garments produced in 7 months = 150000 + 921000 = 1071000 2 Number of months left in a year = 12 – 7 = 5 3 Number of garments produced in the next 1 month = half of 1,50,000 = 1,50,000 ÷ 2 = 75,000

1

+

4 Number of garments produced in the next 5 months = 75,000 × 5 = 3,75,000

1

0

7

1

0

0

0

1

4

4

6

0

0

0

3

7

5

0

0

0

5 Total number of garments produced in the year = 10,71,000 + 3,75,000 = 14,46,000 So, the total number of garments produced in a year is 14,46,000.

Example 13

A teacher has 7 packets of 12 pencils each, and 2 packets of 54 pencils each. The teacher puts these pencils into 8 pencil stands. How many pencils will be in each stand?

Type 1

Type 2

Number of packets of pencils = 7

Number of packets of pencils = 2

Number of pencils in each packet = 12

Number of pencils in each packet = 54

Total number of pencils = 12 × 7 = 84

Total number of pencils = 54 × 2 = 108

Total number of pencils of both the types = 84 + 108 = 192 Number of pencil stands these pencils are put into by the teacher = 8 Number of pencils in each stand = 192 ÷ 8 = 24 Chapter 2 • Estimation and Operations on Large Numbers

29

Do It Together

Nikhil and Preeti went to a video game parlour. Nikhil won 152 tokens. Preeti won 84 tokens. They want to put their tokens together to buy a large toy monkey that costs 300 tokens. How many more tokens do they need?

Total number of tokens they have altogether = __________________ Number of tokens needed to buy a large toy monkey = _______________ Number of tokens required by Nikhil and Preeti to buy the toy monkey = _______________

Do It Yourself 2F 1

A florist arranges roses into bunches of 12 flowers each. He buys 9 bunches, each with 180 roses from one place

2

You have a collection of flash cards. They are to be kept in a folder with 12 cards on each page. You have 36

3

A printer can print 350 birthday cards. For an order, the printer needs to print the cards in a certain way.

and 4 bunches, each with 165 roses from another. How many bunches of 12 roses can be made?

complete sets of 15 cards, and another 75 cards. How many pages will be needed to store the cards?

a 270 packets of 4 cards

b 200 packets of 10 cards

How many sheets of cards will be needed to print these cards?

4

A farmer has tomato plants in his backyard. This year, the plants produced 1270 tomatoes. Birds ate 190 of the

5

Shreya bought some fabric and paid ₹3520 for it. When she got home she measured the length of the fabric and

tomatoes. 230 tomatoes had been ruined by bugs. He picked the rest. How many tomatoes did the farmer pick? found that it was half a metre less in length than what the shopkeeper had told her. She found that she had lost ₹220 because of this. What is the length of the fabric she brought home?

Word Problem 1

A local charity has 3 fundraising events. The events raise ₹17600, ₹81000, and ₹30900 each. After costs of ₹9200 on repairing of stalls etc., are deducted, the money is shared equally among three local children’s groups. How much does each group receive?

Points to Remember •

Rounded off numbers are the nearest values of the exact numbers.

•

To find the estimated sum, difference, product or quotient, we round off each of the numbers and then perform the required operation.

•

•

30

While rounding off a number to the nearest ten, hundred, or thousand we always check the digit at the next place. If the digit is equal to or greater than 5, we round up. We round down if the digit is less than 5.

BDMAS stands for Brackets, Division, Multiplication, Addition and Subtraction.

Math Lab Mystery Number Math Setting: In groups of 4 Materials Required: Flashcards with math riddles written on it, pencils, A4-size sheet, erasers, chalk and duster, set of 2 dice

Method: All 4 members of each group must follow these steps.

(A number table will be drawn up to 20 like a

1

2

3

5 4

6

8 7

9

10

12 11

13

14

16 15

17

19 18

20

hopscotch game (as shown below) before the game begins.

Flashcards with riddles will be kept at random numbers to solve. One example of a riddle is given below.

I am a 9-digit number. If you divide me by 85, then add 7, and finally multiply by 6, you'll get 16,422,990.

What number am I?)

1

Choose your captain and he/she rolls the set of dice.

2

Add the numbers you get on the dice and move that many steps forward on the number table.

3

If you find a flashcard at any number while moving in the table, solve it. You may use a sheet of

4

If you get the answer right, you are allowed to move two extra steps forward as a reward. If you

paper, a pencil and an eraser to solve the riddle.

get the answer wrong, you will have to move 5 steps back as a penalty from the number that you get on the dice.

(Every time a riddle is solved, it is replaced by a new one by the teacher.)

5

The group that completes 1 round (to and fro) from 1 to 20 first wins a small reward.

Chapter Checkup 1

2

3

Round off the numbers to the nearest ten, hundred, thousand.

Ten

7,65,378

65,408

57,67,883

8,76,537

Hundred

2,34,568

43,336

1,23,35,032

4,25,489

Thousand

9,03,426

83,211

47,69,865

2,56,79,067

Round off the numbers as given. Find the actual and estimated answer. a 76,426 + 23,143 (to the nearest 10)

b 4,36,476 − 23,134 (to the nearest 100)

c 9,05,042 × 321 (to the nearest 1000)

d 7653 ÷ 43 (to the nearest 1000)

Simplify the sentences. a [72 – 36 ÷ 6 of 2] + (30 – 22) ÷ 5

b 9 [8{12 – 7 + 5}] of 8

c 888 × [370 ÷ {65 + (18 ÷ 2)}]

d 60 + 13 – {(5 ×

Chapter 2 • Estimation and Operations on Large Numbers

1 of 10) – 75 ÷ (17 – 2)} 2

31

4

Find the error. Solve correctly. a 50 – [30 + {40 – (20 – 10)}] = 10

5

Expand using brackets to find the product. a 6 × 92

6

b [81 – 36 ÷ 6 of 2] + (30 – 22) ÷ 4 = 62

b 8 × 112

c 9 × 162

d 3 × 87

Which operations (+, −, ×, or ÷) would you use in the blank boxes in the given number sentence to get the answer 50?

50 – [140 ______ 2 ______ 5 + 50 ______ 6]

7

Find the value of: a 67589 – 23543 + 13678

8

9

b 90800 + 32552 – 45765

Which of the following is the correct option? a 24369 ÷ 3 = 8121, 8121 × 4 = 32,490

b 24360 ÷ 3 = 8210, 8210 × 4 = 32480

c 24369 ÷ 3 = 8112, 8112 × 4 = 32,448

d 24369 ÷ 3 = 8123, 8123 × 4 = 32,492

What numbers would you choose in the blanks to make the given statement true? [_____ ÷ (____ × _____ – ___ × 5)] – [____ ÷ (____ × 5 – ____ × 5)] = 3

10 Write any two possible number sentences with all the operations that have a value of 100.

Word Problems 1

An e-commerce company sells 76,578 books in the first month of the year 2000.

Estimate the number of books that the company will sell in the whole year if the company sells an equal number of books each month.

2

A factory manufactured 7,65,875 toys on Monday and 4,36,586 toys on Tuesday. Find an estimated number of more toys manufactured on Monday.

3

In the school library, there are 12,536 academic books, 15000 storybooks, and 12,549 children’s magazines. Each student can borrow up to 3 magazines and 2 storybooks at the same time. a b c

32

How many books are there in total?

There are 4 shelves for academic books. To divide the academic books equally among the shelves, how many academic books should be put on each shelf?

1018 children’s magazines are on loan. How many are left in the library?

4

A shipping company has a fleet of 35 cargo ships, each capable of carrying 27,850 tons of cargo. These ships are divided into three fleets: Fleet A, Fleet B, and Fleet C. Fleet A has 9 ships, Fleet B has 12 ships, and Fleet C has the remaining ships. Each year, the total cargo capacity of all the ships in Fleet C is filled to its maximum, and the cargo capacity is divided equally among the ships in Fleet C. Using the information, find:

a

The total cargo capacity of Fleet A and Fleet B combined.

b

The total cargo capacity of Fleet C.

c

The amount of cargo that each ship in Fleet C carries every year when it's filled to its maximum capacity.

3

Whole numbers

Let's Recall Imagine we are in an apple orchard. Each tree in the orchard has a number starting from 1, 2, 3, and so on...

These are called natural numbers.

The natural numbers are the counting numbers that start from 1 and go on infinitely without any end.

Let us see natural numbers on a number line.

Natural Numbers 1

2

3

4

5

6

7

8

9

We can skip count by numbers on a number line. Let us skip count by 2 on a number line. +2 0

1

+2 2

3

+2 4

+2

5

6

7

+2 8

+2

+2

9 10 11 12 13 14 15

Let's Warm-up

Help the squirrel to jump 4 times with the help of skip counting.

1 By 3

2 By 4

3 By 5

4 By 6

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

I scored _________ out of 4.

Whole Numbers and Natural Numbers Real Life Connect

Robin and Riya were counting flowers of each colour in their backyard. They wanted red, yellow and pink flowers.

Robin: There are 10 red flowers and 19 yellow flowers but there are no pink flowers.

Natural and Whole Numbers

0

The numbers 10 and 19 are natural numbers. The numbers 1, 2, 3, 4… are called natural numbers. All natural numbers along with 0 are called whole numbers.

1, 2, 3, 4, 5,...

Natural Numbers Whole Numbers

Predecessor and Successor Predecessor comes before the given number, i.e., Given number – 1.

Think and Tell

Successor comes after the given number, i.e., Given number + 1.

9

Predecessor of 10

Example 1

10

11

Are all natural numbers whole numbers?

Successor of 10

Write the successor and predecessor of 12499. Write the next 3 consecutive numbers after 12499.

The successor of 12499 = 12499 + 1 = 12500 The predecessor of 12499 = 12499 – 1 = 12498 The three consecutive numbers after 12,499 are 12,500, 12,501 and 12,502.

Do It Together

Remember!

Every whole number

Write down three consecutive numbers just preceding 87453.

except zero has an

The predecessor of 87453 = 87453 – 1=87452

immediate predecessor.

The predecessor of _______ = _______ – 1= _______ The predecessor of _______ = _______ – _______ = _______ Therefore, the three consecutive numbers before 87453 are _______, _______ and _______.

Representing Numbers on a Number Line Natural Numbers 1

2

3

4

5

6

7

Whole Numbers 8

9 10

0

1

2

3

4

5

6

7

8

9 10

The distance between two corresponding points on a number line is called a unit distance.

Example 2

Represent 7 on the number line.

0

34

1

2

3

4

5

6

7

8

Think and Tell 9 10

What is the largest whole number?

Do It Together

Imagine you're at a train station with a platform divided into sections, each labelled with whole numbers.

0

1

2

3

4

5

6

7

8

9 10

1 If you're standing in section 3, the whole number that represents the section just before the one you're in is ____________.

2 If you take a step to the right and move to section 5, the whole number that represents the section you were previously in is ____________.

Operations on a Number Line Add: 5 + 3

Subtract: 9 – 4

Jump 3 steps of unit distance forward

Jump 4 steps of unit distance backward

+1 +1 +1 0

1

2

3

4

5

6

7

–1 –1 –1 –1

8

9 10

0

Thus, 5 + 3 = 8

1

2

3

+3

+3 2

3

7

8

9 10

Start from 15 and Jump 3 units backward till we reach 0.

4

+3 5

6

7

+3 8

–3

9 10 11 12

Therefore, 4 × 3 =12

Example 3

6

Divide: 15 ÷ 3

Start from zero and jump 3 units forward, 4 times.

1

5

Thus, 9 – 4 = 5 Multiply: 4 × 3

0

4

0

1

–3 2

3

4

–3 5

6

7

–3 8

–3

9 10 11 12 13 14 15

Therefore, 15 ÷ 3 = 5

Each student in a class has 3 pencils. How many pencils do 6 students have?

Number of pencils each student has = 3; +3 0

1

Total number of students = 6

+3 2

3

4

5

+3 6

7

+3 8

+3

+3

9 10 11 12 13 14 15 16 17 18 19 20

Total number of pencils = 6 × 3 pencils. Therefore, 6 students have 18 pencils.

Do It Together

Vansh bought 4 candies for ₹20. How much did one candy cost? Show your answer on the number line.

0

1

2

3

4

5

6

7

Cost of 4 candies = `_______________;

8

9 10 11 12 13 14 15 16 17 18 19 20

Cost of 1 candy = ________ ÷ ________

Therefore, the cost of 1 candy is `____________.

Chapter 3 • Whole numbers

35

Do It Yourself 3A 1

Find the predecessor and successor of the numbers. a 66778

2

3

4

b 54557

c 86457

d 89665

What is the distance between the points on the number line? a 0 and 12

b 6 and 16

c 3 and 20

d 4 and 18

e 12 and 18

f 2 and 14

Locate the missing points on the number line below.

0

10

20

0

25

50

40

75

60

80

125

100

175

Write the results of the number line representation in each case. a

b 0

5

e 97564

10 20 30 40 50 60 70 80 90 100

0

3

6

9 12 15 18 21 24

Evaluate using a number line. a 15 + 7

b 14 – 9

c 7+5

d 18 ÷ 3

e 2×5

f 20 ÷ 5

6

Write the next 5 whole numbers after 13214.

7

What is the smallest whole number? How many whole numbers are there between 67 and 75?

8

On dividing a whole number by 5, the quotient and the remainder are the same. Find such a whole number.

Word Problems

36

1

Adi was baking cookies. Out of 15 cookies that he baked, 7 got burnt. Find the number of

2

A class of 20 students decides to contribute ₹5 each for a class party. Find the total amount

cookies that weren’t burnt using a number line. of money contributed using a number line.

Operations in Whole Numbers The fruit aisle in a supermarket has a variety of fruits.

Real Life Connect

Robin decided to buy 4 tomatoes, 3 oranges and 5 guavas.

Addition of whole numbers The total number of fruits that Robin bought is 4 + 3 + 5 = 12.

Properties of addition

Closure property

Sum of 2 Whole Numbers = Whole number +

If 4 and 3 are two whole numbers then 4 + 3 = 7 is also a whole number.

Commutative property

Two whole numbers can be added in any order. =

+

+

If 4 and 3 are two whole numbers then,4 + 3 = 3 + 4 = 7 Additive property zero Adding 0 to any number results in the number itself. Zero is the identity element and is referred to as additive identity. +

0 = 0 +

=

3+0=0+3=3 Associative property

When three or more whole numbers are added, the sum remains the same regardless the grouping of the addends. +

+

=

+

+

(4 + 3) + 5 = 4 + (3 + 5) = 12 Example 4

Find the sum of 137 and 295 in two ways.

137 + 295 = 432;

295 + 137 = 432

⇒ 137 + 295 = 432 = 295 + 137 Do It Together

Vaani has ₹478 in her bank account and she deposits ₹22 more. Her sister has ₹589 in her bank account and she deposits ₹53 more. How much money do Vaani and her sister have in total? Find using the associative law.

Money in Vaani’s bank account = `_________

Money on her sister’s bank account = `_________

Money Vaani deposited = `_________

Money her sister deposited = `_________

Total money they have = ______ + ______ + ______ + ______ Chapter 3 • Whole numbers

37

Do It Yourself 3B 1

Add the numbers and check if the sum is a whole number. What property is being used? State the property. a 4 and 7

2

b 9 and 13

4

5

6

d 30 and 40

e 51 and 39

Check if the numbers for addition satisfy the commutative property of addition. a 7+8

3

c 25 and 15

b 9+10

c 13+7

d 21+18

e 49+1

Determine the sums using suitable arrangements. a 736 + 280 + 356

b 837 + 208 + 363

d 1 + 2 + 3 + 4 + 496 + 497 + 498 + 499

e 647 + 142 + 858 + 253

c 453 + 647 + 1538

Add the amounts using a short method. a 7845 + 999

b 9878 + 999

d 98789 + 9999

e 72145 + 9999

c 45456 + 9999

State true or false. a 4570 + 0 = 0

______________

b 8953 + 782 = 782 + 8953

______________

c If x is a whole number so that x + x = x, then x = 0.

______________

d Whole numbers are commutative under addition.

______________

Find the sum of the greatest and the smallest 6-digit numbers formed by the digits 2, 0, 4, 7, 6, 5 using each digit only once.

Word Problems 1

Jyoti puts ₹154 in her money box and her mother gives her ₹50. If she already has saved ₹146, how much money does she have? Does the situation satisfy the closure law and commutative law?

2

Seema got 99 marks in maths, 69 marks in English, and 91 in science. Another student Rita got 92 marks in maths, 33 in English and 84 in science. What are their total marks?

Subtraction of Whole Numbers Robin's friend Riya bought 7 apples from the fruit aisle. She gave 7 apples to her mother, 4 to her brother and ate the remaining 3.

38

Properties of Subtraction 1 Closure Property The difference between two whole numbers may not always be a whole number. a and b are two whole numbers then: Case 1: a > b –

=

Since, 14 is greater than 7 14 – 7 = 7 is a whole number Case 2: a = b –

=

0

Since, 7 and 7 are equal then 7 – 7 is a whole number.

Case 3: a < b –

=

Since, 7 is less than 14 then, 7 – 14 is not a whole number as the subtraction is not possible if the subtrahend is smaller than the minuend.

Therefore, closure property does not hold true for subtraction.

2 Commutative property If we change the order of subtraction, then the difference between the two whole numbers will not be the same.

–

≠

–

Since, 7 and 4 are whole numbers then, 7–4≠4–7 3 Property of zero When 0 is subtracted from a whole number, then the difference is the number itself.

–

0 =

Then, 7 – 0 = 7 Therefore, the property of zero holds true for subtraction.

Chapter 3 • Whole numbers

39

4 Associative property

Two whole numbers cannot be regrouped and subtracted first. Hence, the order of subtraction is an important factor.

–

–

–

–

Let 14, 7 and 4 be three whole numbers.

Therefore, the associative law does not hold true for subtraction.

14 – (7 – 4) π (14 – 7) – 4 5 Inverse Operations Inverse operations are the pairs of mathematical operations in which one operation undoes the other operation.

–

=

Addition and subtraction are inverse operations. We can say that addition undoes subtraction and vice versa.

Let 7, 4 and 3 be three whole numbers, then If 7 – 4 = 3; then 4 + 3 = 7 Example 5

Use a suitable property to solve.

(958 – 445) – 510 Also check if associative law exists for subtraction. ⇒ (958 – 445) – 510 ⇒ (513) – 510 ⇒ 3 Using associative law, 958 – (445 – 510); but 445 < 510 Therefore, the associative law does not hold true for subtraction.

Do It Together

A company made a profit of `9,85,610 in a year. They distributed `5,54,000 in total among all the employees. How much money did the company keep? Check the results.

Profit made by the company = `9,85,610 Amount distributed among the employees = `5,54,000 Money the company had left = __________ – __________ = __________

Do It Yourself 3C 1

40

Fill in the blanks. a 452 – 0 = ________

b 78,856 – ________ =0

c ________ – 1=3999

d 7893 – ________ =7893

e 10,000 – ________ =7456

f 2365 – 1256 = ________

2

3

Write two subtraction sentences. a 12 + 13 = 25

b 350 + 150 = 500

c 500 + 500 = 1000

d 786 + 14 = 800

e 9989 + 11 = 10,000

f 2365 + 3659 = 6024

Subtract (i) 9 (ii). 99 (iii) 999 (iv)9999 from the given numbers. a 12,345

4

b 23,965

c 45,684

d 74,586

e 84,126

f 99,999

‘x’ is a whole number. Find the value of x when: a x + 9 = 25

b 8 + x = 18

c 45 + x = 45

5

Find the difference between the smallest natural number and the smallest whole number.

6

Find the difference between the smallest 4–digit number and the largest 3–digit number.

Word Problem 1

A person had `10,00,000. He purchased a colour T.V. for `16,580, a motorcycle for `45,890

2

Chinmay had `6,10,000. He gave ₹87,500 to Jyoti, `1,26,380 to Javed and `3,50,000 to John.

3

Vansh wants to sell 78,956 pens in a year. He sold 9999 of them in 4 months. How many

and a flat for `8,70,000. How much money did he have left?

How much money did he have left? more pens does he have to sell?

Multiplication of Whole Numbers Richa gave 7 apples to her mother, 4 to her brother and kept 3 for herself. Each apple costs 5. The total cost of the apples = 14 × 5 = 70

Properties of Multiplication 1 Closure Property When we multiply two whole numbers, the product is always a whole number. =

×

Then 3 × 5 = 15 is also a whole number.

2 Commutative property Two whole numbers can be multiplied in any order. ×

=

×

=

If 3 and 5 are two whole numbers.

Then, 3 × 5 = 5 × 3.

Chapter 3 • Whole numbers

41

3 Multiplicative property of zero When a whole number is multiplied by zero, the resulting product is always zero. × 0 = 0 ×

=0

3×0=0=0×3 4 Existence of multiplicative identity When a whole number is multiplied by one, the product is equal to the number itself. × 1 = 1 ×

=

3×1=3=1×3 5 Associative property When three or more whole numbers are multiplied, the product remains the same regardless the grouping of the numbers. ×

×

=

×

×

=

i.e., if 1, 2 and 3 are three whole numbers then, (3 × 2 ) × 1 = 3 × (2 × 1 ) = 6

6 Distributive property of multiplication over addition Multiplying the sum of two addends by a whole number will give the same result as multiplying each addend individually by the number and then adding the products together. ×

=

+

×

=

1 × (2 + 3) = 1 × 5 = 5 (1 × 2) + (1 × 3) = 2 + 3 = 5 1 × (2 + 3) = (1 × 2) + (1 × 3) 7 Distributive property of multiplication over subtraction Multiplying the difference of two whole numbers by another whole number will give the same result as multiplying each number individually by the number and then subtracting the products together. ×

–

=

×

=

1 × (3 – 2) = 1 × 1 = 1 (1 × 3) – (1 × 2) = 3 – 2 = 1 1 × (3 – 2) = (1 × 3) – (1 × 2) Example 6

42

Solve and state the properties used.

1 544 × 6 + 544 × 9

2 823 × 5 – 823 × 3

= 544 × (6 + 9)

= 823 × (5 – 3)

= 544 × 15 ⇒ 8160

= 823 × (2) ⇒ 1646

The property used is the distributive property of multiplication over addition.

The property used is the distributive property of multiplication over addition.

Do It Together

A restaurant ordered 21 refrigerators and 21 microwave ovens. If each refrigerator costs ₹30,000 and each microwave cost ₹20,000, find the total amount spent.

Cost of 1 refrigerator = _________

Cost of 1 microwave = _________

Cost of 21 refrigerators = _________ × _________

Cost of 21 microwaves = _________ × _________

Hence, the total amount spent by the restaurant is _____________.

Do It Yourself 3D 1

State the property in the cases. a 452 × 0 = 0

2

3

Find the product. a 5 × 356 × 8

b 7 × 2354 × 9

c 32 × 4512 × 3

d 45 × 9875 × 12

e 0 × 1213 × 5 × 542

f 123 × 51 × 21 × 321

b 256 × 10 + 256 × 14

c 437 × 13 + 437 × 17

Find the value of. a 132 × 15 + 132 × 12

4

5

b 154 × 125 = 125 × 154

Using the distributive property, find the product of the numbers. a 458 × 105

b 125 × 1002

d 0 × 99

e 5014 × 999

c 137 × 1005

Find the product of the greatest 5-digit number and the smallest 6-digit number.

Word Problems 1

A businessman owns 20 large buildings and 17 small buildings. Each of the large buildings has 18 floors with 5 apartments on each floor. Each small building has 8 floors with 4 apartments on each floor. Find the total number of apartments.

2

A fruit vendor sells 33 kg of apples in one week and 65 kg apples in the next two weeks. If

the cost of 1 kg apples is `40. Find the amount of money the vendor spent on apples in the three weeks using the properties.

3

Akash bought 18 packs of green shirts and 17 packs of blue shirts for his football team.

Both shirts come in packs of 15 shirts. How many shirts did Akash buy in total?

Chapter 3 • Whole numbers

43

Division of Whole Numbers Mother divided the 7 apples between Rita and her brother.

Rita's brother divided his 4 apples between his two friends.

So, for 7 apples divided between 2 is 7 ÷ 2 = 31 2

further, 4 apples divided between 2 friends is 4÷2=2

Properties of Division 1 Closure property When one whole number is divided by another whole number, then the quotient may not always be a whole number. ÷

=

7 ÷ 2 = 31, which is not a whole number.

2 2 Commutative property If the order of the dividend and divisor is changed, the quotient will not be the same. ÷

÷

=

=

6 ÷ 3 = 2 and, 3 = 1 6 2 So, 6 ÷ 3 ≠ 3 ÷ 6 3 Associative property When three or more whole numbers are divided, the result does not remain the same if the order of the grouped numbers is changed.

÷

÷

=

÷

≠

÷

=

(18 ÷ 3) ÷ 6 = 6 ÷ 6 = 1

18 ÷ (3 ÷ 6) = 18 ÷ 1 = 9 2 (18 ÷ 3) ÷ 6 ≠ 18 ÷ (3 ÷ 6) 4 Division by 1 When a non–zero whole number is divided by one, the quotient is the whole number itself.

If 6 is a whole number, then 6 ÷ 1 = 6 ÷ 1 =

44

5 Division by a number itself When a non–zero whole number is divided by itself, it gives a quotient of 1.

6÷6=1 = 1

÷

6 Zero divided by a whole number When zero is divided by a non–zero whole number, it gives the quotient zero.

0÷6=0 0÷

=0

7 Division by zero A non–zero whole number cannot be divided by zero. Division by zero is not defined. 6 ÷ 0 Not Defined Division of a whole number by zero is not possible. ÷ 0

8 Division Algorithm According to the division algorithm, Dividend = (Divisor × Quotient) + Remainder

10 2 21 –2 01 –0 1

When we divide 21 by 2 then the quotient and the remainder are 10 and 1 respectively Which can be written by division algorithm as: 21 = 2 × 10 + 1 ÷

Example 7

≠

Divide 75,850 by 450 and check the result by division algorithm.

We are given that, divisor = 450 and dividend = 75,850.

On dividing 75,850 by 450, we get: So, quotient = 168 and remainder = 250.

According to the division algorithm: Dividend = (Divisor × Quotient) + Remainder = (450 × 168) + 250 = 75,600 + 250 = 75,850 → Dividend

Chapter 3 • Whole numbers

168 450 75850 – 450 3085 – 2700 3855 – 3600 250

Error Alert!

The remainder can never be greater than the divisor.

1 7 100 – 7 30

14 7 100 – 7 30 – 28 2

45

Example 8

Find the largest 5–digit number that is exactly divisible by 340.

The largest 5–digit number is 99,999.

On dividing 99,999 by 340, we get: According to the division algorithm: Dividend = (Divisor × Quotient) + Remainder 99,999= (340 × 294) + 39 So, the required number = 340 × 294 = 99,999 – 39 Therefore, the required number is 99,960.

Do It Together

294 340 99999 – 680 3199 – 3060 1399 – 1360 39

56,458 books were distributed among 233 schools. Find the number of books each school received and check the result using the division algorithm.

Total number of books = ____________ Number of schools the books were distributed in = ____________ Number of books each school received = ____________ ____________ Therefore, each school received ____________ books.

–

9

9

9

9

9

9

9

9

6

0

3

9

2__ 233 56458 – 4__ ___ – ___ ___ – ___ __

Do It Yourself 3E 1

2

3

46

Find the value of the divisions. a 9875 ÷ 1

b 0 ÷ 789

d 999 + (45 ÷ 45)

e (1450 ÷ 50) – (999 ÷ 3)

c 448 ÷ (224 ÷ 28)

Divide and check the result by the division algorithm. a 2468 ÷ 617

b 685 ÷ 100

d 27,380 ÷ 120

e 1,21,878 ÷ 999

c 2765 ÷ 35

Divide and find the quotient and remainder. a 29,812 ÷ 100

b 56,846 ÷ 45

d 65,412 ÷ 346

e 94,335 ÷ 820

c 99,999 ÷ 123

4

The greatest 4–digit number made with the digits 1, 0, 9, 2 is divided by the smallest 3–digit number made with

5

Find the smallest number that should be added to the smallest 4–digit number so that 35 divides the sum exactly.

6

What number should be subtracted from 367 to make it divisible by 3?

7

What is the greatest number which can be divided by 24, 56 and 18?

1, 9, 2. Find the quotient and the remainder.

Word Problems 1

An engine pumps at 850 litres of water is one minute. How many hours will it take to pump

2

A shirt can be made out of 3 m cloth. If there is a roll of 1350 m cloth, how many shirts can

3

The number of students in each class is 25. The fees paid by each student are ₹900 per

out 1,27,500 litres of water? be made out of it in total?

month. If there are 50 classes in a school, what is the total fee collection in a month?

Patterns in Whole Numbers Real Life Connect

Jiya and her friends were climbing a staircase.

Jiya climbed two steps at a time. Her friend climbed one step, skipped a step and then skipped two steps.

Her other friend skipped two steps every time she climbed.

Jiya noticed they were forming a pattern while climbing.

Counting by ones

Counting in twos

+1 +1 +1 +1 +1 0

1

2

+3 0

1

3

4

5

3

4

7

8

9 10

0

1

2

3

+2 4

5

+2 6

7

+2 8

Pattern: 2, 4, 6, 8, 10,….

Counting in threes

Counting in fours

+3 5

6

7

+3 8

+3

9 10 11 12 13 14 15

+4 0

2

+4 4

6

+4

+4

8 10 12 14 16 18 20

Pattern: 4, 8, 12, 16, 20,…

Counting in fives

Counting in tens

+5 5

+5 10

+5 15

+5 20

Pattern: 5, 10, 15, 20, ….

Chapter 3 • Whole numbers

+10 25

0

+10 10

9 10

+4

Pattern: 3, 6, 9, 12, 15, 18, ….

+5 0

6

+2

Pattern: 1, 2, 3, 4, 5…

+3 2

+2

+10 20

+10 30

+10 40

50

Pattern: 10, 20, 30, 40,…

47

Even and Odd Whole Numbers Even numbers are numbers that form pairs with no leftover. Even numbers end with 0, 2, 4, 6 or 8. Some examples of even numbers are 432, 134, 456, 978, 590.

Did You Know?

Pythagoras was the first man to come up with the idea of

Odd numbers are numbers that have one left over after forming pairs. Odd numbers end with 1, 3, 5, 7, or 9. Some examples of odd numbers are 521, 133, 945, 707, 959.

odd and even numbers.

Whole numbers can also be represented in the form of shapes based on their arrangement.

Every number that is greater than 1 can be represented in the form of a line.

The number 3 can be shown as:

The number 2 can be shown as:

The number 4 can be shown as:

The number 5 can be shown as:

Similarly, we can arrange the rest of the numbers in a line.

We can represent some whole numbers by triangles.

The number 3 can be shown as:

The number 1 can be shown as:

The number 6 can be shown as:

The number 10 can be shown as:

These whole numbers are called triangular numbers.

We can also represent some whole numbers by squares.

The number 1 can be shown as:

The number 4 can be shown as:

The number 9 can be shown as:

The number 16 can be shown as:

These whole numbers are called square numbers or perfect squares.

We can represent some whole numbers by rectangles.

The number 6 can be shown as:

Or

48

The number 10 can be shown as:

Or

We can also simplify the mathematical calculations by just observing certain patterns, which include addition, subtraction or multiplication of certain numbers.

Addition of 9, 99, 999, etc. to a whole number

Subtraction of 9, 99, 999, etc. from a whole number

Multiplication of a whole number by 9, 99, 999, etc.

1000 – 9

112 × 9 = 112 × (10 – 1) = 112 × 10 – 112 × 1 = 1120 – 112 = 1008

100 + 9

= 100 + 10 – 1

= 1000 – (10 – 1)

= 109

= 990 + 1

= 110 – 1

= 1000 – 10 + 1 = 991

100 + 99

1000 – 99

= 100 + 100 – 1

= 1000 – (100 – 1)

= 199

= 900 + 1

= 200 –1

= 1000 – 100 + 1 = 901

100 + 999

1000 – 999

= 100 + 1000 – 1

= 1000 – (1000 – 1)

= 1099

=1

= 1100 – 1

Example 9

112 × 99 = 112 × (100 – 1) = 112 × 100 – 112 × 1 = 11,200 – 112 = 11,088 112 × 999 = 112 × (1000 – 1) = 112 × 1000 – 112 × 1 = 1,12,000 – 112 = 1,11,888

= 1000 – 1000 + 1

For an event, 4569 invitations have to be sent out by Friday. If on Wednesday 999 invitations have been sent out, then how many more invitations have to be sent out?

Number of invitations to be sent out = 4569; Number of invitations sent out = 999 Number of invitations that are yet to be sent out = 4569 – 999 = 4569 – (1000 – 1) = 4569 – 1000 + 1 = 3569 + 1 = 3570 Therefore, 3570 more invitations need to be sent out.

Example 10

Study the pattern and write the next two steps.

We can see that the whole numbers are represented in the form of a triangle.

So, the next two terms will be:

1 Do It Together

3

6

10

1

3

6

15

There are 4525 employees in a company. Each employee gets a bonus of ₹999 at the end of the month.

Use the shorter method to find out the total bonus handed out by the company.

Total number of employees

= ______________

Bonus each employee got

= ______________

Total bonus handed out

= ______________ × ______________ = ______________

Chapter 3 • Whole numbers

49

Do It Yourself 3F 1

Find the missing numbers. a 5, 7, 12, 14, 19, ____, 26

2

Simplify using the shorter method. a 4452 + 99

3

b 2, 3, 6, 18, 108, ____

b 2356 + 9999

c 7821 – 999

d 98,658 – 9999

Study the pattern and write the next two steps.

1×1=1 11 × 11 = 121 111 × 111 = 12321

4

Solve and establish a pattern. a 45 × 9

5

6

b 45 × 99

1×8+1=9

1 × 9 + 1 = 10

12 × 8 + 2 = 98

12 × 9 + 2 = 110

123 × 8 + 3 = 987

123 × 9 + 3 = 1110

Add 9, 99, 999, 9999 and 9,999 to the numbers. b 7852

c 9545

d 8455

e 78,425

f 9,65,452

d 654

e 751

f 987

Multiply 9, 99, 999 and 9,999 by the numbers. a 245

8

d 45 × 9999

Study the patterns and write the next 4 steps.

a 1125

7

c 45 × 999

b 356

c 445

Write the greatest 7–digit number using the digits 4, 6, 9 if digits may be repeated.

Word Problems 1

There are 7 students standing in a line. Each student has some pencils, so that the number of pencils each student has is double the number of pencils the student in front of him has. How many pencils does the 7th student have?

2

50

Aman does tasks and for every 1 task he gets ₹99. For 3 tasks he gets ₹999, for 5 tasks he gets ₹9999. How much will he get for 13 tasks?

Points to Remember •

Numbers 1, 2, 3, 4, and so on are called natural numbers.

•

Whole numbers are a set of numbers that includes all the natural numbers (positive counting numbers) along with zero.

•

Numbers can be represented in the form of a line, triangle, square and a rectangle.

Math Lab Aim: Understanding whole numbers, their order, and their recognition.

Setting: Individual Material: Cards and markers.

Method:

1

Create Bingo cards with grids of whole numbers from 1-25.

2

Distribute one Bingo card and a set of chips or markers to each student.

3

Draw a number from the container or bag and call it out (e.g., "Number 4").

4

Students should check their Bingo cards for the called number and strike out the number if they have that number on their card.

5

Continue drawing numbers and calling them out one by one.

6

The goal for students is to complete a row (horizontally, vertically, or diagonally) on their Bingo card. When they do, they should call out "Bingo!".

Chapter Checkup 1

Write the successor and predecessor of the numbers.

2

Solve on a number line.

3

a 1,54,698

a 9–6

b 2,54,895

b 5+5+5

c 5,45,977

c 10 – 2 – 2

d 6,45,712

d 20 ÷ 5

e 4×5

e 8,91,453

f

6+5–7

Write true or false. a The smallest even natural number is 0.

______________

b Commutative law and associative property hold true only for addition and multiplication.

______________

c 1 is referred to as multiplicative identity.

______________

d 1 is referred to as additive identity.

______________

Chapter 3 • Whole numbers

51

4

Solve using the distributive property. a 550 × 45 – 550 × 15

b 865 × 12 + 865 × 45

5

Find the number which when divided by 65 gives 9 as a quotient and 5 as a remainder.

6

Find the product of the largest 5–digit number and the largest 2–digit number.

7

Which whole number should be multiplied with 6,54,875 to get the number itself?

c 420 × 36 – 420 × 23

Word Problems 1 A school ordered 96 chairs and 48 tables. The cost of each chair is `200 and the cost of each table is `150. If the school gave `3500 as an advance, what amount is to be given now?

2 The budget for an event is `82,360. Out of this `15,500 was spent on the decorations, `10,000 was paid to the music band and `6500 was spent on refreshments. How much money is left after these expenses?

3 A librarian purchased 45 English books and 45 Hindi books. If the cost of one English book is `100 and one Hindi book is `75, find the total amount he has to pay.

4 Dev lives in a hostel that charges `60 for dinner and `50 for lunch. Find the money that he has to pay for 30 days.

5 Out of 19,700 food boxes, 9999 are distributed among the people in a village. Find the number of remaining food boxes.

6 Bhanu ordered 15 cartons of oranges to distribute as charity. Each carton has 10 boxes and each box has 12 oranges. How many oranges did Bhanu order?

7 In a village there are 45,895 people. It is found that 2 out of 50 persons are illiterate. How many illiterate persons live in that village?

8 A playground is 324 m long and 220 m wide. How much distance will Vicky cover if going 4 times around it?

52

4

Playing with Numbers: Factors and Multiples

Let's Recall Shreya arranged 12 tiles in different ways as shown. She arranged them in rows so that each row has the same number of tiles in it. She writes the multiplication sentence for each arrangement and finds that 12 can be written as the product of two numbers in different ways.

2 × 6 = 12

12 × 1 = 1

4 × 3 = 12

3 × 4 = 12

6 × 2 = 12

1 × 12 = 12 1 and 12, 2 and 6, and 3 and 4 are the numbers that when multiplied give the same product. They are the factors of 12. We can divide 12 by any of these factors without a remainder.

Let's Warm-up

Fill in the blanks with any two factors of numbers given below: 1 10: ______________ , ______________ 2 15: ______________ , ______________ 3 16: ______________ , ______________ 4 18: ______________ , ______________ 5 20: ______________ , ______________

I scored _________ out of 5.

Mean, Median and Mode Reviewing Factors and Multiples Real Life Connect

Neerja and her mom went to the nearby grocery store to purchase fruit.

Mother: Neerja, when we’re shopping for fruit for the week, we want to make sure we get the right amount without wasting money. Let’s say we need 4 apples every day. How many apples will we need for a week?

Neerja: We will need 28 apples.

Mother: You are absolutely correct!

Factors are whole numbers that can be multiplied to give another number as a result.

is a factor of

For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

is a factor of

We can also use division to find the factors of a number as a factor of a number divides it completely without leaving any remainder.

4

Multiples are numbers that can be obtained by multiplying a number by other whole numbers. For example, we can multiply 4 by different numbers to get the multiples of 4 as 4, 8, 12, 20 and so on.

×

3

=

is a multiple of is a multiple of

Let us now learn about the properties of factors and multiples.

Properties of Factors

54

Properties of Multiples

Every number is a factor of itself, and 1 is also a factor of every number.

Every number is a multiple of 1 and itself.

Division of a number by its factor leaves no remainder.

A multiple can be expressed as a product of its factors.

A factor of a number is always less than or equal to the number.

Multiples of a number are always equal to or more than the number.

The number of factors of a number is fixed.

The number of multiples of a number is infinite.

A number can be a factor of more than one number.

Any number can be a multiple of many different numbers.

12

Example 1

Write all the factors of 48.

1 × 48 = 48 2 × 24 = 48 3 × 16 = 48 4 × 12 = 48 6 × 8 = 48 (We stop here as 8 and 6 are already covered) So, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Example 2

Find two numbers whose difference is 3 and the product is 54.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54.

Clearly, 6 and 9 are the required numbers as their difference is 3 and the product is 54.

Do It Together

List all the factors and any three multiples of the numbers given below. a 16

b 21

Factors = _________

Factors = 1, 2, 4, 8, 16

Multiples = _________ c

Multiples = 16, 32, 48

27

d 39

Factors = _________

Factors = _________

Multiples = _________

Multiples = _________

Do It Yourself 4A 1

2

Write all the factors of the given numbers. a 20

b 26

c 36

d 88

e 94

f 45

g 72

h 22

Write the first six multiples of each of the given numbers. a 37

b 62

c 84

3

The product of two numbers is 24. Their sum is 14. Find the numbers.

4

Find two numbers of which the difference is 1 and the product is 72.

5

Find two numbers of which the difference is 2 and the product is 63.

6

Without actual division, show that each of the given numbers is divisible by 9. a 9999

b 9099

Chapter 4 • Playing with Numbers: Factors and Multiples

c 990099

d 99

d 900099

55

Word Problem 1

Raj claims to have 100 rupees, all in 2-rupee coins. Roy claims to have 67 rupees, all in 5-rupee coins. Who is wrong? Give a reason for your answer.

Mean, and Mode Types of Median Numbers Real Life Connect

Ankita got a bag of square coloured blocks from her mother. The bag had 12 red blocks and 13 blue blocks. When Ankita and Naman tried to share the blocks equally, they found that they could split the red blocks evenly but not the blue ones. They were left with one blue block, which made them think about how they could divide the red blocks evenly but not the blue ones.

Let’s understand the mystery behind these numbers.

Perfect Number A perfect number is a number that is the sum of all its factors (excluding itself).

For example, 6: The factors of 6 are 1, 2, and 3.

1+2+3=6 Example 3

Check whether 496 is a perfect number or not.

Factors of 496 = 1, 2, 4, 8, 16, 31, 62, 124, 248, 496 Sum of factors (excluding itself) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 Therefore, 496 is a perfect number.

Example 4

Is 16 a perfect number?

Factors of 16 = 1, 2, 4, 8, 16 Sum of factors (excluding itself) = 1 + 2 + 4 + 8 = 15 Since 16 does not meet the criterion (15 ≠ 16), it is not considered a perfect number.

Do It Together

Check whether 8128 is a perfect number.

The factors of 8128 are 1, 2, 4, 8, ___, 32, ___, 127, ________________________________________ Sum of factors (excluding itself) = 1 + 2 + 4 + 8 + ______________________________________________ = ___________ Therefore, ___________________ perfect number.

56

Even and Odd Numbers Even Number

Odd Number

Any number that can be divided exactly by 2 is called an even number. These numbers always end with digits 0, 2, 4, 6, or 8.

Odd numbers are numbers that are not divisible by 2, meaning they cannot be divided exactly by 2 and always leave a remainder of 1 when divided by 2. These numbers always end with one of the digits: 1, 3, 5, 7, or 9.

Example: 26, 30, 18, 42, 64 etc.

Example 5

Example: 15, 19, 27, etc.

Which of the numbers is even? a 35

b 47

c

59

d 62

Answer: Even numbers end with 0, 2, 4, 6 or 8. Hence, 62 is an even number.

Example 6

Which of the numbers is odd? a 64

b 56

c

32

d 41

Answer: Odd numbers end with 1,3, 5, 7 or 9. Hence, 41 is an odd number.

Do It Together

Classify the numbers as even or odd. a 91 — Odd c

79 — ________________

b 61 — ________________ d 7 — ________________

e 45 — ________________

f

82 — Even

g 39 — ________________

h 19 — ________________

Prime and Composite Numbers Prime Number A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.

Example: 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Here 2 has 2 factors; 1 and 2; 17 has 2 factors 1 and 17.

Composite Number A composite number is a natural number greater than 1 that has more than two factors. Composite numbers are formed by multiplying prime numbers.

Examples of composite numbers are 4, 6, 9, and so on.

Factors of 4 - 1, 2, 4 (more than 2 factors)

Prime Numbers

Composite Numbers

Factors of 9 - 1, 3, 9 (more than 2 factors) Factors of 15 - 1, 3, 5, 15 (more than 15 factors) Chapter 4 • Playing with Numbers: Factors and Multiples

57

Note: • 1 is neither a prime nor a composite number.

• 2 is the smallest prime number.

• 2 is the only even prime number.

Twin Primes

Prime Triplets

Co-Primes (Relatively Prime)

Twin primes are pairs of prime numbers that have a difference of 2 between them, such as (3, 5), (5, 7), and (11, 13).

Prime triplets are sets of three prime numbers that have a difference of 2 between each consecutive pair, such as (3, 5, 7) and (11, 13, 17).

Co-primes, also known as relatively prime or mutually prime numbers, are two or more numbers that share no common factors other than 1, such as (8, 9), (15, 28), (7, 11), etc.

Example 7

Which of the numbers below is a prime number? a 37

b 42

c

26

d 18

Answer: Even numbers end with 0, 2, 4, 6 or 8. Hence, (b), (c) and (d) have 2 as factor too. On the other hand, 37 has only 2 factors, 1 and itself. So, 37 is a prime number.

Example 8

Which of the numbers given below would be a co-prime of 15? a 12

b 5

c

10

d 17

Answer: 17 is a prime number. Hence, it would only share 1 as a common factor with 15. The number 12, 5 and 10 have 3, 5 and 5 respectively as common factors with 15. Hence, (15, 17) are co-primes.

Do It Together

Determine if the given pairs of numbers are co-prime (relatively prime) or not. Justify your answers. a 15 and 28

b 9 and 16

c

21 and 40

d 7 and 18

Answer:

Factors of 15: 1, 3, 5, 15

Think and Tell

Factors of 28: 1, 2, 7, 14, 28

How many even prime numbers are there?

Only 1 is the common factor.

Hence, 15 and 28 are co-prime.

Do It Yourself 4B 1

Identify if there are any perfect numbers amongst the numbers given below. a 57

2

58

b 63

c 94

d 83

Determine whether the given numbers are even or odd. Write “E” for even and “O” for odd next to each number. a 42

b 57

c 64

d 89

e 76

f 93

g 50

h 67

3

Write all the prime numbers between the ranges given below. a 40 to 50

4

b 60 to 70

c 10 to 30

d 0 to 10

Write True or False for the given statements. a 25 is an even number.

b 17 is a prime number.

c 6 is a composite number.

d 13 and 17 are twin prime numbers.

e 20 and 35 are co-prime numbers

5

Write all pairs of twin primes between 60 and 90.

6

Make a list of all composite numbers below 49.

7

Make a list of all prime numbers below 40.

8

Express the numbers below as the sum of two odd primes. a 36

9

b 58

c 76

d 42

Express the numbers given below as the sum of three odd primes. a 91

b 77

c 83

d 65

Word Problem 1

A maths teacher asks students to organise themselves into two groups based on whether their

roll numbers are prime, or composite. How many students are there in each group if there are a total of 36 students in the class?

Mean, Median and Mode Divisibility Rules Real Life Connect

Abha and Priya picked 36 flowers from a garden. They attempted to create bunches with 4, 6, 8, and 10 flowers each, but the only even bunches they could make were with 4 and 6 flowers. Why do you think they were not able to make equal bunches with 8 and 10 flowers?

Let’s see some rules that will help us figure out whether a number is divisible without actually dividing.

Chapter 4 • Playing with Numbers: Factors and Multiples

59

Example 9

Divisible by

Divisibility Rules

Examples

2

the digit at the ones place of the number is 0, 2, 4, 6 or 8

10, 32, 54, 76, 98 and so on

5

the digit at the ones place of the number is 0 or 5

25, 70, 135, 160 and so on

10

the digit at the ones place of the number is 0

20, 50, 70, 100 and so on

3

the sum of the digits of the number should be divisible by 3

36: Sum of digits = 3 + 6 = 9 (divisible by 3) 315: Sum of digits = 3 + 1 + 5 = 9 (divisible by 3)

6

the number should be divisible by both 2 and 3

18: divisible by both 2 and 3.

72: divisible by both 2 and 3.

4

the number formed by the last two digits should be divisible by 4

424: 24 is divisible by 4 736: 36 is divisible by 4

8

the number formed by the last three digits should 2016: 16 is divisible by 8 be divisible by 8 4032: 32 is divisible by 8

9

the sum of digits of a number should be divisible by 9

108: Sum of digits = 1 + 0 + 8 = 9 (divisible by 9) 2169: Sum of digits = 2 + 1 + 6 +9 = 18 (divisible by 9)

11

the difference of the sum of the digits in the odd place and the sum of the digits in the even place is a multiple of 11

847: Sum of digits at odd places – sum of digits at even places = (7 + 8) – 4 = 11 (divisible by 11) 935: Sum of digits at odd places – sum of digits at even places = (5 + 9) – 3 = 11 (divisible by 11)

Which of the numbers is divisible by 3? a 572

b 648

c

793

Answer: A number is divisible by 3 if the sum of its digits is divisible by 3. Let’s calculate the sums: For 572: 5 + 7 + 2 = 14 (not divisible by 3); For 648: 6 + 4 + 8 = 18 (divisible by 3)

Did You Know?

Lakshit Pusri from India made a world record by

For 793: 7 + 9 + 3 = 19 (not divisible by 3); For 921: 9 + 2 + 1 = 12 (divisible by 3)

solving 100 division sums of 3-digit by 1-digit numbers in just 1 minute 54 seconds.

So, the numbers 648 and 921 are divisible by 3.

Example 10

d 921

Which of the numbers given below is divisible by 8? a 237 d 1724

b 416

c

589

Answer: A number is divisible by 8 if its last three digits form a number that is divisible by 8.

For 237: The last three digits are 237, which is not divisible by 8.

60

For 416: The last three digits are 416, which is divisible by 8. Therefore, the number 416 is divisible by 8.

For 589: The last three digits are 589, which is not divisible by 8.

For 1724: The last three digits are 724, which is not divisible by 8.

So, only the number 416 is divisible by 8 because its last three digits (416) form a number divisible by 8.

Do It Together

Find which of these numbers are divisible by 4, 6, 9 and 11. a 5628

b 4356

c

4986

d 8172

a For 5628: __________

b For 4356: __________

c

For 4986: __________

d For 8172: __________

Do It Yourself 4C 1

Find which of these numbers are divisible by 2, 3, 4, 5, 6, 8, 9 and 10.

2484, 5376, 9243, 3028, 8164, 5732, 9120480, 932716, 1432, 41873, 990, 727272, 540, 7269, 7269, 89120, 5482, 76319, 1247, 801632, 123456789

2

3

Identify the prime numbers from the list: a 161

b 167

c 217

d 367

e 223

f 397

g 241

h 128

b both 6 and 8

c both 11 and 9

d 3 but not 2

Give a number divisible by: a 2 but not 6

4

Prove that the result of multiplying any three consecutive numbers will always be divisible by 6, and provide

5

If a number is divisible by both 5 and 9, what other number will it always be divisible by?

6

Given that two numbers are divisible by a certain natural number, demonstrate that both their sum and their

7

examples to support your explanation.

difference are also divisible by that same number. a 66 and 42 are divisible by 6.

b 646 and 390 are divisible by 13.

c 72 and 99 are divisible by 9.

d 442 and 306 are divisible by 17.

Replace ‘__ ‘ with the smallest digit to make the number divisible by 8. a 24__16

b 400__6

Word Problem 1

Alex was born on 17/ 7 /2010, Ben on 23 /11/2010, and Chloe on 29/ 12 /2011. They add the digits of their birthdates to create numbers. They want to know if these numbers are divisible by 2, 3, and 6. Determine if each number is divisible by 2, 3, and 6.

Chapter 4 • Playing with Numbers: Factors and Multiples

61

Prime Factorisation Real Life Connect

Maya runs a factory that produces cardboard boxes. The factory uses prime factorisation to figure out how many different-sized boxes can be created from a certain size of cardboard sheet to minimise waste.

Let us see how prime factorisation helps us!

Prime factor: When a prime number is a factor of a given number, it’s referred to as a prime factor. Here is one of the methods for finding the factors of the number 36.

36

36

2

18

36

4

9

6

36

6

3

12

Every composite number can undergo further factorisation. This factorisation process continues until all the factors derived from it are prime, and these prime factors are highlighted.

36

36

2

18

36

4

2

9

2

3

9 2

36

6

3

3

3

6

2

3

2

12 2

3

6

3

2

3

Hence, 36 has been expressed as a product of prime factors. This is prime factorisation of 36.

The prime factorisation of a number can be done by these methods. a Factor tree

Let‘s factorise 75 using both of these methods.

Factor tree

5

75 = 3 × 5 × 5

Example 11

75

5

5

25 1

75 = 3 × 5 × 5

Express the each number as the product of its factors by using a factor tree. a

48

b 56

Factor tree of 48 2

c

Factor tree of 56

48

64 Factor tree of 64

56 24

2

2 12

2

64 28

2 6

2

2 14

2

32 2

7

3

48 = 2 × 2 × 2 × 2 × 3 62

3 5

25 5

The prime factors may be presented in varying orders, yet they remain the same.

Repeated Division

75 3

Remember!

b Repeated division

16 2

8 2

56 = 2 × 2 × 2 × 7

4 2

2

64 = 2 × 2 × 2 × 2 × 2 × 2

Example 12

Express each number as the product of its factors by using repetitive division. a 45

b 72

Prime factors of 45

c

Prime factors of 72

68 Prime factors of 68

3

45

2

72

2

68

5

5

2

18

17

17

3

15

2

1

3 3

36

2

9

34 1

3 1

72 = 2 × 2 × 2 × 3 × 3

45 = 3 × 3 × 5 Do It Together

68 = 2 × 2 × 17

Express each number as the product of its factors by using both factor tree and repeated division. a 76 Prime factors of 76

Factor tree of 76 76 2

38 2

76 = 2 × 2 × 19

b 99 Prime factors of 99 3

99

11

11

3

19

Factor tree of 99

33 1

99 = 3 × 3 × 11

Do It Yourself 4D 1

Show the prime factorisation of each of the given numbers. a 126

b 882

c 6241

d 192

f 5720

g 360

h 1369

i

e 1265

10000

2

Find the prime factors of 345. Arrange them in ascending order. Find a relation between two consecutive

3

State whether they are prime factorisations or not.

prime factors.

a 189 = 3 × 3 × 21

4

b 252 = 2 × 6 × 3 × 7

c 385 = 5 × 7 × 11

Express the each number as the product of its factors by using a factor tree. a 210

b 293

Chapter 4 • Playing with Numbers: Factors and Multiples

c 816

d 952

e 756

63

5

Find the prime factors of 252. Arrange them in ascending order. Determine the difference between two

6

Find the prime factors of 560. Arrange them in ascending order. Calculate the product of the two smallest

7

a Write the smallest 5-digit number and express it in terms of its prime factors.

consecutive prime factors. prime factors

b Write the largest 4-digit number and express it in terms of its prime factors.

Word Problem 1

Jane wants to buy a bouquet of flowers with 36 roses, 24 lilies, and 18 tulips. She wants to arrange them in identical bouquets with the same number of each type of flower. Suggest at least three numbers that will help her do so.

Mean, Median and Mode Working with LCM and HCF Real Life Connect

Maya: Ansh, I have 36 red balloons and 48 blue balloons. Can you tell me what is the maximum number of identical groups that I can make so that each group has both types of balloons?

Ansh: We can find the common factors of 36 and 48. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, while the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

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

So you can make 12 identical groups, each having 3 red balloons and 4 blue balloons.

Let us revise common factors and common multiples and learn how finding the HCF and LCM can help us solve such problems in our daily lives.

Revisiting HCF and LCM Common Factors and Common Multiples We’ll first revisit the concept of common factors and multiples to get started.

Common Factors Common factors are numbers that can exactly divide two or more numbers without leaving a remainder.

Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 20: 1, 2, 4, 5, 10, 20 64

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The common factors of 12, 20, and 24 are 1 and 2.

Common Multiples Common multiples are numbers that are multiples of two or more numbers.

Multiples of 24: 4, 48, 72, 96, 120, 144, 168, 192, 216, 240, ...

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...

Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, ...

The common multiples of 12, 20, and 24 are 120, 240, 360, …

Example 13

List the common factors of 48, 36 and 56.

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 So, the common factors of 48, 36, and 56 are 1 and 2.

Example 14

List two common multiples of 27, 36 and 45.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72,...

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...

Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180...

So, two common multiples are 36 and 72.

Do It Together

Fill in the blank with two common factors of the numbers given below: 1 12, 18, 24: 6, _______

2 18, 27, 36: ______

Highest Common Factor HCF stands for “Highest Common Factor,” and it refers to the largest number that evenly divides two or more numbers without leaving a remainder.

Let us learn how to find the HCF of two numbers using three different methods.

By Listing Factors Step 1: List the factors of 24 and 36.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Identify the common factors.

Common factors of 24 and 36: 1, 2, 3, 4, 6, 12

Step 3: Determine the largest common factor.

The largest common factor is 12.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

So, the HCF of 24 and 36 is 12.

Now, let’s find HCF of 324 and 540 using the prime factorisation method.

Chapter 4 • Playing with Numbers: Factors and Multiples

65

By Prime Factorisation Step 1: Find the prime factorisation of each number.

2

324

2

540

Step 2: Identify the common prime factors.

3

81

3

135

2 3

Step 3: Select one prime factor from each group

of shared prime factors and multiply these selected prime factors to obtain the HCF.

162 27

2 3

45

3

9

3

15

3

3

5

5

1

So, the HCF of 324 and 540 is 2 × 2 × 3 × 3 × 3 = 108.

270

1

540 = 2 × 2 × 3 × 3 × 3 × 5

Now, let us see how to find the HCF of two numbers using the long division method.

324 = 2 × 2 × 3 × 3 × 3 × 3

By Long Division Step 1: Start by dividing the larger number by the smaller number. If the remainder is zero, the divisor is the HCF. Step 2: If the remainder is not zero, divide the divisor by the remainder.

Step 3: Repeat this process until the remainder becomes zero. The last divisor used is the HCF. So, the HCF of 216 and 324 is 108.

Example 15

–

324 216

2

108

216

–

216 0

Find the HCF of the numbers using prime factorisation: 1 168 and 252 2

168

2

252

2

84

2

126

2

42

3

63

3

21

3

21

7

7

7

7

1

1

168 = 2 × 2 × 2 × 3 × 7 252 = 2 × 2 × 3 × 3 × 7 Hence, HCF of 168 and 252 = 2 × 2 × 3 × 7 = 84

2 280 and 360 2

280

2

540

2

140

2

180

2

70

2

90

5

35

3

45

7

7

3

15

1

5

5 1

66

216

1

168 = 2 × 2 × 2 × 5 × 7 252 = 2 × 2 × 2 × 5 × 3 × 3 Hence, HCF of 168 and 252 = 2 × 2 × 2 × 5 = 40

Find the HCF of the numbers using repeated division:

Example 16

1 504 and 720 504

720

–

504

2 126, 198 and 315 1

216

504

–

432

2

72

216

–

216

198

315

–

198

1

9

117

198

–

117

3

0

Hence, HCF of 504 and 720 is 72.

–

1

81

117

–

81

126

126

14

0

1

36

81

–

72

2

9

36

–

36

4

0

Hence, HCF of 198, 315 and 126 is 9.

Do It Together

Fill in the blank with the HCF of the numbers: 1 72, 108, 96

2

72

2

108

2

96

72 = ___________________ 108 = ___________________ 96 = ___________________ Hence, LCM of 72, 108 and 96 = 2 ______________________________

Lowest Common Multiple The lowest common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the given numbers without leaving a remainder. In other words, it’s the lowest common multiple that can be divided exactly by all the provided numbers.

Let us learn the different methods to find the LCM.

By Listing Multiples Find the LCM of 24 and 36 by listing the multiples.

Multiples of 24: 24, 48, 72, 96, 120, 144...

Multiples of 36: 36, 72, 108, 144, ...

The common multiples of 24 and 36 are 72, 144…

The lowest common multiple of 24 and 36 is 72.

So, the LCM of 24 and 36 is 72.

Chapter 4 • Playing with Numbers: Factors and Multiples

67

By Prime Factorisation Step 1: Find the prime factorisation of each number.

Step 2: Identify common prime factors from all numbers.

Step 3: Select one shared prime factor from each set of prime factors and multiply them and all the other common prime factors that are not common to find the LCM.

By Division Method Step 1: Arrange the provided numbers in a line, separated by commas.

2

16, 24, 30, 36

2

4, 6, 15, 9

2

Step 2: Find a number that divides at least two of them exactly.

Step 3: Divide the divisible numbers and note quotients below these numbers. Keep the numbers that cannot be divided.

2 3

Step 4: Repeat until you have co-prime numbers.

8, 12, 15, 18 2, 3, 15, 9 1, 3, 15, 9 1, 1, 5,

Step 5: Multiply divisors and co-prime numbers for the LCM.

3

Hence, LCM of 16, 24, 30 and 36 is 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720.

Properties of LCM and HCF 1 The Highest Common Factor (HCF) of the numbers is always smaller than or equal to the smallest number.

2 The Least Common Multiple (LCM) of the numbers is never less than any of the numbers.

3 Two numbers that are co-prime have an HCF of 1.

4 For two or more co-prime numbers, their LCM is equal to the product of those numbers.

5 The HCF of the numbers is always a factor of their LCM. 6 For any two numbers, their product is equal to the product of their HCF and LCM. Let’s understand this property through an example.

Error Alert!

A random pair of numbers cannot be LCM and HCF of two numbers. HCF must be a factor of LCM.

Consider the numbers 12 and 18. Let’s find their HCF and LCM and compare their product with the product of their HCF and LCM. 2 2 3

12 6 3 1

2 3 3

18

12 = 2 × 2 × 3

9

18 = 2 × 3 × 3

1

HCF = 2 × 3 = 6 LCM = 2 × 2 × 3 × 3 = 36

3

Product of numbers = 12 × 18 = 216 HCF × LCM = 6 × 36 = 216 Example 17

Find the LCM 135 and 246 using prime factorisation: 3

135

2

246

3

15

41

41

3 5

45 5 1

68

3

123 1

135 = 3 × 3 × 3 × 5 246 = 2 × 3 × 41 LCM = 2 × 3 × 3 × 3 × 5 × 41 = 11070

Example 18

Find the LCM 40, 48 and 45 using the division method.

Hence LCM = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720 2

40, 48, 45

2

20, 24, 45

2

10, 12, 45

2

5, 6, 45

3

5, 3, 45

3

5, 1, 15

5

5, 1, 5 1, 1, 1

Example 19

The HCF of two numbers is 9, and their LCM is 108. One of the numbers is 27. What is the other number?

Solution: Let’s denote the two numbers as A and B. Given: HCF = 9; LCM = 108 Let’s say one of the numbers A = 27 We can use the relationship between HCF, LCM, and the numbers: HCF × LCM = A × B Substitute the values: 9 × 108 = 27 × B B=

9 × 108 = 36 27

So, the other number (B) is 36.

Example 20

Is it possible for two numbers to have 25 as their highest common factor (HCF) and 600 as their lowest common multiple (LCM)? Explain your answer.

The HCF of the given numbers is always a factor of their LCM. Since 600 is divisible by 25, it is possible to have two numbers whose LCM and HCF are 600 and 25 respectively.

Do It Together

Fill in the blank with the LCM of the numbers below: 1 60, 84: 420

Chapter 4 • Playing with Numbers: Factors and Multiples

2 126, 154, 189: ________

69

Do It Yourself 4E 1

Find the HCF of the numbers by listing factors. a 24, 36

2

b 30,45

b 30, 45

b 63, 105

b 15, 35

c 72, 96,120

d 65, 132

c 4, 8, 12

d 12,16

Find the LCM of the numbers by the prime factorisation method. a 16, 30, 42

6

d 54, 90, 126

Find the LCM of the numbers by listing their multiples. a 60, 20

5

c 45, 75

Find the HCF of the numbers by repeated division. a 48, 60

4

d 36, 72, 108

Find the HCF of the numbers by prime factorisation. a 20, 30

3

c 18, 27

b 28, 44, 132

c 16, 28, 40, 77

d 20, 25, 30, 50

c 9, 12, 36, 54

d 102, 170,136

Find the LCM of the numbers by the division method. a 11, 22, 36

b 96, 128, 240

7

The Highest Common Factor (HCF) of two numbers is 18, and their Least Common Multiple (LCM) is 126. One of

8

If 25 is the HCF of two numbers, can the LCM of those numbers be 780? Explain.

the numbers is 54. What is the other number?

Word Problem 1

Imagine you have two factories, Factory A and Factory B. Factory A produces products every 18 hours, and Factory B produces similar products every 24 hours. The products produced

by both factories are required to be shipped together to a distribution centre. Is it possible for these factories to coordinate their production schedules in such a way that the time

interval between shipments to the distribution centre is exactly 36 hours (HCF), and the products arrive together every 432 hours (LCM)?

Word Problems in LCM and HCF The knowledge of HCF and LCM can come in handy in many real-life situations. Let us explore some word problems!

Amy is planning to distribute chocolates (24 in a pack), balloons (30 in a pack), and pencils (18 in a pack) equally among her friends for a party. What is the maximum number of gift bags she can create without any items left over?

70

As we need to find the maximum number of gift bags without any left over, we will find the HCF of the given numbers.

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24 Factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30 Factors of 18 = 1, 2, 3, 6, 9, and 18

Remember!

LCM is always greater than HCF.

So, the Highest Common Factor (HCF) of the numbers is 6.

Hence, Amy can create a maximum of 6 gift bags without any items left over.

Example 21

At a bakery, there are three ovens used for baking different types of bread. Oven A bakes a batch in 15 minutes, Oven B in 20 minutes, and Oven C in 25 minutes. How often will all three ovens finish baking their batches simultaneously, and what is the minimum time it will take for them to do so again?

To find out how often all three ovens will finish baking their batches simultaneously and the minimum time it will take for them to do so again, we need to determine the Least Common Multiple (LCM) of the baking times for each oven.

The baking times for the ovens are: Oven A: 15 minutes; Oven B: 20 minutes; Oven C: 25 minutes LCM (15, 20, 25) = 2 × 2 × 2 × 3 × 5 × 5 = 300

2

15, 20, 25

3

15, 5, 25

2 5

15, 10, 25 5, 5, 25

5

1, 1, 25

5

1, 1, 1

So, all three ovens will finish baking their batches simultaneously every 300 minutes.

Do It Together

Emily is organising a bake sale fundraiser for her school. She has 24 chocolate chip cookies, 36 oatmeal cookies, and 48 peanut butter cookies. She wants to make cookie packs with the same number of each type in each pack, with no cookies left over. How many cookies should Emily include in each pack?

To find the number of cookies that Emily should include in each pack we need to calculate the _______________ of 24, 36, and 48.

Do It Yourself 4F 1

John has 36 marbles, and Emily has 48 marbles. What is the maximum number of marbles they can evenly

2

A bus service operates every 15 minutes, and a train service operates every 20 minutes. How often do the bus

3

A farmer has 24 cows and another farmer has 36 cows. What is the largest number of cows they can evenly divide

4

A gardener needs to water his plants every 12 days, and another type of plant needs watering every 18 days.

5

In a school, there are 90 students, and each class can accommodate 30 students. What is the largest number of

distribute among their friends without any remainder? and the train services coincide?

into separate pens with the same number of cows in each pen?

How often does he have to water both types of plants on the same day?

students that can be accommodated in each class without any extra students?

Chapter 4 • Playing with Numbers: Factors and Multiples

71

6

A toy store has three toy-making machines. Machine X produces toys every 15 minutes, Machine Y every 20

minutes, and Machine Z every 25 minutes. After how much time will all three machines complete their work simultaneously?

7

A factory produces batches of 18, 24, and 36 products at a time. What is the largest number of products that can

8

In a school, there are three classes that need to line up for an assembly. Class P takes 12 minutes to line up,

be produced so that each batch is fully utilised?

Class Q takes 18 minutes, and Class R takes 24 minutes. How often will all three classes be ready for assembly together?

9

A carpenter has wooden planks of lengths 54 cm, 72 cm, and 90 cm. What is the maximum length he can cut these planks into, with no wastage?

10 Three different buses depart from a bus station. Bus X departs every 8 minutes, Bus Y every 12 minutes, and Bus Z every 16 minutes. When will all three buses depart simultaneously from the station again?

Word Problem 1

Three friends, Alice, Bob, and Carol, have 30, 45, and 60 candies respectively. What is the greatest number of candies they can distribute equally among themselves without any leftovers?

Points to Remember •

A factor of a number is a number that can divide the number and leave no remainder.

•

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.

•

• • • •

72

A multiple of a number is a number that can be obtained by multiplying that number by another number.

Co-prime numbers (or relatively prime numbers) are two or more numbers that have no common positive integer divisors other than 1.

Twin primes are pairs of prime numbers that have a difference of 2.

The LCM of two or more numbers is the smallest multiple that is divisible by each of them.

The HCF of two or more numbers is the largest positive integer that evenly divides each of them without leaving a remainder.

Math Lab Discovering Prime Numbers with the Sieve of Eratosthenes Setting: Each student should attempt it on their own.

Materials needed: Large sheet of grid paper with numbers from 1 to 100 pre-drawn, Markers or coloured pencils, Copies of the grid paper for each student Instructions:

1

Start by explaining the definition of prime numbers.

2

Provide each student with a copy of the grid paper, which has numbers from 1 to 100.

3

Explain that 1 is neither a prime number nor a composite number, so students should put a

4

Ask students to circle prime numbers like 2 and 3, then cross out their multiples

5

Instruct students to continue this process for the numbers 5, 7, 11, and so on. Encircle each

6

Continue the process until all the numbers in the grid are either encircled (prime) or crossed out

7

After completing the activity, gather the class and discuss the prime numbers they found.

cross over it.

(e.g., 4, 6, 8 for 2; 6, 9, 12 for 3).

prime number and cross out its multiples. (composite).

Emphasise the pattern of prime numbers and how the Sieve of Eratosthenes helps identify them systematically.

Chapter Checkup 1

2

List all the factors of the numbers. a 24

b 30

c 48

e 63

f 56

g 90

d 76

Write the first six multiples of each of the numbers. a 16

b 48

c 92

d 105

3

The product of two numbers is 36, and their sum is 13. Find the numbers.

4

If two numbers are divisible by a certain natural number, prove that both their sum and their difference are also divisible by that same number.

a 54 and 36 are divisible by 9.

b 546 and 364 are divisible by 13.

c 81 and 135 are divisible by 27.

5

Write the each number as the product of its factors by using a factor tree. a 315

6

b 539

c 267

d 714

e 672

Find the prime factors of 675. Arrange them in descending order. Calculate the product of the two smallest prime factors.

Chapter 4 • Playing with Numbers: Factors and Multiples

73

7

8

9

Find the HCF of the numbers by prime factorisation. a 36, 48

b 42, 56

c 60, 90

d 72, 120, 168

e 90, 135

f 105, 147, 245

g 64, 96

h 70, 105, 140

Find the HCF of the numbers by repeated division. a 54, 72

b 77, 121

c 84, 112, 140

d 80, 160

e 88, 176, 264

f 120, 160

g 110, 154, 198

h 150, 200, 250

Find the LCM of the numbers by the prime factorisation method. a 18, 35, 49

b 24, 36, 120

c 15, 27, 45, 88

d 25, 30, 35, 55

e 72, 90, 108

f 48, 32, 54, 63

g 288, 96, 180

h 140, 210, 31

10 Find the LCM of the numbers by the division method. a 14, 28, 42

b 80, 112, 200

c 18, 24, 54, 72

d 120, 200, 160

e 66, 78, 132

f 72, 48, 81, 54

g 60, 90, 150, 200

h 96, 144, 168,224

11 Find the greatest number of four digits which is exactly divisible by 8, 12, 16, 20, and 24.

12 Find the greatest number of five digits which is exactly divisible by 7, 14, 21, 28, and 35.

13 Find the smallest number from which if 15 is subtracted, the result is exactly divisible by 9, 12, 18, 21, 27, and 36.

14 Find the smallest number that becomes divisible exactly by 15, 25, 35, and 45 when 17 is added to it.

15 Determine the largest three-digit number divisible by 8, 12, 18, 24, and 32.

Word Problems 1

2

3

4

5

74

A library has 30 history books, 45 science books, and 60 literature books. They want to arrange these books on shelves so that each shelf contains an equal number of each type of book. What is the maximum number of books they can place on each shelf?

A garden has three sprinklers that operate at intervals of 20 minutes, 25 minutes, and 30 minutes, respectively. What is the minimum amount of time one would have to wait to observe all three sprinklers operating simultaneously in the garden?

A farmer has 36 apples, 48 oranges, and 60 bananas. He wants to create identical fruit baskets with the same number of each type of fruit in each basket. What is the greatest number of fruit he can put in each basket while achieving this balance?

A busy intersection has traffic lights that change every 40 seconds, 60 seconds, and 80 seconds respectively. What is the shortest time interval during which all three traffic lights will be green at the same time, allowing uninterrupted traffic flow in all directions?

Sarah has 18 erasers, 24 pencils, and 30 pens. She wants to distribute them equally among her friends for a school project. What is the largest number of friends among whom she can distribute these supplies, with each friend receiving the same number of each item?

5

Line Segments and Angles

Let's Recall Every day we come across different types of lines. Let us discuss parallel and intersecting lines.

m l

g h

P

Parallel lines are like best friends who walk side by side but never cross each other. They stay the same distance apart everywhere they go, just like railroad tracks that run alongside each other without ever meeting.

On the other hand, intersecting lines are like friends who meet at a special point. They cross each other at one spot, like roads that cross at an intersection. After they meet, they go their separate ways.

Let's Warm-up 1 Classify the following based on whether they are examples of intersecting lines or parallel lines.

Write I for intersecting lines and P for parallel lines. a

b

d

e

c

I scored

out of 5.

Mean, Median and Mode Line Segments Real Life Connect

One day, Anju and her dad wanted more room to dry their clothes.

She took some rope and showed it to her father.

Anju: Dad, can we make a clothesline out of this rope?

Dad: Of course, Anju. Did you know the poles of a clothesline are examples of the geometrical concept known as a ‘line segment’?

Anju: Is it?

Dad: Yeah! From simple poles to massive pyramids all are based on simple geometrical ideas that help us think about shapes.

Key Geometrical Concepts Let’s explore some key geometrical concepts in detail.

Point

Line Segment

Ray

Y

Y

Y X

Z N

Point: A point denotes a location. A point has no dimensions like length, breadth or height. Can be denoted as.

Plane

X

X

Line Segment: A Line segment has a fixed length.

The distance between the two points is the length.

Can be denoted as XY or YX.

X

Y X

A ray is obtained by extending a line segment indefinitely in one direction.

It doesn’t have a fixed length.

Represented as XY, where X is the initial point.

Ray XY is not the same as ray YX.

Plane is a surface on which any two given points can be connected by a line so that the line lies wholly on the surface itself.

Y

It is a smooth flat surface extending indefinitely in all directions.

It has length and breadth but no height or thickness.

The walls of a room, a football ground, a sheet of paper, and a tabletop are some examples of parts of a plane that extends indefinitely in all directions.

76

Types of Lines X

Y

Parallel lines are two or more straight lines that do not intersect each other, no matter how far they are extended.

A line is obtained by indefinitely extending a line segment in both directions.

It has no endpoints and no fixed length.

It is denoted by small letters like l, m, n, etc.

They maintain a constant distance from each other.

or also simply as XY.

Perpendicular lines are two lines that intersect at a right angle.

The walls of a room, a football ground, a sheet of paper, and a tabletop are some examples of parts of a plane that extends indefinitely in all directions.

Football ground

Table top Example 1

Walls in a room

Look at the figure. List all the points, lines, line segments and rays.

P

Points – A, B, P, M, Z

B A

M

Z

Lines – AB Line segments ‐ PM, AB, AZ Rays ‐ AZ

Did You Know?

Pythagoras was the first to

Think and Tell

suggest that our planet has a

Is the surface of a volleyball part of a

spherical shape, and not flat.

plane? What about a volleyball court?

Do It Together

Sheet of paper

Try relating the real‐life examples with their corresponding geometrical elements. Name them and draw to show.

Real‐life example Geometry

Football field

Stars in the sky

Straw pipe

Light emitted by the sun

Plane

Figure

Chapter 5 • Line Segments and Angles

77

Do It Yourself 5A 1

Look at the figure and answer the questions.

N

a Name the points in the given figure. b How many lines and rays are there in the figure? c Name the initial point of the ray in the figure.

2

3

5

M

Which of these is true about rays? a They lack thickness.

b They have a fixed length.

c They only extend in one direction.

d They have only two end points.

Which of these represents a line? a H

4

Y

X

b GH

c OP

d KL

Check whether the statements are true or false. a A line extends in two directions.

b A plane has a definite size.

c A line segment is part of a ray.

d A tay has more endpoints than a line.

Count the number of line segments in the shapes. a

b

c

d

Word Problem 1

Disha saw two roads crossing each other. What is the type of line she’s seeing?

Measuring and Drawing Line Segments Anju and her father are putting up a clothesline.

Anju: Dad, can we find the length of the rope needed?

Dad: Good question, Anju! We'll have to measure the distance from one end to the other where we want to hang the line.

The clothesline is a practical example of a line segment.

Now, let us learn how to measure and draw line segments.

78

Measuring and Drawing Line Segments Drawing a line: Place one end of the ruler at the paper's edge, line up the "0" mark Draw along the ruler's edge.

A ruler is split into 15 units.

Each of these sections is exactly 1 cm long.

Each centimetre is further split into 10 smaller units, called mm.

Correct way of taking a reading from the scale For precise measurements, align your vision directly above the mark in a vertical line; viewing from an angle can lead to error in the measurement due to the viewing angle.







Other Units for Measuring Lengths Units like millimetre and centimetre are suitable for measuring shorter lengths only. We need bigger units like metre, kilometre, etc. to measure longer distances.

Bigger unit to smaller unit

× 10 kilometre (km)

Multiply by 10

× 10 hectometre (hm)

÷ 10

× 10

× 10

decametre (dam)

÷ 10

metre (m)

÷ 10

× 10 decimetre (dm)

÷ 10

× 10

centimetre (cm) ÷ 10

millimetre (mm) ÷ 10

divide by 10

Smaller unit to bigger unit Example 2

Measure the length of the line segments.

D

B E A

AB = 3.5 cm Do It Together

F

C

DC = 5 cm

EF = 4.5 cm

Draw a line segment OP of length 6.5 cm. Draw two more line segments 1 cm and 2 cm longer than OP respectively.

6.5 cm O P

Comparison of Line Segments The length of line segments can be compared by first measuring the line segments and then comparing them. There are multiple ways to compare the length of line segments.

Chapter 5 • Line Segments and Angles

79

Comparison by Observation B

N

Y

Comparison by Tracing By observation XY > MN > AB. Comparison by observation is useful when differences in length are quite obvious.

A M

P

C

S D

It is difficult to compare lengths of line segments PQ, SR and CD just by looking at them.

X

R

Q

It involves the use of tracing paper to compare line segments. Let’s say you want to compare XY and MN. Trace MN using tracing paper and place it on XY in alignment. You would be able to tell which line segment is longer.

Comparison using ruler and divider A divider is a tool used for various geometrical purposes. It typically consists of two pointed legs attached to a hinge. Dividers are adjustable, allowing you to set the distance between the two points and helping measure more accurately.

To measure a line segment using the ruler and divider, follow these steps: 1 Open the divider and place its arms on points A and B. 2 Carefully lift the divider, put it on the ruler, with one end at zero.

3 Check where the other divider end points on the ruler.

4 Read the mark on the ruler to know the length.

A

Example 3

B

A

B

A 4 cm B

Name the line segment with the shortest length in the figure shown below.

O

Q

P Do It Together

C

By observation, we can conclude that CD is the shortest line segment

D S

Measure the line segments using a ruler and divider. Write the lengths.

P X

80

Q Y

PQ = ______ ; CD = _______ ; OS = _______

Do It Yourself 5B 1

Draw line segments of the lengths given. a 5 cm

2

b 7.7 cm

c 5.2 cm

d 35 mm

e 61 mm

Count the number of line segments in the shapes. a

3

O

P

b

Q

R

c

S

T

Which of the following is the shortest length? a 2000 mm

4

b 150 cm

c 100 dm

d 10 m

A road of length 23 km was constructed. What is the length of the road in these units: a metres

5

b centimetres

c decametres

d hectometres

Draw a line segment which is 1 cm longer than the line segment in each case. a

L

b

M

F

G

Word Problem 1

Anna is cutting ribbons into lengths of 120 cm and 150 cm for wrapping gifts. What is the length of the ribbons in decimetres?

Angles Real Life Connect

Alex and Emma are gathered with their toy cars in their study. They had built a makeshift ramp from wooden planks and start the race.

Emma: Alex, I wonder why our cars go faster when the ramp is steeper.

Alex: I think it's because of the angle, Emma.

Emma: What are angles, Alex?

Basics of Angles To help understand how much a line segment, line or ray is inclined on another, we have the geometrical idea of angles.

An angle is formed when two rays (or lines) meet at a arm vertex Q

P

angle R

arm

Chapter 5 • Line Segments and Angles

•

common point called the vertex.

•

Representation: ∠PQR, ∠RPQ, ∠Q

•

Symbol: ∠

81

Interior and Exterior of an Angle

A

As you can see in the figure, if the arms of the angle were extended indefinitely, they would divide the plane into two regions – interior and exterior. Points A, B and C lie on the exterior. Points D, E and F lie in the interior.

E

exterior

interior

D C

B

Adjacent Angles

F

Adjacent angles are a pair of angles that share a common vertex and a common arm. ∠XOY and ∠ZOY are adjacent angles.

X

Y O common vertex Example 4

Z

Observe the diagram given below and list the points that lie in the interior and exterior of angles ∠PON and ∠QON. A Interior Points

Exterior Points

∠PON: E, F, D, G

∠PON: A, B, C

∠QON: D, G

∠QON: A, B, C, E, F

P

B

F

E O

Q

D C

Example 5

G N

Draw two adjacent angles to the angle, ∠POQ. M P

O

P

Q

O

Adjacent angles drawn to ∠POQ are MOP and NOQ.

Q N

Do It Together

Draw an adjacent angle to ∠NMO so that:

A

1 Point B lies in the interior of the new angle but not A. 2 Both points A and B lie in the interior region of the new adjacent angle.

82

M

B

O

N

Do It Yourself 5C 1

Classify the points based on whether they lie in the interior or exterior region of the angle.

M

B

2

In the fig shown here, recognise the adjacent angle(s).

T

Q R

b ∠BQC

c ∠BQB

d ∠CBQ

Draw more points in a way that the angles ∠YMX, ∠XMK and vKMX have an equal number of points in their interior region.

Y A

S

N X

M

W

E

K

R

X

Draw adjacent angles to ∠NMZ so that:

A

a B lies in the interior of the new angle but not A and D.

A B

N

b Both A and B lie in the interior region of the new adjacent angle. c Only C lies in the interior region of the new adjacent angle.

6

R

C

Which of the following is an incorrect representation of an angle? a ∠O

5

P Q

N

O

4

S

P

M

3

A

M

Consider two angles ∠XOY and ∠AOB. Mark four points in each case so that a all the points are interior to ∠XOY only.

C

Z

b all the points are interior to ∠AOB only.

c all the points are exterior to both ∠XOY and ∠AOB.

Word Problems Alice is an artist working on a mural project. She's marking the interior and exterior regions of an angle shape on the mural using symbols.

1

Inside the angle, she marks the region with small 'x' symbols, creating a unique pattern.

2

Outside the angle, she marks the region with dots, forming an intricate design.

Draw how Alice is marking the interior and exterior regions of the angle in her mural project.

Chapter 5 • Line Segments and Angles

83

Measuring Angles Remember how Emma and Alex were racing cars on the ramps.

Emma: Alex, I wonder why our cars go faster when the ramp is steeper.

Alex: I think it's because of the angle, Emma. Let's measure the angles to see which angle measures more.

Measuring Angles Using a Protractor A protractor is a tool shaped like a half‐circle with markings from 0 to 180 degrees (or sometimes 0 to 360 degrees) that helps us measure angles in circles. Let us learn how to measure angles using the protractor.

Steps to measure an angle using a protractor:

1 Put the centre of the protractor right over the vertex of the angle (the point where the two lines of the angle meet).

2 Make sure that the baseline of the protractor (the straight line at the bottom) is aligned with one of the angle's arms.

3 Look at where the other arm of the angle crosses the markings on the protractor. The number where the arm crosses is the measurement of the angle in degrees.

A

A Measure of the angle in degrees

B

C

C

Centre of the protractor Example 6

∠A = 60° ∠B = 120° B

A

Remember!

Always align the centre and baseline of the protractor with the vertex and arm of the angle while measuring the angle.

Do It Together

Measure the angles and write which one has the smallest measure. a

b

c

P

∠P = __________

d

S

Q

∠Q = __________

The smallest angle is __________.

84

R

∠R = __________

∠S = __________

Do It Yourself 5D 1

Which of the following units is used to measure angle? a mm

2

b cm

b Divider

d Protractor

Q

R

Write the difference between the measure of angles formed by blue arm and red arm with the black arm in each case. a

5

c Compass

Measure the given angles. Write their measure.

P

4

d degree

Which instrument is used to measure the angles? a Ruler

3

c km

b

c

d

Study the figure given below.

B 2

3

C a Identify all the angles that have adjacent angles in the figure.

1

4

A

b Measure all the adjacent angles in the figure.

5 D

c Name the angle that has the greatest measure.

Word Problem 1

Two electric poles near your home got bent after being hit by a thunderstorm. Would you be able to tell which one is more bent without actually measuring their inclination?

Chapter 5 • Line Segments and Angles

85

Types of Angles Angles can be classified on the basis of their measure.

Acute Angle

Right Angle

Obtuse Angle

Straight Angle

Reflex Angle

Complete Angle

More tha 0º and less than 90º

90º

More than 90º and less than 180º

180º

More than 180º and less than 360º

360º

Perpendicular

The 90° angle is also called a perpendicular. It forms an ‘L’ shape, like when a vertical and a horizontal line meet. We see perpendiculars in the corners of a sheet of paper, windows and doors, or buildings.

90º

Right and Straight Angles Imagine you're standing, arms extended, facing east. Your back would be to the west, left arm indicating north, and right arm indicating south. In the diagram, you can see that the lines joining the opposite directions are perpendicular to each other.

The child is facing east. What direction will he face if he: 1 turns clockwise by one right angle?

______________

2 turns anticlockwise by one straight angle?

______________

3 turns clockwise by two right angles?

______________

4 turns anticlockwise by two straight angles?

______________

N W

E S

To be able to correctly predict the directions when you turn, you need to take care of two things: •

Amount of turn viz. no. of right angles or straight angles

•

Sense of rotation or turn (clockwise or anti‐clockwise)

Notice the changes in the direction depending on the sense of rotation and the amount of turn.

Turning in terms of right angles can also be expressed in terms turning or rotation by portion of a circle.

Let us see how.

1 On turning clockwise by one right angle, the child will face south.

2 On turning anticlockwise by one straight angle, the child will face west.

3 On turning clockwise by two right angles, the child will face west.

4 On turning anticlockwise by two straight angles, the child will come back to his original position and face east.

86

Angles in a Clock Now let us understand angles using the turns that the hands of a clock made. As the hour hand moves 3 1 1 numbers, it makes of the turn. On every 6 hours, it takes turn.

2 4 1 right angle or

1 4

turn

2 right angles or

1

turn 2 Or 1 straight angle

12

3 right angles or

12 3

9

4

turns

4 right angles or 2 straight angles or one full turn

12 3

9

3

12 3

9

3

9

6

6

6

6

Takes 3 hours

Take 6 hours

Takes 9 hours

Takes 12 hours.

Name the diagonals of the given quadrilaterals.

Example 7

1 2 to 5

When the hour hand moves from 2 to 5, it covers one‐fourth turn.

2 6 to 12

When the hour hand moves from 6 to 12, it covers one‐half turn.

3 12 to 9

When the hour hand moves from 12 to 9, it covers three‐fourths turn.

Alice is standing facing the North. She turns clockwise through 2 right angles. She then turns anticlockwise through 3 right angles.

What direction is Alice facing?

Example 8

1 Clockwise through 2 right angles – New position: Faces south 2 Anticlockwise through 3 right angles – Final position: Faces West Do It Together

Remember!

1 full rotation = 2 straight angles = 4 right angles

The starting position of Jane is given below. She takes different turns clockwise and anticlockwise. Find the final position in each case.

Initial Position

Angle of Turn

Sense of Rotation

Final Position

East

1 right angle

Clockwise

2

North

Two

South

3 right angle

West

Two

1 2

1 4

turns

turns

Chapter 5 • Line Segments and Angles

Clockwise Anti‐clockwise Anti‐clockwise

Error Alert!

Be cautious with rotation direction in problem solving; mixing up clockwise and anti‐ clockwise can cause mistakes.

87

Do It Yourself 5E 1

Classify the following angles as acute, obtuse or reflex angles. a 36o

2

b 93o

c 192o

d 89o

e 126o

f 53o

Which of the following objects is perpendicular to the ground? a Bat kept on the floor

b Stumps in a cricket match

c A boy leaning against the wall

d A book kept on a table

3

Suppose you were initially facing south. You were made to turn anti‐clockwise by three right angles. In which

4

What fraction of full rotation should the minute hand of a clock travel to go from the numbers:

direction are you facing now?

a 5 to 8

5

b 3 to 7

c 11 to 5

d 9 to 6

In which direction would you be facing if your initial position is: a north and you take a

1 2

b south and completing

turn, anti‐clockwise?

1 4

of a turn, clockwise?

c east and turn three‐fourths, clockwise?

Word Problem 1

Rohit has a job interview scheduled at 3:00 p.m. He wants to make sure that he leaves an hour early.

What angle will the minute and hour hand make on the clock when he should leave his house?

Points to Remember •

A point is a mark of location. It has no dimensions.

•

Lines, planes, and rays extend indefinitely.

• • • • •

88

A line segment has a fixed length.

A line segment can be measured using a ruler.

Angles are measured in degrees, using a protractor.

One full rotation is equal to 360°.

One straight angle is equal to two right angles.

Math Lab Angle Hawk Materials required: Protractor, Notebook, Pencil Setting:

1

Divide the students into pairs or small groups. Each group will need a protractor, a notebook, and

2

Provide each group with a list of objects or situations they need to find, each representing a

3

Allow the groups to move around the school or classroom to find the objects or situations on their

4

Gather the groups back together and give them time to discuss their findings within their groups.

a pencil.

specific type of angle.

list. Encourage them to measure the angles using the protractor.

Encourage them to compare the angles they found and share any interesting discoveries.

Chapter Checkup 1

Count the number of line segments in the figures. a

2

c

d

c Plane

d Line segment

c 6.9 cm

d 2.2 cm

Which of the following does not extend indefinitely? a Ray

3

b

b Line

Draw line segments of these lengths. a 4.6 cm

b 77 mm

4

How much is 4 km in decimeters?

5

Identify the points in the exterior and interior region of the ∠AOB. A

D

F E

G

D B Chapter 5 • Line Segments and Angles

C

89

6

Which of the following does not extend indefinitely?

B

A

Only A, E, D and F are in the interior of the angle.

Only B, C, D and E are in the interior of the angle.

D

F

Only C and D are in the interior of the angle.

Only A, F and G are in the interior of the angle.

G

Only B and E are in the interior of the angle.

7

C

E

Draw two adjacent angles to the given angle ∠XOY. Y

X

O

8

Classify the angles as acute, obtuse or reflex. a 26o

9

b 256o

c 79o

d 129o

e 165o

f

189o

Identify the angles as acute, obtuse, reflex or straight without actually measuring them. a

b

c

d

e

10 A clock shows the time as 12 p.m. What is the angle that the hour hand moves through when the time is: a 3:00 pm

b 6:00 pm

C 12:00 am

11 What orientation will you have after starting to face: a west and making

1

of a revolution clockwise?

2 1 b north and completing of a revolution counterclockwise?

4 3 c east and rotating of a revolution clockwise?

4 d south and making a full revolution counterclockwise?

12 If we had defined degree by dividing our circle into 300 slices instead of 360 slices, what would be the angle between the minute and hour hands of the clock at 6:00 p.m.?

Word Problem 1

90

You're standing at the entrance of a building, and you're facing south. Now, you want to head in the direction of east, which is to your left. How many right angles do you think you need to turn anticlockwise to find yourself facing east from your initial south‐facing position?

6

Triangles, Quadrilaterals and Polygons

Let's Recall One day Reena’s teacher came into the class with a box full of shapes.

She asked them to run their fingers around the sides of the shapes.

Teacher: Start with one corner and say corner. Move your finger over the edge and say side. When you reach a corner, say corner. When you reach a side, say side. Count the number of corners and sides in each shape.

Let us form some 2-D shapes using matchsticks. How many sides does each have?

3 sides

4 sides

5 sides

7 sides

Let's Warm-up Fill in the blanks with the number of sides in the shapes.

1

2

4

5

3

I scored _________ out of 5.

Mean, Median and Mode Understanding Triangles Real Life Connect

Raj and Priya were looking at the pictures of various historical buildings in their scrapbook.

Raj: Priya, look at this pyramid. I can see triangles in it.

Priya: That's fascinating, Raj. It's amazing how geometry is intertwined with architecture.

Triangles and Their Features As Raj and Priya were looking at the faces of the pyramids, they noticed that the faces were bound on the three sides by line segments and looked like triangles. Let’s look into the details of triangles and their features.

Did You Know?

The pyramids of Giza are around 4500 years old.

Parts of a Triangle A triangle has 3 vertices, 3 sides and 3 angles.

P

Vertices: P, Q, R

Vertex

Sides: PQ, QR, RP

Angles: ∠PQR or ∠Q, ∠QRP or ∠R, ∠RPQ or ∠P.

Angle Q

R

Side

Naming a Triangle

A triangle is denoted by the uppercase delta symbol followed by the names of all its vertices in any order. For example: ∆PQR, ∆ABC, etc.

The part of the plane enclosed by the triangle is called the interior of the triangle. The part of the plane outside the area enclosed by the triangle is called the exterior of the triangle.

Example 1

Exterior of a

List the vertices, sides and angles of the triangle given below.

M

Vertices: M, N, O Angles: ∠MNO, ∠NOM, ∠OMN N

Example 2

Sides: MN, NO, OM

O

Identify the points lying on the interior and exterior of the given triangle.

B F

A E

G

C D

92

Interior: E, F, G Exterior: A, B, C, D

triangle

Exterior of a Interior

triangle

of a triangle Exterior of a triangle

Do It Together

Use the points A, B, C, S, E, F and G and draw triangles so that: 1 Only F and G are in the interior of the triangle.

A

D

2 Only D, E, F and G are in the interior of the triangle.

A

G

B

F

E

Classification of Triangles on the Basis of Sides Triangles can be classified on the basis of their sides by measuring the length of their sides.

Equilateral Triangle

Isosceles Triangle

A

X

B

C

Y

All the sides are equal in length.

N

Z

The angles opposite to the equal sides are also equal.

Here, ∠XYZ = ∠XZY.

The angles of an equilateral triangle are also equal.

O

M

Any two out of three sides are equal. Here, XY = XZ.

AB = BC = CA

Example 3

Scalene Triangle

All the sides are unequal.

MN ≠ NO ≠ OM All the angles are unequal. ∠MNO ≠ ∠NOM ≠ ∠OMN.

It is known that two out of three sides of a scalene triangle ∆XYZ measure 7.8 cm and 8.7 cm. Which of the side lengths given below CANNOT be the measure of the third side?

1 6 cm

2 7. 1 cm

3 8.7 cm

4 5 cm

8.7 cm can’t be the side length of the third side as the triangle is a scalene triangle.

Example 4

Label the triangles as equilateral, scalene or isosceles.

4 cm

Do It Together

11 cm

4 cm

11 cm

5 cm

11 cm

As two sides of the triangle are equal, it is an isosceles triangle.

As all the sides of the triangle are equal, it is an equilateral triangle.

6 cm 10 cm

16 cm

As none of the sides of the triangle are equal, it is a scalene triangle.

Measure the lengths of the sides of the given triangles. Label them as equilateral, isosceles or scalene.

1

2

Chapter 6 • Triangles, Quadrilaterals and Polygons

3

93

Do It Yourself 6A 1

2

3

List the vertices, sides, and angles in the triangle.

T R

Classify the triangles as equilateral, isosceles or scalene given the length of their sides. a 6 cm, 7 cm, 9 cm

b 15 cm, 15 cm, 15 cm

c 19 cm, 23 cm, 29 cm

d 9 cm, 11 cm, 9 cm

e 18 cm, 31 cm, 27 cm

f 3 mm, 3 mm, 3 mm

Identify the scalene, equilateral and isosceles triangle in the figure given below.

3 cm

E

5 3.

5

m

4 cm

B

3 cm

C

Classify the triangles according to the measure of angles. a 60°,40°,80°

b 65°,85°,30°

c 130°,20°,20°

d 90°,40°,50°

e 110°,25°,45°

f 60°,60°,60°

Classify the triangles according to their sides and angles simultaneously. a

b

3 cm

4 cm

c

9 cm

5 cm

6

6c

cm

cm

D

5 3.

3 cm

A

4

S

9 cm 9 cm

d 15 cm

10 cm

2 cm

6 cm

Analyse the sides of the triangles given below and spot the angles in each triangle that would be equal in measure.

a PQ = 3 cm, QR = 4 cm RP = 4 cm

b AB = 12 cm, BC =12 cm, CA = 13 cm

c MN = 6 cm, ON = 10 cm, OM = 6 cm

d XY = 7 mm, YZ = 7 mm, ZX = 7 mm

Word Problem 1

Sam and Maria have triangular-shaped gardens outside their homes. Sam’s garden measures

7 feet, 7 feet and 10 feet while Maria’s garden measures 5 feet, 6 feet and 7 feet. Identify the types of triangle formed by their gardens.

94

2 cm

Mean, Median and Mode Understanding Quadrilaterals Real Life Connect

Father: Sukhi, look what I’ve bought for you!

Sukhi: What is it, Dad? Father: I have bought a book on shapes for you. This book has shapes with 3 or more sides.

Quadrilaterals and Their Features Sukhi sees shapes made of 4 sides in the book. Such shapes are called quadrilaterals. A quadrilateral is a plane figure that has 4 sides or edges, 4 angles, 4 corners/vertices.

Parts of a Quadrilateral A triangle has 3 vertices, 3 sides and 3 angles.

S

Adjacent Sides: Adjacent sides of a quadrilateral are sides that share a common vertex (corner). They are next to each other in the sequence of sides.

P

E.g. – (PS, PQ), (SR, RQ) Opposite sides: Opposite sides of a quadrilateral are sides that do not share a common vertex. They are located on opposite sides of the quadrilateral.

Q

R

E.g. – (PQ, SR), (PS, QR)

Diagonals: These are the line segments joining opposite vertices of a quadrilateral.

Opposite angles: These are the angles in a quadrilateral with no common arm.

Adjacent angles: Adjacent angles of a quadrilateral are angles that share a common side.

E.g. – PR, QS

E.g. – (∠P, ∠R), (∠Q, ∠S)

E.g. – (∠P, ∠Q), (∠Q, ∠R), (∠R, ∠S), (∠S, ∠P),

Interior of a Quadrilateral: The interior of a quadrilateral refers to the region enclosed by the four sides of the quadrilateral.

Exterior of a

Exterior of a Quadrilateral: The exterior of a quadrilateral refers to the region outside the four sides of the quadrilateral.

quadrilateral Exterior of a

Exterior of a

quadrilateral

quadrilateral Exterior of a

quadrilateral Example 5

Name the diagonals of the given quadrilaterals.

1

A

B

2

E

F

3

P S

B

C

AC, BD Chapter 6 • Triangles, Quadrilaterals and Polygons

H

EG, FH

G

Q R

PR, SQ 95

Example 6

List the adjacent sides, adjacent angles and opposite angles in the given quadrilateral.

S

Adjacent Sides: (RU, UT), (UT, TS), (ST, SR)

R

Adjacent Angles: (∠R, ∠U) , (∠T, ∠U), (∠T, ∠S), (∠S, ∠R) Opposite Angles: (∠R, ∠T) , (∠S, ∠U)

Do It Together

U

T

Draw quadrilaterals to show points P, Q, R, S, T and U according to the rules.

1

2 Only points Q, R and S are in the

Only points P and R are in the

3 Only points P and T are in the

interior.

interior.

exterior.

Q

P

R U

S T

Do It Yourself 6B 1

List the names of sides and vertices in a quadrilateral EFGH.

2

Identify and write all the adjacent and opposite sides in the given quadrilaterals. a

P

Q

S 3

A

B

D

C

b

R

K

b Only B, D, E and F are in the exterior.

List the names of diagonals of the quadrilaterals given below. b

A B

D C

96

M

Draw quadrilaterals using points A, B, C, D, E and F so that:

a

5

J

L

a Only A, B and C are in the interior.

4

c

T

S

J

c

V U

M

Draw quadrilaterals with the names. Draw diagonals and label them.

K

L

Word Problem 1

A quadrilateral garden has four trees inside it near each corner and four decorative stones outside it near the middle of each side of the garden. Draw a quadrilateral with trees and stones as exterior points respectively to represent the layout of the garden.

Classification of Quadrilaterals Quadrilaterals can be in things around us like windows and doors, kites, buildings etc. They can be classified on the basis of the properties of their sides and angles.

Convex and Concave Quadrilaterals B

A

Convex Quadrilaterals In these quadrilaterals the measure of each angle is less than 180º. Both diagonals lieinside it.

D

C Q

Concave Quadrilaterals In these quadrilaterals the measure of one of the angles is more than 180º. One of the diagonals lies outside it.

R S

P

Types of Quadrilaterals A

B

A

B

D

C

D

C

Square All the sides are equal.

All the angles are right angles.

Opposites sides are parallel.

Diagonals are equal and bisect each other at right angles.

Square All the angles are right angles.

P

S

Q

R Parallelogram

Opposite sides and angles are equal.

Opposite sides are equal and parallel.

Diagonals are equal and bisect each other.

Diagonals bisect each other.

Opposite sides are parallel.

Chapter 6 • Triangles, Quadrilaterals and Polygons

Opposite angles are equal.

97

A

B

P A

D

C

D

C

Rhombus

S

Opposite sides are parallel.

Opposite angles are equal.

Diagonals bisect each other at right angles.

Q

S

R

Trapezium

All the sides are equal.

Example 7

Q

P

B

R Kite Opposite sides are unequal.

One pair of opposite sides is parallel.

Here, AB ∥ CD.

In a trapezium, when the non-parallel sides are equal, it is known as isosceles trapezium. Here, PS = QR.

Here, PS ≠ RQ; PQ ≠ RS Two pairs of adjacent sides are equal. Here PS = PQ, RS = RQ

Which of the quadrilaterals given below satisfies the conditions?

1 None of the sides may be equal to another other side: Trapezium 2 All the sides are equal but all its angles are not equal. Rhombus

Example 8

Which of the angles given below CANNOT be an angle in a convex quadrilateral?

1 61°

2 93°

3 113°

4 191°

No angle in a convex quadrilateral can be more than 180°. Hence, 191° is the correct answer.

Do It Together

PQRS is a parallelogram whose diagonals bisect at T. Label the parallelogram and list the line segments that are equal in length.

PQ = SR; _____________________________________

Do It Yourself 6C 1

Name the quadrilateral with the features given. a All sides of equal length and all angles measure 90°. b Opposite sides are parallel and opposite angles are equal. c Two pairs of adjacent sides are of equal length, and one pair of opposite angles is equal.

2

What angles are formed by the intersection of diagonals in a square? a 90°,60°

3

98

b 120° ,90°

c 60° ,60°

d 90° ,90°

LMNO is a rectangle whose diagonals bisect at Q. List the line segments which are equal in length.

4

Look at the figure. Answer the questions.

J

a Identify the quadrilateral.

K

b Name the parallel and non-parallel sides of this quadrilateral. c What is this quadrilateral known as when JM is equal to KL?

5

M

L

LMNO is a rhombus whose diagonals bisect at O. List the equal sides and angles.

Word Problem 1

In an archaeological survey a mysterious patch of land was found with four statues at the corners.

The Archaeologist, curious about the shape of the land, measures the angles between the adjacent statues. ∠ABC = 80°, ∠BCD = 100°, ∠CDA = 75°, and ∠DAB = 105°. a Is the patch of land a convex or a concave quadrilateral? b Can the quadrilateral land be a rectangle or a square? c Can the quadrilateral land be a parallelogram? Why?

Mean, Median and Mode Understanding Polygons Real Life Connect

Rita’s father bought her a beautiful pendant. Rita’s eyes lit up when looking at the gift.

Rita called her friends from the neighbourhood and showed it to them.

Rita: Hey everyone, guess what? My father just gave me this amazing pendant!

Alex: Wow! I’ve never seen such a shape before!

Polygons and Their Features Rita’s pendant has six sides. It is a polygon. Let us learn more about polygons.

Chapter 6 • Triangles, Quadrilaterals and Polygons

99

Parts of a Polygon A polygon is a plane figure that has 3 or more straight sides. It has the same number of vertices and angles as the sides. In a polygon, every line segment intersects with exactly two other line segments. One is its adjacent side and the other a diagonal.

Adjacent Vertices: The endpoints of any given side are called adjacent vertices. (P, Q) and (P, T) are 2 examples.

Adjacent Sides: Adjacent sides of a polygon are sides that share a common vertex (corner). They are next to each other in the sequence of sides.

(PQ, QR) and (SR, RQ) are 2 examples.

Diagonals: These are the line segments joining two nonadjacent vertices of a polygon.

For example, PR, QS, QT, RT, SP

Q

P

R

Interior of a polygon: The interior of a polygon refers to the region enclosed by the sides of the polygon.

S

T

Exterior of a polygon: The exterior of a polygon refers to the space outside the polygon's boundaries.

Check whether the given figures are polygons or not.

Example 9

1

2

3

4

c is NOT a polygon since it does not have straight sides. All others are polygons.

List all the vertices and sides of the given polygon. Which line segments can be drawn to form its diagonals?

Example 10

A F

B

E

C

Vertices: A, B, C, D, E, F Sides: AB, BC, CD, DE, EF, FA Diagonals: AC, AD, AE, BB, BE, BF, CE, CF, DF

D Do It Together

Label the vertices and draw diagonals in the polygon. List the points which are lying in the interior and and the exterior of polygon.

Y W

Point in the interior: _________________

X Z

100

Vertices: _________________ Points in the exterior: _________________

Types of Polygons Regular and Irregular Polygons A regular polygon is a polygon in which all sides are of equal length and all angles are of equal measure. Regular polygons are highly symmetrical and have a uniform appearance.

An irregular polygon is a polygon that does not have all sides of equal length and/or all angles of equal measure. Irregular polygons lack the uniformity and symmetry of regular polygons.

Convex and Concave Polygons Concave Polygons have the measure of one of the angles more than 180°. Diagonals lie in the exterior.

Convex Polygons have a measure of each angle of less than 180°. The diagonals lie inside the polygon.

Let us classify polygons on the basis of their sides.

Error Alert!

Think and Tell

Is right-angled triangle a regular polygon?

Example 11

A polygon with equal sides alone may not be regular; it requires equal angles as well to be considered regular.

Triangle - 3 sides

Quadrilateral - 4 sides

Pentagon - 5 sides

Hexagon - 6 sides

0 diagonals

2 diagonals

5 diagonals

9 diagonals

Heptagon - 7 sides

Octagon - 8 sides

Nonagon - 9 sides

Decagon - 10 sides

14 diagonals

20 diagonals

27 diagonals

35 diagonals

How many sides are there in the polygons?

5 1 Pentagon: ______________________

9 2 Nonagon: _______________________

Chapter 6 • Triangles, Quadrilaterals and Polygons

8 3 Octagon: ______________________ 101

Example 12

Which of the list of the angles could be of a regular polygon?

1 60°, 100°, 130°, 70°

2 90°, 90°, 90°, 90

3 90°, 100°, 120°, 90°

4 45°, 60°, 75°, 90° Do It Together

Name the polygon and list its vertices. Draw diagonals. Write the number of diagonals.

Vertices: _______________________ Diagonals: _______________________

Do It Yourself 6D 1

2

List the names of all the sides and vertices of the polygon.

A

3

G

D F

b Rectangle

E

c Scalene Triangle

d Right-angled Triangle

b Nonagon

c Decagon

d Pentagon

e Hexagon

d 7

e 6

Draw an irregular polygon with the number of sides given. a 5

5

C

How many line segments form the polygons? a Heptagon

4

H

Which of these is a regular polygon? a Rhombus

B

b 4

c 3

Classify the polygons as regular or irregular with the measures given. Give reasons for your answer. a Side lengths = 6 cm, 6 cm, 6 cm, 9 cm, 6 cm

b Angles = 90°,90°,90°,90°

c Side lengths = 2 cm, 2 cm, 2 cm, 2 cm; Angles = 90°,90°,90°,90°

6

Raj chose to build a logo in the shape of a polygon for his company. Read the clues and guess the type of polygon and the number of sides in it.

a All the sides and angles in his polygon are equal.

b He can draw 9 diagonals in his polygon.

Word Problem 1

Sarah has a garden in the shape of a hexagon. She plants two sunflower plants on each side of

her garden. Alex has an octagon-shaped garden. He plants one sunflower plant on each side of the garden. Who has more sunflower plants and how many more?

102

Points to Remember •

A triangle is a 2-D shape with 3 sides.

•

A quadrilateral is a 2-D shape with 4 sides, 4 angles, and 4 vertices.

• • •

Triangles can be classified on the basis of their sides and angles.

A polygon is a 2-D shape with 3 or more sides. It has the same number of sides, vertices and angles. Triangles and quadrilaterals also come under polygons.

A polygon can be regular or irregular, concave or convex.

Math Lab Shape Sorting and Classifying Setting: In a group of 3–4 students Materials Required: Paper/cardboard shapes (different types of polygons), markers/labels, Method:

1

Create and label shapes. Use colours for distinction.

2

Participants sort shapes into piles for triangles, quadrilaterals, and other polygons.

3

Within groups, further classify shapes (e.g., triangles into equilateral, isosceles, scalene).

4

Discuss how they identified shapes, common properties, and differences.

5

Participants make composite shapes using sorted shapes.

6

Each presents their composite shape, explaining types used and arrangement.

Chapter Checkup 1

2

3

Classify the triangles as equilateral, isosceles or scalene. a 7 cm, 7 cm, 10 cm

b 14 cm, 15 cm, 16 cm

c 29 cm, 29 cm, 29 cm

d 10 cm, 12 cm, 10 cm

e 20 cm, 31 cm, 26 cm

f

4 mm, 5 mm, 3 mm

Classify the given triangles according to the measure of angles. a 70°, 30°, 80°

b 45°, 85° 50°

c 120°, 20°, 40°

d 90°, 40°, 50

e 60°, 60°, 60°

f

Identify the scalene, equilateral and isosceles triangle in the figure.

100°, 50°, 30° A

7 cm 9 cm

12 cm

B

Chapter 6 • Triangles, Quadrilaterals and Polygons

F

8 cm

E

8 cm 8 cm

6 cm C 5 cm 7 cm

D

8 cm

103

4

Identify the points which are interior to: a Both ΔDEF and ΔPQR

D

A 3

b Both ΔABC and ΔDEF

2

B

c Only ΔDEF

4

E

d Only ΔPQR

Q

1

5 9

C

10 6

5

7

11

P

12

13 F

R

5

6

RSTU is a square whose diagonals bisect at Q. Find as asked. a Name the diagonals of this square.

b Name any four right angles found.

c Is UQ = TQ? Give a reason.

d Name the pair of sides which are parallel.

Identify the polygon with the length of sides. a 4

7

b 8

c 5

d 7

c 8

d 10

Draw an irregular polygon with the given number of sides. a 3

b 6

8

Draw a hexagon and its diagonals. Write the number of diagonals.

9

Consider the parallelogram given below. Answer the questions.

P

a Are PQ and RS parallel to each other?

Q

b Name any four pair of line segments in the figure which are equal. c Is PR equal to SQ? Give a reason.

S

R

10 The figure MNOP is a kite. Answer the questions. Give reasons.

N

a Is MN equal to PM? b Is ∠MNQ equal to ∠MPQ? c Is ∠NMQ equal to ∠PMQ?

M

Q

O

P

Word Problem 1

104

Lisa is designing a necklace with a pendant in the shape of a heptagon. She wants to attach different gemstones, one on each corner of the pendant. Mark is creating a bracelet in the form of a decagon, and he plans to add two beads along each side of the bracelet. a

How many gemstones will Lisa need for her heptagon pendant's corners?

b

How many beads does Mark need for his decagon bracelet's sides?

c

If Lisa wants to use the same number of gemstones as Mark has beads, how many more gemstones should she acquire?

About This Book Imagine Mathematics seamlessly bridges the gap between abstract mathematics and real-world relevance, offering engaging narratives, examples and illustrations that inspire young minds to explore the beauty and power of mathematical thinking. Aligned with the NEP 2020, this book is tailored to make mathematics anxiety-free, encouraging learners to envision mathematical concepts rather than memorize them. The ultimate objective is to cultivate in learners a lifelong appreciation for this vital discipline.

Key Features • Let’s Recall: Helps to revisit students’ prior knowledge to facilitate learning the new chapter • Real Life Connect: Introduces a new concept by relating it to day-to-day life • Examples: Provides the complete solution in a step-by-step manner • Do It Together: Guides learners to solve a problem by giving clues and hints • Think and Tell: Probing questions to stimulate Higher Order Thinking Skills (HOTS) • Error Alert: A simple tip off to help avoid misconceptions and common mistakes • Remember: Key points for easy recollection • Did You Know? Interesting facts related to the application of concept • Math Lab: Fun cross-curricular activities • QR Codes: Digital integration through the app to promote self-learning and practice

About Uolo Uolo partners with K-12 schools to provide technology-based learning programs. We believe pedagogy and technology must come together to deliver scalable learning experiences that generate measurable outcomes. Uolo is trusted by over 10,000 schools across India, South East Asia, and the Middle East.

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