Imagine_Maths_CB_Grade6_Sales_Sample by Uolo

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.

MATHEMATICS

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

Imagine Mathematics

About This Book

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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IM_G06_MA_MB_Cover.indd All Pages

Gurugram

Bengaluru

hello@uolo.com `XXX

NEP 2020 based

NCF compliant

CBSE aligned

27/10/23 3:35 PM

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

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

21 question 22

6 × 2 = ____

6 × ____ = ____

6 × ____ = 24

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

TTh

Thinkofand Tell I scored out12,344 of 5. + 1115 = _____________. The first five multiples 6 are 6, ____, ____, 24, ____._________ So, 17 What do you think do

even number?

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

____

=31 ×

multiplicand

TTh

multiplier

= 62

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

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

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?

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

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

to the concept

Think and Tellhave in common?

Fun fact, related Story Sums

all the circled numbers

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

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

Add 12,344 and 1115. 6 × ____ = ____

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

Real Life Connect

UM24CB_G4.indb 49

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

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

d 8974 ÷ 7

is 8.

e 5555 ÷ 5

What do we know?

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

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

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:

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

a 14

b 21

c 36

19 L

g 48

h 50

20 30 TTh

approximately how many tents will be needed for the camp?

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

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

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

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

n 20

•

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.

o TTh 25

________

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. • • •

UM24CB_G4.indb 83

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.

________

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

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.

Chapter 4 • Division

UM24CB_G4.indb 77

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

UM24CB_G4.indb 92

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

Divide the entire class into groups of 4.

classroom

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

Lakhs Period

activity

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

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

Lakhs

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.

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

Divisor) + Remainder.

•

Chapter 1 • Numbers up to 6-digits

9/11/2023 4:25:13 PM

7,81,005   

  

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 Real Life Connect

Ajay: Hello daddy, I found this letter, it is for you. 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 us 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

Let us learn more about 5-digit numbers! 0

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 2

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

FM.indd 2

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.

10/19/2023 4:10:52 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 ÷ 2

____

Yes

36 ÷ 4

____

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

Yes

Step363÷ 2

____

Common 36 ÷ 3Factors

____

Do It Yourself 5C

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

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

____

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

Setting:bIn groupsc of 4

a 9

d 13

e 15

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.

8, 10 of 64? Explainbyour 12, answer. 15 IsMethod: 6a a factor

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

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

a 16 and 24

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

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

UM24CB_G4.indb 95

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

6 10

aChapter 45 5 • Multiples andbFactors 66 2

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

UM24CB_G4.indb 92

f 818

92 8

Chapter Checkup

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

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.

Chapter 5 • Multiples and Factors

Gradual Release of Responsibility UM24CB_G4.indb 97

9/11/2023 4:26:2

C o nt e nt s

Large Numbers ................................... 1 • Understanding Roman Numerals • Understanding Large Numbers

on Large Numbers ............................ 16 • Operations on Large Numbers

• Patterns in Whole Numbers

• Prime Factorisation

• Divisibility Rules

• Working with LCM and HCF • Line Segments

• Understanding Polygons

101

Curves and Circles ........................... 107 • Curves

• Understanding Circles

108 110

165

• Application of Decimals

179

• Organising and Representing Data

• Perimeter and Area Problems

213

• Algebra and Patterns Equations

• Lines of Symmetry

• Reflection and Symmetry

Integers ............................................ 129 • Operations on Integers

134

251 255

and Angles ....................................... 263

• Construction of Angles

130

242

Construction of Line Segments • Construction of Line Segments

• Understanding Integers

235

Symmetry ......................................... 250

3-D Shapes ....................................... 119

120

223

and Proportion ................................ 234 • Understanding Proportion

219

Introduction to Ratio • Understanding Ratio

207

Introduction to Algebra ................. 218 • Algebraic Expressions and

185

204

• Understanding 3-D Shapes

173

• Understanding Perimeter • Understanding Area

Polygons ............................................. 93 • Understanding Triangles

• Understanding Decimals

Mensuration .................................... 203

Triangles, Quadrilaterals and

• Understanding Quadrilaterals

Decimals ........................................... 164

Line Segments and Angles............... 77 • Angles

154

Data Handling ................................. 184

145

and Multiples ..................................... 55 • Types of Numbers

• Reviewing Fractions

• Operations on Decimals

Playing with Numbers: Factors • Reviewing Factors and Multiples

• Whole Numbers and Natural Numbers

Fractions .......................................... 144 • Operations on Fractions

Whole Numbers ................................ 35 • Operations in Whole Numbers

Estimation and Operations • Estimation in Large Numbers

• Construction of Perpendiculars and Bisectors

264 267 274

Answers .................................................... 286

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 various 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

Each digit of a number has a place value and a face value. The place values of all digits 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×

100

700

1000

10000

= 3 × 100000

= 2 × 1000000

= =

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 Numerals 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 and clock faces. In this system, we use seven basic symbols to write diﬀerent numbers. This system does not have the concept of place value, so there is no representation for 0. Roman Numbers

Numbers in the Indian System

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. XXX = 10 + 10 + 10 = 30

CC = 100 + 100 = 200

Rule 2: If a symbol of a smaller value is written to the right of a symbol of the greater value, the resulting number is equal to the sum of the symbols. VI = 5 + 1 = 6 CI = 100 + 1 = 101 Rule 3: If a symbol of smaller value is written to the left of (before) a symbol of greater value, the resulting numeral is equal to the diﬀerence of the symbols. IX = 10 – 1 = 9 and XC = 100 – 10 = 90 Rule 4: If a smaller numeral is placed between two larger numerals, it is always subtracted from the larger numeral to the right. XIV = 10 + (5 – 1) = 14 and MCD = 1000 + (500 – 100) = 1400 2

Remember! The symbols V, L and D can never be repeated or subtracted, whereas 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 only from

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

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

215 = 100 + 100 + 10 + 5 = CCXV

Do It Together

symbol C can be subtracted only from D and M.

Match the numbers with their Roman numbers. 1 45

MCCCXC

2 98

3 110

XLV

4 185

XCVIII

5 1390

CLXXXV

Chapter 1 • Large Numbers

Do It Yourself 1A 1

Write Roman numbers for the numbers. a 99

b 184

d 438

e 1564

f 1287

d CCCLXV

e MDCCCLXIV

f CMXCIX

Write number form for the given Roman numbers. a CDLXVI

c 285

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 _______

_______ 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 each of the countries that was given in the data is a 9-digit number. Indonesia

277534122

Russia

144444359

United States 339996563

9-digit Numbers

Let us see how to read and write large numbers using the 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 the ‘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 1

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

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 on 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 =

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

Example 4

Find the sum of the place value and the 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 the 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 When the value of the digits of a number are written according to their place values, separated by a plus sign, it is called the expanded form of that 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

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

  

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, using the Indian number system?

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

Do It Yourself 1B 1

Write the face value and the 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

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 number and the original number?

In 15,87,12,528 how many times is the value of 5 in the crores place to that of the value of 5 in the hundreds 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

Thousands

Ones

Hundred Ten Millions Millions Hundred Ten Thousands Thousands Hundreds Tens Ones Millions (HM) (TM) (M) Thousands (HTh) (TTh) (Th) (H) (T) (O)

periods

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

place

3 places Chapter 1 • Large Numbers

100,000

10,000 3 places

1,000

100

3 places

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 the 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×

=4×

3 ten thousands

1000000

= 7 × 10000000

100

4000

30000

500000

7000000

= 70000000

= 200000000

Write the face value and the 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 consider the population of Indonesia (277534122). 277,534,122 Millions period

Thousands period

  

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 the number name for the given number. 902,564,875 = _________________________________________________________________________________________________ Number name: ________________________________________________________________________________________________ ________________________________________________________________________________________________

Do It Yourself 1C 1

Write the face value and the 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

In a number 764,156,229, the digits 7 and 1 are interchanged to form a new number. Answer the following 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 the place values of 7 in the new and original numbers?

Chapter 1 • Large Numbers

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 various countries and arrange them. Comparing Numbers To compare two numbers, let us 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

L 4 8

TTh 7 7

Th 6 0

H 8 5

T 2 3

O 7 9

TTh

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

We see that 0 < 9. So, 21,04,76,827 < 21,98,70,539.

1=1

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 less?

greater.

Ordering Numbers

e sc

de or

smallest to greatest

De 5

or de

greatest to smallest

To arrange the population of 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

have 9 digits and 1 < 2 < 3.

1<2<3

So, let us compare numbers, starting with digit 1.

On comparing, we get, 4 > 2.

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

1=1

TTh

4>2

Step 2

Think and Tell

Compare the remaining numbers. Write the next smallest number.

Can you now arrange the population

As 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. As 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 want 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 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! 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?

Chapter 1 • Large Numbers

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

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!

This number 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 descending order.

Example 10

Do It Together

Form the greatest and smallest 9-digit numbers, 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.

Think and Tell

Arrange the following numbers in both ascending and descending order.

Why did we choose to

126,856,384; 165,121,349; 860,778,653; 400,897,234

repeat the greatest digit?

Ascending Order: _____________________________________________________________________________________________ Descending Order: ____________________________________________________________________________________________

Do It Yourself 1D 1

Arrange the following 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

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

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

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

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 any digit. a 2, 7, 1, 0, 8, 6, 4, 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 various numbers.

•

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

•

• • • • •

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

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 the greatest or the smallest number with any given digits, arrange the digits in the descending or the ascending order, respectively.

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.

In groups, collect the index cards from the teacher.

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

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

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

Continue until all teams have written the numbers in words.

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

its number names.

Chapter 1 • Large Numbers

Chapter Checkup 1

Write Roman numbers for the numbers given. a 427

b MXLXXV

d 9015

e 215

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 forms. a 350427681

c 635

Write the number forms of the following Roman numbers. a CCDXVII

b 2087

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

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

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

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

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

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 answers in Roman numbers. a XLXX + CDXV

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

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.

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

On day 3, the restaurant set a goal to make ₹90,000. By lunch time, 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 So, the restaurant needs to make ₹354 more to reach its goal.

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 inconvenient to read and speak 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' are added to convey that the number is close to exact. We can round off numbers to various 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

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

4<5

Example 1

154376000

356873660

rounded off to

6>5

356874000

Round off 86,45,00,679 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

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 ____________________.

was 1,43,16,62,681, as of

The population of India 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 7675

c 6352

d 5650

e 94,267

f 40,536

g 76,567

h 2,54,765

20,45,546

Round off the numbers to the nearest hundred. a 545

b 3098

c 4576

d 4723

e 86,850

f 43,607

g 8,09,757

h 7,97,005

23,98,790

k 3,80,08,023

89,74,53,542

8,76,746

49,46,826

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 j

4,65,45,758

h 65,43,567

31,41,457

k 1,03,09,095

30,47,57,698

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

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

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 567790. What are the smallest and 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 lakh = ₹52,00,00,000 Amount raised through the concert in 2021 = ₹52,00,00,000 + ₹50,00,000

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, we need 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 Chapter 2 • Estimation and Operations on Large Numbers

So, the estimated sum of 6584 + 2431 = 9000. 1

The estimated difference of 6584 – 2431 = 4200.

–

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

Thus, the estimated sum of 79,643 and 10,056 when rounded off to the nearest thousand is 90,000. Example 4

What is the estimated difference of 42,431 and 20,564 if they are rounded off to the nearest ten? 42,431 rounded off to the nearest ten = 42,430 20,564 rounded off to the nearest ten = 20,560

–

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

Solve and give the approximate answer. 1 765 + 213 (to the nearest hundred) 765

rounded off to

__________

213

rounded off to

__________

So, estimated sum of 765 and 213 = ______ 2 984 – 356 (to the nearest ten) 984

rounded off to

__________

356

rounded off to

__________

So, estimated difference of 984 and 356 = ______

Estimating the Product Like the estimated sum and difference, we can also find the estimated product of two numbers. For example, we need to find the estimated product of 84 and 28 to the nearest ten. 84 20

rounded off to nearest ten

So, the estimated product of 84 × 28 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

rounded off to ten

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).

Estimating the Quotient Like the estimated product, we can also find the estimated quotient. Say we need 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 = 20 260 260 ÷ 20 = = 13 20

Chapter 2 • Estimation and Operations on Large Numbers

13 20 260 –20 60 –60 0

Example 6

Find the estimated quotient of 5634 ÷ 231 by rounding off each number to the nearest hundred. 5634 rounded off to the nearest hundred = 5600

28 200 5600 –400 1600 –1600 0

231 rounded off to the nearest hundred = 200 5600 5600 ÷ 200 = = 28 200 Thus, the estimated quotient of 5634 ÷ 231 is 28. Do It Together

Estimate the quotient of 332 ÷ 25. 332

rounded off to the nearest hundred

rounded off to the nearest ten

__________

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

Do It Yourself 2B 1

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

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

b 6786 × 9123

c 8174 × 3,589

d 759 × 422

e 215 × 956

f 406 × 202

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

b 4784 × 120

c 5437 × 235

d 2154 × 78

e 8973 × 705

f 7146 × 54

a 67 ÷ 13

b 4356 ÷ 1025

c 9174 ÷ 2008

d 1256 ÷ 1032

e 715 ÷ 115

f 668 ÷ 122

Find the estimated quotient.

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

b 4356 ÷ 920

c 8914 ÷ 358

d 1547 ÷ 97

e 715 ÷ 56

f 468 ÷ 42

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

Chirag purchased 45 calculators for his stationery shop. If each calculator costs ₹560, then

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

shopkeeper, about how much did she pay for the bedsheet? estimate the total cost of the calculators. estimated number trees in each row.

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 To find the number of garments produced on the third day, we first add the production of garments in two days and then subtract it from the total production. 1

–

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

Chapter 2 • Estimation and Operations on Large Numbers

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 Number of copies of newspaper printed on Tuesday = 45,784 Total number of copies printed in two days = 47,489 + 45,784 1

4 7 4 8 9 + 4 5 7 8 4 9 3 2 7 3 So, the total number of copies of newspapers printed in two days is 93,273. Example 8

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 Weight of the empty cylinder = 14 kg 500 g = 14.5 kg Weight of the gas in the cylinder = 30.100 kg – 14.5 kg 2

3 0 – 1 4 1 5

. . .

Think and Tell

How do you think 30 kg 100 g and 14 kg 500 g is written as

1 5 6

30.100 kg and 14.500 kg, respectively?

So, weight of the gas in the cylinder is 15.6 kg or 15 kg 600 g. Do It Together

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 were females, and the rest were children. How many children were there in the city? Total population of the city in India = 37,65,392 Number of males in the city = 18,90,898 Number of females in the city = 16,54,976 Number of children in the city = ______________ – (______________ + 16,54,976) So, the number of children in the city was _________.

Do It Yourself 2C 1

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

1 9

7 9

– 3

1 4

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

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

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

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.

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

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

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?

The distance at a particular time between Mercury and Earth was 9,63,77,000 km and the distance

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?

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

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

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

previous 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?

Chapter 2 • Estimation and Operations on Large Numbers

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 every 3 days = 15,000

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

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 × +

2 2

1 5 7

3 5 2

2 0 0

So, distance covered by the car in 32 hours is 2720 km. Example 10

Do It Together

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

Solve. a 54,76,587 × 40

b 9,87,98,706 ÷ 9

c (3,24,45,670 ÷ 5) × 13

d 65,78,598 ÷ 3

e 3,24,543 × 120

f (79,69,889 × 6) ÷ 2

Find the missing digits. a 56,48,756 × 21 = 11

623

b 22,63,657 × 56 = 12

c 4,53,76,575 ÷ 15 = 30

d 763658890 ÷ 2 = 381

2944

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

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

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

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

792

Word Problems 1

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

A vessel contains 8 L 650 mL of water. How many glasses of capacity 500 mL each will be filled

In a company, each person in the HR department sent 2150 emails to their customers

with the total quantity?

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

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

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

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

A car‐racing event attracted 15,690 race‐car fans. After the race, the spectators were leaving

5 years?

in a day?

are there in one set?

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 Chapter 2 • Estimation and Operations on Large Numbers

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.

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 (), {} and []. 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 ÷, ×, + and −. We do not have the ‘÷’ and ‘×’ signs. So, we skip these operations. Next, we solve for the ‘+’ sign.

= 150000 + 921000

= 150000 + [153500 × 6] = 10,71,000

As 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 Method 2 (with use of brackets):

The number of roses Sanya and Aakash purchased = 8 × (3 + 5) = 8 × 8 = 64 28

Think and Tell

Which method is easier to use and find the answer?

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 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.

Error Alert! 44 – 8 + 5 × 3 = 36 + 5 × 3 = 41 × 3 = 123

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

5 × 102 = 5 × (100 + 2) = 5 × 100 + 5 × 2 = 500 + 10 = 510 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 8 × 105 = 8 × (______ + 5) = 8 × ______ + 8 × ______ = ______ + ______ = ______

Do It Yourself 2E 1

Find the value of the number statements. a 402 – 118 + 180 ÷ 45 × 162 3 c 15 × 25 ÷ of 21 + 20 7

Chapter 2 • Estimation and Operations on Large Numbers

b 168 ÷ 14 × 22 – 210 + 185 d 35 ÷ 5 × 9 + 7 × 30 ÷ 3 of 6

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

b 89 × 112

c 63 × 142

d 34 × 7

Simplify. a 250 + [24 – {4 of 3 + (8 – 5)}]

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. a 6 + 5 × 3 – 1 = 20

b 3 + 4 × 8 – 4 = 28

c 10 + 8 – 4 + 1 + 5 = 8

Insert signs (+, −, ×, ÷, =) to make the statements true. a 8 ____ 3 ____ 4 ____ 5 ____ 44 ____ 12

b 15 ____ 5 ____ 72 ____ 8 ____ 27 ____ 18

Using brackets in different places, find the possible ways to solve the given expressions? a 4+6×5–2

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 that shows how many slices each of them will get. i

(3 × 8) + (1 + 5)

3 × ( 8 + 1) + 5

iii

(3 + 8) × (1 + 5)

b How many slices of pizza does Damini get?

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, say, 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 a year. Number of garments produced in the first month = 1,50,000

Number of garments produced in the next 6 months = 9,21,000 30

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

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

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 in 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

Total number of pencils = 12 × 7 = 84

Total number of pencils = 54 × 2 = 108

Number of pencils in each packet = 12

Number of pencils in each packet = 54

Total number of pencils of both the types = 84 + 108 = 192 Number of pencil stands these pencils are put in by the teacher = 8 Number of pencils in each stand = 192 ÷ 8 = 24 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

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

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?

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

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?

Chapter 2 • Estimation and Operations on Large Numbers

Word Problem 1

A local charity has 3 fundraising events. The events raise ₹17,600, ₹81,000 and ₹30,900 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. 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. To find the estimated sum, difference, product or quotient, we round off each of the numbers and then perform the required operation. BDMAS stands for Brackets, Division, Multiplication, Addition and Subtraction.

Math Lab Mystery Number Math

Setting: In groups of 4

Materials Required: Flash cards 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 is drawn up to 20 like a hopscotch game (as shown below) before the game begins. 1

5 4

8 7

12 11

16 15

19 18

Flash cards 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 will get 16,422,990. What number am I?)

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

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

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

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 need 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.)

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

Chapter Checkup 1

Round off the numbers to the nearest ten, hundred or 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 indicated. Find the actual and estimated answers. 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 problems. 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 ×

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

1 of 10) – 75 ÷ (17 – 2)} 2

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

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

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]

Find the value of: a 67,589 – 23,543 + 13,678

b 90,800 + 32,552 – 45,765

Which of the following is the correct option? a 24,369 ÷ 3 = 8121, 8121 × 4 = 32,490

b 24,360 ÷ 3 = 8210, 8210 × 4 = 32,480

c 24,369 ÷ 3 = 8112, 8112 × 4 = 32,448

d 24,369 ÷ 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.

Chapter 2 • Estimation and Operations on Large Numbers

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.

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

In the school library, there are 12,536 academic books, 15,000 storybooks and 12,549

estimated number of more toys manufactured on Monday.

children’s magazines. Each student can borrow up to 3 magazines and 2 storybooks at the same time.

a How many books are there in total? b 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?

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

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. Find:

a the total cargo capacity of Fleet A and Fleet B combined. b the total cargo capacity of Fleet C. c the total cargo capacity of all 3 ﬂeets.

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

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

+2 2

+2 4

+2 6

+2 8

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

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.

Whole Numbers on a Number Line

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 a given number, i.e., Given number – 1.

Think and Tell

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

Predecessor of 10

Example 1

Are all natural numbers whole numbers?

Successor of 10

Write the successor and predecessor of 12,499. Write the next 3 consecutive numbers after 12,499. The successor of 12,499 = 12,499 + 1 = 12,500 The predecessor of 12,499 = 12,499 – 1 = 12,498 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 87,453.

except zero has an

The predecessor of 87,453 = 87,453 – 1 = 87,452

immediate predecessor.

The predecessor of _______ = _______ – 1 = _______ The predecessor of _______ = _______ – _______ = _______ Therefore, the three consecutive numbers before 87,453 are _______, _______ and _______.

Representing Numbers on the Number Line Natural Numbers 1

Whole Numbers 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

Think and Tell 9 10

What is the largest whole number?

Do It Together

Imagine you are at a train station with a platform divided into sections, each labelled with whole numbers. 0

9 10

1 If you are 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 the unit distance forward.

Jump 4 steps of the unit distance backwards.

+1 +1 +1 0

–1 –1 –1 –1

9 10

Thus, 5 + 3 = 8

+3 2

9 10

Start from 15 and jump 3 units backwards till we reach 0.

+3 5

+3 8

–3

9 10 11 12

Therefore, 4 × 3 =12

Example 3

Divide: 15 ÷ 3

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

Thus, 9 – 4 = 5 Multiply: 4 × 3

–3 2

–3 5

–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 in total do 6 students have? Number of pencils each student has = 3; +3 0

Total number of students = 6

+3 2

+3 6

+3 8

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.

Cost of 4 candies = `_______________;

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

Do It Yourself 3A 1

Find the predecessor and successor of the numbers. a 66,778

b 54,557

c 86,457

d 89,665

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

125

100

175

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

b 0

e 97,564

10 20 30 40 50 60 70 80 90 100

9 12 15 18 21 24

Evaluate using the number line. a 15 + 7

b 14 – 9

c 7+5

d 18 ÷ 3

e 2×5

f 20 ÷ 5

Write the next 5 whole numbers after 13,214.

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

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

Word Problems

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

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

cookies that were not burnt using the number line. of money contributed using the number line.

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

Real Life Connect

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

Addition of Whole Numbers The total number of fruits that Robin buys 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 of 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 in her sister’s bank account = `_________

Money Vaani deposited = `_________

Money her sister deposited = `_________

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

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

b 9 and 13

d 30 and 40

e 51 and 39

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

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 98,789 + 9999

e 72,145 + 9999

c 45,456 + 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 smallest 6-digit numbers formed by the digits 2, 0, 4, 7, 6 and 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?

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 14 apples from the fruit aisle. She gave 7 apples to her mother, 4 to her brother and ate the remaining 3.

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 –

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

As 7 and 7 are equal, then 7 – 7 is a whole number. Case 3: a < b –

As 7 is less than 14, then 7 – 14 is not a whole number, as 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. –

≠

–

As 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

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 the 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

Fill in the blanks. a 452 – 0 = ________

b 78,856 – ________ = 0

c ________ – 1 = 3999

d 7893 – ________ = 7893

e 10,000 – ________ = 7456

f 2365 – 1256 = ________

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 and (iv) 9999 from the given numbers. a 12,345

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

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

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

Word Problems 1

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

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

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? pens does he have left 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, 5 × 3 = 3 × 5. Chapter 3 • Whole Numbers

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

3×0=0×3=0 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=1×3=3 5 Associative property When three or more whole numbers are multiplied, the product remains the same regardless of 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 gives 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 So, 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 gives 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 So, 1 × (3 – 2) = (1 × 3) – (1 × 2) Example 6

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.

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

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

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.

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.

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

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 =

5 Division by a number itself When a non–zero whole number is divided by itself, it gives the quotient 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÷

7 Division by zero A non–zero whole number cannot be divided by zero. Division by zero is not deﬁned. 6 ÷ 0 is 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 3850 – 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

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

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

–

Therefore, the required number is 99,960. Do It Together

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.

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

Do It Yourself 3E 1

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

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

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

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

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

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

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

+3 0

9 10

+2 4

+2 6

+2 8

Pattern: 2, 4, 6, 8, 10,….

Counting in threes

Counting in fours

+3 5

+3 8

9 10 11 12 13 14 15

+4 0

+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

+10 10

9 10

Pattern: 3, 6, 9, 12, 15, 18, ….

+5 0

Pattern: 1, 2, 3, 4, 5…

+3 2

+10 20

+10 30

+10 40

Pattern: 10, 20, 30, 40,…

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 and 590. 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 and 959.

Did You Know? Pythagoras was the first man to come up with the idea of 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:

The number 10 can be shown as:

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

= 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

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

Do It Yourself 3F 1

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

Simplify using the shorter method. a 4452 + 99

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

Solve and establish a pattern. a 45 × 9

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

d 45 × 9999

Study the patterns and write the next 4 steps.

a 1125

c 45 × 999

b 356

c 445

Write the greatest 7–digit number using the digits 4, 6 and 9 if digits can 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?

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:

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

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

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

Students should check their Bingo cards for the called number and strike out the number if they

have that number on their card.

Continue drawing numbers and calling them out one by one.

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.

Solve on a number line.

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

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

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

b 865 × 12 + 865 × 45

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

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

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 going 4 times around it?

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 fruits. Mother: Neerja, when we’re shopping for fruit for the week, we want to make sure we get the right amount without wasting money. 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.

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, 16 and so on.

is a multiple of is a multiple of

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

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 numbers.

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. 1 16

2 21

Factors = 1, 2, 4, 8, 16

Factors = _________

Multiples = 16, 32, 48

Multiples = _________

3 27

4 39

Factors = _________

Multiples = _________

Do It Yourself 4A 1

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

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

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

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

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 9,90,099

d 99

d 9,00,099

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 us 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.

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? 1 35

2 47

3 59

4 62

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

Which of the numbers is odd? 1 64

2 56

3 32

4 41

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. 1 91 — Odd

2 61 — ________________

3 79 — ________________

4 7 — ________________

5 45 — ________________

6 82 — Even

7 39 — ________________

8 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 25 factors) Chapter 4 • Playing with Numbers: Factors and Multiples

Note: • 1 is neither a prime number nor a composite number. • 2 is the smallest prime number.

Think and Tell How many even prime numbers are there?

• 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).

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? 1 37

2 42

3 26

4 18

Even numbers end with 0, 2, 4, 6 or 8. Hence, (2), (3) and (4) 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? 1 12

2 5

3 10

4 17

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. 1 15 and 28

2 9 and 16

3 21 and 40

4 7 and 18

Factors of 15: 1, 3, 5, 15 Factors of 28: 1, 2, 7, 14, 28 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

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

Write all the prime numbers between the ranges given below. a 40 to 50

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

Write all pairs of twin primes between 60 and 90.

Make a list of all composite numbers below 49.

Make a list of all prime numbers between 40 and 60.

Express the numbers below as the sum of two odd primes. a 36

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 us see some rules that will help us figure out whether a number is divisible without dividing it.

Chapter 4 • Playing with Numbers: Factors and Multiples

Example 9

Divisible by

Divisibility Rules

Examples

The digit at the ones place of the number is 0, 2, 4, 6 or 8.

10, 32, 54, 76, 98 and so on

The digit at the ones place of the number is 0 or 5.

25, 70, 135, 160 and so on

The digit at the ones place of the number is 0.

20, 50, 70, 100 and so on

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)

The number should be divisible by both 2 and 3.

18: divisible by both 2 and 3. 72: divisible by both 2 and 3.

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

The number formed by the last three digits should be divisible by 8.

2016: 16 is divisible by 8 4032: 32 is divisible by 8

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)

The difference of the sum of the digits in the odd place and the sum of the digits in the even place is 0 or 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? 1 572

2 648

3 793

A number is divisible by 3 if the sum of its digits is divisible by 3. Let us 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

4 921

Which of the numbers given below is divisible by 8? 1 237

2 416

3 589

4 1724

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.

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 or 11. 1 5628

2 4356

3 4986

4 8172

Do It Yourself 4C 1

Find which of these numbers are divisible by 2, 3, 4, 5, 6, 8, 9 or 10. 2484, 5376, 9243, 3028, 8164, 5732, 9,12,048, 9,32,716, 1432, 41,873, 990, 7,27,272, 540, 7269, 7269, 89,120, 5482, 76,319, 1247, 801,632, 12,34,56,789

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

Prove that the result of multiplying any three consecutive numbers will always be divisible by 6, and provide

If a number is divisible by both 5 and 9, what other number will it always be divisible by?

Given that two numbers are divisible by a certain natural number, demonstrate that both their sum and their

examples to support your explanation.

difference are also divisible by that same number. a 66 and 42 are divisible by 6.

b 650 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

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 is referred to as a prime factor. Here is one of the methods for finding the factors of the number 36. 36

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

9 2

6 3

12 2

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. 1 Factor tree Let‘s factorise 75 using both of these methods. Factor tree

75 = 3 × 5 × 5

Example 11

25 1

75 = 3 × 5 × 5

Express each number as the product of its prime factors by using a factor tree. 1 48

2 56

Factor tree of 48 2

3 64

Factor tree of 56

Factor tree of 64

56 24

2 12

64 28

2 6

2 14

32 2

48 = 2 × 2 × 2 × 2 × 3 64

3 5

25 5

The prime factors may be presented in varying orders, yet they remain the same.

Repeated Division

75 3

Remember!

2 Repeated division

16 2

8 2

56 = 2 × 2 × 2 × 7

4 2

64 = 2 × 2 × 2 × 2 × 2 × 2

Example 12

Express each number as the product of its prime factors by using repetitive division. 1 45

2 72

Prime factors of 45

3 68

Prime factors of 72

Prime factors of 68

3 3

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. 1 76 Prime factors of 76

Factor tree of 76 76 2

38 2

76 = 2 × 2 × 19

2 99 Prime factors of 99 3

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

e 1265

10,000

Find the prime factors of 345. Arrange them in ascending order. Find a relation between two consecutive

State whether they are prime factorisations or not.

prime factors.

a 189 = 3 × 3 × 21

b 252 = 2 × 6 × 3 × 7

c 385 = 5 × 7 × 11

Express each number as the product of its prime factors by using a factor tree. a 210

b 293

Chapter 4 • Playing with Numbers: Factors and Multiples

c 816

d 952

e 756

Find the prime factors of 252. Arrange them in ascending order. Determine the difference between two

Find the prime factors of 560. Arrange them in ascending order. Calculate the product of the two smallest

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. Let us find the common factors of 12, 20 and 24. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 20: 1, 2, 4, 5, 10, 20 66

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The common factors of 12, 20 and 24 are 1, 2 and 4. Common Multiples Common multiples are numbers that are multiples of two or more numbers. Let us find the common multiples of 12, 20 and 24 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, ... Multiples of 24: 4, 48, 72, 96, 120, 144, 168, 192, 216, 240, ... 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, 2 and 4.

Example 14

List two common multiples of 6, 12 and 18. 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, the two common multiples of 6, 12 and 18 are 36 and 72.

Do It Together

Fill in the blanks 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 us find HCF of 324 and 540 using the prime factorisation method. Chapter 4 • Playing with Numbers: Factors and Multiples

By Prime Factorisation Step 1: Find the prime factorisation of each number.

324

540

Step 2: Identify the common prime factors.

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

So, the HCF of 324 and 540 is 2 × 2 × 3 × 3 × 3 = 108.

270

324 = 2 × 2 × 3 × 3 × 3 × 3

Now, let us see how to find the HCF of two numbers using the long division method.

540 = 2 × 2 × 3 × 3 × 3 × 5

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

Find the HCF of the numbers using prime factorisation: 1 168 and 252 2

168

252

126

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

360

140

180

5 1

280 = 2 × 2 × 2 × 5 × 7 360 = 2 × 3 × 2 × 3 × 2 × 5 Hence, HCF of 280 and 360 = 2 × 2 × 2 × 5 = 40

216 –

324 216

108

216

–

216 0

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

198

315

–

198

216

–

216

117

198

–

117

Hence, HCF of 504 and 720 is 72.

–

117

–

126

–

Hence, HCF of 126, 198, 315 is 9. Do It Together

Find the HCF of the given numbers using prime factorisation. 1 72, 108, 96

108

72 = ___________________ 108 = ___________________ 96 = ___________________ Hence, HCF 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 is 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

By Prime Factorisation

16 = 2 × 2 × 2 × 2

Step 1: Find the prime factorisation of each number.

24 = 2 × 2 × 2 × 3

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.

30 = 2 × 3 × 5 36 = 2 × 2 × 3 × 3

By Division Method Step 1: Arrange the provided numbers in a line, separated by commas.

16, 24, 30, 36

4, 6, 15, 9

Step 2: Find a number that divides at least one of them exactly. Step 3: Divide the divisible numbers and note quotients below these numbers. Keep the numbers that cannot be divided.

Step 4: Repeat until you have co-prime numbers. Step 5: Multiply divisors and co-prime numbers for the LCM. Hence, LCM of 16, 24, 30 and 36 is 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720.

8, 12, 15, 18 2, 3, 15, 9

1, 3, 15, 9

1, 1, 5,

1, 1, 1,

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 us 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 us find their HCF and LCM and compare their product with the product of their HCF and LCM. 2

6 1

12 = 2 × 2 × 3

18 = 2 × 3 × 3

HCF = 2 × 3 = 6 LCM = 2 × 2 × 3 × 3 = 36

Product of numbers = 12 × 18 = 216 HCF × LCM = 6 × 36 = 216 Example 17

Find the LCM of 135 and 246 using prime factorisation: 3

135

246

3 5

45 5 1

123 1

135 = 3 × 3 × 3 × 5 246 = 2 × 3 × 41 LCM = 2 × 3 × 3 × 3 × 5 × 41 = 11,070

Example 18

Find the LCM of 40, 48 and 45 using the division method. 2

40, 48, 45

20, 24, 45

10, 12, 45

5, 6, 45

5, 3, 45

5, 1, 15

5, 1, 5 1, 1, 1

Hence LCM of 40, 48 and 45 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720 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 us denote the two numbers as A and B. Given: HCF = 9; LCM = 108 Let us 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: ________

Do It Yourself 4E 1

Find the HCF of the numbers by listing factors. a 24, 36

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

d 54, 90, 126

Find the LCM of the numbers by listing their multiples. a 60, 20

c 45, 75

Find the HCF of the numbers by repeated division. a 48, 60

d 36, 72, 108

Find the HCF of the numbers by prime factorisation. a 20, 30

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

The Highest Common Factor (HCF) of two numbers is 18, and their Least Common Multiple (LCM) is 126. One of

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? 72

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 × 3 × 5 × 5 = 300

15, 20, 25

15, 5, 25

2 5 5

15, 10, 25 5, 5, 25 1, 1, 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 HCF _______________ 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

A bus service operates every 15 minutes, and a train service operates every 20 minutes. How often do the bus

A farmer has 24 cows and another farmer has 36 cows. What is the largest number of cows they can evenly divide

A gardener needs to water his plants every 12 days, and another type of plant needs watering every 18 days.

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?

Chapter 4 • Playing with Numbers: Factors and Multiples

In a school, there are 90 students, and each class can accommodate 30 students. What is the largest number of

A toy store has three toy-making machines. Machine X produces toys every 25 minutes, Machine Y every

students that can be accommodated in each class without any extra students?

35 minutes, and Machine Z every 40 minutes. After how much time will all three machines complete their work simultaneously?

A factory produces batches of 18, 24 and 36 products at a time. What is the largest number of products that can

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?

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 multiple of a number is a number that can be obtained by multiplying that number by another number.

•

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.

•

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:

Start by explaining the definition of prime numbers.

Provide each student with a copy of the grid paper, which has numbers from 1 to 100.

Explain that 1 is neither a prime number nor a composite number, so students should put a

Ask students to circle prime numbers like 2 and 3, then cross out their multiples

Instruct students to continue this process for the numbers 5, 7, 11 and so on. Encircle each

Continue the process until all the numbers in the grid are either encircled (prime) or crossed out

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

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

The product of two numbers is 36, and their sum is 13. Find the numbers.

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.

Write each number as the product of its factors by using a factor tree. a 315

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

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

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?

Line Segments and Angles

Let's Recall Every day we come across various types of lines. Let us discuss parallel and intersecting lines.

m l

g h

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 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. 1

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 us explore some key geometrical concepts in detail. Point

Line Segment

Ray

Y X

Z N

Point: A point denotes a location. A point has no dimensions like length, breadth or height. Can be denoted as X, Y, Z, N.

Plane

Line Segment: A Line segment has a fixed length. The distance between two points is the length. Can be denoted as XY or YX.

Y X

A ray is obtained by extending a line segment indefinitely in one direction. It does not have a fixed length.

Represented as XY, where X is the initial point. Ray XY is not the same as ray YX.

A 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 table top are some examples of parts of a plane that extend indefinitely in all directions. 78

Types of Lines X

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. Point P is the point of intersection.

The walls of a room, a football ground, a sheet of paper and a tabletop are some examples of parts of a plane that extend 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

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 is 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

Do It Yourself 5A 1

Look at the figure and answer the questions.

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.

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

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 ray has more endpoints than a line.

Count the number of line segments in the shapes. a

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 clothes line. 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 clothes line is a practical example of a line segment. Now, let us learn how to measure and draw line segments. 80

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 and kilometre to measure longer distances. Bigger unit to smaller unit

× 10 kilometre (km)

Multiply by 10

× 10 hectometre (hm)

÷ 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 a bigger unit Example 2

Measure the length of the line segments.

B E A

AB = 3.5 cm Do It Together

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 lengths of line segments. Chapter 5 • Line Segments and Angles

Comparison by Observation B

Comparison by Tracing By observation, XY > MN > AB. Comparison by observation is useful when differences in length are quite obvious.

A M

S D

It is difficult to compare lengths of line segments PQ, SR and CD just by looking at them.

It involves the use of tracing paper to compare line segments. Let us 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.

Example 3

A 4 cm B

Name the line segment with the shortest length in the figure shown below. O

P Do It Together

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

Q Y

PQ = ______ ; CD = _______ ; OS = _______

Do It Yourself 5B 1

Draw line segments of the lengths given. a 5 cm

b 7.7 cm

c 5.2 cm

d 35 mm

e 61 mm

Count the number of line segments in the shapes. a

Which of the following is the shortest length? a 2000 mm

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

b centimetres

c decametres

d hectometres

Draw a line segment which is 1 cm longer than the line segment in each case. a

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 is 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: arm vertex Q

angle R

arm

Chapter 5 • Line Segments and Angles

• • •

Is formed when two rays (or lines) meet at a common point called the vertex. Has the symbol: ∠

Has the representation: ∠PQR, ∠RPQ, ∠Q 83

Interior and Exterior of an Angle

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.

exterior

interior

D C

Adjacent Angles

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

Observe the diagram given below and list the points that lie in the interior and exterior of angles ∠PON and ∠QON. A P

Interior Points

Exterior Points

∠PON: E, F, D, G, Q

∠PON: A, B, C

∠QON: D, G

∠QON: A, B, C, E, F, P

E O

D C

Example 5

G N

Draw two adjacent angles to the angle, ∠POQ. M

P O

Adjacent angles drawn to ∠POQ are ∠MOP and ∠NOQ. Q N

Do It Together

Draw an adjacent angle to ∠NMO so that:

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.

Do It Yourself 5C 1

Classify the points based on whether they lie in the interior or exterior region of the angle. M

In the figure shown here, recognise the adjacent angle(s).

Q R

b ∠BQC

c ∠BQB

d ∠CBQ

Draw more points in a way that the angles ∠YMX, ∠XMK and ∠KMV have an equal number of points in their interior region.

Y A

N X

Draw adjacent angles to ∠NMZ so that:

a B lies in the interior of the new angle but not A and D.

A B

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.

Which of the following is an incorrect representation of an angle? a ∠O

P Q

Consider two angles ∠XOY and ∠AOB. Mark four points in each case so that: a all the points are interior to ∠XOY only.

b all the points are interior to ∠AOB only.

c all the points are exterior to both ∠XOY and ∠AOB.

Word Problems 1

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.

a Inside the angle, she marks the region with small 'x' symbols, creating a unique pattern.

b 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

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 is because of the angle, Emma. Let us 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 a 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

C Centre of the protractor

Example 6

Baseline

Measure the given angles. Write their measurements. ∠A = 60° ∠B = 120° B

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. 1

∠P = __________

∠Q = __________

The smallest angle is __________. 86

∠R = __________

∠S = __________

Do It Yourself 5D 1

Which of the following units is used to measure angle? a mm

b cm

b Divider

d Protractor

Write the difference between the measure of angles formed by blue arm and red arm with the black arm in each case. a

c Compass

Measure the given angles. Write their measure.

d degree

Which instrument is used to measure the angles? a Ruler

c km

Study the figure given below. B 2

C a Identify all the angles that have adjacent angles in the figure.

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 measuring their inclination?

Chapter 5 • Line Segments and Angles

Types of Angles Angles can be classified on the basis of their measures. 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 a paper, windows and doors, or buildings.

90º

Right and Straight Angles Imagine you are 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 of 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. 88

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

turn 2 Or 1 straight angle

3 right angles or

12 3

turns

4 right angles or 2 straight angles or one full turn

12 3

Takes 3 hours

Take 6 hours

Takes 9 hours

Takes 12 hours.

What fraction of a full turn will the hands of a clock make if the hour hand moves from:

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 anti-clockwise 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 various 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

South

North

Two

South

3 right angle

West

Two

1 2

1 4

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.

Do It Yourself 5E 1

Classify the following angles as acute, obtuse or reflex angles. a 36°

b 93°

c 192°

d 89°

e 126°

f 53°

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

Suppose you were initially facing south. You were made to turn anti‐clockwise by three right angles. In which

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

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 be leaving his house?

Points to Remember •

A point is a mark of location. It has no dimensions.

•

Lines, planes, and rays extend indefinitely.

• • • • •

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:

Divide the students into pairs or small groups. Each group will need a protractor, a notebook and

Provide each group with a list of objects or situations they need to find, each representing a

Allow the groups to move around the school or classroom to find the objects or situations on their

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

c Plane

d Line segment

c 6.9 cm

d 2.2 cm

Which of the following does not extend indefinitely? a Ray

b Line

Draw line segments of these lengths. a 4.6 cm

b 77 mm

How much is 4 km in decimeters?

Identify the points in the exterior and interior regions of ∠PQR. P A D Q B

Chapter 5 • Line Segments and Angles

F E

D C

Look at the points marked. Create new angles such that:

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.

Only C and D are in the interior of the angle. Only A, F and G are in the interior of the angle.

Only B and E are in the interior of the angle.

Draw two adjacent angles to the given angle ∠XOY. X

Classify the angles as acute, obtuse or reflex. a 26°

b 256°

c 79°

d 129°

e 165°

189°

Identify the angles as acute, obtuse, reflex or straight, without actually measuring them. a

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

of a revolution clockwise? 2 1 b North and completing of a revolution anti-clockwise? 4 3 c East and rotating of a revolution clockwise? 4 d South and making a full revolution anti-clockwise?

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

You are standing at the entrance of a building, and you are 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 anti-clockwise to find yourself facing east from your initial south‐facing position?

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 shape 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

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 is 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 us 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

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

Identify the points lying on the interior and exterior of the given triangle. B F

A E

C D

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

2 Only D, E, F and G are in the interior of the triangle.

Classification of Triangles on the Basis of Sides Triangles can be classified on the basis of their sides by measuring the lengths of their sides. Equilateral Triangle

Isosceles Triangle

All the sides are equal in length.

The angles opposite to the equal sides are also equal. Here, ∠XYZ = ∠XZY.

The angles of an equilateral triangle are also equal.

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 ∆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

Chapter 6 • Triangles, Quadrilaterals and Polygons

Do It Yourself 6A 1

List the vertices, sides and angles in the triangle.

T R

Classify the triangles as equilateral, isosceles or scalene, given the lengths 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 triangles in the figure given below. 3 cm

5 3.

4 cm

3 cm

Classify the triangles according to the measures 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. a

3 cm

4 cm

9 cm

5 cm

5 3.

3 cm

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, whereas Maria’s garden measures 5 feet, 6 feet and 7 feet. Identify the types of triangle formed by their gardens.

2 cm

Mean, Median and Mode Understanding Quadrilaterals Real Life Connect

Father: Sukhi, look what I have 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 and 4 corners/vertices.

Parts of a Quadrilateral 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.

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 the opposite sides of the quadrilateral.

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 the 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

P S

AC, BD Chapter 6 • Triangles, Quadrilaterals and Polygons

EG, FH

Q R

PR, SQ 97

Example 6

List the adjacent sides, adjacent angles and opposite angles in the given quadrilateral. S

Adjacent Sides: (RU, UT), (UT, TS), (ST, SR), (SR, RU)

Adjacent Angles: (∠R, ∠U) , (∠T, ∠U), (∠T, ∠S), (∠S, ∠R) Opposite Angles: (∠R, ∠T) , (∠S, ∠U)

Do It Together

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.

exterior.

R U

S T

Do It Yourself 6B 1

List the names of sides and vertices in a quadrilateral EFGH.

Identify and write all the adjacent and opposite sides in the given quadrilaterals. a

S 3

b only B, D, E and F are in the exterior.

Draw and name the names of diagonals of the quadrilaterals given below. b

A B

D C

Draw quadrilaterals using points A, B, C, D, E and F such that:

a only A, B and C are in the interior.

V U

Draw a quadrilateral and label its vertices. Draw diagonals and label them.

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 and buildings. They can be classified on the basis of the properties of their sides and angles.

Convex and Concave Quadrilaterals B

Convex Quadrilaterals In these quadrilaterals, the measure of each angle is less than 180º. Both diagonals lie inside it.

C Q

Concave Quadrilaterals In these quadrilaterals, the measure of one of the angles is more than 180º. One of the diagonals lie outside it.

R S

Types of Quadrilaterals A

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.

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.

P A

Rhombus

Trapezium

Opposite sides are parallel. Opposite angles are equal. Diagonals bisect each other at right angles.

R Kite Opposite sides are unequal.

A trapezium whose non-parallel sides are equal 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 satisfy the given conditions? 1 None of the sides may be equal to any 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.

What angles are formed by the intersection of diagonals in a square? a 90°, 60°

100

b 120°, 90°

c 60°, 60°

One pair of opposite sides is parallel. Here, AB CD.

All the sides are equal.

Example 7

d 90°, 90°

LMNO is a rectangle whose diagonals bisect at Q. List the line segments which are equal in length.

Look at the figure. Answer the questions.

a Identify the quadrilateral.

b Name the parallel and non-parallel sides of this quadrilateral. c What is this quadrilateral known as when JM is equal to KL?

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 on 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

101

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. 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

Interior of a polygon: The interior of a polygon refers to the region enclosed by the sides of the polygon.

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

Figure d is NOT a polygon as 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

Vertices: A, B, C, D, E, F Sides: AB, BC, CD, DE, EF, FA Diagonals: AC, AD, AE, BD, 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 to the exterior of polygon. Y W

Point in the interior: _________________

X Z

102

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 to the exterior.

Convex Polygons have a measure of each angle of less than 180°. The diagonals lie inside the polygon.

Error Alert!

Think and Tell

Is a right-angled triangle a regular polygon?

A polygon with equal sides alone may not be regular; it requires equal angles as well to be considered regular.

Let us classify polygons on the basis of their sides.

Example 11

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: ______________________ 103

Example 12

Which of these options can be the angles of a regular polygon? a 60°, 100°, 130°, 70°

b 90°, 90°, 90°, 90°

90°, 100°, 120°, 90°

d 45°, 60°, 75°, 90° Since all the angles in a regular polygon are equal so, option b is correct. Do It Together

Name the polygon and list its vertices. Draw diagonals. Write the number of diagonals. Vertices: _______________________ Diagonals: _______________________

Do It Yourself 6D 1

List the names of all the sides and vertices of the polygon. A

D F

b Rectangle

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

How many line segments form the polygons? a Heptagon

Which of these is a regular polygon? a Rhombus

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°

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?

104

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:

Create and label shapes. Use colours for distinction.

Participants sort shapes into piles for triangles, quadrilaterals and other polygons.

Within groups, further classify shapes (e.g., triangles into equilateral, isosceles, scalene).

Discuss how they identified shapes, common properties, and differences.

Participants make composite shapes using sorted shapes.

Each presents their composite shape, explaining types used and arrangement.

Chapter Checkup 1

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

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°

Identify the scalene, equilateral and isosceles triangle in the figure.

100°, 50°, 30° A

7 cm 9 cm

12 cm

Chapter 6 • Triangles, Quadrilaterals and Polygons

8 cm

8 cm 8 cm

6 cm C 5 cm 7 cm

8 cm

105

Identify the points which are interior to:

a Both ΔDEF and ΔPQR

b Both ΔABC and ΔDEF

c Only ΔDEF

d Only ΔPQR

5 9

10 6

13 F

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

b 8

c 5

d 7

Draw an irregular polygon with the given number of sides. a 3

b 6

c 8

Draw a hexagon and its diagonals. Write the number of diagonals.

Consider the given parallelogram. Answer the questions.

d 10

a Are PQ and RS parallel to each other?

b Name any four pair of line segments in the figure which are equal. c Is PR equal to SQ? Give a reason.

10 The figure MNOP is a kite. Answer the questions. Give reasons.

a Is MN equal to PM? b Is ∠MNQ equal to ∠MPQ?

c Is ∠NMQ equal to ∠PMQ?

Word Problem 1

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?

106

Curves and Circles

Let's Recall ‘Do you know that circles are everywhere?’ Kate asks Mary. Mary looks confused. ‘Circles, like bicycle wheels?’

Kate explains, ‘Circles have no corners, but a central point, and all its edge points are at an equal distance from this point.’ The tyres of our bicycle and the sun are examples of a circle. Let us think of other things that are shaped like a circle.

Let us see some examples of circular objects:

Let's Warm-up Fill in the blanks with ‘Yes’ or ‘No’, based on whether the object is circular or not. 1

I scored _________ out of 5.

Curves Real Life Connect

On a warm summer morning, Emily and Liam go to their favourite beach. Armed with sticks and surrounded by endless sand, they are ready to be creative. Emily: Hey Liam, I’m drawing these wavy lines that ﬂow like waves! Liam: Great idea, Emily! I will draw lines that don’t meet at the ends!

Types of Curves Emily and Liam create various types of curves with their drawings. Some form loops, others cross and a few are wavy. These distinctions allow curves to be grouped into various categories. Let us explore a few of these categories. Simple and non-simple curves A curve that does not cross itself is known as a simple curve. Simple Curves

Non-simple Curves

Curve crosses itself once.

Curve crosses itself twice. Open and closed curves Curves can be open or closed, based on whether they start and finish at the same point. Open curves don't begin and end at the same point, whereas closed curves do.

Open Curves

Remember! Both open and closed curves can have crossings.

Closed Curves Simple closed curves The curves that start and finish at the same point without crossing themselves are known as simple closed curves.

108

Position with respect to a closed curve Given below is a simple closed curve with points X, Y and Z marked at diﬀerent places. Boundary of the curve

Interior of the curve: The region inside the boundary of the curve.

Y Interior of the curve

Exterior of the curve

Exterior of the curve: The region outside the boundary of the curve.

Here, X lies in the interior, Y lies on the boundary and Z lies in the exterior of the curve. Which of the curves is NOT a simple curve?

Example 1

Figure 4 is not a simple curve, as it crosses itself many times. Which of these is an open curve?

Example 2

Figure 4 is an open curve. Count the number of crossings in the curves.

Example 3

Categorise the points based on whether they are in the interior, the exterior or on the boundary of the curve.

Example 4

Interior of the curve: X, Y, Z Exterior of the curve: A, B, C Boundary of the curve: P, Q, R Do It Together

Draw 2 of each type of curve. Simple Open Curve

Chapter 7 • Curves and Circles

Simple Closed Curve

P Q

B Y

Non-simple Open Curve Non-simple Closed Curve

109

Do It Yourself 7A 1

Tick( ) the simple curves. a

Classify the curves as open or closed curves. a

Draw 2 simple and 2 non-simple curves.

Draw closed curves that cross themselves the number of times given. a 1

b 2

c 3

d 4

Mark any three points on the interior, exterior and boundary of the curve.

Word Problem 1

While walking on a loopy curved track in the park, Rajiv realises he has crossed the same path

three times. Is the curvy track being described here a simple closed curve? Explain your reasoning.

Mean, Median and Mode Understanding Circles Real Life Connect

After receiving circular bangles as a birthday gift, Maya notices round objects around her home. Maya: Dad, look at these beautiful bangles you gave me for my birthday. They are all round and shiny! Father: I'm glad you like them, Maya. Circles are special shapes. They appear in many things around us. Let us see the parts of a circle!

110

Parts of a Circle Chord r

ete

m Dia

Radius

Centre Secant

Diameter

Centre

• A line segment passing through the centre of a circle, connecting two points on the circle.

• A point within the circle that is equidistant from all points on the circle’s circumference. It is often denoted by the letter ’O’.

• It is the longest chord of the circle. • The diameter is twice the length of the radius.

Chord

Radius

• A line segment that connects two points on the circle.

• The distance from the centre of a circle to any point on the circle.

Secant • A straight line that intersects a circle at two distinct points and extends beyond the circumference of the circle.

Minor Arc

The Parliament House in

Major segment

Major Arc

Did You Know?

Major sector

Minor segment

India has a Central Hall in a circular shape, representing

Minor sector

the Ashoka Chakra. Q

Segment of a Circle: A circle segment is a defined area enclosed by a chord and the corresponding arc positioned between the end points of the chord.

Arc of a circle: An arc of a circle is a part of the circumference between two distinct points.

Chapter 7 • Curves and Circles

Sector: The sector of a circle is the region bounded by two radii and the arc they intercept.

Think and Tell

Can there be a chord longer than the diameter of a circle?

111

Semi-circle means half of a circle. When a circle is cut into two equal parts, you get two semi-circles.

Concentric circles are circles that share the same centre but have diﬀerent radii.

Shade the major and minor segments in the circles. Label the major arc with a cross ( ).

Do It Together

In the circle shown below, draw and label the radius, diameter, secant and chord.

nt me seg

Ma jor

no rs

me n

Example 5

Remember! Multiple circles within a circle may or may not all be concentric circles.

Do It Yourself 7B 1

Shade the major and minor segments in the circles with diﬀerent colours. a

b arc

c diameter

d secant

The region bounded by two radii and arc is known as the ____________. a major segment

112

Which of these does not have a finite length? a chord

Mark the minor arc in the circles with a cross ( ). a

b minor segment

c sector

d secant

Which of the following represents concentric circles? a

Word Problem 1

Three friends, Rajiv, Vishakh and Justin, see their teacher on the other side of the circular park. Rajiv goes in a straight line to reach his teacher. Vishakh follows the major arc along the track to reach his teacher, whereas Justin follows the minor arc. Who travels the least distance to reach his teacher?

Constructing Circles Fascinated by circles, Maya removes her bangles and uses them to sketch circles on paper. She wishes to create circles of various sizes, yet her bangles allow only for one fixed size. Her father steps in and tells her that she can draw circles of diﬀerent sizes using a compass. We can also construct circles using a compass . A compass is a drawing tool used for making circles and arcs. It consists of two arms connected by a hinge, one with a pointed end and the other with a holder for a pencil or pen. Steps to construct a circle using a compass Step 2

Use your pencil to make a small

Place the needle of the compass on the zero mark

centre of your circle.

take the pencil to the required radius length.

dot on your paper. This will be the

of the ruler and open the arms of the compass to

Step 3 Put the pointed end of the compass on the dot you made in step 1. Example 6

Radius

Step 1

Step 4 Centre of the circle

While holding the centre of the compass steady

on the dot, carefully rotate the pencil end around the centre. This will draw the circle.

Using a compass, construct a circle with a radius of 3.7 cm. Show the radius of the circle through a line segment and mark its end points. Open the compass to 3.7 cm with the help of a ruler and draw the circle, as shown below.

Chapter 7 • Curves and Circles

3.7cm

113

Example 7

Using a scale of 3 m on the ground to 1 cm on paper, illustrate how to make a scaled-down representation of a circular swimming pool, which has a diameter of 9 m. Label the diameter. Solution: Scale is 3 m = 1 cm. So, 9 m = 3 cm Diameter = 3 cm, so radius = 3 ÷ 2 = 1.5 cm. O

Do It Together

3 cm

Error Alert! When constructing a circle, we always open the compass according to the required radius and not the diameter.

Using a scale of 2 m on the ground to 1 cm on paper, demonstrate how to create scaled-down representations of two circular objects in a garden. 1 A circular ﬂower bed with a diameter of 8 metres

2 A circular fountain with a diameter of 4 metres

Do It Yourself 7C 1

Using a compass, construct circles with the following radii. a 3 cm

b 3.8 cm

c 5.6 cm

d 4.7 cm

c 9 cm

d 8.6 cm

Construct circles with the diameters given. a 8 cm

b 6 cm

Construct two circles from the same centre. Make one circle of radius 4.7 cm and the other of radius 32 mm.

Using a compass and ruler, construct a circle with a diameter of 45 mm. After constructing the circle, draw a

Construct a circle with a radius of 6.2 cm. Once the circle is drawn, draw a secant that intersects the circle at two

Construct 2 concentric circles with radii measuring 3 cm and 4 cm.

Label the radii.

chord inside the circle and label its end points.

distinct points.

Word Problem 1

Using a scale of 4 metres on the ground to 1 centimetre on paper, construct a scaled-down

model of a circular garden, which has a diameter of 20 metres. After drawing the scaled circle, draw its diameter and label the end points of the diameter.

114

Circumference of a Circle Maya notices that her bangles are smaller than her mom’s because her bangles won’t fit her mom’s wrist. She then thinks about how she will know which of her circles are smaller or bigger. She needs a way to measure the size of her circles. The circumference of a circle helps you understand the size of the circle. It is the length of its outer edge or boundary. In simpler terms, it is the total length of the circle's curved path. It is equivalent to the perimeter of other shapes, but the term ‘circumference’ applies only to circles. Circumference = 2pr, where r is the radius and p =

22 7

. 22

as a substitute to simplify The symbol ‘π’ here is a constant whose value is 3.14. We can also use π = 7 calculations. What is the circumference of a circle with a radius of 6 cm? Provide your answer rounded to two decimal places. Use the value of π = 3.14.

Example 8

Given, radius, r = 6 cm. Using, C = 2πr = 2 × 3.14 × 6 cm = 37. 68 cm Hence, circumference of the circle is 37.68 cm. Find the radius of a circle whose circumference is 44 cm. Use π =

Example 9

Given, circumference, C = 44 cm. C = 2πr. So, 44 cm = 2 × r = 7 cm

22 7

×r

Sarah wants to put lace around her circular tablecloth. How much lace is needed if the length of the radius is 28.7 cm? Use π = 22 .

Example 10

Radius of the garden = 28.7 cm Circumference = 2πr

= 2 × 22 × 28.7 = 180.4 cm 7

So, the length of lace needed is 180.4 cm. Do It Together

Fill in the blanks with the correct diameter or circumference. Use π = 22 . 7

Diameter Circumference

Chapter 7 • Curves and Circles

10.2 cm 47.12 cm

115

Do It Yourself 7D 1

Find the circumference of the circles with the following radii. Take π = a 7 cm

b 8.5 cm

c 9 cm

Find the value of the radius for the following circumference. Take π = a 154 cm

b 88 cm

c 110 cm

22 7

. d 49 cm

e 14 cm

d 66 cm

e 352 cm

Imagine that you are designing a circular garden for a park. The park management has specified that the

Rohan takes 2 rounds around a circular park. The distance from one end of a park to the other through the centre of the park is 28 m. What is the distance that Rohan covers? Take π = 22 .

circumference of the circular garden should be 94 metres. What is the radius of the garden?

There are 3 concentric circular tracks in a sports complex. The innermost track has a radius of 49 m, and each

successive track has a radius 7 m larger than the previous one. What is the diﬀerence in the length of the first, second and third tracks? Take π = 22 . 7

Word Problem 1

Sia is planning to bake a circular cake for her friend’s birthday. She wants the cake to have a circumference of 36 centimetres. Find the radius of the cake to the nearest whole number. Use π = 3.14.

Points to Remember

116

•

Simple curves do not cross themselves.

•

Open curves do not start and end at the same point, whereas closed curves do.

•

A secant is a straight line that intersects a circle at two distinct points.

•

An arc of a circle is a portion of a circle bounded by two distinct points.

•

A chord splits a circular region into major and minor segments.

•

A chord splits the circle into major and minor arcs.

•

A compass can be used to construct a circle from the centre.

•

The circumference of a circle is the distance around its outer edge or boundary.

Math Lab Creating Rangoli Designs with Circles Materials needed: Large sheets of paper, pencils, markers or coloured pencils, compass Steps:

Distribute sheets of paper and art supplies to each student or group.

Instruct the students to create rangoli designs by constructing circles using the compass.

Encourage the students to try constructing concentric circles to make the design. They can then colour the design.

After the drawings are complete, ask the students to share their rangoli designs with rest of the class.

Chapter Checkup 1

Identify simple curves among the figures. a

Identify open and closed curves among the figures. a

Draw closed curves that cross themselves the number of times given. a 3

b 4

d 5

Draw open curves that cross themselves three times.

Construct a circle of any radius. Draw and label a secant, a chord and a diameter of the circle.

Construct a circle with the given radius. a 2 cm

b 3.8 cm

c 9.5 cm

Find the diameter of the circles that were drawn in Question 6.

Find the radius of the circles from their diameters. a 7.2 cm

Chapter 7 • Curves and Circles

b 3.6 cm

c 9.8 mm

d 7.1 cm

e 4.6 cm

d 10 m

e 11.2 km

117

Find the circumference of the circles with the given radius. a 10.5 cm

b 6.2 cm

c 15.7 cm

d 28.5 cm

e 20.1 cm

d 10.5 cm

e 56 mm

10 Find the value of the radius for each of the circumferences. a 24.5 cm

b 14 cm

c 17.5 cm

11 Using a compass and ruler, construct a circle with a diameter of 8.5 cm. After constructing the circle, indicate its centre and label its circumference. 12 Construct a circle with a radius of 38 mm, using a compass. Ensure that the circle is accurately drawn and labelled. 13 Construct 2 concentric circles with radii of lengths 6.8 cm and 2.5 cm. 14 Look at the figure. Find the shortest path, the longest path and two paths that are equal in length. Q

Path L: Z

Path M: P to Q via the chord PQ Path N: A

Path R: B

Z via a semi-circle

Z P

O B

Path S: P to Q via the major arc A

Word Problems 1

At a botanical garden, there are 5 circular flower beds. The smallest flower bed has a diameter of 8 metres, and each successive flower bed has a diameter 4 metres larger than the previous one. Find the difference in the circumference of the smallest and the largest flower bed.

Era wants to put a ribbon around her circular cards each of diameter 14 cm. How much ribbon would she need, if she wants to make 10 cards?

118

3-D Shapes

Let's Recall Rahul and Preeti are playing a game of shapes. They have to put the solid shapes inside the box, based on their faces. Let us see if they can identify the shapes.

A dice can be put inside a box from a square-shaped hole.

A ball can be put inside a box from a circular hole.

Let's Warm-up

Match the shapes with 1 or more faces. 1

Circle

Rectangle

Triangle

Square

I scored _________ out of 4.

Understanding 3-D Shapes Real Life Connect

Jimmy owns a gift and bakery shop. The gifts are in different 3-D shapes. Let us see how these gifts are different from each other.

Features of 3-D Shapes Jimmy likes one gift in particular, as he can see a lot of 3-D shapes in it. Face Vertex Edge

A 3-D or a three-dimensional shape has three dimensions (length, breadth and height). Such shapes are also called solid shapes. A 3-D shape has faces, edges and vertices. Let us look at various 3-D shapes in Jimmy’s gift. Cylinder: 0 vertices, 2 edges, 2 plane faces + 1 circular face Cube: 8 vertices, 12 edges, 6 plane faces Cone: 1 vertex, 1 edge, 1 plane face + 1 circular face Sphere: 0 vertices, 0 edges, 1 curved face Cuboid: 8 vertices, 12 edges, 6 plane faces 3-D shapes with straight sides can be classified as prisms and pyramids.

Prisms: Prisms are 3-D shapes with straight sides and identical opposite bases that can be polygons. The lateral faces of prisms are rectangles and the opposite faces are parallel. Cubes and cuboids are also prisms, as they have identical opposite faces that are squares in a cube and squares or rectangles in a cuboid. Square Prism

Triangular Prism Identical triangular bases

Square opposite faces

Hexagonal Prism Identical hexagonal bases 120

Rectangular lateral faces

Octagonal Prism Identical octagonal bases

Pyramids: A pyramid is a 3-D shape with one vertex and one polygonal base. It has the same number of lateral triangular faces as the number of sides in the polygonal base. The lateral triangular faces meet at the vertex.

3 la

4 faces

s ace

f ral

late

4 la t er al f ace s

Triangular lateral faces

ac al f ter

ate

ral

6 la

ter al f

ace

fac

5 faces

6 faces

7 faces

Triangular base

Square base

Rectangular base

Pentagonal base

Hexagonal base

Triangular pyramid

Square pyramid

Rectangular pyramid

Pentagonal pyramid

Hexagonal pyramid

A triangular pyramid is also called a tetrahedron.

Think and Tell

What are the numbers of faces, edges and vertices in a decagonal prism?

Example 1

Identify the shape. Write the number of vertices, edges and faces in the shape. The base of the shape has 8 sides so this is an octagonal prism. It has 16 vertices, 24 edges and 10 faces (2 octagonal and 8 rectangular).

Example 2

Name the 2 shapes. Write 1 difference and 1 similarity between the shapes. Shape A is a triangular pyramid and Shape B is a triangular prism. Difference: Shape A is a pyramid and Shape B is a prism. shape A

Similarity: Both shapes have a triangular base. Do It Together

shape B

Complete the table. Name

Cube

Hexagonal Prism

Faces

Edges

Vertices

Shape

Chapter 8 • 3-D Shapes

2 1

121

Do It Yourself 8A 1

Tick the correct option for the questions. a Which of the figures is not a solid figure? i

cube

ii circle

iii pyramid

iv sphere

iii 5

iv 6

iii cylinder

iv cone

b How many faces does a triangular pyramid have? i

ii 4

c What is the shape of a playing dice? i

cuboid

ii cube

d How many vertices does a heptagonal pyramid have? i

iii 9

iv 10

Name the shape of the objects. a

ii 8

Fill in the blanks. a A cube has _______ faces, _______ edges and _______ vertices. b A _______ has 2 circular edges. c All the faces of a _______ are identical. d A pentagonal prism has _______ more edges than vertices. e A hexagonal pyramid has _______ triangular faces. f A _______ and a _______ have square bases.

Match the shapes with their features. a 6 faces

Octagonal pyramid

b 12 edges and 7 vertices

Cone

c 1 vertex and 2 faces

Cuboid

d 16 edges

Pentagonal prism

e 7 faces and 10 vertices

Hexagonal pyramid

How is a nonagonal pyramid different from a hexagonal pyramid? How are these shapes similar?

Word Problem 1

122

John is playing with his geometric shapes. He has an octagonal pyramid and an octagonal prism. Write 2 differences and 2 similarities between the 2 shapes.

Nets of 3-D Shapes Remember, Jimmy has a bakery. He gets an order for 4 cakes of various shapes. The cakes have to be packed in boxes of the same shapes as those of the cakes. Let us look at some of the boxes.

Remember! A net is a 2-D figure that can be folded to form a 3-D figure.

He folds the cardboard to form the boxes.

Nets of Cubes and Cuboids A net is a set of 2-D shapes drawn on a sheet of paper or cardboard that can be folded to form a 3-D shape. Let us see the nets of some 3-D shapes.

Cube

Cuboid

Remember! The cube can be opened in 11 ways, whereas a cuboid can be opened in 54 different ways.

Error Alert! Always look at the opposite faces in the net of a cuboid. They can never be different.

Did You Know? A solid shape with 6 or more plane faces is called a polyhedron.

Nets of an open box An open box does not have the top face of the box. Let us see some examples.

Chapter 8 • 3-D Shapes

123

Example 3

Which of the nets can form a cube whose opposite faces will have the same colour?

Net A

Net B

Net C

In net B, the opposite faces will have the same colour. Example 4

Which of the following nets will form a cube when folded? a

The nets in option (a) and option (b) will form a cube when folded. Do It Together

A net for an open box is shown. Where should the square(s) be placed so that the figure becomes the net of a closed cube? Draw the net. A

Nets of Cones and Cylinders Let us look at the nets of cones and cylinders. Cone: 1 plane face, 1 curved face

Example 5

How many faces does the net of a cone have? The net of a cone is

Example 6

Cylinder: 2 curved faces, 1 plane face

. It has 2 faces.

Draw the net of a cylinder which is open at both ends.

The net of a cylinder which is

124

open at both ends is a rectangle.

Do It Together

Draw any three nets for a cylinder. 1

Nets of Pyramids Jimmy also packs some cakes in pyramid-shaped boxes. Let us see what the box looks like when we open it. Triangular pyramid 1 square face, 4 triangular faces

Let us learn about the nets of pyramids: Triangular Pyramid 1 triangle face, 4 rectangular faces

Example 7

Pentagonal Pyramid 1 pentagonal face, 5 rectangular faces

Draw the net of a heptagonal pyramid. A heptagon has 7 sides. Heptagonal Pyramid

Example 8

Net of a Heptagonal Pyramid

What 3-D shape does the net form? The base of the net is a rectangle and the other 4 faces are triangles. So, it forms a pyramid with a rectangular base. This is called a rectangular pyramid.

Chapter 8 • 3-D Shapes

125

Do It Together

Name the shape. Draw its net.

Name of the shape: _____________

Do It Yourself 8B 1

Write the names of the solid shapes for which the nets are shown. a

Fill in the blanks. a A ___________ has no net. b The net of a cylinder has ___________ faces. c A cubical box is open from the top. The net of the box has ___________ faces. d A square pyramid has ___________ squares and ___________ triangles in its net.

Draw the nets for the shapes given. a

Draw the nets for the shapes given. a Cuboid

b Triangular pyramid

c Cone

d Pentagonal pyramid

Which of these is the net of an octagonal pyramid?

Word Problem 1

126

Madhav is making boxes from nets. He wants to pack a gift box of a tetrahedron shape. What is the net of a tetrahedron?

Points to Remember •

All 3-D objects have faces, edges and corners.

•

Two-dimensional shapes that are folded to make three-dimensional shapes are called nets.

Math Lab Aim: To reinforce understanding of how faces of 3-D shapes can be used to draw their nets Setting: In groups of 2 Materials required: Coloured paper, scissors, glue or tape, markers Method:

Students discuss the faces of various pyramids and prisms.

They then draw the nets of those 3-D shapes on coloured paper.

Students carefully fold and glue/tape the nets to create their 3D shape.

Each student displays the net formed.

Chapter Checkup 1

Identify the number of faces in the shapes given. a Cube

c Pentagonal prism

d Hexagonal pyramid

What are the number of edges and vertices in the shapes given? a

b Sphere

Fill in the blanks. a A pentagonal pyramid has _____ edges and _____ vertices. b The difference between the number of vertices and number of faces of a decagonal prism is _____. c The total number of faces, edges and vertices in a heptagonal prism is _____.

Which of these are NOT prisms? a Cone

Chapter 8 • 3-D Shapes

b Cube

c Cuboid

d Cylinder

127

Write the number of faces and their types in each of the shapes. a Square prism

b Cube

c Cylinder

d Triangular pyramid

iii 9

iv 10

iii 5

iv 6

iii 10

iv 11

Draw the polygonal base for the nets of the 3-D shapes given. a Nonagonal pyramid

Draw the nets for the shapes given. a Octagonal pyramid

c Hexagonal prism

Write the names of the given nets. a

b Heptagonal pyramid

b Hexagonal pyramid

Choose the correct option. a The net of a heptagonal pyramid has _______ faces. i

ii 8

b The net of a cube has _______ faces. i

ii 4

c The net of a nonagonal pyramid has _______ faces. i

ii 9

10 Choose the correct shape of the nets given. Triangular pyramid

Cone

Cylinder

Square Pyramid

Word Problems

128

Jenna is participating in an origami competition. She wants to create a paper model of a

David is a packaging designer. He is tasked with creating a unique gift box in the shape

triangular prism, using a net. Draw the net of a triangular prism. of a rectangular prism. Draw the net of a rectangular prism.

Integers

Let's Recall Natural numbers are the numbers that begin with 1 and go up to infinity, whereas whole numbers are numbers that begin with 0 and go up to infinity. They can be shown on a number line as:

Natural Numbers Whole Numbers The number to the left of a natural/whole number is its predecessor and the number to the right of a natural/whole number is its successor. For example, the predecessor and successor of 636 are 635 and 637 respectively. –1 630

631

632

633

634 635 636 637 638 Predecessor Successor

639

640

Let's Warm-up

Fill in the blanks with the predecessor and successor of the following whole numbers. S. No.

Predecessor

Whole Number

106

Successor

I scored _________ out of 5.

Understanding Integers Real Life Connect

Simran’s father bought her a new video game. Both Simran and her brother Jay are excited to play the game. When the siblings are playing the game, they notice some strange numbers with (+ and −) signs coming on the screen each time they achieve or miss their targets. Let us see what these numbers are!

Player 1

Player 1 2

Round 1 points

+2, –1, –1, +2, –1, +2, –1, +2

–1, –1, –1, +2, –1, –1, –1, –1

Positive and Negative Integers The numbers that the siblings see with the + and – sign are called integers. All negative and positive numbers including 0 are called integers. For example, −1, 2, −20 and 25 are all integers. Let us see some real-life examples of integers. –250 kal

500 ft

+150 kal

450 ft

100 ft

50 ft

–5

0 ft

–10

–15

–20

400 ft 350 ft

300 ft 250 ft

200 ft 150 ft

–50 ft –100 ft –150 ft

–25

–200 ft

–25

–250 ft –300 ft

Calories taken

Heights above

Temperature above 0°

and calories burnt

and below sea level

and temperature below 0°

We know how to represent natural numbers and whole numbers on a number line. Let us see how we can show integers on a number line. Middle point

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Negative Integers

Zero

9 10

Remember! Positive integers can be represented without

Positive Integers

its sign as well.

Representation of integers on a number line Let us mark –5, 9, –8 and 6 on a number line. Step 1: Draw a line and mark numbers at equal intervals. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Step 2: Mark 0 at the centre point.

Step 3: Mark positive numbers to the right and negative numbers to the left of 0.

130

9 10

Step 4: Circle the integers to be shown on the number line.

An opposite of an integer is the number that is the same distance from zero but with the opposite sign. The integers –1, +1; –2, +2; –3, +3 and so on... are considered opposites of each other. Some other opposite integers are: Situation

Opposite

Depositing ₹500 = +500

Withdrawing ₹500 = −500

Rise by 100 points = +100

Fall by 100 points = −100

5 km towards the north = +5

5 km towards the south = −5

Successor and predecessor of an integer The integer to the left of an integer on a number line is called its predecessor and an integer to the right of an integer on the number line is called its successor. For example, the predecessor and successor of –2 and +2 can be given as: –5 –4 –3 –2 –1

–5 –4 –3 –2 –1

Predecessor Successor of –2 of –2

Predecessor Successor of +2 of +2

Error Alert! The predecessor of a negative integer is greater and its successor is smaller than the integer in terms of the numerical value. For integer –7 Predecessor = –6; Successor = –8

Predecessor = –8; Successor = –6

Comparing and ordering integers Smaller to greater –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Greater to smaller

1 A positive number is always greater than a negative number. 2

a A positive integer with more digits is always greater than an integer with fewer digits. b If two positive integers have the same number of digits, then compare the left-most digit of two

numbers until we come across unequal digits.

3 A negative integer having a smaller numerical value is always greater than the one having a greater numerical value.

Chapter 9 • Integers

131

Absolute value of an integer The absolute value of an integer is the actual distance of the integer from zero on the number line. In other words, the absolute value of an integer is the numerical value of the integer regardless of its sign . The absolute value of a number is denoted by the sign ||. 4 units –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

2 units 0

|–4| = 4

Example 2

9 10

|2| = 2

Distance of −4 from 0

Example 1

Distance of +2 from 0

Write the situations in the form of integers. 1 25 cm to the right = +25

2 A fall of 70 points = –70

3 25 m below ground level = –25

4 15 km north = +15

Mark –3, +5, +7 and –6 on the number line. Also, give the absolute value of each of the integers. The integers can be marked on the number line as: –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

The absolute values can be given as: |–3| = 3 Example 3

|+5| = 5

|+7| = 7

|+6| = 6

Arrange the integers in ascending order. –3, 5, 8, –7, –9, 3, 6 To arrange the integers in ascending order, we move from left to right on the number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Hence, the integers can be arranged as, –9 < –7 < –3 < 3 < 5 < 6 < 8 Do It Together

Rudra lives on the 5th floor and Suhani lives on the 2nd floor of the same building. Rudra parks his car in basement 2 and Suhani parks her car in basement 3. Mark the floors and basements on a vertical number line.

+5 +4 +3 +2 +1 0 –1 –2 –3 –4

132

Rudra’s car parking

Do It Yourself 9A 1

Write the situations in terms of integers. Write their opposites as well. Going 10 km to the east

Spending ₹1000

Payal gets 5 apples

3 kg decrease in weight

Win by 3 runs

Be late by 10 minutes

Represent the given integers on a number line. a −7

b 10

c 25

d –88

e 125

d –9 and –4

e –15 and –8

c –12, –7

d 12, –1

e –15, –25

c 25

d –35

e –65

Write all integers between the integers given. b –5 and 0

c 1 and 6

Which of the two integers falls on the right, on the number line. a –4, 4

b –12

a –3 and 4

e 2

Write the predecessor and successor of the following integers. a 17

d 8

Given below is a number line representing integers. Write the integers corresponding to the points. B

c −1

b 10, 7

Write the absolute value of the integers. a 8

b –10

Write any 5 integers less than –6.

Write any 7 integers greater than –10.

10 Compare using the >, < or = sign. a –26

–33

b –15

–12

c 47

–47

d 14

e –51

–15

11 Write the integers in ascending order. a –2, 5, 9, –8, –10

b –6, –11, 16, 20, –7

c 14, –9, 13, –15, 23

d –11, 15, 35, –45, 63

e –32, –18, 47, –63, 95

f –36, 25, –45, 63, –69

a –6, –8, 12, 4, –9

b –1, 6, 0, –5, –11

c 12, 15, –14, 20 –17

d –3, 17, 25, –15, 32

e –12, 14, –30, 52, –28

f –25, 36, –52, 59, 63

12 Write the integers in descending order.

Chapter 9 • Integers

133

Word Problems 1

A weather report showed the temperatures of certain cities as 5°C, –3°C, 12°C, –7°C, 18°C

Kanak deposited ₹1000 in her bank account on Monday and withdrew ₹750 on Wednesday.

A submarine is at a depth of 15 km below sea level. A parachute is 35 km above sea level

and 2°C. Arrange the temperatures in ascending order. Show the deposit and withdrawal in terms of integers.

and a helium balloon is 10 km above sea level. Which of these is between the other two?

Operations on Integers Real Life Connect

Simran scored +4 points and Jay scored –4 points in the first round. During the second round, Simran scored +2 points and Jay scored +6 points. They want to know who the final winner is after the two rounds. Let us help them out!

Addition of Integers We already know about the addition of whole numbers. Let us now learn to add two or more integers.

Rules for Addition of Integers Addition of two integers Let us add Simran’s score during the two rounds which is +4 and +2 using a number line. Step 1: Start from 0

Step 3: Again move 2 steps right as the second digit is

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Move 4 steps to the left of 0 reaching –4, then move 2 steps to the left. Hence, (–4) + (–2) = –6

134

+2 3

9 10

For two positive integers, move towards the right to add.

Hence, (+4) + (+2) = +6

What if Simran’s score was –4 and –2

+4 and +2

Step 2: Move 4 steps to the right as we need to add +4.

Simran’s Score:

–2

–4

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

From the above discussion, we can conclude that: To add two integers with the same sign: • add their absolute values and

For two negative integers, move towards left to add.

• place the common sign before the sum Therefore, (+4) + (+2) = |+4| + |+2| = 4 + 2 = 6 And, (–4) + (–2), |–4| + |–2| = 4 + 2 = 6 (–4) + (–2) = –6 Addition of integers with different signs

Let us now add Jay’s score, which is – 4 and +6 using a number line. Jay’s Scores: –4 and +6

Step 1: Start from 0 Step 2: Move 4 steps to the left, as we need to add – 4. Step 3: Move 6 steps right, as the second digit is +6.

Hence (–4) + (+6) = +2135

–10 –9 –8 –7 –6 –5 – 4 –3 –2 –1

9 10

OR Move 6 steps to the right of 0 reaching +6, then move 4 steps to the left. Hence (+6) + (–4) = +2

–10 –9 –8 –7 –6 –5 – 4 –3 –2 –1

From the above discussion, we can conclude that: To add two integers with different signs: • find the difference of their absolute values and • place the sign of the integer with greater numerical value Therefore, (–4) + (+6) = |+6| – |– 4| = 6 – 4 = 2 Example 4

Show 5 more than –4 on the number line. 5 more than –4 = –4 + 5 It can be shown on the number line as:

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Hence –4 + 5 = 1 Example 5

Add the given integers. 1 (–2345) and (1852) As –2345 and 1852 are of opposite signs, |–2345| – |1852| = 493 Place the '–' symbol in front of 493 as 2345 has greater numerical value than 1852. So, –2345 + 1852 = –493

Chapter 9 • Integers

2 (– 430), (–360) and (1236) (–430) + (–360) = –790 Add the sum –790 with 1236 –790 + 1236 = |1236| + |–790| = 446 (–430) + (–360) + (1236) = 446

135

Example 6

Renu parked her car in the basement, 2 floors below ground level. She got in the elevator and went up 12 floors. At what floor did she get off the elevator? 2 floors below ground level = –2 As Renu went 12 floors up the basement, the floor she gets off = –2 + 12 = 10 Hence, she left the elevator at the 10th floor.

Do It Together

A research team aboard an underwater research vessel descends 1200 feet beneath the surface of the water. They then rise 635 feet and descend 455 feet again. Where are they currently located? 1200 feet beneath surface water = –1200 Rise of 635 feet = +635 Descend again 455 feet = _________ Current location of vessel = (–1200) + (635) + _________ = _________ Hence, the vessel is _________ feet _________ the surface of the water.

Properties of Addition of Integers Closure Property

Commutative Property

If a and b are two integers, then

For example,

1 (+2) + (+8) = +10

a+b=b+a

(a + b) + c = a + (b + c)

This means that the sum of two integers is the same regardless of the order in which they are added.

This means that when adding three or more integers, the order in which additions are performed does not change the final sum.

For example,

2 (+5) + (–9) = – 4

(−5) + (+8) = (+8) + (−5) = +3

Existence of Additive Identity For any integer a, we have: a+0=0+a=a This equation states that adding zero to any integer results in the same integer. For example,

1 5+0=5

Example 7

For any three integers a, b and c,

For any two integers a and b,

a+b=c where c will always be an integer.

Associative Property

2 –5 + 0 = –5

For example, (−5 + 3) + (−6) = −5 + (3 + (−6)) = −8 Existence of Additive Inverse

For every integer, there exists an additive inverse so that when the integer and its additive inverse are added together, the result is zero. In simpler terms, for any integer a, another integer exists –a so that a + (–a) = 0 For example, the additive inverse of 10 is –10 and vice versa.

Which of these represent the additive inverse property? 1 18 + 3 = 3 + 18

2 30 + 0 = 30

3 15 + (–15) = 0

Option 3 represents the additive inverse property. Example 8

Which property of the addition of integers is used in (–12 + 13) + 25 = –12 + (13 + 25). The associative property is used in the given addition sentence.

136

Do It Together

Fill in the blanks and name the property used. 1 –25 + 0 ___________

Existence of additive identity

2 –15 + 35 = 35 +(–15)

_______________________________

3 39+ (−12) = ___________

Closure property

4 26 + (–13 + 9) = ___________

_______________________________

Do It Yourself 9B 1

Use the number line and add the integers. a 6 and 4

b –3 and –6

c 8 and –5

d –6 and –4

e –9, 3 and 5

Write the addition sentence shown by the number lines. a –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Fill in the blanks. a 12 + (–14) = ____________

b –16 + (–18) = ____________

c –25 + 63= ____________

d 52 + (–45) = ____________

e 125 + (–256) = ____________

f (–365) + (–145) = ____________

Fill in the boxes with the <, > or = sign. a 5 + (–2) d 32 + (–52)

–7 + 4 –62 + 10

b 10 + 12

20 + (–5)

e –65 + (–42)

–57 + (–34)

c –12 + 25

–36 + 41

f 58 + (32)

–87 + 41

Simplify. a (–9) + 12 + (–15)

b 18 + (–25) + 28

d 36 + (–78) + (–14) + 63

e –175+ (–48) + 259 + 88

Add the successor of –1025 to the predecessor of 923.

Write the additive inverse of the given integers. a –8

Chapter 9 • Integers

b 12

c –14

c –32 + (–14) + (–35)

d –36

e 45

137

Find an integer a so that a 5+a=5

b 7+a=0

c 6+a=3+6

Find three integers of different signs whose sum is –158.

Word Problems 1

Richa’s football team loses 7 yards on one play and gains 12 yards on another play. What is

The temperature of a city is recorded as 42°C on Monday. The temperature falls by 15°C

the score of the team after two plays?

on Tuesday and falls by a further 7°C on Wednesday. Find the temperature recorded on Wednesday.

Subtraction of Integers Jay now wants to know the difference between the number of points he scored in the second round as compared to the first round. Let us help him!

Rules for Subtraction of Integers Step 1: Take the additive inverse of the integer to be subtracted.

Jay’s Scores: –4 and +6

Additive inverse of −4 = +4 Step 2: Add it to the other integer 6 – (–4) = 6 + (+4)

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

+4 3

9 10

Hence, Jay scored 6 – (–4) = 6 + (+4) = 10 more points in the second round compared to the first round. From the above discussion, we can conclude that: To subtract integer b from a:

Did You Know?

• change the sign of b and

on the Earth was 58°C in the Libyan

• add to a Example 9

The highest temperature ever recorded desert. The coldest temperature ever measured was –88°C in Antarctica.

Subtract. 1 85 – (–53) = 85 + (additive inverse of –53) = 85 + 53 = 138 2 –269 – (136) = –269 + (additive inverse of 136) = –269 + (–136) = –405 3 –2445 – (–1196) = –2445 + (additive inverse of –1196) = –2445 + 1196 = –1249

Example 10

Subtract the sum of –215 and 635 from –856. Sum of –215 and 635 = –215 + 635 = 420 Subtract 420 from –856 –420 = –856 + (–420) = –1276

138

Example 11

The temperature of a city is 10°C on Monday. It drops by 15°C on Tuesday. What is the temperature of the city on Tuesday? Temperature on Monday = 10°C Drop in temperature on Tuesday = 15°C Temperature on Tuesday = 10°C – (15°C) = –5°C

Do It Together

Subtract. 1 –36 from 58 = 58 – (–36) = 58 + 36 = _____ 2 –145 from –365 = –365 – (–145) = _____ + _____ = _____ 3 –247 from 354 = 354 – (_____) = 354 + _____ = _____ 4 3654 from –5214 = –5214 – _____ = _____ + _____ = _____

Properties of Subtraction of Integers Closure Property

Commutative Property

If a and b are two integers, then

While subtracting integers, the order in which they are written is important.

a – b = c,

For any two integers a and b,

where c will always be an integer.

a–bπb–a

For example,

This means subtraction is not commutative for integers.

1 (+12) – (+8) = +4

For example, (+22) – (+13) = +9;

2 (+15) – (–9) = +24

⸫ (+22) – (+13) ≠ (+13) – (+22)

Associative Property

Subtraction Property of Zero

For any three integers a, b and c,

The subtraction property of zero in integers states that subtracting zero from any integer leaves the integer unchanged.

(a – b) – c π a – (b – c) This means that integers are not associative under subtraction.

For an integer a,

For example, (5 – 3) – 4 = –2;

For example,

5 – (3 – 4) = +4

⸫ (5 – 3) – 4 π 5 – (3 – 4)

Example 12

(+13) – (+22) = –9

a–0=a (–3) – 0 = –3

Show that the integers 12, –18 and –34 are not associative under subtraction. The associative property of subtraction of integers states that

(a – b) – c π a – (b – c)

(a – b) – c = (12 – (–18)) – (–34) = 12 + 18 – (–34) = 30 + 34 = 64 a – (b – c) = 12 – (–18 – (–34)) = 12 – (–18 + 34) = 12 – (16) = –4 64 π –4, hence (a – b) – c π a – (b – c) Do It Together

Fill in the blanks. 1 –125 – _______ = –125

2 523 – 365 = _______; 365 – 523 = _______

3 (125 – 523) – 363 = _______; 125 – (–523 –363) = _______ Chapter 9 • Integers

139

Do It Yourself 9C 1

Subtract the given integers using a number line. a –4 from –3

c –8 from 15

b –48 from –42

c –52 from 69

d –6 from –2

Subtract. a 15 from –36

b 5 from –10

Subtract the integers and match the answers. a –125 – (–253)

777

b 236 – (–541)

–790

c –365 – (425)

–157

d –414 – (–257)

128

Find the sum of 254 and −146 and subtract it from 367.

Subtract −389 from the sum of −745 and −425.

Subtract the sum of 623 and –584 from the sum of –795 and 475.

Write true or false. a Integers are associative under subtraction.

___________

b The difference of two integers is always an integer.

___________

c Zero subtracted from an integer results in the additive inverse of the integer.

___________

d Subtraction is not commutative for integers.

___________

Simplify: 256 + (–365) – 124 + (–475) – 321

The sum of two integers is –423. If one of them is 89, find the other.

Word Problems

140

The melting point of mercury is –38.9°C and its boiling point is 356.7°C How many degrees is the

Dev drove her car for 125 km to the north on Monday. He drove 182 km south on Tuesday. How

Pooja had a balance of `(–1250) in her bank account due to some penalty. If she wants to make the

A submarine is 2514 feet below sea level. An aeroplane is 10,500 feet above sea level. What is

melting point below the boiling point? far is Dev from his initial point?

balance in her bank account as `5000, how much money does she need to deposit in her account? the distance between the submarine and the aeroplane?

Points to Remember •

The numbers (…. −4, −3, −2, −1, 0, 1, 2,3,4, ….) are called integers.

•

Negative integers lie on the left of the number line, whereas positive integers lie on the right of the number line.

•

Integers on the right of the number line are greater than integers on the left of the number line.

•

0 is neither a positive nor a negative integer.

•

The absolute value of an integer is its numerical value regardless of its sign.

•

To add two integers with like signs, we add their values and place the common sign before the total.

•

To add integers with unlike signs, we take the difference of their absolute values and put the sign of the integer with the greater absolute value.

•

To subtract an integer b from integer a, we change the sign of b and add it to a.

Math Lab Setting: In two groups Materials Required: Paper and pencil, index cards with expressions written on them Method:

Prepare index cards with various integer expressions written on them.

1 member from each team draws an index card and solves the expression.

After solving the expression, both players take turns explaining their solution to the class.

If both Player A and Player B provide the correct solution, they each earn a point for their team. If one or both provide incorrect solutions, no points are awarded.

Repeat the process for as many rounds as the time allows.

Chapter Checkup 1

Circle the integer which is to the left of the other on the number line. a –2, 5

b 13, 18

c 8, –8

d –13, –17

e 25, 36

The table shows the temperatures of some places in India. Place

Temperature

Place

Temperature

Chennai

31° above 0°

Pune

18° above 0°

Ladakh

5° below 0°

Sonamarg

8° below 0°

Chapter 9 • Integers

141

a Write the temperatures in the form of integers. b Name the coldest and the hottest place.

Which point will be marked as −2 on the number line?

Tick the integers that are given in their absolute values. a 9

b +12

c –36

b 15

d 45

e –57

c –26

d –38

e 45

Which of the statements is true? +8 is larger than –16

+16 is smaller than +8

How many positive integers lie between –5 and 0?

Which set of integers is written in ascending order? a –25, –65, 28, 63, 100

Write the predecessor and successor of the following integers. a –8

–8 is larger than +16

b –22, –18, –15, 16, 36

c 29, 15, –9, –16, –32

+16 is smaller than –8

d –32, –17, –19, 42, 48

Add the integers. –25 and 36

–142 and –48

–157 and 96

452 and –152

10 Among the integers given, find the sum of the largest and the smallest integers. 32, –7, 361, 143, –423

11 Fill in the boxes with the <, > or = sign. a 7 – (–5)

–5 – 9

b 21 – 12

c –36 – 17

48 – 96

d 132 – (–152)

–262 – 204

658 – (–132)

–387 – 841

e –365 – (–542)

–357 – (–534)

–30 – (–15)

12 Subtract the predecessor of –526 from the successor of 658. 13 Write three integers with different signs whose difference is –48. 14 Write true or false. a The sum of two negative integers is always negative.

____________

b The sum of an integer and its inverse is zero.

____________

c The difference of two integers is always an integer.

____________

d The sum of 4 different integers can never be zero.

____________

15 Write a number smaller than −858 but greater than −861 and is odd.

142

Word Problems 1 Nilesh ended round one of a quiz with 325 points. In round two, he scored –130; and in the third round; he gained 78 points. What was his total score at the end of the third round?

2 Muskan burnt 480 calories working out on a treadmill and 260 calories by skipping. She

indulged in a sweet that had a calorie count of 500. How many calories did she lose or gain?

3 On a hot summer day, the temperature in a desert was 132°F. It dropped to –250°F on a windy day. What is the difference between the two temperatures?

4 Priya uses an online wallet for shopping which allows her to shop for anything up to `750 even if she has no balance in the account. She wanted to purchase a dress for `1000, and

her account balance was –125. She added `1500 to her account. What is the balance in the account after the purchase of the dress?

Chapter 9 • Integers

143

Fractions

Let's Recall A fraction is a number that represents a part of a whole. It has two parts, a numerator and a denominator. Let us say Riyanshi has a strawberry pie and she divides it into 7 equal parts. So, here a strawberry pie is considered as a whole and each piece of it is a part. She eats 3 parts out of the total parts. To represent the pie eaten by her as a fraction Riyanshi shades 3 parts out of 7. The 3 shaded part represents . Here, 3 is the numerator and 7 is the denominator. 7

Let's Warm-up Match the following. 1

2 6

4 7

5 11

4 9

3 8

I scored _________ out of 5.

Reviewing Fractions Real Life Connect

Miya is baking an oat cake today. She divided the delicious oat cake into 10 equal parts.

Representing Fractions We know that Miya has divided her cake into 10 equal parts.

 1  The whole cake here represents a whole (1) and each piece of cake    10  represents a part of a whole cake. 1 Each piece of cake is of the cake. These can be added to make one whole. 10 1 So, if we add ten times, we will get 1. 10 Representing fractions on a number line There are various types of fractions which we can represent on a number line. Types of Fractions Proper Fraction

Improper Fraction

A proper fraction is a fraction whose numerator is smaller than the denominator. A proper fraction is always less than 1. 1

1 3

2 3

An improper fraction is a fraction whose numerator is equal to or greater than the denominator. An improper fraction is always greater 7 than or equal to 1. Example: 6

Improper fraction: 7 5

Proper fraction: 2 3 0

Mixed Number

4 3

5 3

0 1 2 5 5

3 5

4 5

A mixed number is a combination of a whole number and a proper fraction. A mixed number is another way of representing an improper fraction. 2 It is always greater than 1. Example: 2 5

4 9 Mixed number: 1 = 5 5 6 5

7 5

8 5

9 5

1 2 3 5 5 5

4 1 5

6 7 5 5

8 5

9 5

Remember! The gap between two whole numbers on a number line should be divided into the same number of equal parts as the denominator of the fraction.

Chapter 10 • Fractions

145

Example 1

Classify the fractions as proper, improper or mixed numbers. 3 8 2 3 12 20 , , , 5 , , , 11 2 5 7 2 5 23 13 7

Example 2

Did You Know?

Proper Fractions

Improper Fractions

Mixed Number

3 12 , 5 23

8 2 20 , , 7 2 13

3 2 5 , 11 5 7

The human body is primarily composed of water. On average, 3 water makes up about of an adult 5 human's body weight.

Represent the fractions on a number line. 1

4 5

7 5

3 5

4 5 Do It Together

7 5

3 5

Write the fractions represented on a number line marked with black dots. 0

A A=

B ;B=

C ;C=

Do It Yourself 10A 1

Write the fraction of the shaded part. a

Colour the fraction. a

2 3 146

15 25

1 2

1 3

Classify the fractions as proper fractions, improper fractions or mixed numbers.

4 1 6 51 47 4 16 15 , , , , , 3 1, 4 2, , , 6 7, 1 1, 5 22 4 89 14 5 5 9 3 8 3 14 4

Represent the fractions on a number line. a

1 6

4 7

4 3

d 14

3 7

1 8

Write the fractions represented on the number line. a

Word Problem 1

James wanted to cut a ribbon of length

on a number line?

2 2 m to decorate a box of cake. How do you represent m 3 3

Comparing and Ordering Fractions

1 2 Remember, Miya was baking a delicious cake? If she gave fraction of cake to her father and fraction of 5 5 cake to her mother, who got a bigger slice of cake? 1 2 and are like fractions. 5 5 To compare like fractions, compare the numerators.

We know,

Greater the numerator, greater the fraction. 2>1

1 2 1 5 Thus, > . Thus, her mother got a bigger slice. 5 5 2 2 What if Miya gave part to her sister and to her brother. Who received the larger portion? 7 8 Comparison of unlike fractions

2 5

For unlike fractions with the same numerator, we compare the denominators. The fraction with the greater denominator is smaller. Compare

2 2 and 7 8

7<8

2 2 > 7 8 But what if both numerator and denominator are diﬀerent?

Thus,

LCM Method: Make the fractions like, using the LCM method. Compare

1 2 and 4 3

Chapter 10 • Fractions

147

Step 2: Find the equivalent fractions of 1

Step 1: Find the LCM of the denominators.

and

2 such that their denominators 3

are 12. The LCM of 3 and 4 is 12.

1 3 3 × = 4 3 12

2 4 8 × = 3 4 12

1 4

2 3

Step 3: Now that the fractions are like fractions, compare the numerators and identify which fraction is larger.

LCM of 3 and 4 = 12

3 12

3<4

So, 3 < 8

12 12 Thus, 1 < 2 4 3

8 12

Cross-multiplication method Compare

1 2 and 4 9

Step 1

Cross multiply the denominators with the numerators. 1 × 9 = 9 and 4 × 2 = 8 9>8 Thus,

1 2 > 4 9

1 4

2 9

Ordering of fractions To arrange fractions in ascending order (least to greatest) or descending order (greatest to least), remember to compare the fractions and then arrange. 1 3 2 4 , Arrange , , in ascending order. 2 5 10 15 Step 1

Convert all the fractions into like fractions. LCM of 2, 5, 10 and 15 is 30. 1 1 15 15 = × = 2 2 15 30 3 3 6 18 = × = 5 5 6 30 2 2 3 6 = × = 10 10 3 30 4 4 2 8 = × = 15 15 2 30

148

Step 2

Since denominators are the same, compare the numerators and order the fractions. 6 < 8 < 15 < 18 So,

6 8 15 18 < < < 30 30 30 30

Thus,

2 4 1 3 < < < 10 15 2 5

Example 3

1 3 or ? Represent on a number line. 2 5 Convert both the fractions into like fractions. Which fraction is bigger,

LCM of 2 and 5 = 10 1 1×5 5 = = 2 2 × 5 10 0

1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10

6 > 5, so

Example 4

3 3×2 6 = = 5 5 × 2 10 1

1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10

6 5 3 1 > Thus, > 10 10 5 2

1 5 5 2 Order , , , in descending order. 2 6 12 3 LCM of 2, 6, 12, 3 = 12 1 1×6 6 = = 2 2 × 6 12

5 5 × 2 10 = = 6 6 × 2 12

5 5 = 12 12

2 2×4 8 = = 3 3 × 4 12

10 > 8 > 6 > 5 Thus, Do It Together

5 2 1 5 > > > . 6 3 2 12 5 3 and . Identify the larger fraction. 7 6

Compare

LCM of 7 and 6 = ______

5 = 7 42 Thus, the larger fraction is ______.

3 = 6 42

Do It Yourself 10B 1

Circle the larger like fraction and represent these on a number line. a

11 13 and 14 14

5 8 and 9 9

8 9 and 6 7

3 4 and 5 7

7 11 and 8 14

4 12 and 5 20

11 12

7 1 3 21 , , , 8 2 4 16

Compare the fractions using appropriate symbols (>, < or =). a

2 6 and 8 8

Tick the smaller unlike fractions. a

5 3 and 7 7

1 2

4 7

8 7

7 8

4 6

2 3

12 15

Arrange the fractions in ascending order. a

4 6 8 7 , , , 19 19 19 19

Chapter 10 • Fractions

15 4 3 14 , , , 28 14 7 28

1 4 6 21 , , , 3 18 9 27

149

Write the shaded part as fractions and arrange them in descending order. a

Word Problems 3 1 hours in a day and Preeti studied 2 hours in a day. Who studied for a longer time? 4 2

Richa studied 2

In school A, 480 students passed out of 500, whereas in school B 550 students passed out of

600. Write the fraction of students who passed in both schools. In which school did the greater fraction of students pass?

5 Jacky and Pritam are jogging in a park. Jacky completes rounds of the park in 10 minutes, 8 7 rounds of the park. Who covers the longer whereas in the same time, Pritam completes 12 distance in 10 minutes? Sana purchased diﬀerent lengths of a cloth. She purchased 2 the lengths of the cloth in descending order.

1 1 3 2 m, 2 m, 2 m and 2 m. Arrange 4 3 4 8

Converting Fractions Converting a mixed number into an improper fraction Convert 2 Step 1

2 into an improper fraction. 5

Multiply the denominator with the whole number. Step 2

Add the result to the numerator. Step 3

+ ×

2 12 = 5 5

Keep the denominator the same and write the result as the numerator.

Convert an improper fraction to a mixed number Write

24 as a mixed number. 5

Step 1 Divide the numerator by the denominator. Step 2 The quotient becomes the whole number part. Step 3 The remainder becomes the numerator and divisor denominator.

150

4 5 24 – 20 4

Example 5

Example 6

Convert the following mixed numbers into improper fractions. 1

3 2 × 5 + 3 10 + 3 13 = = 2 = 5 5 5 5

7 3 × 9 + 7 27 + 7 34 = = 3 = 9 9 9 9

Convert the following improper fractions into mixed numbers. 1 42 5

8 5 42 – 40 2 Thus, Do It Together

23 4

Error Alert! The fractional part of a mixed number cannot have a numerator bigger than the denominator.

8 4 23 – 20 3

42 2 =8 2 5

Thus,

23 3 =5 4 4

24 9 =3 5 5

24 4 =4 5 5

Convert. 1 10 10

4 into an improper fraction. 5

2 Convert

4 (5 × __) + 4 = = 5 5

31 into a mixed fraction. 6

31 =5 6

Do It Yourself 10C 1

Write the reciprocal of the fractions. a 17 6

d 5

e 1 9

3 7

b 38 7

c 51 8

d 81 4

e 56 5

f 121 13

Convert the mixed numbers into improper fractions. a

c 11 18

Convert the improper fractions into mixed numbers. a 15 4

8 15

1 5

2 6

7 8

1 8

5 6

1 2

Compare using appropriate signs (>, < or =). a

5 6

29 6

Chapter 10 • Fractions

214 7

1 7

319 12

5 12

47 5

4 5

151

a A farmer harvested 5 and b Tim used

1 bushels of apples. Write the quantity as an improper fraction. 4

18 feet of wood to make a table. Write the length of the wood as a mixed number. 5

Word Problems 1

Renu has baked 100 cookies and wants to divide them equally among her 3 siblings. What will be the

Eva and Sarah baked a chocolate and a vanilla cake, respectively. Eva's chocolate cake was cut into

share of each sibling? Express your answer as a mixed fraction.

8 equal slices. Sarah's vanilla cake was cut into 12 equal slices. If Eva gave away 3 slices of her cake, and Sarah gave away 5 slices of her cake, which of them had the greater fraction of their cake left?

Equivalent Fractions and Simplest Form

While preparing, Miya added 3 tablespoon of baking soda and 6 4 8 tablespoon of baking powder to the cake mixture.

Baking Soda =

We can see from both the figures that the amounts of baking soda and baking powder Miya used are equal in quantity. Hence 3 = 6. Therefore, they are equivalent fractions. 4 8

3 4

Baking Powder =

6 8

Remember! Equivalent fractions are the fractions which may have different numerators or denominators, but they represent the same value.

Finding an equivalent fraction of any fraction ×2 1 = 5

2 10

1 5

2 10

10 15

2 3

×2 ÷5 10 2 = 15 3

÷5 152

Identify equivalent fractions

Check whether 3 and 9 are equivalent or not. 7 21 Step 1

Cross multiply the denominator of one fraction with the numerator of the other fraction and vice versa.

3 7

3 × 21 = 63 ; 7 × 9 = 63 Step 2

Check if the products of both the multiplications are the same or not. If the products are the same, then

�

9 21

the fractions are equivalent, else they are not.

Thus, 3 and 9 are equivalent 7 21 Simplest form of a fraction A fraction is in its simplest form if the HCF of the numerator and denominator is 1.

Think and Tell

What is the reciprocal of 0?

For example, 3 is in its simplest form. 5

Do It Yourself 10D 1

Write the equivalent fraction for the given fractions. a

5 with denominator 91 9

17 with numerator 91 19

18 with denominator 4 24

36 with numerator 3 72

Write two equivalent fractions for the fractions. a

36 48

3 7

228 144

10 11

4 5

16 28

12 35

7 13

Answer the questions. 6 a Write 2 equivalent fractions for , using multiplication. 7

b Write 2 equivalent fractions for

50 , using division. 60

Identify whether each pair of fractions is equivalent or not. a

4 12 and 5 15

1 5 and 3 21

7 14 and 8 24

7 35 and 11 44

2 12 and 12 72

7 21 and 13 26

2 12 and 9 54

14 28

Reduce the following fractions into their simplest forms . a

12 15

Chapter 10 • Fractions

6 28

16 144

153

Word Problems 1

Minal asked her brother to guess the fraction based on the given clues: a The numerator is three less than twice the denominator.

2 b When you add 1 to both the numerator and the denominator, the fraction becomes . 3

Riya was eating chocolates. She had a total of 24 chocolates. She gave 10 chocolates to her friend and 10 chocolates to her sister. What fraction of the chocolates does she have left? Express your answer in the simplest form.

Operations in Fractions Lisa and Maya run a fruit squash café. They are making a fruit punch. The list of ingredients used by them are shown in the table.

Real Life Connect

Ingredients

Lisa

Blueberry juice

1 of a cup 5

Strawberry juice

1 of a cup 2

Orange juice

1 of a cup 3

Sugar

2 1 of a cup 5

Apple juice

3 of a cup 4

Maya 1 of a cup 5 1 of a cup 3 1 of a cup 3 1 3 of a cup 3 3 of a cup 4

Addition and Subtraction of Fractions Lisa wants to find the fraction of blueberry and strawberry juice she used to make the fruit punch.

Addition and Subtraction of Unlike Fractions 1 1 Fraction of blueberry juice used by Lisa = ; fraction of strawberry juice used by Lisa = 5 2

Remember! Sum or difference of like fractions =

154

Sum or difference of numerators Common denominator

Total =

1 1 + 5 2

Step 1: Find the LCM of 2 and 5.

Step 2: Find the

1 5

equivalent fractions with denominators of 10.

LCM of 5 and 2 = 10

1 2

Step 3: Add the fractions.

2 10 Thus, Lisa used total of

5 10

7 10

7 of a cup of blueberry and strawberry juices in her recipe. 10

Lisa wanted to know how much extra strawberry juice she used in comparison to the blueberry juice. –

Step 1: Find the LCM of

1 2

2 and 5.

LCM of 5 and 2 = 10

Step 3: Subtract the

1 5

fractions.

–

Step 2: Find the

5 10

equivalent fractions with denominators of 10.

2 10

3 10

It is clear that more strawberry juice was used compared to the blueberry juice by Example 7

2 5 and 9 12 LCM of 9 and 12 = 36 Add

2 2 4 8 = � = 9 9 4 36

Example 8

Do It Together

3 3 2 6 = � = 4 4 2 8

7 6 7–6 1 – = = ; 8 8 8 8

8 15 8 + 15 23 + = = ; 36 36 36 36 Thus,

7 3 – 8 4 LCM of 8 and 4 = 8 Find

7 7 1 7 = � = 8 8 1 8

5 5 3 15 = � = 12 12 3 36

3 of a cup. 10

2 5 23 + = 9 12 36

Thus,

7 3 1 – = 8 4 8

1 2 fraction of a cake and her sister Riya ate of it. Find the total fraction of the cake consumed 9 3 and the fraction of the cake left.

Radha ate

Fraction of the cake eaten by Radha = LCM of 9 and 3 = ___________

1 2 ; Fraction of the cake eaten by Riya = 9 3

Fraction of cake eaten by Radha and Riya = Fraction of cake remaining =

Chapter 10 • Fractions

–

155

Addition and Subtraction of Mixed Numbers While preparing fruit punch, Maya aimed to determine the total quantity of sugar used by both of them altogether.

2 1 Fraction of sugar used by Lisa = 1 of a cup; Fraction of sugar used by Maya = 3 of a cup 5 3 2 1 Total = 1 of a cup + 3 of a cup 5 3 2 1 To find the total, we will add = 1 + 3 5 3 Step 1

Ingredients

Lisa

Maya

Blueberry juice

1 of a cup 5

Strawberry juice

1 of a cup 2

1 of a cup 3

Orange juice

1 of a cup 3

Sugar

1 of a cup

2 5

3 of a cup

Apple juice

3 of a cup 4

Add the whole number and the whole number.

1+ 3 = 4 Step 2

Add the fractional part and the fractional part.

2 1 2 3 1 5 6 5 + = � + � = + 5 3  5 3   3 5  15 15 =

6 + 5 11 = 15 15

Step 3

Add the results of step 1 and step 2. 4+

11 11 =4 15 15 2 5

1 3

Thus, 1 + 3 = 4

1 3

11 15

Subtracting unlike fractions from mixed numbers

1 1 What if Maya had used 6 of a cup of sugar to make the same fruit punch instead of 3 of a cup? How 5 3 much extra sugar would it have been then? 1 1 We subtract 3 from 6 . 3 5

Step 1

Step 2

Convert the mixed numbers into

Subtract the improper fractions.

1 (6 � 5) + 1 30 + 1 31 = = 6 = 5 5 5 5

31 10  31 3   10 5  – = � – �  5 3  5 3   3 5  93 50 93 – 50 43 – = = = 15 15 15 15

improper fractions.

1 (3 � 3) + 1 9 + 1 10 3 = = = 3 3 3 3 Step 3

Convert the improper fraction into a mixed number.

43 (2 � 15) + 13 2 � 15 13 13 = = + =2 5 15 15 15 15 1 1 13 Thus, 6 – 3 = 2 5 3 15

156

Example 9

1 3 Add 1 and 2 7 7 Add the whole numbers first 1 + 2 = 3. 1 3 1+3 4 + = = 7 7 7 7 4 4 1 3 4 3 + = 3 ; Thus, 1 + 2 = 3 7 7 7 7 7

Example 10

2 1 Find the diﬀerence: 4 – 3 6 4 2 (4 � 6) + 2 24 + 2 26 4 = = = 6 6 6 6

1 (3 � 4) + 1 12 + 2 13 3 = = = 4 4 4 4

26 13  26 2   13 3  � – �  – = 6 4  6 2  4 3

52 39 52 – 39 13 – = = 12 12 12 12

13 (1 � 12) + 1 1 � 12 1 1 = = + =1 12 12 12 12 12

2 1 1 Thus, 4 – 3 = 1 6 4 12 Do It Together

2 1 Subtract 5 from 7 . 6 4 On converting to improper fractions, we get: 2 5 = 6

1 4 = 4

1 2 7 –5 = 4 6

Do It Yourself 10E 1

Add the fractions and reduce the fractions, if necessary. a

5 2 – 17 17

3 2 + 7 14

11 3 + 10 10

1 2 d 1 +2 6 6

1 2 e 2 +4 3 3

12 3 – 11 11

2 1 d 5 –2 3 3

3 2 e 9 +6 5 5

9 3 + 10 11

2 1 d 5 +2 3 3

3 2 e 9 +6 5 5

11 2 –1 15 15

3 2 + 5 3

c 2

b 3

Subtract the fractions. 9 1 14 11 – a b 3 – 13 2 11 22 4 7 e 7 –4 9 12

13 2 + 5 5

Add the fractions. a

Subtract the fractions and reduce the fractions, if necessary. a

1 2 + 7 7

5 3 c 2 –1 7 14

2 1 d 17 – 15 3 12

2 7 f 6 –3 3 15

1 1 2 1 What do we get if we subtract the diﬀerence between 4 and 2 from the diﬀerence between 8 and 5 ? 5 10 3 9

Chapter 10 • Fractions

157

Word Problems 1

6 2 An empty box weighs 2 kg. An iron pan weighing 5 kg is placed inside it. What is the total weight 7 3 of the box?

Sophia is baking cookies for a school fundraiser. She's making two batches of cookies. In the first 3 1 batch, she uses cup of sugar, and in the second batch, she uses cup of sugar. Find 5 5 how much sugar Sophia used in total for both batches of cookies?

Jennifer went shopping for her birthday and bought a total of 9

3 4

2 kg of chocolate candies, 1

1 kg of candies. She purchased 2

1 1 kg of gummy bears, 3 kg of jellybeans and some lollipops. 10 2

What was the weight of the lollipops that she purchased?

Priya bought some cookies from the market. She generously gave three-fifths of them to her

sister, distributed one-fourth of the cookies among her neighbours and kept the rest for herself. a What fraction of cookies did she give away?

b What fraction of the cookies did Priya have left?

Multiplication and Division of Fractions and Mixed Numbers 2 Lisa was thinking of trying a new fruit punch recipe which requires only cup of the available quantity of 5 apple juice. Refer to the table of ingredients above.

Multiplying Fractions or Mixed Numbers

3 of a cup. 4 2 Fraction of apple juice she wants to use in her new fruit juice recipe = of the quantity that Lisa has. 5

Fraction of apple juice Lisa has =

What is

2 3 of ? 5 4

2 3 2�3 6 3 � = = = 5 4 5 � 4 20 10

Thus, Lisa will use only a fraction

2 5

3 4

6 20

3 cup of apple juice of the available quantity. 10

Remember! When the fraction is used with the word “of”, it acts as an operator. Ideally “of” means multiplication.

2 1 Remember Lisa used 1 cups of sugar for the fruit punch. What if she used 1 cups of what she was using 5 6 earlier? 1 This time Lisa was thinking, what if she used 1 cups of sugar of what she was using earlier? 2 158

2 1 2 Fraction of sugar Lisa had = 1 cups; Fraction of sugar Lisa used = 1 cups � 1 cups 5 2 5 Step 1

Step 2

Step 3

Convert both mixed fractions

Multiply both improper fractions by multiplying

Convert

into improper fractions.

2 3 1 = 5 2

the numerators and the denominators.

2 7 1 = 5 5

21 into mixed fraction. 10

21 1 =2 10 10

3 7 3 � 7 21 � = = 2 5 2 � 5 10

Thus, Lisa used 2

1 cups of sugar. 10

Remember! To multiply one fraction with another fraction, multiply the numerators, multiply the denominators and then write the answer in the lowest form.

Example 11

Multiply. 1

5 3 and 9 5

5 3 5 � 3 15 1 � = = = 9 5 9 � 5 45 3

2 3 3 and 1 5 4 3

2 17 3 7 � and 1 = 5 5 4 4

17 7 17 � 7 119 19 � = = =5 5 4 5�4 20 20 Do It Together

Multiply 5 13 = 5 14

13 1 and 2 14 2

Think and Tell What is the product of

1 and 2 = 2

5 and its reciprocal ? 8

Dividing Fractions or Mixed Numbers

1 cup of orange juice equally to 3 diﬀerent 3 glasses? How much juice would go in each cup? What if Lisa adds

Is there a diﬀerent way to do this mathematically? Let us see how. Dividing Keep 1 3

1 1 by 3 can also be written as ÷ 3 3 3 Change �

What if Lisa wants to add

Chapter 10 • Fractions

Flip 1 3

Answer

1 3

Divide

1 into 3

3 equal parts

1 9

1�1 1 = 3�3 9

1 1 of a cup equally to fill of each glass? 3 9

159

1 1 by 3 9

Divide 1 3

1 1 fits in three times 9 3

1 9

1 9 1 9

1 3 Keep 1 3

So;

1 9

Change

Flip 9 1

�

Answer

= 1�9=9=3 3�1

1 1 ÷ = 3 3 9

Remember!

To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.

Example 12

While dividing two fractions always change the divisor into its reciprocal.

Divide. 1

2 10 by 9 9 2 10 2 × 9 18 1 ÷ = = = 9 9 9 × 10 90 5

9 4 by 2 10 5

9 29 4 14 = and 2 = 10 10 5 5

29 14 29 5 29 1 ÷ = � = =1 10 5 10 14 28 28 Do It Together

Divide 2 2

3 15 by 11 11

3 and 11 42

3 15 ÷ = 11 11

�

Remember! Always convert a mixed fraction into an improper fraction when performing multiplication or division.

160

Do It Yourself 10F 1

Multiply the fractions and reduce the fractions if necessary. a

5 2 ÷ 6 6

1 7 � 10 10

1 1 d 1 �2 2 2

1 2 e 5 �4 3 3

10 12 ÷ 15 15

2 7 c 1 ÷2 9 10

1 1 d 1 ÷2 2 2

1 2 e 5 ÷4 3 3

3 2 � 5 9

9 3 � 12 9

1 1 d 1 �2 3 9

1 1 e 3 �4 4 16

2 1 d 1 ÷1 7 14

4 3 e 3 ÷1 5 10

Divide the fractions and reduce the fractions if necessary. a 3

c 2

Multiply the fractions. 3 2 a 2 �1 7 14

3 2 � 15 15

Divide the fractions and reduce the fractions if necessary. a

1 2 � 3 3

9 1 ÷1 10 2

14 10 ÷ 12 22

7 9 ÷ 3 14

3 The perimeter of an equilateral triangle is 2 m. What is the length of each side? 4

Word Problems 1 2 3

9 1 area of a field and Atif ploughed area of the field ploughed by Ananya. Find 12 3 the area ploughed by Atif. 1 Maya is making porridge. She has 75 grams of raisins, but she uses only of it. Find the amount of 5 raisins used by Maya. 2 1 Emma is baking cookies and needs cup of flour for each batch. She has 4 cups of flour in 3 2 total. How many batches of cookies can Emma bake with the flour she has, and how much flour

Ananya ploughed

will be left over?

Points to Remember a b

•

Fractions are numbers of the form , where a and b are whole numbers and b ≠ 0.

•

When two like fractions are compared, then the fraction with the greater numerator will be the greater.

• • • • • •

Improper fractions are always greater than 1, whereas proper fractions are less than 1.

To add or subtract unlike fractions, find the LCM of the denominators, convert them into like fractions and then perform addition or subtraction.

To multiply one fraction with another fraction, multiply the numerators, multiply the denominators and then write the answer in the lowest form. A fraction is said to be in its lowest form if the HCF of the numerator and denominator is 1.

A reciprocal of a fraction can be obtained by interchanging the numerator and denominators.

To divide two or more fractions, multiply the first fraction with the reciprocal of the other fraction.

Chapter 10 • Fractions

161

Math Lab Fraction Bingo cards: Create Bingo cards with diﬀerent fractions (both proper and improper) Fraction chip/markers: A set of fraction chips/markers (small tokens or stickers) to cover the fractions on their Bingo cards

Caller's cards: Prepare a deck of cards with multiplication problems, involving fractions. Instructions:

Distribute Fraction Bingo cards and fraction chips/markers to each student.

The goal is to cover five squares in a row (horizontally, vertically or diagonally) to win.

To cover a square, students must correctly solve the multiplication problem on the caller's card.

Students should call out 'Bingo!' when they have five covered squares in a row.

Start the game by having one student act as the caller.

Continue playing until a student calls out 'Bingo!'

Chapter Checkup 1

Convert the mixed numbers into improper fractions. 1 2

b 5

2 3

c 4

8 23

3 8

6 11

g 5

2 19

a 10 e 4

146 7

87 17

5 7

2 9

11 121

4 5

16 24

24 42

108 132

75 195

32 56

45 125

66 94

62 134

2 10 and 5 25

1 16 and 3 48

Check whether the each pair of fractions is equivalent or not. a

162

43 5

Write the following fractions in their simplest form. a

Find any three equivalent fractions, using division. a

15 4

Find any three equivalent fractions, using multiplication. a

7 8

Convert the improper fractions into mixed numbers. a

d 1

10 40 and 8 32

3 28 and 7 32

Check whether the each pair of fractions is equivalent or not. a 3

4 5

b 5 and

35 4

21 and 8 3

54 1 and 2 9 3

6 12 � 18 15

d 1

Multiply and reduce into their simplest forms. a

1 13 and 2 5

2 3 � 9 4

3 10 � 5 15

1 18 � 9 36

Divide the following fractions and reduce if required. a 2

9 1 ÷1 27 57

50 10 ÷ 33 22

7 4

c 1 ÷2

9 4

d 1

2 1 ÷1 7 14

10 Answer the following questions.

3 1 1 2 and 3 or sum of 2 and 1 . 8 4 4 15 2 4 1 4 b Which is less? The product of 1 and or the sum of and 2 . 3 5 4 5 a Which is more? The diﬀerence of 8

c What is the sum of the product of 3 and d What is

1 of a dozen? What is 10 dozen? 5

4 1 1 and the sum of and . 5 4 3

Word Problems 1 2 3 4 5 6

3 5 of the cookies, whereas Sarah took of the 5 8 cookies. Who took a greater fraction of the cookies. 4 3 Shweta has kg of sugar. She uses kg to make a sweet dish. How much sugar does she 5 4 have left? 2 1 of them are girls. A school has 1000 students and teachers altogether. of them are boys and 5 10 Find the number of teachers. 2 3 Radha purchased 40 m of cloth. She used m cloth for the curtains and 3 m cloth for bed 5 4 sheet. How much cloth does she have left? 4 7 A tank is full. When 65 litres of water are drawn from it, it is full. Find how much water is left 5 12 in the tank. Riya and Siya are sharing cookies. Riya took

A group of students created 2 diﬀerent types of tote bags: X and Y. 1 of the proceeds to the food bank. For every 4 tote bags of type X sold, they would donate 13 2 For every 2 tote bags of type Y sold, they would donate of the proceeds to the food bank. 5 During the event they sold a 24 tote bags of type X for $40 each. b They sold 18 tote bags of type Y for $60 each.

Calculate the total amount donated to the food bank from the proceeds of each type of a tote bag sale.

Chapter 10 • Fractions

163

311

Decimals

Let’s Recall Decimals are numbers that are used in real-life situations where we need to be more exact than whole numbers. For example, when we weigh ourselves, our weight might not be a whole number on the scale, so decimals are used to show the exact weight. Decimals can be used to write money amounts, lengths, temperature etc. A decimal number is a number which consists of a whole and a decimal part. The decimal part is always less than 1. Decimal point

Whole number part

Decimal part

Yashu takes part in a sprint race at his school. The time taken to finish the race is represented in decimal form. Yashu takes 15.53 seconds, Vyom takes 15.6 seconds, Aman takes 17.3 seconds and Sukant takes 15.45 seconds to finish the race.

START

The increasing order of the time is 15.45 seconds < 15.53 seconds < 15.6 seconds < 17.3 seconds. Thus, Sukant is the fastest followed by Yashu, Vyom and then Aman secured the fourth position.

Let's Warm-up Fill in the blanks using the >, < or =. 1 15.61

15.6

2 14.5

14.500

4 17.32

17.3

5 16.5

16.45

3 15.33

15.4

I scored _________ out of 5.

Understanding Decimals Shraddha takes her cat to a vet clinic. The vet prescribes cough syrup for her cat.

Real Life Connect

0.8 ml of cough syrup is to be given to the cat every morning.

Representing Decimals on Number Line Shraddha wonders how she can measure 0.8 mL and give this dosage to her cat. Let us understand how to represent decimal numbers on a number line. 0.8 is more than 0 but less than 1. 0

So, 0.8 lies between 0 and 1.

0.8

Number of tenths between 0 and 1 = 10 tenths Number of tenths in 0.8 = 8 tenths To represent 0.8 on number line, divide the unit length between 0 and 1 into 10 equal parts and represent 0.8 at the 8th division.

Let us now represent 0.54 on the number line. 1 one is divided into 10 tenths and 1 tenth is further divided into 10 hundredths. So, 0.54 lies between 0.5 and 0.6 Example 1

0.5

0.6

0.7

0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.49 0.51 0.53 0.55 0.57 0.59 0.61 0.63 0.65

Represent 1.5 on number line. 1.5 lies between 1 and 2.

1.5

1.5 = 1 + 0.5 = 1 + 5 tenths Divide the unit length into 10 equal divisions and then mark 1.5 five marks after 1. Example 2

Represent 3.61 on a number line. 3.6 3.61

3.61 lies between 3.6 and 3.7 Do It Together

3.7

Mark 12.6 on the number line. 12

Do It Yourself 11A 1

Between which two whole numbers do the decimal numbers lie? a 1.6

Chapter 11 • Decimals

b 6.3

c 4.8

d 5.36

e 12.71

f 65.02

165

Mark the numbers on a number line. a 5.2

b 6.6

c 81.5

d 12.28

e 51.97

f 125.42

Locate the points on a single number line. 0.4, 0.5, 1.3, 1.6, 1.9

Identify the shaded part and represent it on a number line. a

Write the decimal number represented by A, B, C and D. B

–1

Word Problems 1

Read the temperatures of 5 days in a city. Plot these temperatures on a number line. Day

Monday

Tuesday

Wednesday

Thursday

Friday

34°C

33°C

33.2°C

34.5°C

35°C

Temperature

A boy throws a pebble in water. The points at

which the pebble hits the water are shown using a number line. What are the points

at which the pebble hits the water before sinking? (All distances are in metres.)

–1

Place Values in Decimals Shraddha took her cat to Dr. Awasthi after a week. The dosage of cough syrup was increased to 0.925 mL. Let us understand how to represent this number. 0.925

Place value

Thousands

Hundreds

Tens

Ones

Decimal point

Tenths

Hundredths

Thousandths

1000

100

1 10

1 100

1 1000

The place value increases by 10 times while moving from right to left. 166

The place value decreases by 10

times while moving from left to right.

Represent 1532.178 in a place value chart. Thousands Hundreds

1532.178 Place value

1000

100

Value

1 × 1000 = 1000

5 × 100 = 500

Tens

Ones

Decimal Point

Tenths

Hundredths

Thousandths

1 10

1 100

1 1000

3 × 10 = 2 × 1 = 30 2

1×

1 = 0.1 10

7×

1 = 0.07 100

8×

1 = 0.008 1000

Expanded form for 1532.178 = 1000 + 500 + 30 + 2 + 0.1 + 0.07 + 0.008 or 1532.178 = 1000 + 500 + 30 + 2 + One thousand five

1 7 8 + + 10 100 1000

One hundred seventy-

and

hundred thirty-two

Did You Know?

eight thousandths

The world record for men's 100 m race is 9.58

1532.178 One thousand five

Example 3

Place 97.502 in a place value chart and write in expanded form. Tens

Ones

Decimal Point

Tenths

Hundredths

Thousandths

Place value

1 10

1 100

1 1000

Value

9 × 10 = 90

7×1=7

97.502

97.502 = 90 + 7 + 0.5 + 0.002 Example 4

one seven eight

point

hundred thirty-two

seconds by Usain Bolt.

5×

1 = 0.5 10

97.502 = 90 + 7 +

0×

1 =0 100

2×

1 = 0.002 1000

5 2 + 10 1000

Consider the decimal 213.018. Write it in words. 213.018 is written as two hundred thirteen and eighteen thousandths or two hundred thirteen point zero one eight.

Do It Together

Consider the number 256.003. Write the number in expanded form and in words. Expanded form = ______________________________________________________________________________________________ Number name = _______________________________________________________________________________________________

Do It Yourself 11B 1

State true or false. a four hundred thirty and forty thousandths = 430.400 b three thousand twenty-one and five thousandths = 321.005

Chapter 11 • Decimals

167

c three hundred four point three zero two = 304.302 d eleven and twenty-three hundredths = 11.023

Write the given numbers in a place value table. a 1.021

b 54.325

b 12.021

c 81.123

Write the given numbers in decimal form.

e 1702.012

d 180.506

e 812.004

4 6 + 10 100

b 800 + 40 +

8 7 + 100 1000

e 5000 + 500 + 50 + 5 +

a 600 + 6 + d 30 + 9 +

d 71.035

Write the given numbers in expanded form. a 16.45

c 813.003

2 1 + 10 1000

c 1000 + 200 + 40 + 8 +

1 6 + 10 100

5 5 5 + + 10 100 1000

Find the sum of the place values of the digits which the numbers have in common. (For example: Consider 42.345. (Find the sum of place values of digit 4.)) a 15.053

b 164.483

c 143.763

d 817.864

e 3264.215

Word Problems 1

A maths book costs $26.98. Write the number in expanded form.

Frank purchased three lengths of the cloths of 1.56 m, 21.5 m and 14.5 m respectively. Write the numbers in words and also write the expanded form of the numbers.

Conversion Between Fractions and Decimals On the next visit to the vet, Shraddha’s cat had to be tested for FCoV. Dr. Awasthi had to perform a blood test for it. Dr. Awasthi drew 1 mL blood in the syringe. 2 Converting fractions to decimals Step 2: Write a fraction with the denominator as 10, 100 or 1000.

Step 1: Multiply the denominator and the numerator so that we can get 10, 100 or 1000 in the denominator.

168

1 = 1 × 5 = 5 = 0.5 2 2 × 5 10

Step 3: Insert a decimal point before the number of places equal to zeros in the denominator from the rightmost side.

Converting decimals to fractions Step 1: Write the decimal as the denominator of 1.

Step 2: Convert the denominator into 10, 100 or 1000 to eliminate the decimal point.

0.75 =

Example 5

Step 3: Express the fraction in its simplest form.

0.75 0.75 × 100 75 75 ÷ 25 3 = = = = 1 1 × 100 100 100 ÷ 25 4

Convert the fractions to decimals. 1

15 20

15 15 5 15 × 5 = × = 20 20 5 20 × 5 75 = 0.75 100

2102 500

3 4

1 5

2102 2102 2 = × 500 500 2

= 1.05

2102 × 2 4204 = 500 × 2 1000

1 21 21 5 105 = = × = 5 20 20 5 100

= 4.204

Example 6

Convert the decimals into fractions. 1 0.06

2 23.15

0.06 = 0.06 × =

Do It Together

3 50

100 0.06 × 100 6 = = 100 100 100

23.15 = 23.15 × =

100 23.15 × 100 = 100 100

2315 2315 ÷ 5 463 3 = = = 23 100 100 ÷ 5 20 20

Convert each of the decimals to a mixed fraction. 1 3.25 = 3.25 ×

100 = 100

2 4.14 = 4.14 ×

100 = 100

Do It Yourself 11C 1

Identify the number by which you need to multiply the denominator of the fraction to convert it into a decimal number. a

7 4

13 2

78 4

d 5

Convert the fractions into decimals. a

3 8

18 5

97 20

37 25 7 8

12 5

e 23

17 40

89 40

f 235

11 25

Express the decimals as fractions in their simplest form. a 0.24

Chapter 11 • Decimals

b 0.59

c 0.65

d 0.125

e 0.225

f 0.512

169

Write the decimals as mixed fractions. a 3.24

b 5.12

c 2.052

d 4.075

e 52.002

f 25.06

Express the amount using decimals. a In kilograms – 3 kg 68 g, 14 kg 50 g, 5 kg 5 g, 725 g, 45 g b In kilometres – 4 km 15 m, 8 km 40 m, 3 km 4 m, 750 m, 15 m

Word Problems 2 of the students participated in a discussion. What fraction of students 5 did not participate in the discussion? Express your answer as a decimal.

In a class of 60 students,

Mr. Rathore uses 0.125 cup of sugar in his pudding. What fraction of a cup of sugar did he use?

More on Decimals Shraddha went to collect the reports on FCoV of her cat. The cat had to be given 1.2 mL of cough syrup in the morning and 0.95 mL of cough syrup in the evening.

Like and Unlike Decimals Did you notice that the decimals 1.2 and 0.95 look different? Let us figure out how! Decimal numbers that have the same number of digits after the decimal point are called like decimals. For example, 2.56 and 8.14 are like decimals. Decimal numbers that have different numbers of digits after the decimal point are called unlike decimals.

2.56

For example, 1.2 and 0.95 are unlike decimals.

Example 7

8.14

Same number of digits after the decimal point.

1.2

0.95

Different number of digits after the decimal point.

Separate groups of like decimals from the decimal numbers. 4.48, 4.51, 1.1, 87.3, 148.484, 81.654, 781.925, 4.354, 15.68, 48.61, 14.3, 54.975

Do It Together

Group 1 (with 1 decimal place)

Group 2 (with 2 decimal places)

Group 3 (with 3 decimal places)

1.1, 87.3, 14.3

4.48, 4.51, 15.68, 48.61

148.484, 81.654, 781.925, 4.354, 54.975

Check whether 15.45 and 814.012 are unlike decimals or not. Number of digits after decimal point in 15.45 = 2

170

Number of digits after the decimal point in 814.012 = ____________ So, the number of digits after the decimal point ____________ equal. Thus, 15.45 and 814.012 are ____________ decimals.

Equivalent Decimals Equivalent decimals are the decimal numbers which have the same value.

Remember! six-tenths

six-hundredths

0.6

0.60

6 10

Adding 0 at the end of a decimal number does not change the value of the number.

60 = 6 100 10

1.2

These types of decimals are also called equal decimals. For example, 0.6 = 0.60 = 0.600 = 0.6000; 1.54 = 1.540 = 1.5400 The concept of equivalent decimals is used to convert unlike decimals to like decimals.

1.20

0.95

Write any two equivalent fractions for the numbers given. 1 3.6

2 7.81

3.6 = 3.60 = 3.600 Example 9

0.95

Same number of digits after the decimal point.

Let us convert 1.2 and 0.95 to like decimals. Example 8

Add 1 zero

7.81 = 7.810 = 7.8100

Convert each of these to like decimals. 4.5, 6.63, 12.250, 14.02 4.5 = 4.500

6.63 = 6.630

12.250 = 12.250

14.02 = 14.020

Thus, 4.500, 6.630, 12.250, 14.020 are like decimals. Do It Together

Write the numbers as like decimals. 1.546, 53.2, 10.02, 18.3, 17.5, 67 In the numbers, the highest number of decimal places is 3. ___________________________________________ ___________________________________________

Comparing Decimals Compare 4.5 and 4.503. Step 1: Convert the decimal numbers into like decimals.

Chapter 11 • Decimals

4.5

Add 2 zero

4.500

4.503

Same number of digits after the decimal point.

171

Step 2: Start comparing the whole number part. If the whole number part is the same then move to the tenths place. If the digits at the tenths place are the same, then compare the digits at the hundredths place.

Decimal point

4 . 5 0 0

4=4 5=5

Follow the same process until you find unequal terms.

0=0 0<3

Thus, 4.5 < 4.503 Example 10

4 . 5 0 3

Which is greater, 3.64 or 3.46? The numbers are already like decimals so let us put these in a place value chart to compare the digits. O 3 3

3.64 3.46

. . .

t 6 4

H 4

Remember!

Start comparing from the ones place; 3 = 3

We always start comparing numbers starting from the leftmost place.

Move to the tenths place. Compare the tenths place; 6 > 4 Thus, 3.64 > 3.46; Hence, 3.64 is the greater number. Example 11

Arrange 1.23, 1.2, 1.235, 1.3 in ascending order. 1.23 = 1.230

1.2 = 1.200

1.235 = 1.235

1.3 = 1.300

Put the numbers in a place value chart. 1.230 1.200 1.235 1.300

O 1 1 1 1

. . . . .

t 2 2 2 3

h 3 0 3 0

Th 0 0 5 0

On comparing, the ascending order for the numbers is: 1.200 < 1.230 < 1.235 < 1.300 Do It Together

Arrange the numbers in descending order. 48.023, 48, 48.1, 48.02 O

. . . . .

On comparing, the descending order of the numbers is:

Do It Yourself 11D 1

Tick () if the decimals are like and cross () if the decimals are unlike. a 0.154 and 2.315

172

b 31.2 and 153.145

c 1.302 and 8.024

d 1.111 and 2.002

Convert the given decimals into like decimals. a 1.54 and 15.021

d 18.054 and 97

b 178.1, 178.2, 1.78, 178.15

c 12.810, 12.82, 12.815, 12.825

Fill with the correct symbol <, >, or =. a 84.23 ___ 84.32

c 1.1 and 4.65

Identify the greatest decimal number. a 15.4, 15, 15.14, 15.41

b 54 and 48.021

b 15.74 ___ 15.7

c 6.122 ___ 6.120

d 0.97 ___ 0.87

Arrange the given decimal numbers in ascending order. a 7.3, 8.37, 7.23, 8.32

b 58.37, 58.73, 58.45, 58.54

c 97.08, 97.18, 97.2, 97.8

Word Problems 1

The table shows the distance covered by Shantanu on different days. Day

Distance

Monday

15.48 km

Tuesday 12.3 km

Wednesday 19.457 km

Thursday 17.5 km

Convert the decimals into like decimals.

Rajesh buys 5 kg 70 g of sugar and 5.65 kg of rice. What item does he buy in greater quantity?

Reena bought 5 m 15 cm of cloth from the first shop and 5.25 m of cloth from the second shop.

Kunal likes swimming. He swims 1.58 km on day one, 2.53 km on day two, 2.39 km on day three

From which shop did she buy less cloth?

and 3.13 km on day four. Arrange the distances in ascending order and find on which day Kunal swam the most.

Renu has 4 bottles of water. The quantity of water in each bottle is shown. Water bottle Quantity

2.54 L

1.75 L

2.45 L

2.3 L

Arrange the quantities in descending order.

Operations on Decimals Real Life Connect

Xavier had to transport some luggage from one place to another. He had an almirah weighing 49.75 kg and a box to pack it of 3.5 kg.

Addition and Subtraction of Decimals Addition of decimals What is the total weight of the luggage that Xavier has to transport? Chapter 11 • Decimals

173

We find the total weight of luggage by adding 49.75 kg and 3.5 kg. Write the digits in the column method and align the decimal point and the digits.

T O

Remember to convert the decimals to like decimals, if required. +

Perform addition as it is done. Use regrouping, if required. Put the decimal point in the answer at the same place as in the numbers above it.

Thus, Xavier has to transport 53.25 kg of luggage. Subtraction of decimals

Error Alert!

Xavier can carry 65 kg weight with him. If the weight of luggage that he is carrying is 53.25 kg, then how much more weight can he carry?

Always align the decimal point while adding decimal numbers. 15.3 + 2.35 = 38.8

Total weight that can be carried = 65 kg; Weight carried by Xavier = 53.25 kg

15.3 + 2.35 = 17.65

Extra weight that Xaview can carry = 65 kg – 53.25 kg

Write the digits in the column method and align the decimal point and the digits. Remember to convert the decimals to like decimals if required.

T O

– 5

Perform subtraction as it is done. Use regrouping if required. Put the decimal point in the answer at the same place as in the numbers above it.

Thus, Xavier can carry 11.75 kg of weight more in his truck.

Think and Tell

If Xavier carries a parcel of 5.2 kg more, then how much more weight can he carry in his truck?

Example 12

Add 15.984 + 34.457 + 12.96 T O

+ 1

3 6

4 3

. .

4 4

5 0

7 1

Thus, 15.984 + 34.457 + 12.96 = 63.401 Do It Together

Example 13

Subtract: 51.47 – 12.072 T

– 1

5 3

Thus, 51.47 – 12.072 = 39.398

The height of Aman and Rajneesh is 1.54 m and 1.71 m respectively. What is the difference between their heights? Height of Aman = 1.54 m Height of Rajneesh = 1.71 m Difference between the heights = 1.71 m – 1.54 m Thus, the difference between the heights of Aman and Rajneesh is ______________.

174

–

Do It Yourself 11E 1

Add the decimals. a 5.7, 4.6, 8.97 and 5.35

b 14.2, 15.3, 14.51, 18.33

d 81.25, 64.48, 97 and 101.225

e 3, 17.64, 81.989 and 106

c 1.1, 2.22, 3.333 and 4.444

Subtract. a 15.4 – 12.9

b 14.51 – 12.79

c 1.215 – 0.986

d 153.04 – 15.042

Match the following. a 147.6 – 48.8

(i) 63.604

b 51.45 + 46.69

(ii) 98.8

c 817.48 – 753.876

(iii) 98.013

d 453.021 – 355.008

(iv) 98.14

Compare using >, < or =. a 96.321 + 53.412

147.733

b 148 – 20.35

c 851.02 + 25.999

879.019

d 756.3 – 98.608

127.65 657.702

Simplify the decimals. a 53.64 + 18.3 + 51.97

b 17.9 + 97.74 + 56.47

c 814.64 – 15.903 + 12.17

d 71.023 + 41.846 – 12.998

e 51.97 + 87.2 – 97.807

f 35.143 − 23.5 + 64.52

Word Problems 1

Rakhil has 25 m 12 cm of cloth. He uses 12.56 m of the cloth. What length of cloth does he have

A warehouse stores sugar in its rooms. It has 584 kg 80 g, 888.045 kg and 748 kg sugar in its 3

It rained 4 mm on Monday, 3.5 mm on Tuesday, 3.75 mm on Wednesday and 4.2 mm on

The distance between Ruhi’s home and the school is 6.98 km. She moved to a new place so that

left?

rooms. What is the total amount of sugar stored at the warehouse? Thursday. How much did it rain altogether on the 4 days?

her new home is 9 km away from the school. How much extra distance does she have to travel each day?

Pawan goes to a jewellery store. He buys a 20 g 45 milligram gold chain and a ring of 12.5 g in

Rajul had ₹587.2 in his bank account. He deposited ₹300 and ₹950 in his bank account. After a few

weight. What weight of jewellery did he buy? Write your answer in grams.

days, he withdrew ₹450.5 from his account. What is the total amount of money he has left in his bank account?

Monica spent ₹148.6 on stationery, ₹1500.5 on clothes and ₹238.6 on transport. She had a total of ₹2000 with her. How much money does she have left?

Chapter 11 • Decimals

175

Multiplication of Decimals Xavier charges `2 for every kilogram of luggage that he carries in his truck. How much will he charge for carrying 53.75 kg of luggage? Total weight = 53.75 kg Cost for carrying each kg = ₹2; Total cost = ₹2 × 53.75 Step 1: Ignore the decimal point in the decimal number and multiply the whole number.

Step 2: Place the decimal point in the product to obtain as many decimal places as there are in the decimal number.

Thus, Xavier charges ₹107.50 for transporting the luggage. Example 14

Multiply: 36.56 × 9

Example 15

Multiply: 15.2 × 12.24

Add a decimal point after 2 decimal places from the right.

+ 1 5 1 8

Thus, 36.56 × 9 = 329.04

1 3 0 2 6

2 2 0

0 0

4 0 4

0 0 0 8

Add the decimal point after 3 (1 decimal point + 2 decimal points) places from the right. Thus, 15.2 × 12.24 = 186.048

Do It Together

Manuel purchases 18 books of ₹156.56 each. How much money does he have to pay in total? Cost of each book = ₹156.56

Total number of books = 18

Total cost of 18 books = _______________________________

Do It Yourself 11F 1

Multiply the decimal number with the whole number. a 51.2 × 4

b 2.6 × 23.5

d 123.5 × 14

e 348.9 × 21

c 12.3 × 51.31

d 2.36 × 1.24

Fill in the blanks. a 31.24 × 6 = _________

176

c 81.7 × 11

Multiply the decimal number with the decimal number. a 1.2 × 5.3

b 12.6 × 9

b 32.6 × 1.2 = _________

c 97.87 × 1.3 = _________

d 157.41 × 8

Compare using >, < or =. a 51.3 × 12 ___ 618.6

b 32.65 × 1.3 ___ 42.145

c 103.4 × 14___ 1446.6

Answer the questions. a What is the product of 15.3 and 81.36?

b Find the product of 87.64 and 23.6.

Word Problems 1

The length of a rectangular vegetable garden is 5.8 feet and the width is 3.6 feet. What is the area

Anuj purchases 3.6 gallons of milk. How many quarts of milk does he purchase? (1 gallon =

The cost of a computer is $190.64. What will be the cost of 6 such computers?

The length of the side of a square-shaped field is 50.23 m. What is the area of the field?

A drawing is 24.75 inches wide. It has to be scaled down to

of the vegetable garden?

4 quarts)

of the painting after re-sizing?

1 of the original size. What is the width 5

Division of Decimals If 5 people equally distributed ₹107.50 and gave it to Xavier, how much money did each of them give to Xavier? Total money given = ₹107.50 = ₹107.5 Number of people = 5 Share by each person = ₹107.5 ÷ 5

21.5 Step 2: Divide the whole number part.

Step 1: Place the decimal point directly above the decimal point in the dividend.

5 107.5 – 10 07 –5

Step 3: Divide the tenths.

25 –25 0

As 107.5 ÷ 5 = 21.5, each person contributed ₹21.5. Chapter 11 • Decimals

177

Dividing a decimal number by a decimal number Divide 48.15 by 1.5. Step 2: Divide using long division.

Step 1: Move the divisor towards the right until the divisor becomes a whole number and write the dividend and the divisor. 1.5

1 decimal point

Move 1 decimal place

32.1

48.15

15 481.5 – 45 31 –30

481.5

15 481.5

–15 0

Thus, 48.15 ÷ 1.5 = 32.1

Example 16

Divide 577.06 by 11.

Example 17

Divide 138.72 by 1.7. 138.72 ÷ 1.7 = 1387.2 ÷ 17

52.46 11 577.06

81.6

– 55

17 1387.2

– 136

–22

–17

–44

102

–102

–66

577.06 ÷ 11 = 52.46 Do It Together

138.72 ÷ 1.7 = 81.6

Rahul purchases 15 packets of paint for ₹3547.5. What is the cost of each packet of paint? Cost of 15 packets of paint = ₹3547.5

Total number of packets = 15

Cost of each packet of paint = ₹3547.5 ÷ 15 = __________________

Do It Yourself 11G 1

Divide the decimal number with the whole number. a 106.4 ÷ 7

c 495.6 ÷ 12

d 247.5 ÷ 15

e 1635 ÷ 25

d 5.52 ÷ 1.5

e 162.84 ÷ 6.9

Divide the decimal number with the decimal number. a 2.76 ÷ 1.2

178

b 191.2 ÷ 8

b 14.24 ÷ 1.6

c 31.5 ÷ 2.5

Fill in the blanks. a 338.88 ÷ 8 = ______

c 890.4 ÷ 15 = ______

b 189.52 ÷ 8 ___ 23.59

c 338.52 ÷ 7 ___ 12.4 × 3.9

Compare using >, < or =. a 1030.4 ÷ 14 ___ 73.6

b 1110.45 ÷ 15 = ______

Answer the questions. a Find the quotient when 111.24 is divided by 9. b Find the quotient when 466.44 is divided by 5.2.

Word Problems 1

Bijoy distributes ₹1622.4 equally among three friends. How much money does each friend get?

A chef has a recipe for 10 servings that calls for 0.75 cups of a special sauce. If she wants to make 35 servings, how many cups of the sauce does she need?

Application of Decimals Naresh starts jogging every morning. He wears his new smart watch and aims to complete 2.5 km on day 1.

Real Life Connect

After running for 1.275 km, he takes a break. What distance more does he have to run to achieve his aim? Total distance to be covered = 2.5 km

O .

Distance covered before the break = 1.275 km

Distance to be covered after the break = 2.5 km – 1.275 km

– 1

Thus, Naresh has to run 1.225 km more to achieve his aim. Example 18

0 5

Vedica walks for 1.65 km. She takes an auto and travels 3.4 km. If she has to travel a total distance of 11.6 km, how much more distance does she have to cover? Distance walked = 1.65 km Distance travelled by auto = 3.4 km = 3.40 km Total distance covered on foot and by auto = 1.65 km + 3.40 km = 5.05 km

Chapter 11 • Decimals

O .

+ 3

179

Total distance to be covered = 11.6 km = 11.60 km

T O

Distance yet to be covered = 11.60 km – 5.05 km

Thus, Vedica has to travel 6.55 km more.

Example 19

–

Virat earns ₹250.5 for each hour he works. If he works for 6.5 hours in a day, how much money does he earn? Money earned each hour = ₹250.5 Total hours worked = 6.5 hours Total money earned = ₹250.5 × 6.5 = ₹1628.25

Do It Together

2 1 2

1 5 0 1 6 2

3 8

6 5 0

A cyclist climbs 45.14 m from the camp which is already at 3214.3 m. He climbs another 97.89 m after some time. What height has he reached? Height reached after first stage = 3214.3 m + 45.14 m = 3214.30 m + 45.14 m Total height reached = _____________ Thus, the cyclist reaches a height of = _____________

Do It Yourself 11H

180

Sophie travelled 15 km 3 m on day 1 and 26 km 408 m on day 2. What distance did she travel in total?

Rose has two cats named Mary and Poppy. Mary’s weight is 1.78 kg and Poppy’s weight is 1.59 kg. What is the

Vikas buys a pencil for ₹2.5, an eraser for ₹5.6 and a notebook for ₹34.55. How much does he pay the

Vrisha bought 15 m of ribbon. She used 1.6 m to decorate her notebook and she used 8.75 m more in her project.

Mohanlal purchased 15 kg apples at the rate of ₹96.65 per kg. What is the total cost of the apples?

A person covers 200.8 km in 4 hours. What distance does he cover each hour?

The length of a rectangle is 11.56 m and its breadth is half of the length. What is the perimeter of the rectangle?

Mr. Verma uses 0.25 kg of flour, 0.1 kg of sugar and 0.275 kg of cocoa powder and some milk to bake a 1 kg cake.

total weight of the two cats?

shopkeeper?

What length of ribbon does she have left?

Also, calculate the area of the rectangle.

How much milk does he use to bake a 5 kg cake?

Points to Remember •

Numbers written in decimal form are called decimal numbers.

•

The decimal numbers which have the same number of decimal places are called like decimals, and the decimal numbers with different numbers of decimal places they are called unlike decimals.

•

Unlike decimals can be converted to like decimals by adding 0’s as placeholders at the extreme right side of the decimal number.

•

To add and subtract the decimal numbers, convert the decimal numbers into like decimals.

•

To multiply the decimal numbers, remove the decimal points from the numbers to be multiplied.

•

To divide the decimal numbers, place the decimal point directly above the decimal point in the dividend.

Math Lab Game of Equivalent Fractions Aim: To compare fractions and decimals Materials Required: Pack of cards (20 cards) and an empty box Setting: In groups of 2 Process:

Write the decimal numbers and their equivalent fractions on the cards.

Mix all the cards and give 5 cards to each student. Keep the rest of the cards in the empty box.

The first student asks the other student if he/she has a particular card.

Each student has to match one decimal number with its equivalent fraction.

If the second student has that card, then he/she gives that card to the first student.

If the second student does not have that card, then the student will take a card from the box.

Now, it’s the turn of the second student to ask for the card.

The game continues until all the cards have been used and the student who has the greatest number of matches wins.

Chapter 11 • Decimals

181

Chapter Checkup 1

Mark the decimal numbers on a number line. a 2.6

b 14.9

c 95.3

d 11.64

e 87.59

Identify the shaded part and represent it on a number line. a

d 10 + 3 +

182

100

c 600 + 40 + 9 +

100

1000

b 81.9

c 145.84

d 5.23

e 12.056

Convert the fractions into decimals. 1 2

512 25

145 8

e 21

11 40

507.633

c 12 f

3 5

134

41 50

Express the decimals as fractions in their simplest form. a 5.6

b 81.4

c 1.56

d 51.26

e 6.814

12.508

Arrange the decimal numbers in descending order. a 15.1, 15, 15.05, 14.95

b 900 + 50 +

a 5.3

10 5

Write the expanded form of the decimal numbers.

a 45

Write the numbers in decimal form. a 40 + 3 +

11.05

b 1.23, 1.2, 1.331, 1.303

c 153.32, 153.23, 152.1, 153.233

a 15 × 1.6

b 418.4 ÷ 8

c 1.2 × 6.7

d 5.9 × 47.6

e 492.26 ÷ 5

Solve. 14.65 × 165.7

Word Problems 1

In a long-jump competition, Rachna jumped 3.6 m, Nina jumped 3.65 m and Rohini

Adhvica went to a restaurant. The restaurant bill came to $38.75, and she left a $10.50 tip.

A cyclist covered 15.486 km in the morning and 8.75 km in the afternoon. What was the

Bhoomi wants to buy a laptop. The laptop costs $899.75, and she has $550.50 saved. How

Himani buys a shirt for $24.99 and a pair of shoes for $58.50. She pays with a $100 bill.

A store has 65.75 kg of apples. After selling 42.5 kg, they restock with 28.325 kg of apples.

Sarah earns $10.50 per hour and works for 15.75 hours in a week. How much does she

The length of a rectangular garden is 45.6 m and its width is

A smartphone plan costs ₹45 for the first 100 text messages in a month plus an additional

jumped 3.55 m. Who jumped the farthest? How much did Adhvica spend? total distance covered?

much more money does she need to buy the laptop? How much change should she receive?

What is the total weight of apples now? earn in a week?

perimeter and the area of the garden.

1 of its length. Find the 3

₹0.10 per text message. If you send 275 text messages in a month, what is the total cost?

10 A toy store sells two types of toys: Type A for ₹129.90 and Type B for ₹97.50. If the store sold a total of 60 toys and the total revenue was ₹7567.50, how many of each type of toy were sold?

Chapter 11 • Decimals

183

312

Data Handling

Let's Recall Shreya surveyed her class to know their favourite breakfast food. She listed them in her notebook. juice, toast, fruit, juice, milk, poha, toast, toast, toast, juice, milk, poha, poha, toast, fruit, milk, poha, toast, juice, juice, milk, juice, fruit, juice, milk, juice, milk, poha, juice, juice, milk Let us draw a table to show the data. Favourite Breakfast Breakfast

Number of Students

Toast

Fruit

Poha

Juice

Milk

Shreya concluded that: Juice is liked by the greatest number of students which is 10. Toast is liked by 6, fruit by 3, poha by 5 and milk by 7 students.

Let's Warm-up

Fill in the blanks based on the table above. 1 ____________ students like to have milk for breakfast. 2 Fruit is liked the ____________ (most/least). 3 Juice is liked the ____________ (most/least). 4 The total number of students who took the survey is ____________. 5 ____________ more students like to have poha for breakfast than fruit.

I scored _________ out of 5.

Organising and Representing Data Real Life Connect

Rajveer is fond of collecting data on various things. We know that data is the collection of information and numerical facts. He collects raw data about various countries such as their population, their areas and the number of states in each of them. When the data is collected in its original form, it is called raw data; and each item in the raw data is called an observation. So, Rajveer conducted a survey and listed 4 countries. He wrote the number of states in each country. Austria has 9 states, Malaysia has 13 states, New Zealand has 2 states and South Sudan has 10 states. He represented the data in a tabular form. Name of Country

Number of States

Austria

Malaysia

New Zealand

South Sudan

Organising Data as Tally Marks Rajveer decides to represent the data using tally marks as shown. Name of Country

Tally Marks

Austria Malaysia New Zealand South Sudan

In tally marks, we use ‘ ’ for one item. So, for one state, we will draw one tally mark ‘ ’. For 4 states, we will draw ‘ ’. For 5 states, we will draw and not . Note: For 5 items, we always use . To show 12 items, we will draw tally marks as .

Now, let us look at another example of data and understand how to represent it. The following data gives the number of marks obtained by 30 students. 5, 7, 6, 7, 8, 9, 5, 6, 7, 6, 6, 8, 8, 9, 5, 6, 7, 8, 8, 7, 6, 5, 7, 5, 7, 9, 9, 9, 6, 9 To ﬁnd out how many times a given observation appears in this data, we will use a frequency distribution table. Frequency is the number of times an observation occurs. Frequency distribution is the tabular arrangement of numerical data which shows the frequency of each observation. Let us learn how to draw it. Drawing frequency distribution table First arrange the raw data in ascending order. 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9 When the raw data is arranged in ascending order or descending order, then it is called an array. Chapter 12 • Data Handling

185

Now, draw the frequency distribution table, as shown. Marks Obtained

Example 1

Tally Marks

Number of Students (Frequency)

Total

The table shows the number of children in various families in a colony. Answer the questions. Number of Children

Tally Marks

0 1 2 3 4

1 How many families have 2 children? 7 2 For how many families is the data collected? 6 + 8 + 7 + 4 + 3 = 28 3 How many families have more than 1 child? 7 + 4 + 3 = 14 4 How many families have fewer than 4 children? 28 – 3 = 25 Do It Together

The height of 30 students in a class is given. Make a frequency distribution table for the data. (All heights are in cm.) 135, 134, 136, 135, 132, 130, 136, 136, 133, 135, 134, 132, 135, 133, 133, 136, 134, 134, 135, 132, 132, 133, 136, 132, 134, 136, 135, 136, 132, 135 Height of Students 132 cm 133 cm 134 cm 135 cm 136 cm

186

Tally Marks

Number of Students (Frequency) 6

Do It Yourself 12A 1

State true or false. a Raw data is the organised form of data. b Arranging the numerical numbers in ascending or descending order is called an array. c Five as tally marks is represented as

d Frequency is always the same for every observation in data.

Fill in the blanks. a The collection of numerical facts is called ______________. b In data, the number of times a particular observation is repeated is called its ______________. c The data collected in its original form is called ________________.

Draw tally marks for the given numbers. a 20

b 31

c 42

d 27

Prepare a tally mark table for the favourite fruit of some children. Orange, Apple, Banana, Guava, Litchi, Litchi, Orange, Apple, Banana, Banana, Apple, Guava, Guava, Apple, Litchi, Guava, Banana, Apple, Banana, Apple, Litchi, Orange, Apple, Orange, Litchi, Banana, Banana, Guava, Apple, Litchi, Banana, Orange, Banana, Apple, Litchi, Guava, Orange, Banana, Litchi, Apple, Apple, Apple, Orange, Banana, Litchi, Apple, Guava, Banana

Prepare a frequency distribution table for the favourite sports of the students in a class and answer the given questions. Cricket, Football, Basketball, Tennis, Volleyball, Badminton, Football, Cricket, Tennis, Tennis, Volleyball, Cricket, Badminton, Basketball, Basketball, Volleyball, Basketball, Cricket, Cricket, Football, Badminton, Tennis, Tennis, Football, Cricket, Tennis, Basketball, Volleyball, Basketball, Cricket, Football, Tennis, Cricket, Football, Badminton, Badminton, Basketball, Volleyball, Volleyball, Cricket, Football, Football, Badminton, Tennis, Badminton, Football, Cricket, Football, Tennis, Football, Cricket, Volleyball, Volleyball, Basketball, Cricket, Tennis a How many students like cricket? b How many students like basketball? c What is the diﬀerence between the number of students who like cricket and those who like football? d What is the total number of of students who like both basketball and volleyball? e For how many students is the data collected?

Chapter 12 • Data Handling

187

Word Problems 1

Rashmi was playing ludo with her friend. She noted down the digits that appeared on her dice, and the outcomes were as follows:

5, 6, 2, 3, 4, 1, 5, 6, 4, 6, 3, 6, 6, 4, 5, 2, 1, 1, 3, 4, 2, 1, 3, 6, 6, 4, 5, 2, 3, 4, 2, 4, 3, 2, 2, 1, 5, 6, 6, 4, 2, 3, 4, 2, 1, 3, 4, 5, 5, 6, 6, 2, 6, 2, 1, 1, 1, 4, 3, 6, 4, 2

Prepare a frequency distribution table for the observations and answer the questions. a Which digit appeared the most?

b Which digit appeared the least?

c How many chances were played by Rashmi altogether?

d Did any two digits appear the same number of times?

The ages of 30 students of a school are shown below.

10, 11, 14, 12, 13, 11, 12, 13, 14, 10, 10, 12, 13, 14, 13, 12, 10, 11, 13, 14, 10, 12, 13, 14, 12, 10, 12, 13, 14, 10

Make a frequency distribution table for the data and answer the questions. a How many students are 12 years old?

b How many students are 14 years old?

c How many students are less than 13 years old?

d How many students are more than 11 years but less than 14 years old?

Pictographs Remember Rajveer was collecting data for diﬀerent countries. He drew a pictograph for the population of diﬀerent countries. Country

Population

Austria Malaysia New Zealand South Sudan

Key:

Remember! A pictograph is the representation of data or information with the help of pictures or symbols.

= 2 million and = 1 million

Error Alert! The size of all pictures in a pictograph should always be the same.

Did You Know? Pictographs are used as road signs so that people of different languages can understand them better. indicates pedestrian crossing.

188

Example 2

The data shows the number of houses in 5 villages. Draw a pictograph to represent the data. Let us take

= 200 houses.

Village A

1000

Village A

Village B

1200

Village B

Village C

1400

Village C

Village D

1200

Village D

Village E

1000

Village E

Key: Example 3

= 200 houses

The pictograph shows the number of various types of vehicles by a traﬃc survey. Answer the questions. Car Truck Rickshaw Bike/ Scooter Bicycle

Key:

= 600 vehicles

1 How many cars are there? 2400 2 How many more rickshaws are there than bicycles? 2100 – 1800 = 300 3 What is the total number of trucks and bikes/scooters? Trucks and Bikes/Scooters = 2700 + 3300 = 6000. 4 How many vehicles are there in total? 2400 + 2700 + 2100 + 3300 + 1800 = 12,300 Do It Together

The number of bats sold by a shop in a week is shown using a pictograph. Answer the questions based on the pictograph. Monday Tuesday Wednesday Thursday Friday Saturday

Key:

= 80 bats

1 How many bats were sold on Thursday? ____________________ 2 On which day were the most bats sold and how many? ____________________ 3 On which day were the least bats sold and how many? ____________________ 4 How many bats were sold in the entire week? ____________________ Chapter 12 • Data Handling

189

Do It Yourself 12B 1

The data represents the mode of transport opted for by 34,000 children in a city to go to school. Make a pictograph for the data. Vehicle

Car

Bus

Bicycle

Bike

Auto

Walking

Number of Children

5000

8000

4000

3000

5000

9000

The data represents the number of people who travelled from Delhi to Bangalore on diﬀerent days of a month. Represent the data using a pictograph. Day

Monday

Tuesday

Wednesday

Thursday

Friday

Number of People

20,000

35,000

25,000

52,500

50,000

The data shows diﬀerent people’s favourite brands of soaps. Represent the data using a pictograph. Brand

Brand A

Brand B

Brand C

Brand D

Brand E

1500

2250

2000

2500

2750

Number of People

The data shows the number of children with their favourite sports. Make a pictograph based on the given data. (Round oﬀ the number of children to the nearest 100)

Favourite Sport

Cricket

Football

Basketball

Volleyball

Tennis

Number of Children

3500

3000

2510

1460

1000

The pictograph shows the sale of electric bulbs by a shop on diﬀerent days of a week. Day

Monday

Tuesday

Wednesday

Number of Bulbs Sold

Key:

= 300 bulbs

a How many bulbs were sold on Tuesday? b How many more bulbs were sold on Wednesday than on Thursday? c How many bulbs were sold on Monday and Tuesday together? d On which day were the least bulbs sold and how many? e How many bulbs were sold in total in the week?

190

Thursday

Friday

Consider the pictograph shown below. It shows the number of fans manufactured by a company in the ﬁrst 6 months. Answer the given questions.

Month

January

February

March

April

May

June

Number of Fans

Key:

= 1000 fans

= 500 fans

a How many fans were manufactured in May? b How many fans were manufactured in January? c In which month were the most fans manufactured and how many? d How many more fans were manufactured in April than in February?

The pictograph shows the number of baskets of fruit sold by 5 diﬀerent merchants in a village. Study the pictograph and answer the questions.

Rakesh

Mohan

Surender

Raghu

Rajul

Key:

= 100 baskets

a Which merchant sold the most baskets and how many? b Which merchant sold the least baskets and how many? c How many baskets were sold by Mohan? d What is the total number of baskets sold?

Chapter 12 • Data Handling

191

Word Problem 1

Some people decided to distribute blankets at an orphanage. The pictograph shows the

number of blankets given by diﬀerent people in an orphanage. Study the pictograph and answer the questions. Benu

Mallika Cherry Ena Garima Pari Key:

= 120 blankets

a Who gave the most blankets and how many? b How many blankets were given by Garima? c How many more blankets were given by Mallika than Cherry? d How many blankets were distributed in total?

Bar Graphs Remember Rajveer collected data for various countries. He collected data for the number of states and the population of countries. Now, he collected the data for the total area covered by the countries. Look at the data. Country

Austria

Malaysia

New Zealand

South Sudan

Total Area (in sq. km)

80,000

3,30,000

2,60,000

6,40,000

(The data is estimated and is in accordance with the year 2021).

192

Drawing Bar Graphs Rajveer wanted to represent the data using a bar graph. Let us learn to present data using bar graphs.

Step 2: Write the title as Total Area of Countries.

Step 3: Label both the axes as Total Area and Countries.

Total Area (in sq. km)

Step 1: Draw the x-axis and y-axis.

Total Area of Countries 1 division = 40,000 sq. km

y 680 000 640 000 600 000 560 000 520 000 480 000 440 000 400 000 360 000 320 000 280 000 240 000 200 000 160 000 120 000 80 000 40 000 0

Step 4: Write the scale. Scale is the number that shows the units used.

Step 5: Draw the rectangular bars. The height of the bar graph shows the area of each country.

Austria

Malaysia New Zealand South Sudan Countries

Think and Tell Which is an easier way to represent the data here?

The data shows the number of area of land (in acres) used by a farmer for planting various types of vegetables. Draw a bar graph for the data. Vegetable

Spinach

Potato

Brinjal

Tomato

Carrot

Area of Land (in acres)

Area of Land used by Farmers

30 Number of Acres

Example 4

1 division = 5 acres

25 20 15 10 5 0

Chapter 12 • Data Handling

Spinach

Potato Brinjal Tomato Type of Vegetables

Carrot

193

The data shows the number of electronic appliances sold by various electronic companies in a month. Draw a bar graph to represent the data. Number of Electrical Appliances Sold

Electronic Companies

Number of Electronic Appliances

550

500

450

325

450

100

575

1 division = 100 appliances

600 Number of Appliances

Do It Together

500 400 300 200

Electronic Companies

Interpreting Bar Graphs The bar graph shows the marks obtained by Rajveer in various subjects. What can be understood from the bar graph? Let us see! Marks Obtained by Rajveer

1 division = 10 marks

100 90 80

Marks

70 60 50 40 30 20 10 0

Maths

Science

English

Hindi Social Science

Subjects

Rajveer scored the highest marks in Maths and Social Science. He scored 90 marks in both the subjects. He scored the lowest marks in Hindi. He scored 78 marks in Hindi. He scored 80 marks in Science and 84 marks in English.

194

Example 5

The bar graph shows the number of children in families. Study the bar graph and answer the questions. Number of Children in Families

1 division = 5 families

50 45 Number of Families

40 35 30 25 20 15 10 5 0

Number of Children

1 How many families have only 2 children? 45 2 How many families have more than 1 child? 45 + 25 + 10 = 80 3 How many families have 3 children? 25 Do It Together

The bar graph shows the snowfall at a hill station in various months of a year. Study the bar graph and answer the questions. y

Snowfall at a Hill Station

1 division = 10 cm

150 140 130

Snowfall (in cm)

120 110 100

90 80 70 60 50 40 30 20 10 0

Nov

Dec

Jan

Months

Feb

March

1 In which month did the least amount of snow fall and how much? __________ 2 How much snowfall occurred in the month of January? __________ 3 How much more snowfall was there in December than in February? __________

Chapter 12 • Data Handling

195

Do It Yourself 12C 1

The data shows the number of people who went to various countries from a city. Draw a bar graph to represent the data. Scale: 1 division = 1000 people Country

USA

Japan

Germany

France

Number of People

6000

4500

3000

5500

3250

The data shows the number of family members in various families of a colony. Draw a bar graph to represent the data. Scale: 1 division = 10 families Number of Members

Number of Families

The bar graph shows the expenditure of a family on various items in a month. Study the bar graph and answer the questions.

y 10 000

Family Expenditure in a Month

1 division = `1000

9000

Expenditure (in `)

8000 7000 6000 5000 4000 3000 2000 1000 0

Food

Clothes

Education Items

a What is the expenditure on food? b On which item did the family spend the least and how much? c On which item did the family spend the most and how much? d What was the total expenditure of the family?

196

Rent

Misc.

The bar graph shows the number of saplings planted by a farmer in diﬀerent blocks of a farm. Study the bar graph and answer the questions.

Saplings Planted by a Farmer 1 division = 5 saplings

50 45

Number of Saplings

40 35 30 25 20 15 10 5 0

Blocks

a How many saplings were planted in block C? b How many saplings were planted in block D? c In which block were the most saplings planted? d How many saplings in total were planted by the farmer?

The bar graph shows the population of three nearby towns in the year 1975 and the year 2015. Answer the questions based on the bar graphs.

Population (1975)

1 division = 1000 People

6000 5000 4000 3000 2000 1000 0

Cottage

Arsenal Cities

Britannia

Population (2015)

1 division = 5000 People

30000 Number of People

Number of People

25000 20000 15000 10000 5000 0

Cottage

Arsenal Cities

Britannia

a What was the population of Cottage in 1975? b What was the population of Arsenal in 2015? c What is the increase in population of Britannia? d Which town had the greatest increase in population over the years and by how much?

Chapter 12 • Data Handling

197

In a city, the average power supply from generators, instead of being constant, was ﬂuctuating.

Power supply (in KW)

Average Power Supply (in KW)

12000

1 division = 2000 KW

10000 8000 6000 4000 2000 0

06:00-07:00 07:00-08:00 08:00-09:00 09:00-10:00 10:00-11:00 Time

a The power supply from the generators was at its highest during the time _______________. b The power supply from the generators was at its lowest during the time _______________.

Word Problems 1

A sports event was organised at a school. The data shows the number of students

participating in various sports. Draw a bar graph to represent given data. Take scale: 1 division = 10 participants.

Sport

Yoga

Basketball

Football

Dancing

Taekwondo

Number of Participants

The bar graph shows the population of a city in various years. Study the bar graph and answer the questions given below.

Population of a City

1 division = 20 000 people

160 000

Population

140 000 120 000 100 000 80 000 60 000 40 000 20 000 0

2011

2012

Year

2013

2014

a What was the increase in the population from year 2013 to year 2014? b Which year had the lowest population and how much? c Which year had the highest population and how much? d In which year did the highest increase in population from the previous year take place?

198

Points to Remember •

Numerical ﬁgures collected to convey information are called data.

•

Representation of organised data in the form of pictures or parts thereof is called a pictograph.

•

• •

To get particular information from the collected data, it can be arranged in a tabular form, using tally marks. Such a table is called a frequency distribution table.

A bar graph is a representation of data by a number of rectangular bars of uniform width erected vertically or horizontally and there are equal spaces between two consecutive bars. The scale for making pictographs and bar graphs is chosen according to the need.

Math Lab Favourite Ice-cream Flavours Aim: To collect, organise, and analyse data about the favourite ice-cream ﬂavours of the class Settings: In groups of 5 Materials Needed: Chart paper, markers, a hat or box to draw from, ice-cream scoops and sticky notes or index cards Process:

Distribute sticky notes or index cards to each student.

Ask them to write down their favourite ice-cream ﬂavour on a sticky note or index card.

Collect the cards in a hat or box.

One by one, draw a card from the hat and read out the ﬂavour.

Record each ﬂavour on the chart paper or whiteboard.

Create a frequency table on the chart paper or whiteboard, listing the ice-cream ﬂavours on the left-hand side and the number of students who chose each ﬂavour on the righthand side.

Using the data from the frequency table, create a bar graph on the chart paper or whiteboard.

Discuss the graph with the students.

Label the graph and give the appropriate title.

a Which ﬂavour is the most popular? b How many students chose chocolate as their favourite?

Chapter 12 • Data Handling

199

Chapter Checkup 1

The data shows how people go to their oﬃce. Make a data frequency table for the data. walk, bus, bike, walk, bike, bus, walk, car, walk, bike, bike, bus, walk, walk, walk, car, bus, walk, bus, bus, walk, car, car, walk, walk, train, bike, bus, walk, walk

Draw a tally-marks table for the following shapes.

The frequency data table shows the favourite subject of students. Read the table and answer the questions.

Subject

a Which subject is the most favourite?

Science

b How many students like English?

Maths Technology

c Which is the least favourite subject?

English

d What fraction of the total students like Maths?

The data shows the number of students in various schools. Draw a pictograph to represent the data. School

Number of Students

4500

5000

6500

7500

7000

5500

The data shows the number of mangoes sold by various shopkeepers. Draw a pictograph to represent the data. Shopkeeper

Sohail

Ashraf

Mahmood

Raju

Vivek

Number of Mangoes

9000

4500

6000

7500

10500

The pictograph shows the number of ice creams sold by an ice-cream vendor in diﬀerent months of an year. Read the pictograph and answer the questions. January February March April May June Key:

200

Tally Marks

= 150 ice creams

a How many ice creams were sold in the month of March? b In which month were the most ice creams sold and how many? c What is the diﬀerence between the number of ice creams sold in January and May? d What number of ice creams were sold altogether?

The number of people of diﬀerent age groups in a town are shown using a table. Draw a bar graph to represent the data.

Age Group Number of People

1 – 14

15 – 29

30 – 44

45 – 59

60 or above

1 lakh 50 thousand

2 lakh

1 lakh 40 thousand

1 lakh 30 thousand

1 lakh 10 thousand

The data shows the number of bedsheets manufactured by a power loom factory in diﬀerent weeks of a month. Draw a bar graph to represent the data. Week

First

Second

Third

Fourth

Number of Bedsheets

500

600

750

625

The bar graph shows the favourite colour of students in a school. Read the bar graph and answer the given questions.

Favourite Colour of Students 1 division = 20 students

200

Number of Students

180 160 140 120 100 80 60 40 20 0

Yellow

Green

Red

Blue

Favourite Colour

Black

a Which is the favourite colour among students and how many students like it? b How many students like the least favourite colour? c What is the diﬀerence between the number of students who like black and who like red? d How many students are there in total?

Chapter 12 • Data Handling

201

10 The bar graph shows the number of books sold by a bookstore during 5 consecutive years. Read the bar graph and answer the questions.

Books Sold in Diﬀerent Years

1 division = 5000 books

50 000 45 000

Number of Books

40 000 35 000 30 000 25 000 20 000 15 000 10 000 5 000 0

2017

2018

a How many books were sold in the year 2019?

2019

Year

2020

2021

b How many books were sold in the year 2018? c What fraction of the total books were sold in the year 2020? d What is the ratio of the number of books sold in years 2017 and 2021?

Word Problem 1

The bar graph shows the average temperature of a city in °C for diﬀerent months of a year. Read the bar graph and answer the questions. y

Average Temperature of a City for Diﬀerent Months 1 division = 5°C

50 45

Temperature (°C)

40 35 30 25 20 15 10 5 0

Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec

Months

a Which month has the highest temperature and how much? b Do any two months have the same temperature? If yes, which months and how much? c How many months are warmer than September? d Which month is colder than July but warmer than May?

202

Mensuration

Let’s Recall The perimeter of a shape is defined as the total length of its boundary. We can find the perimeter of a shape using a grid. A grid is made up of unit squares. The perimeter on a grid can be found by counting the number of unit lengths along the boundary of the figure. For example, the perimeter of the shape below can be given as: 9 units

8 units

3 units

1 2 3 4 5 6 7 8 9

5 units

3 units

6 units

Perimeter = 9 + 3 + 3 + 5 + 6 + 8 = 34 units

Let's Warm-up Match the figures with their perimeters. 1

14 units

26 units

16 units

28 units

18 units

I scored _________ out of 5.

Understanding Perimeter Real Life Connect

Raman, a farmer, wants to fence his field to protect the crops from animals. He goes to buy fencing for his field. Raman: I want to fence my field. Shopkeeper: What length of fence do you need? Raman shows the picture of his field with its dimensions.

Perimeter of Plane Figures To find the length of the fencing, we need to find the total length of the sides of the field. Perimeter of the field = Total length of the sides of the field

Perimeter of the field = 5 m + 7 m + 3 m + 10 m = 25 m

Hence, Raman buys 25 m of fencing. Example 1

10 m

Which of the figures has a perimeter of 42 cm? 1

12 cm

5 cm

7 cm

9 cm

6 cm 11 cm

4 cm

7 cm

6 cm

2 cm 13 cm

8 cm

Perimeter of figure 1 = 12 cm + 7 cm + 4 cm + 2 cm + 8 cm + 9 cm = 42 cm Perimeter of figure 2 = 6 cm + 5 cm + 7 cm + 6 cm + 13 cm + 11 cm = 48 cm Hence, figure ‘1’ has a perimeter of 42 cm. Do It Together

Find the perimeter of the figures. 1

7 cm

2 cm 5 cm 6 cm

3m 2m 6m

9 cm

5 cm 6 cm

5 cm 11 cm

Perimeter = 7 cm + 5 cm + 6 cm

Perimeter = ___________________

+ 2 cm = _______________________

________________________________

204

Perimeter of Regular Shapes We can find the perimeter of diﬀerent shapes by using some formulas. Shape

Perimeter

Equilateral Triangle

Perimeter Formula

Perimeter = 7 cm + 7 cm + 7 cm = 21 cm

7 cm

Perimeter = 3 × side Perimeter = 3 × 7 = 21 cm

7 cm Pentagon 5 cm

Length of all 3 sides of an equilateral triangle are equal,

5 cm

Perimeter = 5 cm + 5 cm + 5 cm + 5 cm + 5 cm = 25 cm

Length of all 5 sides of a regular pentagon are equal, Perimeter = 5 × side

5 cm

Perimeter = 5 × 5 = 25 cm

5 cm Hexagon

Perimeter = 4 cm + 4 cm + 4 cm + 4 cm + 4 cm + 4 cm = 24 cm

4 cm 4 cm

4 cm

Perimeter = 6 × side Perimeter = 6 × 4 = 24 cm

4 cm

Remember! Perimeter of regular polygons = Number of sides in the polygon × Length of each side

Example 2

Length of all 6 sides of a regular hexagon are equal,

Think and Tell

What will be the perimeter of a regular octagon with side 7 cm?

Find the length of the side of a square with perimeter 42 cm. Perimeter of the square = 42 cm Perimeter of square = 4 × Length of the side = 42 cm Length of the side = 42 ÷ 4 = 10.5 cm

Example 3

Find the cost of framing a rectangle of length 20 cm and breadth 17 cm at the rate of ₹12 per cm. Length of the frame = 20 cm; Breadth of the frame = 17 cm Perimeter = 2 × (length + breadth) = 2 × (20 + 17) = 74 cm Rate of framing = ₹12 per cm Cost of framing 74 cm = ₹12 × 74 = ₹888

Chapter 13 • Mensuration

205

Example 4

Mohit jogs around a rectangular park of length 41.5 m and breadth 55.5 m. If he jogs 12 rounds of the park each day, what is the distance covered by him (in km) in a day? Length of the park = 41.5 m; Breadth of the park = 55.5 m 1 round of the park = Perimeter of the park = 2 × (41.5 + 55.5) = 2 × 97 = 194 m 12 rounds of the park = 12 × 194 = 2328 m We know that 1000 m = 1 km 2328 m =

Do It Together

1 × 2328 = 2.328 km 1000

The perimeter of a regular heptagon and that of a rectangle are equal. If the side of the heptagon is 8 m and the length of the rectangle is 15 m, find the breadth of the rectangle. Length of the side of heptagon = _____ Perimeter of heptagon = 7 × _____ = 56 m Length of rectangle = _____ Perimeter of rectangle = Perimeter of heptagon 2 × (L + B) = 56

56 – _____; B = 28 – _____ = _____ m 2 Hence the breadth of the rectangle is _____ m.

Do It Yourself 13A 1

Find the perimeter of the figures. a

2 cm 5 cm 11 cm

13 cm 4 cm

4 cm 6 cm

13 cm

3.5 cm 3.5 cm

2 cm 6 cm

2 cm

4 cm

18 cm

6 cm

22 cm

6 cm

Find the perimeter of a rectangle of length 30 cm and breadth 12 cm.

A ribbon is used to form a square figure with perimeter of 164 cm. What is the length of the side of the square?

What is the cost of fencing a rectangular field of 25 m by 17 m at the rate of ₹28 per metre?

Rohan is making a wooden box. The base measures 15 cm by 10 cm. What length of wood does he need to go

A piece of string is 42 cm long. What would be the length of each side if the string is used to form

around all the sides?

a an equilateral triangle?

206

b a regular heptagon?

A rectangular piece of glass measures 3 m 25 cm by 1 m 42 cm. What is its perimeter?

A copper wire is bent to form a shape with each side 4 cm as shown. If the same wire is bent

Pooja’s dining table is in the form of a regular octagon with side 75 cm. She wants to put a frame around it that

4 cm

again to form a regular decagon, what would be the length of each side of the decagon?

costs ₹125 per meter. Find the cost incurred by Pooja.

Word Problems 1

Rohini plans to paint the wall of her room. The height of the wall is 262 cm and the width is 185 cm. Rohini gets some masking tape to put around the edge of the wall. How much masking tape does she need?

The length and breadth of a rectangular football field are 77 m and 52 m respectively. If a player takes two and a half rounds of the field, what is the distance covered by the player?

Understanding Area Real Life Connect

Richa is working on her new craft in which she needs to cover a rectangular sheet of paper of length 8 cm by 10 cm with 1 cm by 1 cm pieces of paper. How many pieces of paper does she need to cover the entire sheet?

Area on Graph Paper We have discussed the outer boundary (perimeter) of a figure. The region enclosed by a closed figure is called its area. We can find the region enclosed with the help of squared paper.

1 cm Count the number of squares enclosed in the figure.

On counting, we found that there are 80 squares enclosed in the figure. So, the number of squares of paper required by Richa is 80. Number of unit squares enclosed = Area of the figure. Hence the area of the rectangular sheet of paper = 80 sq. cm. Chapter 13 • Mensuration

Remember! Unit of area depends on the unit of the dimensions of the figure.

207

But the squares do not always fit into the area we want to measure. Can we still measure the area of such figures? Yes! Let us find out how. Look at the figure below and the rules to estimate the area of such figures. Area of 1 fully filled square = 1 sq. unit

Count area = 0 sq units if the portion is less than (<) half a square.

Area =

Count area = 1 sq unit if the portion is greater than (>) half a square.

1 sq. unit if exactly half the 2 square is counted.

The area of the figure can be estimated as: 1

Fully filled squares = 13

Half-filled squares = 1

More than half-filled squares = 16

Less than half-filled squares = 11

Area = 13 + 1 × Example 5

1 1 1 + 16 = 13 + + 16 = 29 sq. units 2 2 2

Find the area of the figures in sq. cm. 1

Example 6

Number of squares enclosed by the figure = 36.

Number of squares enclosed by the figure = 35.

Number of squares enclosed by the figure = 57.

Area of the figure = 36 sq. cm.

Area of the figure = 35 sq. cm.

Area of the figure = 57 sq. cm.

By counting the squares, estimate the area of the figure in sq. units. 1

Fully filled squares = 14

Half-filled squares = 2

More than half-filled squares = 13

Less than half-filled squares = 13 Area = 14 + 2 ×

208

1 + 13 = 14 + 1 + 13 = 28 sq. units. 2

Do It Together

Find the area of the figures in sq. units. 1

Fully filled squares = 33

Fully filled squares = ____

Half-filled squares = ____

Half-filled squares = 0

Half-filled squares = ____

More than half-filled squares = 10

More than half-filled squares = ____

Less than half-filled squares = ____

Area = ______

Area of Squares and Rectangles Remember Richa who was covering her rectangular sheet of length 8 cm by 10 cm with 1 cm by 1 cm pieces of paper. Let us see her data along with the measures of a few more rectangles. 8 cm

6 cm

5 cm

7 cm

10 cm

8 cm

Number of squares = 80

Number of squares = 35

Number of squares = 48

Area = 80 sq. cm = 8 cm × 10 cm = 80 sq. cm

Area = 35 sq. cm = 5 cm × 7 cm = 35 sq. cm

Area = 48 sq. cm = 6 cm × 8 cm = 48 sq. cm

From the above data, we found that

Area of a rectangle = Length × Breadth

Let us now find the area of a square with lengths given. 6 cm

4 cm

Area = 16 sq. cm = 4 cm × 4 cm = 16 sq. cm

Chapter 13 • Mensuration

6 cm

4 cm

Number of squares = 16

Number of squares = 36 Area = 36 sq. cm = 6 cm × 6 cm = 36 sq. cm

209

We can infer from the above data that, Example 7

Area of a square = Side × Side

The area of a rectangle is 252 sq. cm. If the length of the rectangle is 18 cm, then find its breadth. Area = 252 sq. cm

Did You Know?

Length of rectangle = 18 cm; Breadth = ?

Rajasthan is the largest

Area of rectangle = Length × Breadth

area is 3,42,239 sq. km.

state in terms of area. Its

252 = 14 cm 18 Hence the breadth of the rectangle is 14 cm. 252 = 18 × Breadth; Breadth =

Example 8

The floor of Neeta’s room is 5 m long and 4 m wide. She laid a carpet of side 4 m on the floor. What is the area of the floor not carpeted? Length of the room = 5 m; Breadth of the room = 4 m Area of the room = Length × Breadth = 5 × 4 = 20 sq. m. Length of the side of the carpet = 4 m Area of the carpet = Side × Side = 4 × 4 = 16 sq. m Area of the floor not carpeted = Area of the room – Area of the carpet = 20 – 16 = 4 sq. m

Example 9

Find the cost of polishing a table top with dimensions 556 cm by 2.75 m at the rate of ₹62 per sq. m. Length of the table = 556 cm = 5.56 m; Breadth of the table = 2.75 m Area of the table = Length × Breadth = 5.56 × 2.75 = 15.29 sq. m. Cost of polishing the table = Area of the table × Rate per sq. m. = 15.29 sq. m × `62 = `947.98

Do It Together

Error Alert! Convert the dimensions into the same unit before any operation. 200 cm × 3 m = 600 sq. m 2 m × 3 m = 6 sq. m

Jiya wants to cover the ﬂoor of a room 5 m wide and 6 m long with rectangular tiles. If each rectangular tile is 0.2 m wide and 0.3 m long, find the number of tiles required to cover the ﬂoor of the room. Width of the room = 5 m; Length of the room = 6 m Area of the room = Length × Breadth = ____ × ____ = ______ Width of the rectangular tile = ______; Length of the rectangular tile = 0.3 m Area of one tile = Length × Breadth = ______ × ____ = ______ Area of the room = _______ Area of one tile Thus, the number of tiles required is _______.

Number of tiles required =

210

Think and Tell Can a shape be split in more than one way?

Area and Perimeter of Combined Shapes Find the area and perimeter of the shape. Step 1: Split the shape into rectangles or squares and name them.

Step 2: Find the missing dimensions.

Here the shape can be split into two parts named A and B.

25 cm

60 cm 25 cm

50 cm

40 cm

A 40 cm

60 cm

50 – 25 = 25 cm

50 cm

60 – 40 = 20 cm

Step 3: Find the area of each shape.

Step 4: Find the perimeter of the combined shape.

Area of part A = L × B = 60 × 25 = 1500 sq. cm

Perimeter = 60 + 25 + 40 + 25 + 20 + 50 = 220 cm.

Area of part B = L × B = 20 × 25 = 500 sq. cm Area of combined shape = 1500 + 500 = 2000 sq. cm

Example 10

Find the perimeter of the shape. Find the missing dimensions. Perimeter = 12 + 12 + 15 + 3 + 10 + 6 + 7 + 3 = 68 m.

12 m 12 m 12 – 7 = 5 mA 12 – 3 – 3 = 6 m

10 m

Perimeter of the shape = 68 m.

3m 15 m

Example 11

What is the area of the shape?

4 cm

We split the shape into parts. Name them and find the missing dimensions. Area of part A = Side × Side = 4 × 4 = 16 sq. cm Area of part B = (L × B) = (14 × 5) = 70 sq. cm

5 cm

Area of the shape = 16 + 70 = 86 sq. cm

Do It Together

4 cm

4 cm 5 cm

5 cm

5 + 4 + 5 = 14 cm

Find the perimeter and area of the figure. We split the shape into parts. Name them and find the missing dimensions.

2 cm

Area of part A = L × B = 5 × ____ = ____ sq. cm 5 cm

Area of part B = L × B = ____ × 6 = ____ sq. cm Area of the figure = ______ + ______ = ______ Perimeter of the figure = _____________________________________

Chapter 13 • Mensuration

3 cm

6 cm

A 7 cm

211

Do It Yourself 13B 1

Find the area of the shapes in sq. cm. a

Find the area of a square of which the side is 7.2 cm.

Find the length of the rectangle with area 371.2 sq. cm and breadth 25.6 cm.

What is the cost of tiling a square plot with side 15 cm at the rate of ₹15 per square cm?

A square of 10 cm length is cut into tiny squares of 2 cm long. Calculate the number of tiny squares that can be

The area of a rectangular frame is 1125 sq. cm. If its width is 25 cm, what is its length?

What is the cost of tiling a rectangular plot of land 450 m long and 375 m wide at the rate of ₹15 per sq. m?

Find the perimeter and area of the figures.

created.

4 cm

20 cm

8 cm 8 cm

3 3

20 cm 3

3 3 4 cm

Kunal went to purchase a plot. He found two options. The one was a rectangular plot of size 140 m by 30 m and the other square plot of side 6400 cm at the same price. Which one should he prefer and why?

Word Problem 1

The length and breadth of a room are 6 m and 7 m respectively. How many square metres of

carpet are required to cover the ﬂoor of the room completely? If the carpet costs ₹250 per sq. m, how much will it cost to cover the entire ﬂoor?

212

Perimeter and Area Problems Real Life Connect

Rohan wanted to sell his square plot. He was willing to buy a rectangular plot having the same area.

Word Problems on Perimeter and Area The area of Rohan’s square plot was 484 sq. m. Rohan purchased a rectangular plot of breadth 11 m. He thought of fencing the plot. What would be the length of fencing that Rohan would require? Let us solve the problem using CUBES strategy. C = Circle the numbers The area of Rohan’s square plot was 484 sq. m. Rohan purchased the rectangular plot with breadth 11 m and the same area as the square plot. He thought of fencing the plot. What length of fencing would Rohan require?

E = Evaluate and draw

U = Underline the question

B = Box the keywords

Area of square plot = Area of rectangular plot = 484 sq. m

Area = 484 sq. m

Square plot

Rectangular plot

Breadth = 11 m

Breadth of rectangular plot = 11 m

Length of fencing = Perimeter of the rectangle S = Solve We know that Perimeter of rectangle = 2 × (L + B) We also know that Area of rectangle = Length × Breadth = 484 = (L × 11)

484 = 44 m 11 Length of fencing required = 2 × (L + B) = 2 × (44 + 11) = 2 × 55 = 110 m L=

Hence the length of fencing required is 110 m. Example 12

Manisha has a square garden with perimeter 72 m. What is the length of each side of the garden? Also find its area. Let us solve using the CUBES strategy, Chapter 13 • Mensuration

213

Manisha has a square garden with perimeter 72 m. What is the length of each side of the garden? Also find its area. Perimeter of the square garden = Total length of the four sides = 72 m Perimeter of square = 4 × side = 72

Area = ?

Perimeter = 72 m

72 = 18 m 4 Area of square = Side × Side = 18 × 18 = 324 sq. m Length of each side =

Example 13

Mahi has a square sheet of paper of perimeter 48 cm. The area of Shweta’s rectangular sheet is 18 sq. cm more than the area of Mahi’s sheet. If the length of the rectangular sheet of paper is 18 cm, find its breadth. Mahi has a square sheet of paper of perimeter 48 cm. The area of Shweta’s rectangular sheet is 18 sq. cm more than the area of Mahi’s sheet. If the length of the rectangular sheet of paper is 18 cm, find its breadth. Perimeter of the square sheet = 48 cm Area of rectangular sheet = 18 sq. cm more than the area of the square sheet Length of rectangular sheet = 18 cm Area = Area of square sheet + 18 Perimeter = 48 cm

Length = 18 cm

To find the breadth of the rectangular sheet of paper, we first need to find its area. Perimeter of square sheet = 4 × side = 48 cm

48 = 12 cm 4 Area of the square sheet = side × side = 12 × 12 = 144 sq. cm Length of the side of the square sheet =

Area of rectangular sheet = 144 + 18 = 162 sq. cm Also, Area of rectangular sheet = Length × Breadth = 162 = 18 × Breadth Breadth =

162 = 9 cm 18

Hence the breadth of the rectangular sheet of paper is 9 cm. Do It Together

It costs ₹5200 to fence a rectangular park of breadth 8 m at the rate of ₹130 per sq. m. Find the length of the park and its perimeter. Also find its area. It costs ₹5200 to fence a rectangular park of breadth 8m at the rate of ₹130 per sq. m. Find the length of the park and its perimeter. Also find its area. Cost of fencing the rectangular park = ₹5200 Breadth = _____ m; Rate of fencing = ₹130 Total cost of fencing = Perimeter × Rate of fencing per sq. m.

Total cost of fencing = _______________ m Rate of fencing per sq. m Perimeter = 2 × (Length + Breadth) Perimeter Length = – Breadth = ______ – ______ = ______ 2 Area = Length × Breadth = ______ × ______ = ______ Perimeter =

214

Do It Yourself 13C 1

Find the area of a square of which the perimeter is 90 m.

The area of the ﬂoor of a rectangular kitchen is 130 sq. m. If the length of the kitchen is 10 m, then find its

Seema formed a figure by putting two squares of side 10.5 cm side by side. Find the perimeter and area of the

How many metres of carpet are required to cover a rectangular ﬂoor of perimeter 84 m and one of the sides 24 m?

The area of a square and a rectangle are equal. If the side of the square is 3 m 50 cm and the length of the

Pooja is planning to decorate the walls of her room with stickers. How many stickers with dimension 8 cm and

A rectangular park has an area of 3150 sq. m. The length of the park is 90 m. If Rahul walks around the park

perimeter.

new figure obtained.

rectangle is 5 m, find the breadth of the rectangle and its perimeter.

13 cm are required to fit in a region of perimeter 552 cm and length 120 cm? covering 6 km, how many times did he go round the park?

Word Problem 1

If the length of Garima’s room is 6 m and its perimeter is 18 m, then find the cost of tiling it at the rate of ₹50 per sq. m.

Points to Remember •

Perimeter is the distance around the outside of a shape. Shape Perimeter

4 × side

2 × (L + B)

3 × side

•

The region enclosed by a closed figure is called its area.

•

Area can be measured using a square grid.

•

Area of rectangle = Length × Breadth; Area of square = Side × Side

Chapter 13 • Mensuration

5 × side

6 × side

215

Math Lab Exploring Area and Perimeter Setting: In group of 2 Materials Required: Graph paper, Ruler, Pencils, shapes (square, rectangle, triangle, irregular polygon) Method:

Provide each student with a piece of graph paper, a ruler, and a pencil.

The students draw the provided shapes on their graph paper.

The students find the area and perimeter of the shapes and note down their answers.

The group that gives all the answers correctly, wins!

Chapter Checkup Find the perimeter of the given figures. 3 cm

3 cm 4 cm

2 cm

6 cm

2.5 cm 2 cm 2.5 cm

H 2 cm 1 cm F 10 cm

2 cm

10 cm

4 cm

5 cm

1 cm

2 cm

5 cm

3 cm 2 cm

9 cm

216

Find the length of a rectangle of which the breadth is 25 cm and perimeter 110 cm.

If the area of a rectangular garden of length 50 m is 400 sq. m, find its breadth.

A piece of thread is 70 cm long. What will be the length of each side if the thread is used to form a square?

Find the cost of fencing a square ﬂower garden at the cost of ₹10 per m. The side of the ﬂower garden is 47 m.

Find the area of a tabletop with length 21.6 cm and breadth 18.8 cm.

How many square tiles of dimension 12 cm will be needed to cover a square of 3 m?

The length of a square field measures 0.8 km. If each side is to be fenced with three rows of wire, what is the total

Find the cost of painting a rectangular piece of cardboard at the rate of 25 paise per sq. cm. The length and 1 breadth of the cardboard are 80 cm and m respectively. 4

length of wire required?

10 Find the area of the figures.

11 Two pieces of wire of length 42 cm each are bent to form a rectangle and a square. The breadth of the rectangle formed is 5 cm. Which of the two figures has the greater area and by how much?

12 Sunita and Ramesh bought pieces of plots as shown. Whose plot has the greater area and perimeter? 10 m

2m 8m

10 m

Sunita's Plot

Ramesh's Plot

13 Find the cost of tiling a square hall of perimeter 52 m at the rate of ₹45 per sq. m. 14 Each side of a rectangular field is 19 m and the length of the rectangular field is 20 m. What is the area of the field?

15 The base of a swimming pool measuring 5 m by 4 m needs to be covered by rectangular tiles of measure 20 cm by 10 cm. Find the number of tiles required.

1 16 The length of one side of Mihir’s square plot is 25 m. He used of his plot for farming. How much land is vacant? 4

Word Problems 1

Kunal has a rectangular plot, the area of which is 528 sq. m and length 22 m. Find the cost

Sailesh runs around a square park of side 88 m 3 times a day while Renu runs around a

of fencing it at the rate of ₹12.5 per m.

rectangular park of side 30 m by 70 m five times a day. Who runs the longer distance every day?

How many square tiles of side 10 cm are required to fit onto a rectangular ﬂoor of perimeter 24 m and breadth 5 m?

Chapter 13 • Mensuration

217

314

Introduction to Algebra

Let’s Recall We know how to recognise patterns and extend them. Let us recall some patterns.

What shape comes after the circle? A triangle. What shape comes before the triangle? A circle. Extending this pattern in both the directions, we get,

These are called repeating patterns.

In this pattern the shape L is formed using dots. Each figure has 2 more dots than the previous figure. This is the rule of the pattern. Extending this pattern, the next figure in the sequence will have 9 dots. These are called growing patterns.

Let's Warm-up Match the following. Number pattern

Next number in the pattern

1 1, 2, 3, 4…

2 3, 6, 9…

3 1, 3, 5, 7…

4 2, 4, 6, 8…

5 4, 8, 12…

5 I scored _________ out of 5.

Mean, Median and Mode Algebra and Patterns Real Life Connect

In a lecture hall, seats are arranged in such a way that the ﬁrst row contains 3 seats, the second row has 6 seats, the third row has 9 seats and so on. Ravi is sitting in the 8th row. He wonders how many seats there are in his row.

Number Patterns We can ﬁnd the answer by making a table. Row

…

Number of seats

…

The number of seats in the 8th row is 24. Did you analyse any pattern in the table? The number of seats is three times the row number. Let us write the letter ‘x’ to represent the row numbers. Then, the number of seats = 3 × x or 3x. This is the rule for the given pattern. By substituting the row number in the rule, we get the total number of seats in that row. For row 1, x = 1. Number of seats = 3x = 3 × 1 = 3. For row 2, x = 2. Number of seats = 3x = 3 × 2 = 6. For row 8, x = 8. Number of seats = 3x = 3 × 8 = 24.

Think and Tell

How many seats will there be in the tenth row?

Here, x is a variable as x takes diﬀerent values.

A symbol that does not have a fixed value and takes various values is called a variable. In 3x, 3 is called the coefficient of x. A coefficient is a number that is written along with a variable, or it is multiplied by the variable.

Rule: 3x

Variable Coeﬃcient

We can ﬁnd patterns everywhere. Look at this matchstick pattern. Shape 1

Shape 2

Shape 3

How many matchsticks are required to form shape 4, shape 5 and shape 20? Chapter 14 • Introduction to Algebra

219

Step 1: Identify the rule.

Shape number

We can make a table to ﬁnd a rule for the pattern.

Let us write the letter ‘n’ to represent the shape numbers.

2 3

Step 2: Use the rule to extend the pattern.

Number of sticks required 1×3+1=4

2×3+1=7

3 × 3 + 1 = 10

Rule

Number of sticks required = (n × 3 + 1) or (3n + 1)

Number of sticks required to form shape 4 = 4 × 3 + 1 = 13 sticks Number of sticks required to form shape 5 = 5 × 3 + 1 = 16 sticks Number of sticks required to form shape 20 = 20 × 3 + 1 = 61 sticks In the rule (3n + 1), n is a variable, 3 is the coeﬃcient of n and 1 is the constant term. The quantities with ﬁxed numerical values are called constants.

Example 1

What are the next three numbers in the sequence 2, 5, 8, 11, …

Remember!

First term of the sequence = (3 × 1) − 1 = 2

A list of numbers which form a pattern is called a sequence.

Second term of the sequence = (3 × 2) − 1 = 5 nth term of the sequence = (3 × n) – 1 = (3n – 1) Fifth term = (3 × 5) − 1 = 14

Example 2

Sixth term = (3 × 6) − 1 = 17

Seventh term = (3 × 7) − 1 = 20

While counting chairs in the lunch room, Madan noticed that there were 3 chairs in the first row, 7 chairs in the second row, 11 chairs in the third row, and 15 chairs in the fourth row. If this pattern continues, how many chairs will there be in the fifth row? Number of chairs in the first row = (4 × 1) − 1 = 3 Number of chairs in the second row = (4 × 2) − 1 = 7 Number of chairs in the nth row = (4 × n) – 1 = (4n – 1) Number of chairs in the fifth row = (4 × 5) − 1 = 19

Do It Together

Complete the table for the pattern of shapes. Use the rule to ﬁnd the number of matchsticks needed for the shape in the 15th position. Shape 1

Shape 2

Shape 3

Let us write the letter ‘n’ to represent the shape numbers. Shape number

Number of sticks required

Rule

220

Number of sticks required = n × ____

Do It Yourself 14A 1

Write the numerical coeﬃcient of the expressions. a 2x

b 4ab

1 x 3

f –3abc

Find the general rule for each of the number patterns. Then ﬁnd the 18th term. a 2, 4, 6, 8, 10, …

d –xyz

c 10pqr

b 3, 6, 9, 12, 15, …

d 3, 8, 13, 18, 23, …

c 5, 10, 15, 20, …

Pavan wrote the sequence using a certain rule. 1, 5, 9, 13, … Find the rule and the 10th term of the sequence.

Find the rule in terms of variables for the matchsticks pattern and write the number of matchsticks used in the 50th shape. a Shape 1

Shape 2

Shape 3

Shape 4

Shape 1

Shape 2

Shape 3

What is the rule for each of the number patterns? a 3, 5, 7, 9, …

b 5, 8, 11, 14, …

Word Problem 1

Complete the table, write the rule, and answer the question. A bakery makes 10 cupcakes an hour. Hours

Cupcakes

Write the rule to work out the number of cupcakes this bakery produces within a certain amount of time.

How many cupcakes will it make in 1 day?

Chapter 14 • Introduction to Algebra

221

Rules in Patterns Rules from Geometry D

Perimeter = sum of the sides

C A square has 4 equal sides

P = a + a + a + a = 4 times a = 4a Where P is the perimeter and a is each side of square

A D

a l

Opposite sides of rectangle are equal in length

Example 3

Perimeter = sum of the lengths of sides

P = l + b + l + b = 2l + 2b = 2 (l + b)

Where P is the perimeter, l is length and b is the breadth of the rectangle

Write the general rule to ﬁnd the perimeter of an equilateral triangle. An equilateral triangle has all three sides of equal length. a

The perimeter of an equilateral triangle = sum of the lengths of its three sides

So, P = a + a + a = 3 × a = 3a, where P is the perimeter and a is the side

Example 4

Thus, we get the rule as P = 3a.

Write a general rule to ﬁnd the area of a square.

Area of a square = product of the length and width

So, A = a × a = a2, where A is the area and a is the side

Thus, we get the rule for the area of a square as A = a2 Do It Together

Write the perimeter of the regular pentagon using variable a.

Perimeter of a regular pentagon = sum of the lengths of all its sides

So, P = a + a + a + a + a = ____ × a = ____ Thus, we get the rule for the perimeter of a regular pentagon as _______.

a a

Do It Yourself 14B 1

Write the general rule to ﬁnd the perimeter of the shapes. a an isosceles triangle

222

b a rhombus

c a kite

d a parallelogram

Write the general rule to ﬁnd the area of a rectangle.

Write the general rule to ﬁnd the perimeter of the shapes. a a regular hexagon

b a regular heptagon

c a regular nonagon

Write the general rule to ﬁnd the diameter of a circle using its radius.

Write the general rule to ﬁnd the perimeter of a decagon.

d a regular octagon

Word Problem 1

Karan has a piece of string bent into a shape with twelve equal sides. What will be the perimeter of such a shape in terms of variables?

Mean, Median and Mode Algebraic Expressions and Equations Real Life Connect

The charge of a taxi service includes a ﬂat rate of ₹50 plus an additional ₹15 per km.

Forming Algebraic Expressions Let us try to express the total amount charged by the taxi service using variables. Let the total distance covered be x km. So, the required expression is 15x + 50. Coeﬃcient

Variable 15x

Constant Terms

Meaning and deﬁnitions of some of the important words used in the expression above. Words used

Meaning/deﬁnition

variable

A variable is an unknown quantity that may change.

coeﬃcient

A numerical and constant quantity placed before the variable.

constant

A speciﬁc number that is assigned a ﬁxed value.

term

A term can be a number, a variable, the product of two or more variables or the product of a number and a variable.

algebraic expression

An expression which is made up of terms, along with algebraic operations (addition, subtraction, etc.).

Chapter 14 • Introduction to Algebra

223

Forming Algebraic Expressions We can convert a verbal statement of fact into an algebraic expression. For example, ‘3 added to a number’ can be converted into the expression ‘3 + x’, where x is a variable. Some more examples of algebraic expressions: Statement

Example 5

Algebraic expression

Seven more than x

x+7

Four times n

Twice of a added to thrice of b

2a + 3b

Five taken away from the product of x and y

xy – 5

Six subtracted from y

y–6

Did You Know? C. P. Ramanujam was an Indian mathematician who worked in the fields of number theory and algebraic geometry.

The base of a triangle is x cm. The height of the same triangle is 6 times the base of the triangle. What is the height of the triangle? The base of the triangle is x. Height of the triangle = 6 times the base of the triangle = 6x.

Example 6

If the length of a rectangular hall is 4 m more than 2 times the breadth of the hall, then what will its length be, if the breadth is y metres? Given, the breadth of the rectangular hall = y metres Since the length of a rectangular hall is 4 m more than 2 times the breadth of the hall, therefore, length = (2y + 4) m.

Example 7

Rahul is n years old. Narendra is twice as old as Rahul and Renu is 2 years younger than Narendra. Express Renu’s age in terms of variable n. Rahul’s age is n years. Narendra is 2 times older than Rahul. So, he is 2n years old. Renu is 2 years younger than Narendra. So, Renu is (2n – 2) years old.

Do It Together

The age of Anil’s mother is 2 years less than 3 times Anil's age. Express the age of Anil’s mother in terms of Anil’s age. Let Anil’s age be x years. So, the age of Anil’s mother = _______ years

224

Rules in Algebraic Expressions Rules from arithmetic Properties of Addition Closure property If a and b are any two natural numbers, then

Commutative property

Associative property

If a and b are any two natural numbers, then

If a, b, and c are three natural numbers, then (a + b) + c = a + (b + c)

a + b = c, where c is also a natural number a + b = b + a Properties of Multiplication Closure property If a and b are any two natural numbers, then

Commutative property

Associative property

If a and b are any two natural numbers, then

If a, b, and c are three natural numbers, then (a × b) × c = a × (b × c)

a × b = c, where c is also a natural number a × b = b × a

Distributive property of multiplication over addition If a, b, and c are three natural numbers, then a × (b + c) = a × b + a × c Example 8

Find a generalised rule for the distributive property of multiplication over subtraction. Thus, the distributive property of multiplication over subtraction is expressed as: a × (b − c) = a × b − a × c where a, b, and c are any three natural numbers.

Example 9

The additive identity property of zero states that adding 0 to any number results in the number itself. Generalise this property using a variable. According to the additive identity property, a + 0 = 0 + a = a, where a is any number.

Do It Together

Multiplicative identity property states that “Any number multiplied by 1 is the number itself”. Generalise this property using a variable. The multiplicative identity formula is expressed as a × _____ = _____, where a is any number.

Do It Yourself 14C 1

Sort these into arithmetic expressions or algebraic expressions. a 7×2+3−2

b 7x + 5

d 10y – 6x

c 12y

Write down the terms of the algebraic expressions. a x+y

b 2a + 3b – c

e 2x + 5

4 x+2 3

c x + y + 2z

d x – 6z

g 5abc – 2ab + 7ac

h 2ab + 4ac – 6c

Write algebraic expressions for the statements. 3 added to 6m

10 subtracted from n

15 times x

Twice the product of x and y

9 multiplied by y added to 1

Thrice of y added to the diﬀerence of x and 3

Chapter 14 • Introduction to Algebra

225

Fill in the blanks using properties of algebraic expressions. a x + y = y + _______

b (x + y) + _______ = _______ + (_______ + z)

c a × b = b × _______

d (a × b) × _______ = _______ × (_______ × c)

Write a statement for each of the algebraic expressions. a 18 − y

b 3x + 4

c p–3

1 a 4

After sharing x pencils with your friend, you are still left with 7 pencils. What was the total number of pencils

When a larger box of apples is emptied, the apples from it ﬁll three smaller boxes and 5 apples still remain

Out of 25 students in grade 6, x number are performing a group dance and 6 are performing solo. y students are

before sharing?

outside. If the number of apples in a small box is x, what is the number of apples in the larger box?

not performing at all. Express y in an algebraic expression.

Word Problems 1

Mala is Rima’s younger sister. If Mala is ﬁve years younger than Rima, then write Mala’s age in

A teacher gave 4 pens to each student. Find the number of pens the teacher had, if the number of

Karan had m chocolates. He gave his younger brother 5 chocolates and divided the remaining

terms of Rima’s age.

students is n and she has 5 pens left.

chocolates equally between his friend and himself. How many chocolates did his friend get in terms of m?

The number of rooms on the ground ﬂoor of a building is 12 less than twice the number of rooms

Rahul spends ₹y daily and saves ₹z per week. What is his income for three weeks?

Ashutosh purchased tomatoes at a price of ₹29 per kg and cucumbers at the price of ₹38 per kg.

on its first floor. If the first floor has x rooms, how many rooms are on the ground floor?

He purchased 4 kg less cucumbers than tomatoes. If he purchased x kilograms of tomatoes, write an algebraic expression for his total purchase.

Forming Algebraic Equations The expression for the amount to be paid by Gaurav for a cab can be given as 15x + 50. Gaurav paid ₹3800; hence the above expression would change to form an equation as: 15x + 50 = 3,800 Left-Hand Side (LHS)

Right-Hand Side (RHS)

The equation formed above is a linear equation. A linear equation is an equation in which the highest power of the variables involved is 1. 226

Example 10

Translate the statement into an equation: Twice a number increased by 12 gives 24. Let the number be x; Twice the number = 2x Twice the number increased by 12 = 2x + 12. Hence, 2x + 12 = 24 is the required equation.

Remember! An equation remains the same if its LHS and RHS are interchanged. Thus, x + 25 = 47 is same as 47 = x + 25

Example 11

Ramesh is twice as old as his brother Rakesh. The sum of their ages is 18 years. Express the situation as an equation.

Do It Together

Express the statement ‘Adding 3 to 4 times a number gives 21’ in the form of an equation. Let the number be z.

Let Rakesh’s age = x years. Ramesh’s age is twice that of Rakesh = 2x years

Adding 3 to 4 times this number can be written algebraically as ______.

Sum of their ages = 18 years.

So, the required equation is ______.

Hence, x + 2x = 18 is the required equation.

Do It Yourself 14D 1

Tick the linear equations. a 15 – 6 = 9

b 2x + 5 = 7

c x2 + 1 = 5

d 12x < 28

e 4+y=9

f 23 + y > 45

Write the linear equations for each of the statements. a When we add 3 to a number, we get 12.

b When 7 is taken away from a number, it leaves 2.

c When 7 is taken away from a number, it leaves 2.

d 14 less than half a number is 7.

e 2 added to the product of a number and 4 gives 26. f Rohan’s mother is 37 years old. The sum of Rohan’s age and that of his mother is 46 years.

A number divided by 4 and then increased by 3 is 10. Express this statement as an equation.

A shopkeeper has 193 artifacts. He sells x artifacts every day. Write an equation to express the situation if he has

3 The denominator of a fraction exceeds the numerator by 5. If 3 is added to both, the fraction becomes . Write an 4 equation to represent this situation.

Let T represent the total number of plums. Let P represent how many each of 6 friends gets. What equation

Write a statement for each of the equations.

46 artifacts left after a week.

models this situation?

a 4x + 5 = 9

b 2n + 1 = 5

Chapter 14 • Introduction to Algebra

c 12a – 7 = 5

d 3(y + 1) = 12

227

Word Problems 1

Badri and Jai like to collect stamps. Badri has x stamps and Jai has 29 more stamps than Badri.

Seventy-two people signed up for the soccer league. After the players were evenly divided

Together they have a total of 73 stamps. Write an equation to describe this situation.

into teams, there were 6 teams in the league and x people in each team. Write an equation to represent this situation.

Tina thought of a number and then tripled it. After subtracting 7 from the result, she multiplied it by 2. If the product is 52, what equation represents the number Tina thought of ?

Solving Linear Equations in one Variable Gaurav continued using the same taxi service. The taxi charges ₹15 per km. One day he travelled with the taxi and paid ₹105, how many km did he travel? Let the distance travelled by Gaurav = m km Charge per km = ₹15; Charge for m km = 15 × m = 15m As per the given condition, 15m = 105 The value of the variable for which LHS = RHS of an equation is called the solution or root of the equation. When we solve an equation, we ﬁnd the value of the unknown variable.

Remember! There can be only one value or solution to a given linear equation in one variable.

Some methods to solve linear equations are given below. • Trial and error method • Transposition method

Trial and Error Method In this method, we substitute diﬀerent values for the variable and check the equality of LHS with RHS. The value of the variable that satisﬁes the equation is the solution or root of the equation. In the equation 15m = 105. Here, LHS = 15m and RHS = 105 If m = 1, LHS = 15 × 1 = 15 ≠ RHS; If m = 3, LHS = 15 × 3 = 45 ≠ RHS If m = 5, LHS = 15 × 5 = 75 ≠ RHS; If m = 7, LHS = 15 × 7 = 105 = RHS Therefore, for m = 7, LHS = RHS = 105 So, m = 7 is the solution of the equation 15m = 105. Example 12

Solve the equation 8x + 2 = 50 by trial-and-error method. Here, LHS = 8x + 2 and RHS = 50 If x = 1, LHS = 8 × 1 + 2 = 8 + 2 = 10 ≠ RHS; If x = 2, LHS = 8 × 2 + 2 = 16 + 2 = 18 ≠ RHS If x = 3, LHS = 8 × 3 + 2 = 24 + 2 = 26 ≠ RHS; If x = 4, LHS = 8 × 4 + 2 = 32 + 2 = 34 ≠ RHS

228

If x = 5, LHS = 8 × 5 + 2 = 40 + 2 = 42 ≠ RHS; If x = 6, LHS = 8 × 6 + 2 = 48 + 2 = 50 = RHS Therefore, for x = 6, LHS = RHS = 50 So, x = 6 is the solution of the equation 8x + 2 = 50. Do It Together

Solve the equation 3x + 4 = 16 by trial-and-error method. Here, LHS = _____ and RHS = _____ For x = 1, LHS = 3 × 1 + 4 = 3 + 4 = 7 ≠ RHS; For x = 2, LHS = ____________________ For x = 3, LHS = ____________________; For x = 4, LHS = ____________________ Therefore, for x = ______, LHS = RHS = 16 So, x = ______ is the solution of the equation 3x + 4 = 16.

Transposition Method The trial-and-error method is a longer method when the value of the variable is a large number. To save time, we use the method of transposing to solve a linear equation. Transposing means shifting a number or variable from one side of the equality symbol to the other side by reversing the operation. The sign changes as shown below: +

–

Let us solve the equation 15m = 105 by transposition method. To isolate the variable m on the LHS, we transpose 15 from LHS to RHS by reversing the sign. So, m =

Example 13

105 =7 15

Do not forget to change the sign while transposing the digit. –1.5 + k = 8.2

–1.5 + k = 8.2

k = 8.2 + (–1.5)

k = 8.2 + 1.5

k = 6.7

k = 9.7

Solve 2m − 12 = 18 using the transposition method. 2m − 12 = 18

2m = 18 + 12

2m = 30

So, m = 15 is the solution of the equation. Do It Together

Error Alert!

30 = 15 2

Sonal buys 33 balloons for a party. After distributing them among her friends, she has 3 balloons left. Each of her friends gets 5 balloons. Write a linear equation for this problem and ﬁnd how many friends Sonal has. Let Sonal have x number of friends. Number of balloons distributed among friends = 5x. She has 3 balloons left. But the total number of balloons is 33. So, the equation that satisﬁes the condition is 5x + 3 = 33. 5x + 3 = 33

5x = _______

5x = ____

Therefore, Sonal has _____ friends.

Chapter 14 • Introduction to Algebra

___ =6 5

229

Do It Yourself 14E 1

Solve each of the equations by trial-and-error method. a 3x + 5 = 8

b 2(3x + 1) = 8

c 32 – 5x = 12

d 5x – 8 = 7

e 8y = 48

1 x–3=0 3

y = 3 [3, 7, 12, 8] 4

One of the values in the bracket satisﬁes the equation. Find that value. a 6a = 48 [10, 8, 7, 42]

b 3x + 2 = 23 [7, 8, 25, 9]

Complete the table and ﬁnd the solution to the equation. a x–3=4

x–3 b

x =7 2

x x 2

Find the correct solution if the equation is NOT satisﬁed for the given value. a 4a = 20

Value = 5

b 2x + 5 = 11

Value = 3

c 9m – 1 = 8

Value = 9

Verify if the value given is the solution of the equation. If not, ﬁnd the solution. a 4x + 2 = 10; x = 2

b 9y – 8 = 1; y = 3

d 2n – 6 = 2; n = 12 3

c 5x = 25; x = 10 2

e 5z – 6 = 0; z = 1

f 3x + 2 = 5; x = 1

Solve the equations using the transposition method. a x + 6 = 10

b 5=a+2

c m – 12 = 3

d −7 = x + 4

e 4+k–7=2

f 3 – 4x = −13

3 taken away from 4 times a number is 9. Find the number.

8 When two consecutive numbers are added, the sum is 25. Find the numbers. 9 The length of a rectangular field is twice its breadth. If the perimeter of the field is 228 metres, find the dimensions of the field.

10 Rekha is 24 years older than her daughter Ishita. After 8 years she will be twice as old as Ishita. Find their present ages.

230

Word Problems 1

Sahil’s age is 3 years more than 3 times the age of his daughter. If Sahil is 30 years old, ﬁnd the

Sreeja scored x marks in English and 2x marks in mathematics. Her score in science was 70. If her

Rahul is twice as old as Romit. The diﬀerence between their ages is 9. Find their present ages.

Meenal has some money in her purse. If she adds ₹51 to it, it becomes four times the original

Amit is four times as old as his son. Six years ago, he was ten times as old as his son. Find their

age of his daughter.

total score for the three subjects was 220, what was her score in Mathematics and English?

amount. What is the original amount in her purse? present ages.

Points to Remember •

In algebra, we generally use letters of the English alphabet such as a, b, c, d, …, x, y, z to represent numbers. These letters are known as variables.

•

Algebraic expressions are formed by meaningful combinations of variables, constants, and operators.

•

The parts of an algebraic expression joined by ‘+’ or ‘−’ sign are called terms.

•

An equation contains an equality sign (=). The expression on the left-hand side of ‘=’ is called the LHS while that on the right-hand side of ‘=’ is called the RHS.

•

An equation in which the highest power of each variable is 1, is known as a linear equation.

•

The solution or root of an equation is that value of the variable of which LHS = RHS.

•

We can ﬁnd the solution of an equation by trial-and-error or transposition method.

Math Lab Patterns in Algebra Setting: In groups of 4 Materials Required: A4 size sheet, ice-cream sticks, and glue stick. Method:

Form diﬀerent patterns by repeating letters such as V, H and M using ice-cream sticks.

Write the number of ice-cream sticks used to form each shape in the sequence.

Now, observe the number pattern and derive a general formula for each.

Chapter 14 • Introduction to Algebra

231

Chapter Checkup 1

Find the general rule which shows the number of matchsticks required to make the patterns. Use a variable to write the rule. a

If zero is added to a number or a number is added to zero, the result is the number itself. Generalise this property

The order in which three numbers are multiplied will not change their product. Express the property in a general

Write the algebraic expression for the expressions. Also, write the terms for each.

of numbers using a variable.

way using the variables x, y, and z.

a 4 added to 3 times x.

b 2 subtracted from 2 times y.

c 4 less than quotient of x by 3.

d y is divided by 5 and the quotient is added to 6.

State which among the equations are linear. a 2x + 5 = 9

c 2y + 7 = 9

Complete the table. Also ﬁnd the correct solution if the equation is NOT satisﬁed. S. NO.

b 9x + 4 > 5

Equation

Value of variable

3x – 5 = 4

x=1

4y + 3 = 11

y=2

x = 28 7

x=4

n – 12 = 0 6

n = 12

Equation satisﬁed (Yes/No)

The lengths, in cm, of the sides of a triangle are 3x – 5, 2x – 1 and x + 1. a Write down an expression, in terms of x, for the perimeter of the triangle. b If the perimeter of the triangle is 31 cm, ﬁnd out the value of x.

A rectangle has a length of (2x + 3) cm and a width of (x + 5) cm. a Find an expression for the perimeter of the rectangle. b The rectangle has a perimeter of 43 cm. Find the value of x.

Convert the equations into statements. Also, solve the equations using the trial-and-error method. a x + 30 = 50

b 6x – 3 = 9

24 = 4 n

d 4(3x + 7) = 40

10 Ajit has some marbles. Badri has twice as many marbles as Ajit. Charu has 5 more marbles than Badri. In total they have 55 marbles. How many marbles does Charu have?

232

Word Problems 1

If the present age of Rohit is z years, then determine: a his age 8 years from now. b his age 5 years ago. c his grandmother’s age, who is 9 years more than ﬁve times his age. d his grandfather’s age, who is 10 years more than ﬁve times his age.

Sahil has x number of drawing sheets. Rahul has 2 less than Sahil, while Aryan has 6 more than Sahil. Find out how many sheets Rahul and Aryan have in total. Also, ﬁnd the total number of sheets that the three of them have altogether.

The height of a cuboidal box is x cm. Its breadth is two times its height and its length is 4

The price of a muﬃn is ₹y. The price of a cake is 10 times the price of a muﬃn and the cost

cm less than its breadth. Write the length and breadth of the box in terms of its height.

of a cookie is 7 less than 2 times the price of a muﬃn. Express the price of a cake and a cookie in terms of the price of the muﬃn.

A sports teacher orders 42 tennis balls. Each package contains 3 tennis balls. Which of the equations represents the number of packages? a x + 3 = 42

b 3x = 42

x = 42 3

d x= 3 42

The equation P = 2.5m + 35 represents the price P (in rupees) of a bracelet, where m is the cost of the materials (in rupees). The price of a bracelet is ₹115. What is the cost of the materials?

Lakhan is three times as old as Akash. Lakhan is 7 years older than Mala. The sum of their

Anand has x sweets. Bhanu has 7 more sweets than Anand. Kirti has twice as many sweets

ages is 126. What is the age of Akash?

as Bhanu. The total number of sweets is 89.

a Write an expression for the number of sweets that Bhanu has. b Write an expression for the number of sweets that Kirti has.

Chapter 14 • Introduction to Algebra

233

315

Introduction to Ratio and Proportion

Let's Recall

A fraction is a representation of parts of a whole. For example, if we divide a circle into five equal parts and shade three parts of it, we say that 3 of the circle is shaded. 5 Fractions can be classified as like and unlike. Like fractions have the same denominator whereas unlike fractions have diﬀerent denominators. For example, 3 and 7 are like fractions whereas 1 and 2 are unlike fractions. 8 8 4 5 Fractions are represented in their simplest form to make them easier to work with.

4 can be simplified to 1 . 8 2 This shows that half of something is the same as four-eighths of it. Such fractions are also called equivalent fractions. For example, the fraction

Equivalent fractions are fractions that represent the same portion of a whole, even though they may look diﬀerent. 1 2

2 4

3 6

Let's Warm-up Fill in the blanks. 1 Simran has an apple cut into 6 equal slices. She eats 3 slices of the apple. The fraction of apple eaten is ___________.

8 in its simplest form is _____________ 12 3 The equivalent fraction for 2 is _____________ . 5 4 4 and 6 are a pair of _____________ fractions. 7 7 5 Rohan has eaten 4 of 16 slices of a cake. The fraction of cake left is _____________. 2

I scored _________ out of 5.

Understanding Ratio Shane is fond of cooking and is preparing Dosa batter for the next day. He knows that for 3 cups of rice, he requires 2 cups of dal.

Real Life Connect

Introducing Ratios The quantity of rice to the quantity of dal can be shown as :

= 3:2

This is known as the ratio of the two quantities. A ratio is the comparison between two quantities of the same kind and same units. a:b = First term or antecedent

a b Second term or consequent

Did You Know? The ratio of land to water

The order of terms in a ratio is important and the ratio changes when the order of terms changes. That is a : b ≠ b : a as b is the antecedent in the second ratio.

for the whole earth is 1 : 2.

A ratio has no units as it is the comparison of quantities of the same units. Example 1

Ruhi made 7 parathas and 10 cupcakes. What is the ratio of the number of parathas and cupcakes? Number of parathas = 7; Number of cupcakes = 10 The ratio of the number of parathas to the number of cupcakes = 7 : 10

Example 2

Suhani bought 12 carrots and 16 cucumbers from the market. 1

What is the ratio of the number of carrots to the number of cucumbers? Number of carrots = 12; Number of cucumbers = 16 The ratio of the number of carrots to the number of cucumbers = 12 : 16

2 What is the ratio of the number of cucumbers to the total number of vegetables? Number of cucumbers = 16; total number of vegetables = 12 + 16 = 28 Required ratio = 16 : 28 Do It Together

A box contains 20 balls of which 4 are red, 5 black and 9 white. Find: 1

The ratio of the number of red balls to the number of white balls. Number of red balls = 4; Number of white balls = _________ The ratio of number of red balls to the number of white balls = 4 : _________

Chapter 15 • Introduction to Ratio and Proportion

235

2 What is the ratio of the number of black balls to the total number of balls? Number of black balls = 5; Total number of balls = _________ The ratio of the number of black balls to the total number of balls = 5 : _________

Do It Yourself 15A 1

Fill in the blanks. a In a ratio a : b, the quantities a and b are referred to as the __________ of the ratio. b The first term in a ratio is also known as the __________. c The __________ in a ratio is also known as the consequent. d If __________ of terms in a ratio changes, the ratio changes. e A ratio has no __________ because it represents a comparison of quantities of the same units.

Write true or false. a A ratio can have units, depending on the types of quantities being compared. b In 3 : 5, the consequent is 5. c The terms of a ratio a : b are called the antecedent and consequent respectively. d 10 : 2 is the same as 2 : 10. e 10 litres of water can be compared with 10 km.

Apoorv has a basket of apples. He has 16 red apples and 24 green apples. What is the ratio of the number of red

At a pet store, there are 30 dogs, 20 cats, and 10 birds. What is the ratio of the number of cats to the total number

There are 25 boys and 22 girls in a classroom. What is the ratio of the number of boys to the total number of

In Sarah's garden, there are 15 roses, 10 tulips, and 5 lilies. What is the ratio of:

apples to the number of green apples?

of animals?

students in the class?

a The number of roses to the number of tulips. b The number of lilies to the number of roses. c The number of tulips to the number of roses and lilies.

236

In a box, there were 255 toﬀees and 375 chocolates. What is the ratio of the number of chocolates to the total

Sushma has 14 red marbles, 12 green marbles and 15 black marbles. Find the ratio of the number of red marbles

number of sweets?

to the marbles that are not black.

Word Problem 1

Madhav went to a zoo and saw various animals. There were 10 giraﬀes, 3 tigers, 5 bears and 7

rhinoceros in the zoo. Find the ratio of the number of tigers to the number of bears and giraﬀes.

Ratio in Simplest Form Now, Shane took 9 cups of rice and 6 cups of urad dal to make the batter. The ratio of rice to dal taken to make the batter is 9 : 6 We can also say that for every 3 cups of rice, Shane needs 2 cups of urad dal. This means that 9 : 6 can be simplified as 3 : 2. If the HCF of both the terms of a ratio is 1, then we can say the ratio is in its simplest form. Let us learn how to write the ratio in its simplest form. Step 1 Find the HCF of both the quantities.

Remember!

Step 2

HCF stands for the highest common factor, which is the largest number that divides both the numbers.

Divide both the quantities with their HCF. HCF of 9 and 6 = 3 9:6 =

9 = 9 ÷ 3 = 3 = 3:2 6 6÷3 2

Hence the simplest form of 9 : 6 = 3 : 2

Example 3

Express the ratio 16 : 24 in its simplest form.

Example 4

HCF of 16 and 24 is 8.

16 : 24 = 16 = 16 ÷ 8 = 2 = 2 : 3 24 24 ÷ 8 3 Thus, 16 : 24 in its simplest form is 2 : 3.

Do It Together

Find the ratio of 175 litres to 325 litres. HCF of 175 and 325 is 25.

175 : 325 = 175 = 175 ÷ 25 = 7 325 325 ÷ 25 13 The ratio of 175 litres to 325 litres is 7 : 13.

What is the ratio of 1 year 8 months to 5 months? 1 year 8 months = ___ months + 8 months = ___ months The required ratio = ___ months: 5 months =

= 5 The ratio of 1 year 8 months to 5 months is _________.

Remember! In order to ﬁnd the ratio, the two quantities must be in the same unit.

Chapter 15 • Introduction to Ratio and Proportion

237

Do It Yourself 15B 1

Match the ratios with their simplest forms. a 9 : 13

46 : 35

b 120 : 80

3:2

c 138 : 111

18 : 5

d 42 : 280

3 : 13

e 378 : 105

15 : 1

f 480 : 32

3:2

Write the given ratios in their simplest form. a 48 : 72

b 400 cm : 6 m

c 14 km : 21000 m

d 1.5 kg : 750g

e 25 L : 750 mL

f 18 m : 25 cm

g 9 hours : 540 min

h 15 min : 300 sec

₹100 : 10000 paise

If you have 18 litres of water and 12 litres of juice, what is the simplest form of the ratio of water to juice?

Express the ratio of 24 hours to 2 days in its simplest form.

If the HCF of two numbers is 5, and their ratio is 3 : 4, what are the two numbers?

There are 15 boys and 20 girls in a room. Express the ratio of girls to boys in its simplest form.

The cost of two dozen red apples is ₹360 and the cost of 8 green apples is ₹240. Find the ratio of the cost of a

A rectangular field is 80 m long and 60 m wide. Find the ratio of its length to its perimeter.

green apple to the cost of a red apple.

Word Problem 1

A school starts at 8:00 a.m. and finishes at 2:00 p.m., with a lunch break of 30 minutes. What is the ratio of the lunch break interval to the total hours of the school?

Equivalent Ratios We saw that the ratio of rice to dal was 3 : 2 and Shane took the rice and dal in the ratio 9 : 6. Let us learn how to write the ratio in its simplest form. 238

We also saw that on simplifying 9 : 6, we got 3 : 2. The ratios 9 : 6 and 3 : 2 are called equivalent ratios. Equivalent ratios are those that can be simplified to the same value. Let us find the equivalent ratio of 2 : 5 using some steps. Step 1

Step 2

Write the ratio in the form of fractions.

Multiply the numerator and denominator with the

2:5 =

2 5

Hence, 2 : 5 = 4 : 10 = 6 : 15 = 8 : 40

same number.

2×2= 4 2 × 3 = 6 Or 2 × 4 = 8 Or 5 × 2 10 5 × 3 15 5 × 4 20

Comparison of Ratios We can compare two ratios by using some simple steps. Let us compare 5 : 8 and 5 : 10 Step 1

Remember!

Write the ratios in the form of fractions in their simplest form.

5 7

5 : 7 = ; 5 : 10 =

Like fractions have the same denominator.

1 2

Step 2 Convert the fractions into like fractions. LCM of 7 and 2 is 14. 5 7

5 × 2 = 10 1 1×7 7 = and = 2 2 × 7 14 7 × 2 14

Step 3 Compare the like fractions.

Remember!

10 > 7 14 14 Hence,

For like fractions, the fraction with greater numerator is greater.

5> 5 . 7 10

So, 5 : 7 > 5 : 10

Let us now see how we can divide a certain quantity in the ratio. Suhani has ₹1500. She wants to divide the amount among her siblings, Sunita and Ravi, in the ratio 2 : 3. How much will each sibling get? Step 1

Step 2

Find the total number of parts in the ratio.

Find the number of parts given to each.

2 : 3 = 2 parts + 3 parts = 5 parts

2 out of 5 parts are given to Sunita = 3 out of 5 parts are given to Ravi =

Chapter 15 • Introduction to Ratio and Proportion

2 5

239

Step 3 Find the share by multiplying the parts by the amount to be shared. Sunita will get Ravi will get

Example 5

3 5

2 5

of 1500 =

3 5

2 5

Think and Tell

× 1500 = ₹600

How many equivalent ratios can a ratio have?

× 1500 = ₹900

Find two equivalent ratios of 2 : 7.

Example 6

Compare the ratios 3 : 5 and 7 : 8.

2:7 = 2 = 2 × 2 = 4 7 7 × 2 14

As both the fractions are in their simplest form, convert them to like fractions.

or 2 × 3 = 6 = 6 : 21 7 × 3 21

LCM of 5 and 8 = 40 3 = 3 × 8 = 24 and 7 = 7 × 5 = 35 8 8 × 5 40 5 5 × 8 40

Two equivalent ratios of 2 : 7 is 4 : 14 and 6 : 21.

As, 24 < 35 40 40 Example 7

3 < 7 or 3 : 5 < 7 : 8 5 8

Two numbers are in the ratio 2 : 5. The larger number is 15 more than the smaller number. Find the numbers. Let the numbers be 2x and 5x. Given that the larger number is 15 more than the smaller number. Larger number = 15 + smaller number 5x = 15 + 2x 5x – 2x = 15 3x = 15

x = 15 = 5 3

The numbers are 2 × 5 = 10 and 5 × 5 = 25. Do It Together

The teacher gave the children 90 chocolates and asked Manish to divide them in the ratio of 5 : 4 between Group A and Group B. How many chocolates will each group get? Total number of parts in the given ratio = 5 + 4 = 9 Total number of chocolates = 90 Number of parts given to group A = _____ out of 9 parts = Number of parts given to group B = _____ out of 9 parts = Number of chocolates given to group A = Number of chocolates given to group B =

240

9 9

× 90 = _____ chocolates × 90 = _____ chocolates

Do It Yourself 15C 1

Find three equivalent ratios for the ratios. a 5:7

b 4:9

c 12 : 16

d 6 : 15

e 8 : 11

f 9 : 17

g 11 : 25

h 13 : 16

Compare and fill in the blanks with <, > or = sign. a 5 : 3 _____ 3 : 4

b 3 : 5 _____ 2 : 9

c 4 : 7 _____ 5 : 8

d 3 : 5 _____ 5 : 6

e 3 : 7 _____ 7 : 9

f 2 : 7 _____ 4 : 14

g 7 : 11 _____ 5 : 9

h 8 : 14 _____ 4 : 7

12 : 15 _____ 18 : 24

Arrange the ratios 2 : 5, 3 : 7, 5 : 14, 7 : 10 in ascending order.

An artist is mixing two colours of paint to create a new shade. She wants to mix them in the ratio of 1 : 2. If she

In a certain recipe, the ratio of ﬂour to sugar is 3 : 2. If you need 600 grams of sugar, how many grams of ﬂour do

Raj has a bag of marbles with a ratio of 4 blue marbles to 7 red marbles. If he has a total of 132 marbles, how

Sita has a collection of storybooks and comic books in the ratio of 5 : 4. If she has a total of 36 books, how many

The ratio of kids to adults in a party is 5 : 3. If there are 18 adults at the party, how many kids are there?

Father wants to divide ₹1,650 between his sons Manan and Manish in the ratio of their ages. If Manan is 28 years

uses 3 litres of the first colour, how much of the second colour should she use to maintain the ratio?

you need?

many are blue?

storybooks and comic books does she have?

of age and Manish is 49 years old, how much will Manan and Manish get respectively?

10 Sneha splits her pocket money in the ratio of 2 : 3 for spending and saving, respectively. If she saves ₹60 every month, how much money does she spend each month?

11 If the ratio of the number of students in a class who like maths to the number who like science is 5 : 3, and there are 40 students who like maths, how many students are there who like science?

Word Problem 1

In a cricket match, the ratio of runs scored by Team A to Team B is 5 : 3. If Team B scored 120 runs, how many runs did Team A score?

Chapter 15 • Introduction to Ratio and Proportion

241

Understanding Proportion Real Life Connect

John and Fatima go cycling for 15 minutes on weekdays and cover half a kilometre. John: Let us cycle for 30 minutes today as it’s Sunday! Fatima: Sure John! How much distance can we cover in 30 minutes if we pedal at the same speed? Let us see how we can find the distance covered!

Proportion Introducing Proportion We saw that the children usually cover half a kilometre in 15 minutes. So, in the next 15 minutes they will cover the same distance. Distance covered in 15 minutes = half kilometre = 500 m Distance covered in 30 minutes = 500 m + 500 m = 1000 m Ratio of time of cycling = 15 : 30 = 1 : 2; Ratio of distance covered = 500 : 1000 = 1 : 2 Two ratios are said to be in proportion if they are equal. The symbol used to equate the two ratios is ': :' or ‘=’. Here we can say, 15 : 30 : : 500 : 1000 Example 8

Are 11, 33, 22, 66 in proportion? 11 : 33 = 11 = 11 ÷ 11 = 11 = 1 : 3 33 33 ÷ 11 33

22 : 66 = 22 = 22 ÷ 11 = 1 = 1 : 3 66 66 ÷ 11 3

As, 11 : 33 = 22 : 66. Hence, 11, 33, 22, 66 are in proportion. Example 9

Think and Tell

How much distance will they cover by pedalling for 2 hours?

Are the ratios 25 g to 100 g and 5 kg to 25 kg in proportion? 25 g to 100 g = 25 : 100 = 25 = 1 = 1 : 4 100 4 1:4 ≠ 1:5

5 kg to 25 kg = 5 : 25 = 5 = 1 = 1 : 5 25 5

Therefore, the ratios 25 grams to 100 grams and 5 kg to 20 kg are not in proportion. Do It Together

Mansi’s weight is 25 kg and her mother’s is 60 kg whereas Ramesh’s weight is 30 kg and his father’s is 75 kg. Are the weights in proportion? Mansi’s weight = 25 kg; Mother’s weight = 60 kg

Ratio of Mansi’s weight to mother’s weight = 25 : 60 = 25 = 60 Ramesh’s weight = ______; Father’s weight = ______ Ratio of Ramesh’s weight to father’s weight = ____________ = Hence, the given weights are ______ proportion. 242

Proportional Terms Four numbers, a, b, c, and d are said to be in proportion if a : b = c : d. The proportional terms can be given as: a:b = c:d

First Term Second Term

Middle terms or means

Fourth Term

a:b::c:d

Third Term

Extreme terms or extremes

If a, b, c and d are in proportion, then, Product of extremes = Product of means Example 10

a×d=b×c

If 32 : x : : 9 : 36, ﬁnd the value of x.

Error Alert!

We know that, Product of extremes = Product of means Therefore, x × 9 = 32 × 36

The operation changes when transposing the values.

9x = 32 × 36 9x = 1152

x = 1152 = 128 9 Example 11

5x = 25

x = 25 × 5 = 125

5x = 25

x = 25 ÷ 5 = 5

If 4 pencils cost ₹26, how many pencils can you buy for ₹78? Let the number of pencils be x. Using method of proportion, 4 : x : : 26 : 78 x × 26 = 4 × 78 26x = 312 x = 12 Hence, 12 pencils can be bought for ₹78.

Do It Together

A recipe requires 3 cups of flour to make 2 dozen cookies. How many cups of flour would be required to make 15 dozen cookies? Let the number of cups of flour required be x. Using proportion, 3 : _____ : : 2 : _____ Therefore, _____ × 2 = 3 × _____

_____ = _____

Hence, ________ cups of ﬂour are required to make 15 dozen cookies.

Continued Proportion Any three quantities a, b and c are said to be in continued proportion if, First term

a:b::b:c

Third term Middle terms or Means

Chapter 15 • Introduction to Ratio and Proportion

243

Example 12

If x, 15 and 25 are in continued proportion, then ﬁnd the value of x.

Do It Together

As the given quantities are in continued proportion,

If 7 and 14 are in continued proportion, then ﬁnd the third term. As the given quantities are in continued proportion,

x : 15 : : 15 : 25

_____ : 14 : : _____ : x

x × 25 = 15 × 15

__________________________________________

25x = 225

⇒ _______ x = _______ ⇒ x = _______ So, the third term is _________.

x=9 So, the value of x is 9.

Do It Yourself 15D 1

244

Which of the numbers are in proportion? a 4, 5, 12, 15

b 8, 5, 9, 6

c 15, 5, 12, 4

d 9, 5, 18, 10

e 12, 6, 8, 4

f 10, 4, 20, 8

a 17 : 8 : : 1 : 3

b 5 : 15 : : 3 : 9

c 21 : 4 : : 84 : 16

d 7 : 12 : : 21 : 36

e 55 : 60 : : 42 : 7

f 24 : 10 : : 60 : 25

Check and state if true or false.

Create a proportion for each set. Use only 4 numbers from each set. a 3, 13, 9, 16, 39

b 42, 5, 7,3, 9

c 7, 21, 2, 4, 12

d 28, 14, 8, 56, 2

e 1, 3, 7, 21, 9

f 10, 6, 5, 2, 30

Find the missing numbers in the proportions given. a 2 : x : : 5 : 10

b 3:6::9:x

c x : 16 : : 12 : 24

d 1 : 3 : : x : 12

e 6 : x : : 10 : 15

f 4 : x : : 6 : 12

g 2x : 18 : : 6 : 12

h 8 : 16 : : 12 : x

a 12, 4, 9

b 2, 6, 1

c 4, 3, 6

d 32, 8, 24

e 30, 42, 32

f 18, 6, 54

a 9, 3

b 2, 4

c 16, 36

d 36, 42

e 14, 28

f 48, 96

30 : 25 : : 42 : x

Find the fourth proportional to:

Find the third proportion to:

The second, third and fourth term of a proportion are 21, 45, and 63. Find the first term.

If x, 9 and 3 are in continued proportion, then find the value of x.

If the ratio of the length and width of a rectangle is 5 : 3, and the length is 15 metres, what is the width of the rectangle?

10 A certain medicine is given in an amount proportional to a patient’s weight. If a patient weighing 55 kg requires 132 milligrams of medicine. What is the weight of the patient who requires 156 milligrams of medicine?

11 A boat can travel 126 miles on 18 gallons of gasoline. How much gasoline will it need to travel 175 miles? 12 225 g of flour is required to bake 9 cupcakes. Ruchi wants to bake 20 cupcakes and she has 475 g of flour. Is the flour enough? If not, how much more flour does she need?

Word Problem 1

A recipe calls for 2.5 cups of sugar to make 12 cookies. How much sugar is required to make 30 cookies?

Unitary Method Remember John and Fatima who cycle every day. After the cycling session, both of them have some juice. They finish 2 packs of juice in 6 days. Their father needs to stock the juice packs for 30 days. How can he do it? This can be done with the help of the unitary method. In this method we find the value of one unit first and then the value of the required number of units. Step 1: Write the value of the units given. Step 2: Find the value of 1 unit. Step 3: Find the value of the required units.

Example 13

A local train takes 10 hours to cover a distance of 650 km. What distance will it cover in 4 hours? Distance covered in 10 hours = 650 km

Distance covered in 1 hour = 650 10 Distance covered in 4 hours = 650 × 4 = 260 km 10

Chapter 15 • Introduction to Ratio and Proportion

Number of juice packs required for 6 days = 2 2 Number of juice packs for 1 day = 6 2 Number of juice packs required for 30 days = × 30 = 10 packs 6

Example 14

Seven men can paint a house in 12 days. How many men will do the same job in 14 days? In 12 days, the number of men who can paint the house = 7 In 1 day, the number of men who can paint the house = 12 × 7 = 84 (fewer days, more men) In 14 days, the number of men who can paint the house = 84 = 6 (more days, fewer men) 14 Hence, 6 men will be needed to finish the work in 14 days.

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Do It Together

Kunal went to buy some paint to paint his house. The shopkeeper showed him paint A that costs ₹750 for 15 litres and paint B that costs ₹900 for 20 litres. Which paint will be cheaper for him? Cost of 15 litres of paint A = ₹750 Cost of 1 litre of paint A = 750 = ₹_____ Cost of 20 litres of paint B = ₹_____ Cost of 1 litre of paint B =

= ₹_____

Hence paint _____ will be cheaper than paint _____.

Do It Yourself 15E 1

If the cost of 12 pens is ₹108, what is the cost of 20 such pens?

Raj can cycle at a speed of 2 km/h. How long will it take him to cover 8 km?

If 5 kg of rice cost ₹100, how much will 2.5 kg of rice cost?

The cost of 125 postcards is ₹375. How many postcards can be purchased for ₹180?

A bus travels 296 km in 8 hours. How much distance will the bus cover in 13 hours?

If Amit’s salary is ₹1,350 per month, how many months will it take him to earn ₹12,150?

The temperature increased by 25℃ in the last 100 days. If the rate of temperature increase remains the same,

A school needs to transport its students on a field trip. If 3 school buses can carry 120 students. How many school

Manya pays ₹18,000 as interest for 1 year. How much interest does she have to pay for 7 months, if the interest

how many degrees will the temperature increase in the next 16 days?

buses are required to transport 1000 students?

per month remains the same?

10 Raju buys 24 oranges for ₹336 from shop A and Kiran buys 21 oranges for ₹273 from shop B. Who has bought oranges at a cheaper price?

11 If 52 men can do a piece of work in 35 days, in how many days will 28 men do the same work?

246

Word Problem 1

Shreya can type 660 words in half an hour. How many words will she able to type in 10 minutes with the same eﬃciency?

Points to Remember •

Ratio is the comparison between two quantities of the same kind and the same units.

•

In a : b, a is called the antecedent and b is called the consequent.

•

Ratios are expressed in their simplest form when the HCF of the terms is 1.

•

Equivalent ratios are found by multiplying or dividing both terms of the ratio by the same non-zero number.

•

Ratios can be compared by converting them into like fractions.

•

Two ratios are in proportion if they are equal and are represented using ': :' or '='.

•

Three numbers a, b, and c are in continued proportion if a : b and b : c are in proportion.

•

Finding the value of one unit and then calculating the required number of units is called the unitary method.

Math Lab Proportion Picture Challenge Setting: In groups of 4 Materials Required: A set of task cards with diﬀerent scenarios (e.g., "Pizza Slices," "Toy Cars"), Ratio cards (e.g., 1 : 3, 2 : 5, 3 : 4, etc.), Sheets of paper, Coloured pencils, markers Method:

Each team randomly selects a task card with a scenario (e.g., "Pizza Slices") and a proportion card

Teams must use the proportion card to create a simple picture that represents the scenario. For

(e.g., 1 : 3).

example, if they have the scenario "Pizza Slices" and the ratio 1 : 3, they need to draw one slice of pizza divided into three equal parts.

After creating the pictures, the teams can colour them and label the parts to indicate the ratio.

Each team presents their picture, explaining the scenario, the ratio used, and how they visually

(e.g., one red slice and three blue slices). represented it.

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247

Chapter Checkup 1

A supermarket has 45 packets of potato chips, 66 packets of banana chips and 32 packets of corn chips. What is the ratio of:

a Potato chips to corn chips?

b Banana chips to the total number of chips?

Write the ratios in their simplest form. a 8 m : 700 cm

b 360 sec : 7 min

c 360 sec : 7 min

d 5 L : 750 mL

e 500g : 7kg

45 minutes : 4 hours

Find four equivalent ratios for the ratios given. a 3:8

b 5:9

e 9 : 11

10 : 19

c 11 : 15

d 4 : 13

g 13 : 27

h 15 : 17

Compare and fill in the blanks with <, > or = sign. a 4 : 5 ____ 5 : 7

b 2 : 7 ____ 3 : 9

c 3 : 7 ____ 7 : 8

d 8 : 11 ____ 7 : 9

e 9 : 12 ____ 10 : 13

Arrange the ratios 3 : 5, 5 : 7, 8 : 10, 9 : 11 in descending order.

Which of the ratios are not in proportion?

5 : 12 ____ 9 : 14

a 14 : 20 : : 28 : 40

b 16 : 17 : : 20 : 25

c 21 : 19 : : 21 : 34

d 40 : 24 : : 20 : 12

e 3 : 6 : : 22 : 44

24 : 25 : : 29 : 32

If 12 : x : : 6 : 36, find the value of x.

If 9, 12 and x are in continued proportion, then find the value of x.

If a box contains 24 red stamps and 36 blue stamps, what is the ratio of red to blue stamps in its simplest form?

10 If the ratio of the lengths of two rectangles is 3 : 5, and the area of the smaller rectangle is 15 square metres, what is the area of the larger rectangle?

11 The cost of 4 dozen eggs is ₹288. What is the cost of 15 such eggs? 12 The sum of two numbers is 30, and their ratio is 3 : 5. What are the two numbers? 13 A car travels at a constant speed of 60 miles per hour. How long will it take to cover a distance of 180 miles? 14 If the ratio of apples to oranges in a fruit basket is 3 : 2, and there are 15 oranges, how many apples are there? 15 If the ratio of pencils to pens in a box is 4 : 5, and there are 36 pens, how many pencils are there? 16 A machine in a soft-drink factory fills 750 bottles in 5 hours. How many bottles will it fill in the next 3 hours? 248

17 The present age of a mother is 36 years and that of her daughter is 20 years less than the age of the mother. What is the ratio of the age of the mother after 12 years to the age of the daughter 10 years ago?

18 If 18 painters can paint a building in 6 days, how many painters are needed to complete it in 4 days?

Word Problems 1

A school has 4 boys for every 5 girls. If there are 180 students, find the number of girls.

The ratio of John's age to Mary's age is 2 : 3. If John is 24 years old, how old is Mary?

If the ratio of mangoes to apples is 5 : 7 and there are 60 apples, find the number of

A train travels 1,200 km in 6 hours. How much time (in minutes) will the train take to travel

mangoes. 800 km?

Chapter 15 • Introduction to Ratio and Proportion

249

316

Symmetry

Let's Recall When a shape can be folded so that one half of it fits exactly on the other half along the fold line, the shape is said to be symmetrical. For example, when we fold a sheet of paper in two halves, the left side of the paper will overlap with the right side of the paper.

The shape can be called symmetrical.

The shapes that cannot be divided in two halves are called non-symmetrical shapes.

Let's Warm-up Look at the shapes. Tell whether they are symmetrical or non-symmetrical. 1

I scored _________ out of 5.

Mean, Median and Mode Lines of Symmetry Priya and Rishi visited different monuments in India during their summer vacation. They also went to see the Hawa Mahal at Jaipur.

Real Life Connect

Priya: Wow! This monument is so beautiful. Rishi: Y es Priya! Have you noticed something about the structure? Priya: Yes! It looks the same from the left and the right when we look at it from the centre. Rishi: Exactly! I wonder how the workers built it. They noticed the same symmetry in other monuments as well.

Symmetry in Shapes

Think and Tell

When a shape or figure is divided into halves that are identical so that one is a mirror image of the other, it is said to be symmetrical.

Where can you draw a line of symmetry in a line segment?

The line at which a shape or figure is found to be symmetrical is the line of symmetry. Line of symmetry

A shape or figure can have zero, one, two or more than two lines of symmetry. Let us look at some shapes and the lines of symmetry they have.

0 line of symmetry

Example 1

1 line of symmetry

2 lines of symmetry

4 lines of symmetry

How many lines of symmetry do equilateral, isosceles and scalene triangles have? The length of the three sides of an equilateral triangle is the same. Equilateral Triangle (All sides equal in length)

Isosceles Triangle (2 sides equal in length)

Scalene Triangle (All sides unequal in length)

3 lines of symmetry

1 line of symmetry

No lines of symmetry

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251

Example 2

Which of the following shapes have infinite lines of symmetry?

Parallelogram

Star

Circle

Ellipse

Infinite lines of symmetry

2 lines of symmetry

Let us draw the lines of symmetry for all the above shapes.

No lines of symmetry

5 lines of symmetry

We can draw an infinite number of lines of symmetry in a circle. Do It Together

Draw and write the number of line(s) of symmetry for each shape.

Lines of symmetry = 5

Lines of symmetry = ____

Symmetry in Designs or Figures Similar to shapes, we can also find lines of symmetry in different figures and objects around us. Let us look at some figures and objects along with the line(s) of symmetry that they have.

1 line of symmetry

Example 3

2 lines of symmetry

9 lines of symmetry

Draw the lines of symmetry in the letters. How many lines of symmetry does each letter have?

252

Let us draw the lines of symmetry for the given figures.

0 line of symmetry

Example 4

1 line of symmetry

2 lines of symmetry

How many lines of symmetry does the figure have?

Error Alert! The colours of the design should also overlap along the line of symmetry.

Let us draw the possible line(s) of symmetry for the figure.

The figure has 2 lines of symmetry. Example 5

Which of the following figures has more than two lines of symmetry? 1

Let us draw the line of symmetry for the given figures. 1

4 5

1 line of symmetry

5 lines of symmetry

1 line of symmetry

2 line of symmetry

Figure 2 has more than 2 lines of symmetry. It has 5 lines of symmetry. Chapter 16 • Symmetry

253

Do It Together

Draw and write the lines of symmetry for the figures.

1 line of symmetry

_____ line(s) of symmetry

Do It Yourself 16A 1

Draw an isosceles triangle. How many lines of symmetry does it have?

Line 1 and line 2 are drawn to show the lines of symmetry in each figure. Write which is the correct line of symmetry in each figure. a

l2 l1

Draw the lines of symmetry for each of the figures.

4 List and draw any 5 symmetrical objects in your surroundings. Also give their lines of symmetry.

254

Match the figures with the line(s) of symmetry that they have. a

Word Problem 1

Natasha and Pihu are playing chess. Natasha says that the chessboard has 4 lines of symmetry whereas Pihu says that it has only 2 lines of symmetry. Which of them is right and why?

Mean, Median Mode Reflection andand Symmetry Kritika went to the hospital with her mother for an eye checkup. She saw an ambulance at the hospital. She noticed that the spelling of ‘AMBULANCE’ was written as ECNALUBMA Kritika: Look Mom! The spelling on the ambulance is incorrect. Mom: No Kritika the spelling is correct.

Chapter 16 • Symmetry

255

Mom showed Kritika the image of the ambulance in a mirror. Kritika was amazed to see that the spelling appeared correct in the mirror.

NC BULA

ECNA

LUBM

Think and Tell

Why is the spelling of Ambulance written in the form of its mirror image?

After going home, Kritika wrote some letters in her notebook and started looking at them in a mirror. Mirror

S S A A E E

Did You Know? There are some mirrors that can reﬂect sound too. Such mirrors are called acoustic mirrors.

Mirror

The image in the mirror appears due to reflection. The concept of line symmetry is very closely related to the concept of mirror reflection. A line of symmetry is also called mirror line. An object and its image in a mirror are symmetrical along the mirror line. The figure is shown with its reflection along the mirror line. On comparing the lengths of the line segments AB and A’B’, BC and B’C’ we can conclude that: The image of an object or figure is exactly the same in shape and size as the original figure.

Error Alert! A

256

A’

The corresponding line segments of the mirror image should be an equal unit distance as the original image.

B’

C’

Example 6

Which of the figures has an incorrect reﬂection symmetry along the mirror line? 1

Figure 2 has an incorrect reﬂection symmetry.

Example 7

Which letter looks the same in its mirror image?

Let us draw the mirror image of the letters.

NN B B T T The letter T looks the same in its mirror image.

Example 8

Which of the figures is the correct reﬂection symmetry for the figure along the mirror lines m1 and m2? 1

The mirror image of Figure 1 is correct along mirror line m1 but there is no image along mirror line m2 it is incorrect.

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257

In Figure 2 the distance of the image from the mirror line is 3 units while the distance of the mirror image from the mirror line is 4 units. Hence it is not the correct answer. Figure 3 has a symmetrical mirror image along both the mirror lines, hence it is the correct answer. Do It Together

Complete the mirror image of the figures along the mirror line. 1

Do It Yourself 16B 1

Which of the two figures are mirror images of each other?

Draw the reﬂection of the following along the mirror line. a

Draw the reﬂection symmetry of the following on a square grid paper. a

258

Write the mirror image of the following words along vertical lines of symmetry. a MIRROR b SYMMETRY c REFLECTION d MATHEMATICS

Which of the following is the correct mirror image of the figure?

Word Problem 1

Guess the word: Suhani wrote a 3-letter word in her notebook. If you look at the word using a horizontal mirror, you will see the word ‘WOW’. What is the original word?

Points to Remember •

Symmetry is when a shape or figure is divided into halves by a line so that the two halves match exactly when folded along the line.

•

The line at which a shape or figure is found to be symmetrical is the line of symmetry.

•

A figure may have 0, 1, 2 or more than 2 lines of symmetry.

•

A Line of symmetry is also called the mirror line.

•

An object and its image in a mirror are symmetrical along the mirror line.

•

The image of an object or figure is exactly same in shape and size as the original figure.

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259

Math Lab Exploring Reflection and Symmetry Setting: In groups of 4 Materials Required: Pen and Paper, Mirror, Set of words written on paper cutouts. Method: Give the cutouts to the groups. Each group writes the words written on the cutout in its mirror image. The groups check their answer with the help of a mirror. The group with of the most correct answers wins!

Chapter Checkup 1

Draw the lines of symmetry in the letters and numbers.

How many lines of symmetry can be drawn in the figure?

Write if true or false.

a There is 1 one line of symmetry in the letter T. b A regular pentagon has 5 lines of symmetry. c All the quadrilaterals have 4 lines of symmetry. d A semicircle has one line of symmetry.

Draw and write all possible lines of symmetry for the figures. a

260

Complete the figures to make them symmetrical along the line of symmetry. a

Draw the reﬂection of the figures along the mirror line. a

Which of the figures will not look the same along both the mirror lines? a

Draw the reﬂection symmetry of the following on squared paper. a

Chapter 16 • Symmetry

261

Are the lines of symmetry marked on the given figures, correct? If not, draw the correct lines of symmetry. a

10 Draw the mirror image of the figure.

Word Problem 1

Rahul saw a rainbow in the sky. He told his friend

Vaibhav that the rainbow has infinite lines of

symmetry. Is Rahul correct? Explain your answer.

262

Construction of Line Segments and Angles

Let's Recall Line Segments A line segment is a part of a line with two endpoints. Imagine a piece of string between two points – that's a line segment! A line segment can be measured or drawn using a ruler.

We see line segments in different shapes and figures. Angles An angle is the geometric figure formed by two rays or line segments that share a common endpoint, called the vertex, and extend in different directions. We see angles in an inclined tree, pyramids, etc.

angle

Angles can be measured and drawn using a protractor.

Let's Warm-up Fill in the blank with the count of the number of line segments and angles in the shapes given below. 1

Line Segments = _____ Angles = _____

I scored _________ out of 4.

Construction of Line Segments Real Life Connect

In a vibrant town, curious Lily loved to explore. When her handyman aunt, Aunt Sarah, visited, Lily couldn't resist asking about her toolbox. Lily: Aunt Sarah, what's inside your toolbox?

Aunt Sarah: Well, Lily, it's ﬁlled with tools like measuring tapes and pencils for drawing diﬀerent shapes. Want to learn? Lily: Wow, let's start, Aunt Sarah!

Did You Know? The word "geometry" comes from the Greek words "geo" meaning "earth" and "metron" meaning "measure." Therefore, geometry is the measurement of the earth.

Constructing Line Segments and Their Copy Constructing Line Segments We can draw line segments using a ruler. We can also construct a line segment using the ruler and compass. Let us see how! Draw a line segment of length 4.3 cm.

Error Alert! Always measure the reading by viewing it from directly above it.







Using a Ruler 1 Mark a point and label it as X.

2 Place the 0 mark of the ruler on point X and draw a line up to the 4.3 cm mark.

3 Label point Y to form a line

segment XY of length 4.3 cm.

X 4.3 cm Y

Using a Ruler and Compass 1 Draw a line m. 3 Position the

compass' pointer at the zero mark of the ruler, and then adjust the opening until the pencil point aligns precisely with the 4.3 cm mark.

264

2 Mark a point X on the line m.

4 Without

altering the compass' width, position the pointer at X and draw an arc that intersects line m at point Y.

X 4.3 cm

Example 1

Construct a line segment of length 5.6 cm using a ruler and compass. 1 Draw a line x. 2 Mark a point M on the line x. 3 Position the compass' pointer at the zero mark of the ruler, and then adjust the opening until the pencil point aligns precisely with the 5.6 cm mark. 4 With the same compass width, position the pointer at M and draw an arc that intersects line x at point N. MN is the required line segment.

M 5.6 cm

Constructing a Copy of a Line Segment Construct a copy of the line segment MN using a ruler and a compass. Step 1: Begin by drawing a line called l and mark a point X anywhere on it. Step 2: Position the compass with its pointer at point M

of the provided line segment MN and adjust the width so that the pencil tip aligns with point N.

Example 2

Step 3: Without changing the compass' width, move the pointer to point X, and then draw an arc that intersects line l at point Y. This action results in line segment XY, which is an identical copy of MN in terms of length.

P 5.6 cm

Q 5.6 cm R

Construct a line segment of length 5.6 cm. Extend its length to twice the original length using a ruler and compass. What is the total length? 1 Follow the steps to construct a line segment of length 5.6 cm. 2 Keeping the compass width unchanged, position the pointer at Q and draw an arc that intersects line x at point R. PR is the required line segment. The total length of the line segment is 5.6 + 5.6 = 11.2 cm.

Example 3

Construct a line segment XZ equal to the sum of lengths of PQ and QR in the quadrilateral shown below without measuring it. P 1 Draw a line l and mark a point X on it.

2 Position the compass with its pointer at P of the side PQ of the quadrilateral, and adjust the width to align the pencil tip with Q.

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265

3 Without changing the compass' width, move the pointer to point X, and then draw an arc that intersects line l at point Y. 4 Position the compass with its pointer at Q of the side QR of the quadrilateral, and adjust the width to align the pencil tip with R. 5 Without changing the compass' width, move the pointer to point Y, and then draw an arc that intersects line l at point Z. XZ is the required line segment. Do It Together

XZ = PQ + QR

Construct a line segment equal to the sum of the lengths of all the sides of the triangle shown below without measuring it.

Do It Yourself 17A 1

Use a ruler to draw line segments of the given length. a 5.4 cm

b 6.7 cm

c 6.1 cm

d 7.2 cm

Construct line segments of the lengths given using a compass and a ruler. a AB = 4.3 cm

b DE = 5.6 cm

c GH = 3.9 cm

d PQ = 2.7 cm

Construct a line segment CD that is three times the length of line segment EF. Given that EF measures 2.8 cm,

Construct a line segment CD with a length of 8.9 cm. From CD, cut oﬀ CE of length 4.2 cm. What is the length of

Using a ruler, construct a line segment LM with a length of half the line segment NO. If NO measures 9.6 cm,

Given PQ = 2.3 cm and RS = 3. 6 cm, construct the segments so that: line segments such that:

construct CD accordingly.

line segment DE?

construct LM.

a AB = 3RS − 2PQ

c XY = RS + 2PQ

d DE = 2RS + PQ

Construct a copy of the line segments. a A

b BC = 4RS − 3PQ

b P

c M

N A

Copy the length of all the sides in the given shape to a single line segment using a ruler and compass. Measure the length of the line segment.

266

Word Problem 1

An ant travels a zig-zag path to reach a sugar cube placed at some distance. Draw a line segment equal to the length of the path without actually measuring it. Verify that the length of the line segment is equal to the distance travelled by the ant to reach the sugar cube using a ruler.

Construction of Perpendiculars and Bisectors Real Life Connect

Lia’s aunt, an architect, was visiting her. Leo: Aunt Mia, what's inside your architect bag? Aunt Mia: Well, Leo, it has tools like a set square for making perpendicular lines. Want to learn? Leo: Yes, Aunt Mia! But why are they important? Aunt Mia: Perpendicular lines are crucial in construction. They ensure that buildings stand tall, like your door that closes perfectly, perpendicular to the frame.

Constructing Perpendiculars and Their Bisectors Perpendicular lines are lines that form right angles with each other. We can construct perpendiculars using set squares or a compass.

Constructing a Perpendicular Using Set Squares Draw a perpendicular to a given line l at a given point P on it using a ruler and a set square. 1 Draw a line l and mark a point P on it. l

Chapter 17 • Construction of Line Segments and Angles

2 Place the ruler, aligning one of its edges along line l and hold it ﬁrmly. l

267

3 Position a set square so that one of its arms with a right angle touches the ruler.

corner of its right angle precisely matches point P.

4 Slide the set square along the ruler's edge until the

Maintain this set square placement and proceed to draw line PQ along the edge of the set square.

PQ is perpendicular to line l, and it is denoted as PQ ⊥ l. Q

Draw a perpendicular to a given point line l through a point P outside the line using a ruler and a set square. 1 Draw a line l and mark a point P outside it. P l

2 Place a set square beneath the provided line,

ensuring that one of its arms with a right angle aligns with line l. P

Place a ruler alongside the edge opposite to the right angle of the set square. P

4 While keeping the ruler ﬁrmly in place, glide the set square along the ruler until the other (vertical) arm of its right angle makes contact with point P.

5 Starting at point P, draw a line along the vertical edge

of the set square, allowing it to intersect line l at point Q.

P l

6 Consequently, line PQ is perpendicular to line l, and it is denoted as PQ ⊥ l.

268

Example 4

Draw a ΔSRQ with sides of any length using a ruler. Draw the perpendicular to the side QR passing through point X on it using a set square.

S Y

1 Draw ΔSRQ with sides of any length using a ruler. 2 Place the ruler, aligning one of its edges along RS and hold it ﬁrmly. 3 Position a set square so that one of its arms with a right angle touches the ruler. 4 Slide the set square along the ruler's edge until the corner of its right angle precisely matches X. 5 Maintain this set square placement and proceed to draw line XY along the edge of the set square. Consequently, line XY is perpendicular to line RS.

Do It Together

Draw a line segment FG = 6.7 cm. Mark a point X on FG and a point Y above it. Draw the perpendicular to FG passing through points X and Y using a ruler and set square.

Constructing a Perpendicular Using a Compass Let us now see how we can construct a perpendicular to a line from a point on it. Constructing a perpendicular to a line from a point on it Let XY be the given line and P be a point on it. 1 Place the compass needle at point P and draw a

semicircle of any radius that intersects line XY at points A and B.

2 Using point A as the centre and selecting a radius

greater than the distance from A to point P, draw an arc.

3 Using the same radius as used in step 2, draw an

arc from B, cutting the arc drawn from P. Label the point where the 2 arcs cut each other as Q.

Connect points P and Q to form line segment PQ. PQ is perpendicular to XY.

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269

Constructing a line perpendicular to a line from a point outside it Let XY be the line and P be a point outside it. 1 Place the compass needle at point P and draw an arc of any radius that intersects line XY at points A and B.

2 Using point A as the centre and a radius larger than half the length of segment AB, draw an arc. P

3 Using point B as the centre and the same radius as in the previous step, draw another arc that intersects the previously drawn arc at point Q.

B Y

4 Connect points P and Q to form a line segment PQ, which intersects line XY at point L. PQ is the perpendicular to line XY. P

P A

Y Q

Example 5

Y Q

Draw a Δ RST with sides of any length using a ruler. Construct the perpendicular to the side RT passing through point Z on it, using a ruler and a compass. 1 Draw a triangle ΔRST. 2 Place the compass needle at point Z and draw a semicircle of any radius that intersects RT at points A and B. 3 Using point A as the centre and selecting a radius greater than the distance from A to point Z, draw an arc. 4 Using point B as the centre and the same radius as before, draw another arc, which intersects the previously drawn arc at point Y. 5 Connect points Y and Z to form the perpendicular YZ to RT.

270

R A Z S

B T

Do It Together

Construct a line segment UV = 7.1 cm. Mark a point P, 3 cm away from endpoint U. Construct the perpendicular to the side UV passing through points P using a ruler and compass.

Constructing a Perpendicular Bisector Bisectors to a line segment are lines or line segments that divide a line into 2 equal parts.

Perpendicular bisectors are lines that intersect a line segment at a right angle and divide it in half. X

Bisector X

XO = OY

Perpendicular Bisector

Y AO = OB and XY ⊥ AB

Perpendicular bisectors are used to balance objects. Some of the examples may be: • Constructing buildings to ensure that the walls and beams are perfectly perpendicular and centred. • Garden water sprinklers for circular gardens to distribute water evenly to the whole garden. • Wheel alignment in vehicles and machinery. • Navigation to determine a ship’s position in relation to two landmarks.

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271

1 Let us construct a perpendicular bisector to a line segment AB of length 6 cm. Draw a line segment AB of length 6 cm. A

6 cm

3 Now, using point B as the centre and maintaining the same radius as before, draw arcs that intersect the previously drawn arcs at points P and Q, respectively.

6 cm

2 Using point A as the centre and selecting a radius larger than half of AB, draw arcs on both sides of AB.

6 cm

4 Connect points P and Q to form the line segment PQ, which intersects AB at point M. The line segment PM is the perpendicular bisector to AB.

Draw a quadrilateral PQRS with sides of any length using a ruler. Construct the perpendicular bisector of side RS.

Example 6

1 Draw a quadrilateral PQRS using a ruler. 2 Using point R as the centre and selecting a radius larger than half of RS, draw arcs on both sides of RS. 3 Now, using point S as the centre and maintaining the same radius as before, draw arcs that intersect the previously drawn arcs at points X and Y, respectively. 4 Connect points X and Y to form the perpendicular bisector XY to side SR of the quadrilateral. Do It Together

S Y

M X R

Draw the line segments DE = 5 cm and EF = 6 cm at some angle to each other. Construct perpendicular bisectors of both of these line segments. Do these perpendicular bisectors intersect each other if extended in both directions?

272

Do It Yourself 17B 1

Draw a line segment PQ of length 6.5 cm and mark a point R on the line. Construct a perpendicular to PQ

Draw a line segment AB = 4.5 cm. Mark a point X below it. Construct a perpendicular to AB passing through R

Draw a line segment XY = 5.6 cm. Mark a point M on it at a distance of 3 cm from Y. Draw a perpendicular to XY

Draw a line m. Draw a point above the line and label it as C. Draw a perpendicular to m passing through C using a

Draw a line segment FG = 7.2 cm. Mark a point X on it so that the point is more than 3 cm away from both ends

Draw a line segment DE = 6.1 cm. Draw perpendiculars to DE passing through both of its endpoints.

Construct perpendicular bisectors to line segments of the lengths given.

passing through R using a ruler and compass.

using a ruler and compass.

passing through M using a ruler and set square.

ruler and set square.

and draw a perpendicular at this point.

a 5.8 cm

b 9 cm

c 8.2 cm

d 7.6 cm

Draw a ΔABC with any suitable side lengths. Draw perpendicular bisectors of all three sides. Do these 3

Construct a line segment EF of any suitable length.

perpendicular bisectors intersect at the same point?

a Construct the perpendicular bisector of EF. Let the point where the right bisector intersects EF be M. b Draw a line segment MZ on this perpendicular bisector equal to the length of EF. Join ZE and ZF. What kind of triangle is ΔZEF based on the sides of the triangle?

c Draw perpendiculars to EF at the endpoints of EF. Also, draw a perpendicular to MZ passing through Z.

Let the points where these perpendiculars at E and F intersect perpendicular at Z be X and Y. What kind of quadrilateral is EFYX?

Word Problem 1

John is an architect working on a new house design. He needs to draw a perpendicular wall to

the floor that measures 8 metres in length. He wants to ensure that the new 4 m long wall starts from the midpoint of the floor. He wants to draw a plan on paper using a ruler, a set square, and a compass in his toolkit. How can he use these tools to accurately draw the perpendicular wall that starts at the midpoint of the existing 8-metre floor?

Note: John is representing 1 m on the ground as 1 cm on paper.

Chapter 17 • Construction of Line Segments and Angles

273

Construction of Angles Constructing Angles and Their Bisectors Emma is visiting her grandparents in the village. She sees a farmer ploughing his ﬁelds.

Real Life Connect

Emma: Grandfather, why is the plow shaped like that? Grandfather: Emma, the farmer uses angles to plough his field effectively. He adjusts the angle of his plow to control the depth of the furrows, allowing him to plant his crops evenly. Angles are geometric figures formed by two rays or lines that share a common endpoint, known as the vertex. Angles are typically measured in degrees. You can see angles all around us in trees, intersection of roads, tilted roofs, etc.

Constructing Angles of Given Measures Let us draw an angle using a protractor. Draw ∠XYZ of measure 56º. 1 Place the protractor so that Y coincides with the midpoint of the baseline, and the baseline (0-180º. line) aligns with ray YZ.

2 Draw ray YZ . Start with the starting point at Z and mark point X at 56º.

3 Draw ray YZ. Thus, ∠XYZ = 56º.

Y Example 7

56°

Use a protractor to draw ∠DEF measuring 294º. To begin drawing 294º, we will ﬁrst subtract it from 360º. Hence, 360º − 294º = 66º. Now we will draw 66º. D

66° E Do It Together

Draw an ∠RST of measure 216º.

294°

Error Alert! Make sure the extension of the compass isn’t aﬀected while drawing the arc. It may lead to inaccuracies in construction.

274

Constructing a Copy of an Angle ∠ABC is the angle. Construct a copy of the angle. 1 Draw a ray denoted as OX.

2 Draw an arc with its centre at point B, using any desired radius, to intersect line segments BA and BC, marking the points of intersection as P and Q.

3 Using O as the centre and the same radius as before, draw another arc that intersects ray OX at R. P

P Q

4 Now, using point R as the centre and a radius equal to the length of PQ, intersect the previously drawn arc at S.

S Q

5 Draw OX passing through S. Thus, ∠YOX = ∠ABC.

P B

Example 8

S Q

Draw a copy of the ∠ABC of the quadrilateral shown below. D

P B

1 Draw a ray OX. 2 With B as centre and any radius, draw an arc cutting BA and BC at P and Q. 3 With O as centre and the same radius, draw an arc, cutting OX at R. 4 With R as centre and radius as PQ, cut the arc through R, at S. 5 Join OS and extend it to point Y. Then, ∠YOX = ∠ABC.

Chapter 17 • Construction of Line Segments and Angles

275

Do It Together

Draw a copy of the ∠QPR of the triangle shown below. P

Constructing an Angle Bisector An angle bisector is a geometric line or ray that divides an angle into two congruent or equal angles. In other words, it is a line or ray that passes through the vertex of an angle and divides that angle into two smaller angles of equal measure. OT is the angle bisecor to ∠POQ P

T O

∠POT = ∠QOT

Let us construct an angle bisector to ∠AOB. 1 Draw angle AOB of any measure. Using O as the centre and a chosen radius, draw an arc that intersects lines OA and OB, marking the points of intersection as P and Q, respectively.

2 Using point P as the centre and a radius greater than half the length of PQ, Draw another arc.

Q O

O P

3 Using point Q as the centre and the same radius as previously, draw another arc intersecting the previously drawn arc at R.

4 Extend the line segment OR to a point X. OX is the bisector of angle ∠AOB. Verify that ∠BOX = ∠AOX.

B Q O

276

B R

Q A

X R

Remember! The arcs always need to have a radius greater than half the length of the line segment to be bisected; otherwise, they won’t intersect.

Example 9

Construct an angle bisector to ∠QRZ. Q

M Q

Q E

L E

Construct ∠XOY = ∠QRZ using the steps explained previously. With centre R and radius more than half of ED, draw an arc. With centre E and the same radius as before, draw another arc, cutting the previously drawn arc at L. Join RL and produce it to any point M. Then, ray OM bisects ∠QRZ. Do It Together

Construct an angle bisector to the complement of ∠A.

Constructing 60°, 120° and 30° Angles Let us construct an angle of 30º, 60º and 120º using a compass. Steps to construct a 60º angle. 1 Draw the ray OA. O

2 Using point O as the centre and an appropriate radius, draw an arc that intersects OA at point B. A

3 Now, using point B as the centre and maintaining the same radius as previously, draw another arc to intersect the initial arc at point C. C

O B

Chapter 17 • Construction of Line Segments and Angles

O B

4 Join O and C, extending the line to reach point D. ∠DOC is the required 60° angle. D C O B

277

Steps to construct a 120º angle. To construct an angle of 120° we can ﬁrst construct an angle of 60° and then extend it to construct another 60° angle on one of the arms to get the 120° angle. 1 Draw the ray OA. O

2 Using point O as the centre and a suitable radius draw an arc that intersects ray OA at point B.

O B

3 With point B as the new centre and maintaining the same radius, repeat the process to intersect the arc once more at C. Continue by using point C as the centre, again with the same radius, to intersect the arc once more at point D. D

4 Finally, join OD and extend this line segment to reach point E. ∠EOA is the required 120° angle. E

O B

Steps to construct an angle of 30º A 30° angle is half a 60° angle. So, a 30° angle can be constructed by ﬁrst constructing the 60° angle and then constructing its angle bisector to get two 30° angles. 1 Construct an angle ∠XOY = 60°.

2 Draw the bisector OZ of ∠XOY. We get 2 angles of 30°. ∠XOZ = ∠YOZ = 30°.

Example 10

O B

O P

Z 30°

Draw ∠XOY = ∠41° using a protractor. Construct an angle of 60° using OX as the base. 1 Draw ∠XOY = ∠41° using protractor as explained previously. 2 Using point O as the centre and an appropriate radius, draw an arc that intersects ray OX at point P. 3 Now, using point P as the centre and maintaining the same radius as previously, draw another arc to intersect the initial arc at point Q.

41°

4 Join O and Q, extending the line to reach point Z. Hence, ∠ZOX =∠60° and ∠XOY = 41°. Do It Together

Draw ∠PQR = ∠47° using a protractor on the line segment QR. Construct an angle of ∠30° using OX as the base.

278

Constructing 45°, 75°, 150° and 135° Angles To construct the angles 45°, 75°, 150° and 135° we will use the steps of construction of the 90°, 60°, 30°, and 120°. Constructing an angle of measure 45º. A 45° angle is half of a 90° angle. So, to construct a 45° angle, we ﬁrst need to construct a 90° angle and then construct its angle bisector to get two 45° angles. 1 Draw an ∠ABC = 90°.

2 Draw BX, the bisector of ∠ABC. Hence, ∠XBC = 45°.

S R B

90°

Q B

Constructing an angle of measure 75º. To construct angle of 75°, we have to bisect the angle between 90° and 60°. 1 Construct ∠ABC= 90°.

2 With point Q (∠QBC = 60°) as the centre and a radius greater than half of QY, draw an arc in the interior of ∠XBQ.

A X

X R Y

90° B

3 Now, with point Y as the centre and using the same radius as before, draw an arc to intersect the previous arc at point S.

4 Finally, connect points B and S, extending the line to form BZ. Thus, ∠ZBC = 75°.

R Y

Q C

Z S Q

Think and Tell

1 Can you draw an angle of 7 ° by bisecting an angle of 15º? How? 2

Chapter 17 • Construction of Line Segments and Angles

279

Constructing an angle of measure 150°. To construct a 150° angle, we can bisect the angle between the 180° and 120° angles to get the 150° angle. 150 = 90° + 60°. So, to construct an angle of 150°, we can also construct a 90° angle and then a 60° angle on one of the arms. Let us see one of the methods. 1 Draw a line and mark a point B on it. Construct an angle of ∠RBC = 120°.

R l

2 Extend from point S, maintaining a radius larger than half of SR, and draw an arc on the same side as points R and Q.

Q B

3 With point R as the centre and using the same radius as before, draw an arc to intersect the previous arc at point X.

R l

X S

Constructing an angle of measure 135º. To construct angle of 135°, we would have to bisect the angle between 90° and 180°. 1 Construct ∠DBC = 90° using steps explained previously. D R l

Q P

2 Now construct an angle bisector to ∠DBS. ∠ABC = 135°.

280

4 Join B and X, extending the line to A, thus forming BA. Hence, ∠ABC = 150°.

D F R

Q P

Draw an angle ∠ZOY = 51° using a protractor. Also, construct an angle of 45° using OZ as the base (using a ruler and compass).

Example 11

A Z

1 Construct an ∠ZOY = 51° using a protractor as explained previously.

2 Draw an ∠DOZ = 90° as explained above.

3 Draw OA, the bisector of ∠DOZ.

51°

Hence, ∠ZOY = 51° and ∠AOZ = 45°. Do It Together

Construct an angle of 135° on the base of ∠ABC using a ruler and compass.

Do It Yourself 17C 1

Construct the angles with the help of a protractor. a 20°

b 35°

c 50°

d 65°

e 80°

Draw a copy of each of these angles. a

P O

c O

S R

Construct the bisector of all the angles in Question 2 (a) to (d) using a ruler and compass.

Construct the following angles using only a ruler and compass. a 60°

b 90°

c 150°

d 120°

e 75°

f 105°

Construct a copy of each of the angles of the triangles given below. P

Chapter 17 • Construction of Line Segments and Angles

281

Draw an angle of 126° using a protractor and bisect its supplementary angle.

Draw an angle of 71° and bisect its complementary angle.

Word Problem 1

John is an architect designing a blueprint of a complex building with unique angles. His friend

Naman sees him working on the design of the new building shown below. Naman wants to copy the angles in the design used by John and use them in his own projects. Help Naman by copying the angles in John’s shape.

New Building

Points to Remember

282

•

A ruler can be used to draw line segments and a protractor can be used to draw angles.

•

Set squares can be used to draw perpendiculars to a line segment passing through points on or above the line segment.

•

A compass can be used to construct a copy of a line segment or angle without actually measuring it.

•

A perpendicular bisector is a line that intersects a given line segment at a right angle and divides it into two equal parts.

•

An angle bisector is a straight line or ray that divides an angle into two equal parts, creating two smaller angles with the same measure.

Math Lab Geometry Construction Challenge Setting: In a group of 3-4 Materials Required: Rulers, Compasses, Protractors, Set squares, Blank sheets of paper, Pencils and erasers. Method:

Provide a simple architectural ﬂoor plan or sketch on the board, highlighting a room or structure that requires perpendicular walls, perpendicular bisectors, and angle bisectors. Discuss the measurements and angles needed for this design.

Have students work in groups to create their architectural blueprint. Encourage them to design a room, a building, or any structure of their choice while incorporating the following: • At least two perpendicular walls. • A perpendicular bisector for a speciﬁc wall. • An angle bisector for an angle within their design. • At least two measured angles using a protractor.

Allow students to present their blueprints to the class. Their peers can review the designs,

Have a class discussion on the challenges faced and lessons learned during the activity.

identify the perpendicular lines, bisectors, and measured angles.

Chapter Checkup 1

Using a ruler and a compass, construct a line segment: a PQ = 3.6 cm

b AB which is four times the length of line segment XY

c XY with a length of 6.7 cm.

d RS which is twice the length of PQ

Construct a line segment EF with a length of 10.2 cm. From EF, cut oﬀ EG, which measures 3.8 cm. Determine the

Using a ruler, construct a line segment MN that is 9.1 cm long. From MN, cut oﬀ MO, which has a length of 6.5 cm.

Given ST = 4.7 cm and UV = 6.2 cm, construct a line segment as given:

length of FG.

Find the length of NO.

a WX = 2ST – 3UV

b YZ = 5ST – 2UV

c PQ = ST + 4UV

d AB = 3ST + 2UV

Draw a line segment CD of an unknown length. Without measuring CD directly, draw a line segment EF that is twice the length of CD.

Chapter 17 • Construction of Line Segments and Angles

283

Construct a copy of any of the two sides in the given shapes to a single line segment using a ruler and compass. Measure the length of the line segment in each case. a

Draw line segments of the lengths given. Use a ruler and compass to construct a perpendicular on the line segments. a 8 cm

b 9 cm

c 5.8 cm

d 7.5 cm

Draw a line segment CD = 6.8 cm. Mark a point Y above it. Construct a perpendicular to CD passing through Y

Draw a line segment MN = 9.5 cm. Mark a point P on it at a distance of 4 cm from N. Construct a perpendicular to

using a ruler and compass.

MN passing through P using a ruler and set square.

10 Draw a line n. Mark a point K below it. Construct a perpendicular to n passing through K using a ruler and set square.

11 Draw angles of given measures using a protractor. a 27°

b 136°

c 53°

d 73° e. 89° f. 101°

12 Draw an angle of 139° using a protractor and bisect its supplementary angle. 13 Draw an angle of 17° and bisect its complementary angle. 14 Construct an angle of 60° using a ruler and compass. 15 Draw a copy of each of the indicated angles in the shapes given below and bisect them. a

C A

M L

Q J

16 Draw a line segment JK = 5.1 cm. Mark two points X and Y on the line segment so that X is 1 cm away from J and Y is 2 cm away from K.

a Draw perpendiculars to JK passing through X and Y. b Now, choose any point P on a perpendicular passing through X and Q on a perpendicular passing through Y. c Mark any point T on the perpendicular bisector of JK. Is TJ = TK?

17 Draw a quadrilateral ABCD with sides of any suitable length. Construct the perpendicular bisectors of all four sides using a ruler and compass. Do the perpendicular bisectors intersect at a common point? Explain your ﬁndings.

284

Word Problem 1

Ent

ry s

ide

et e sid

Chapter 17 • Construction of Line Segments and Angles

e Str

Samantha wants to construct a gate along the centre of the longest side in her triangular garden. She draws the shape of the garden as shown. She needs to accurately draw the garden's longest side's perpendicular bisector on paper to visualise the gate's placement. Help her construct the perpendicular bisector.

Eastern side

285

MATHEMATICS

Key Features

Imagine Mathematics

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