Imagine_Maths_CB_Grade5 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.

Imagine Mathematics

About the Book

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

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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NEP 2020 based

NCF compliant

CBSE aligned

14/12/23 6:43 PM

MATHEMATICS Master Mathematical Thinking

Grade 5

Maths Grade 5 Book_Chapter 1-6.indb 1

18-12-2023 11:00:56

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 maximises 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

Maths Grade 5 Book_Chapter 1-6.indb 2

18-12-2023 11:00:56

Multiply (×)

Add (+)

× 10 – 20

= 18 + 4 × 10 – 20

= 18 + 40 – 20

0 – 20

= 18 + 40 – 20

= 58 – 20

1 + 1

1 4 6

Subtract (–) O

1150 + 450 = 1600 beads.

Do It Yourself 1C

=058 – 20 = 381600 beads. We have

5 5 0

0 0

This is called addition.

Arrange the following numbers in ascending order. Now, let us say, out of the 1600 beads, we give 200 beads to1 our friend. How many a 1,00,36,782; 5,00,00,367; 8,87,21,460; 93,12,820

g DMAS, the students are left with 38 apples. beads do we have now?

– 200 = 1400 sets of 10 toy cars. His father gives him 5 more He1600 then wants to 1 6 toy 0 cars. 0 So, we have 1400 beads left. 2 0 0 to his friend. How many cars will he be left1– with? 0 4 0 This is called subtraction. Th

b 92,56,890; 36,81,910; 6,92,10,350; 8,26,00,031

c 5,00,21,138; 6,04,50,821; 6,50,24,567; 9,45,21,823

es him 5 more cars = 2 × 10 + 5

division related? Letʼs Warm-up

Introductory

MAS to find out.

Write the greatest and the smallest 8-digit numbers using all the digits but not repeating any digits. of Numbers Up to 6 Digits Concept b 8, 3, 9, 4, 7, 1, 6 Addition and Subtraction a 2, 7, 1, 0, 8, 6, 4 c 7, 5, 2, 0, 4, 9, 3

Match the following.

page with a

Step 2

Arrange the following numbers in descending order.

Think and Tell

s to his friend = 2 × 10 + 5 – 4 cars left = ?

Now what if we say that we have 1000 8,26,34,510; 87,92,345; 6,70,81,256; 4,50,00,921 a How beads and 4 friends. many beads will each friend get?b 42,56,789; 250 8,01,20,450; 92,11,108; 6,78,20,001 How many beads do they have in total? 4 1000 Let us divide 1000 by Let us count. 9,67,28,891; 7,88,21,134; 5,78,20,010 c 4.5,43,33,867; –8 20 1000 ÷ 4 = 250 150 + 150 + 150 + 150 = 600 –20 00 Write the greatest and the smallest 7-digit numbers using all the digits only once. 3 Or, 4 × 150 = 600 –0 Are multiplication and This is called division. 8, 9, 1 a 5, 3, 4, 0, 0 b 5, 7, 6, 2, 1, 3, 8 c 1, 0, 3, 5, 6, 2, 4 This is called multiplication. Division is equal sharing. Imagine there are 4 children and each one of them has 150 beads.

s of 2 = 2 × 10

ove to the

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

he buys 1000

division,

quick warm-up Step 3

Multiply (×)

Add (+)

= 2 × 10 + 5 – 4

= 20 + 5 – 4

= 25 – 4

180 ÷ 2

5500

7000 – 1500

280

2100 + 250

140 × 2

150 × 3

Step 4 Subtract (–)

Father: We made ₹2,50,678 by selling cakes and Write the greatest 8-digit number and the smallest 7-digit number.

₹1,56,240 by selling cookies. with a real-life different digits b Five different digitsSahil: Wow! How much c Four did we make last year?

a Two different digits

example

450

= 25 – 4 = 21

Sahil’s father runs a bakery. At the end of every year, they calculate the total sales for the different types of baked goods they sold.

Real Life Connect

introduction

Father: Last year we made a total of ₹3,17,500. Sahil wonders whether they have made more or less this year!

Rounding Off Numbers

2350

and Subtraction Remember, the number of vaccine doses donatedAddition by India to Bangladesh was I scored _________ out of 5. Sahil wants to know the total sales they made this year. 2,80,82,800. He also wants to know whether they have made more or less this year than last year. will Sahil do that? Let us help him out! But what if we wanted to convey this number to aHow friend? The number 2,80,82,800 is very Adding Numbers Up to 6 Digits inconvenient to read and say out aloud.

Tom is left with 21 toy cars.

If Sahil wants to find the total sales they made this year, he will have to add the numbers

₹2,50,678 and ₹1,56,240. What if we just said that India donated about 2,81,00,00 vaccines to Bangladesh, it still

have already learnt how two numbers are added. When we put two or more numbers 0 flowers of 2 colours each. Her mother gives her 8 more flowers. Lisa gives gives a fair idea of about how many vaccines wereWe donated. iscalled called rounding a together to find theirThis total, it is addition. Let us help Sahil calculate the total sales flowers that her mother gave to her friend, Siya. number off. for this year using the following steps. A quick-thinking Fun fact, related While rounding numbers off, terms like “about” Add: ₹2,50,678 + ₹1,56,240 flowers does she have now? Sahil’s father made ₹4,06,918 this year , and “approximately”to are added to convey that So, question the concept which is more than more than last year. Think and Tell the number is close to the exact. Total number of pensAdd = 9,28,667 Did You Know? 1,98,794 and 52,250 and 21,000. ber of flowers = _________________ TTh

Addend

Sum

TTh

Example 1

If we do not follow DMAS,

ven by mother = _________________

carry

the steps, we can add the numbers as: red pens =Using 58,475 The diameter of the Sun is 1 9 8 7 9 4 We can round off numbers to Number differentofplaces, 1,98,794 + 52,250 + 21,000 = 2,72,044. 5 2 2 5 0 approximately 14,00,000 km. such as tens, hundreds and thousands. Number of Let pensusthat Aare not produced red = Total + − Number 2 1 0of red 0 0pens company 4,56,360 number boxes in 2019.of Thepens same company produced 3,60,780 boxes in 2018 2 7 2 0 4 4 understand how we can do this. 1

will we get the same answer?

Example 2

90,995 boxes in 2017. How many boxes have Therefore, 9,28,667 – and 58,475 = 8,70,192. they produced in three years?

numbers in a place value chart and check the number of digits. C 3

TTh

Do It Together

A store sold 2,85,586 shirts in a year. The sale of shirts for the first month was 4640. How many shirts were in on the remaining 11 months? 19 Chaptersold 2 • Operations Numbers Up to 6 Digits Shirts sold in 1 year or 12 months = ___________________

Chapter 1 • Numbers Up to 8-digits

7 digits

commonly made

Remember!

99,000.

A number with more digits is always greater.

as donated more vaccine doses.

Always write the smaller number below the larger number. We can solve 5 – 2 = 3.

and analytical 4

•

7 multidisciplinary 0 0 5 0

and fun

1=1

8=8

Ravi has 21 kg of rice. He packs the rice in packets of

2 3

1 6 457

3 b kg each. 4 How−many 2 such− 1 4

Ravi has 21 kg of rice. He packs the rice in packets of

L? How many bottles of capacity 3 L can be filled from a container of capacity 4 4

c Subtract the sum of 2

2 4

summary 1 3

10 − 3

___

kg each. How many such

Add the given numbers. • To add or subtract like fractions, simply add or subtract the numerators and keep the a

Aim: To understand fractions and add fractions.

the5same. 1 denominators 1 subtract the fractions. 3 and then 6 add or12

b fractions, first c like fractions d5 23• To+6add into 3 1or subtract −31 unlike 1 convert 8 3 the3unlike 5 fractions 2 9 1

+ convert 8 the 1 mixed 5 numbers 0 7 into improper+ 6 6add7or subtract 2 To mixed numbers, fractions and then perform addition or subtraction. • To multiply 2 fractions, find the product of their numerators and the product of their denominators. Subtract the given numbers. • Reciprocal of a number is that number which when multiplied by the original number gives 1 as the answer. b a c +

–

5•

Method: 1

Create the fraction bingo cards and create the list of fractions that are to be called out.

Each student get a fraction bingo and a coloured pencil or marker.

Call out fraction questions from the prepared list.

The students put a tick or cross over the fraction that is called out.

Once a row, column, or diagonal of the fraction bingo is crossed out, the student adds

Fraction Bingo Aim: To understand fractions and add fractions.

Chapter 2 • Operations on Numbers Up to 6-digits

1 1 1 1 3 . and 1 from the difference 5 andAccess QRofCode: 3 3 2 2

Materials Required: Fraction Bingo cards, markers, or coloured pencils

To divide a given fraction by another fraction, multiply the first fraction with the 8 0 of0the second fraction. – 5 4 2 4 2 – reciprocal

•

Math Lab

1 1 1 1 and 3 from the sum of 5 and 2 . 3 4 4 3

difference of 2 d Subtract theFraction Bingo

Maths Grade 5 Book_Chapter 1-6.indb 3

___

Chapter end

To divide a given fraction by another fraction, multiply the first fraction with the reciprocal of the second fraction.

Chapter 6 • Operations on Fractions

3 L can be filled from a container of capacity 57 L? 2 How many bottles of capacity When 0 is subtracted from a number, the 4difference is zero. 4

to interactive

digital resources

Materials Required: Fraction Bingo cards, markers, or coloured pencils Method: 1

Create the fraction bingo cards and create the list of fractions that are to be called out.

Each student get a fraction bingo and a coloured pencil or marker.

Call out fraction questions from the prepared list.

The students put a tick or cross over the fraction that is called out.

Once a row, column, or diagonal of the fraction bingo is crossed out, the student adds

all the fractions in that line. If the sum of all the fractions is more than 3 wholes, then that student is the winner.

After the winner is declared, students can check in their fraction bingo which row has the sum of more than 3 wholes.

2 1 Chapter 6 • Operations on Fractions all the fractions that line. If the sumstudied of all the fractions is more than 3 wholes, then on Monday. He studied 1 for hours less on Tuesday 1 inNavneet 4 hours that student is the winner. 3 4 6 After the winner is declared, students can check in their fraction bingo which row has the sum of more than 3 wholes. than on Monday. How many hours did Navneet study on Tuesday?

1,86,01,769 or 1,86,04,766? Th

Math Lab

activity

Word Problems

7 = classroom 7 1>0 4=4

TTh

Reciprocal of a number is that number which when multiplied by the original number gives 1 as the answer.

•

3 1 fractions and then perform addition or subtraction. 5 ? be added to 3and theto get b What • To multiply 2 fractions,must find the product of their numerators product of their 4 denominators. 3

Write True (T) or False (F).

then add or subtract the fractions. 1 = 1 • and 2=2 4<6 To add or subtract mixed numbers, convert the mixed numbers into improper

TTh

1 7 • To add likeis fractions, addthan or subtract5 the numerators and keep the What 4 simply less ? a or subtract 3• denominators = 3 the same.3 =33 8 To add or subtract unlike fractions, first convert the unlike fractions into like fractions

Word Problems 1

Points to Remember

TTh

Points to Remember involved in subtraction can be changed. d The order of numbers

2 the 3questions. 6 5 3 Answer

6,47,00,508

Do It Yourself 2A

2 5 3 + − 3 packets 3does 2 he pack? 6 3 9

6=6

4 subtracted from itself, the difference is zero. a When the number is packets does he pack?

Simplify.

Word Problems

c When the order of the addends is changed, the sum remains the same.

mbers 6,47,17,389 and 6,47,00,508.

But we cannot solve 3 – 5.

them

ave the same number of digits, that is, 8 digits.

4,13,23,657.

H 15

Error Alert!

how to avoid

umbers in a place value chart and check the number of digits.

4=4

Th ___

___

mistakes and

want to compare two numbers with the same number of digits? Let us 456 and 4,13,23,657.

he same till the thousandsquestions place. e digits in the hundreds place.

13 8

___

Pointing out

mind has 8 digits, as 7 digits and 3,09,13,200

TTh

–

Therefore, the company sold ____________________ shirts in 11 months.

An important

HOTS:

Shirts sold in 11 months = ____________________

8 digits

s have different number ofto digits. point keep in

ring the digits from the left till we Applicative The number with the greater digits

Shirts sold in 1 month = ____________________

103 3 1 kg wheat. Sanjay uses 10 kg wheat. What quantity of wheat is 4 2 left in the drum?

103

A drum has 50

1 Raghuveer purchased 4 types of fruit weighing 7 3 kg. He purchased 1 kg of 2 4 2 1 apples, 2 kg of pears, 1 kg of oranges and some litchis. What was the weight of 3 4 litchis that he purchased?

iii

18-12-2023 11:00:59

G rad ual R e le ase 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

Perimeter Real Life Connect

Real Life Connect Theoretical Explanation

Students of 5B are decorating their classroom for Independence Day! They want to stick a ribbon around the class blackboard and the bulletin board. How will they do this?

Perimeter of Squares and Rectangles Find the area of the whole shape. Step 5

The design that the students have thought of + 48 sq. cm = 63 sq. cm has the ribbon= 15on all sides of the board and the bulletin board. That is exactly what Error Alert! perimeter is!

Area of the whole shape = area of rectangle A + area of rectangle B

Always calculate the area before comparing the sizes of two shapes. A shape can “appear” to be

I do Example 5

Remember! bigger yetstudents have a smallercan area. find the perimeter of So, ifand the Perimeter is the total distance around the the blackboard and the bulletin, they will be boundary of a closed shape. able find length of ribbon needed. Find the to area of a the rectangle of length 15 cm and breadth 20 cm. Here, l = 15 cm and b = 20 cm

Also, note that the board is a rectangle in shape. Its opposite sides are equal!

Since the area of a rectangle = l × b, the area of the rectangle = 15 × 20. The area of bulletin board = 300 sq. cm Example 6

Find the area of the given shape. 4 cm

Examples

4 cm 3 cm 5 cm

A 3 cm 6 cm

3 cm

5 cm

9 cm

3 cm 6 cm

2 cmsides are equal. The bulletin board, on 2the other hand, is a square. All its cm C

15 cm

If the students find the perimeter of the blackboard (rectangle) and the bulletin board First split the shape into three rectangles A, B, and C and find the missing lengths. (square) they will know the length of the ribbon they need. The area of rectangle A = l × b = 4 × 3 sq. cm = 12 sq. cm

The area of rectangle B = l × b = 9 × 3 sq. cm = 27 sq. cm

Perimeter of a Rectangle

The area of rectangle C = l × b = 15 × 2 sq. cm = 30 sq. cm

If we know the perimeter of any rectangle, we can find the perimeter of the blackboard too. Let us take a rectangle with = (12 + 27 + 30) sq. cm = 69 sq. cm length = l cm and breadth = b cm, as shown.

The area of the whole shape = area of rectangle A + area of rectangle B + area of rectangle C

Do It Together

Find the area of the shape.

The perimeter is sum of all sides = l + b + l + b

The area of rectangle A = _______________________

b So, the Perimeter of a rectangle = 2l + 42cm The area of rectangle B = _______________________

3 cm A

2 cm

b l

The areateacher of the whole shape = _______________________. The and students together measured the size Bof the blackboard. They found the

length of the blackboard as 100 cm and breadth as12 80cmcm.

Maths Grade 5 Book_Chapter 1-6.indb 4

Chapter 12 • Perimeter and Area

209 18-12-2023 11:01:02

Example 7

What is the area of the red triangle? 1 unit

1 unit

1 2

To find the area of this triangle, draw a rectangle around the triangle and split the triangle into two parts.

Did You Know?

Area of the triangle 1 = half of area of rectangle 1 = half of 6 = 3 sq. units

GRR Steps

used the concepts of perimeter

Area of the triangle 2 = half of area of rectangle 2 = half of 6 = 3 sq. units Snapshot

Unit Component

and area to measure and plan fields for agriculture.

Area of the whole triangle = 3 sq. units + 3 sq. units = 6 sq. units Do It Together

We do

The ancient Egyptians

Find the area of the given triangle.

Points to Remember

Do It Together

To find the area of this triangle, draw a rectangle around the triangle and split the Perimeter a rectangle = 2l + 2b and Perimeter of a square = 4 × s. triangle intooftwo parts. Area of a rectangle = l × b and area of a square = s × s. Area of Rectangle 1 = ____________; Area of triangle 1 = ____________ Area is always measured in square units. Area of rectangle 2 =compound ____________; Area ofwe triangle 2 =shapes ____________ To find the area of shapes, split the into squares or rectangles. Area of whole triangle = ____________.

• • • •

211

Chapter 12 • Perimeter and Area

Math Lab

Build a paper house!

Setting: groups 12B of 3 Do It InYourself

Materials Required: Origami sheets, a pair of scissors, glue or tape, rulers, markers or

coloured pencils, templates of squares, rectangles and triangles Which unit would you prefer to find the area of the following? Method: All 3 members of each group must follow these steps. book your school a board b your c classroom of different shapes, such as squares,drectangles, and squares from your 1 Collect templates city lake table farmer’s field e your f g h teacher. These templates will serve as the bases for your a paper houses.

Do It Yourself

Choose shape from and cut it out of the origami sheets. 2 the Find area ofa rectangles ofthe thetemplates given measurements. 2 Word Problems

Word Problems

paper shape to m, represent different of ab house, walls, roof, 15 cm, b =the 22 cm b = 36 m l = 58 m, = 70 m suchd asl = 88 cm, b = 62 cm a 3l = Decorate b l = 47 c parts

doors and windows. 3 kg each. How many such 1 Ravi has 21 kg of rice. He packs the rice in packets of Find the area of the squares of the 4 the andgiven area measurements. of each part of your paper house using the respective 4 Calculate packets does he perimeter pack? 56 m s = 67 cm 24 m of capacityd57 sL?= 40 cm a s 2=formulas. c sa=container How many bottlesbof capacity 3 L can be filled from

You do

Once theofdecoration and done, and assemble the shapes to create 5 each Take side the square oncalculations the squaredare paper as fold 1 unit. Find the area ofpaper the triangles. your house. You may use glue or tape to secure the different parts of the house together.

•

Points to Remember

To add or subtract like fractions, simply add or subtract the numerators and keep the

denominators the same. Chapter • ToCheckup add or subtract unlike fractions, first convert the unlike fractions into like fractions

Find the area of the given compound shapes.

•

5 cm b 3 cm of Find athe perimeter andperform area the following rectangles. fractions and then addition or subtraction.

d l = 70 cm, b = 57 cm 8 cm Reciprocal 8 cm 10 cm of a number is that number which 9 cmwhen multiplied by the original number 3 cm gives 1 as the answer. Find the perimeter and area of the following squares. a

Chapter Checkup

and then add or subtract the fractions. To add or subtract mixed numbers, convert the mixed numbers into improper c

To multiply 2 fractions, find the product of their numerators and the product of their b l = 27 m, b = 21 m c l = 49 m, b = 33 m

•

l = 14 cm, b = 18 cm denominators. •

4 cm

3 cm

To divide a given fraction by another fraction, multiply the first fraction with the 3 cm c s = 45 m d

s = reciprocal 23 cm of the 12second cm fraction. b s = 32 m

6 cm

s = 80 cm

6 cm Complete the given table. 6 Use squared paper to show rectangles or squares with these measurements. Find the measurement

Math Lab

for the following shapes. Sl.that No.is not given Shape Length/side = 36 cm aa Perimeter Rectangle b

Breadth

Perimeter

–

420 m

Fraction Bingo 24 sq. cm b Area 24=cm

Aim: To understand fractions and add fractions.

Square

Materials Required: Fraction Bingo cards, markers, or coloured pencils

Wordc Problems Method:Rectangle

34 m

Area 960 sq. cm

124 m

Create the fraction bingo cards and create the list of fractions that are to be called out.

2 1

Each student getaa new fraction bingofor andher a coloured marker. Advita wants carpet room. pencil If herorroom measures 12 m by 12 m, then how

The students put a tick or cross over the fraction that is called out.

213

Chapter 12 • Perimeter and Area 3

2 5

Call out fraction questions from the prepared list.

much carpet does she need to do to cover her entire floor?

The perimeter ofora diagonal rectangular is bingo 80 m.isIfcrossed the breadth of the adds field is 20 m, find the Once a row, column, of the field fraction out, the student all the of fractions in that line. If the sum of all the fractions is more than 3 wholes, then area the field. that student is the winner.

Pearson, P. D., & Gallagher, G. (1983). Contemporary Educational Psychology. 6 After the winner is declared, students can check in their fraction bingo which row has

sum of more than 3 wholes. Fisher, D., & Frey, N. (2021). Better learning through structured teaching: the A framework for the gradual release of responsibility.

Fisher, D., & Frey, N. (2014). Checking for understanding: Formative assessment techniques for your classroom. Chapter 6 • Operations on Fractions

212

Gradual Release of Responsibility

Maths Grade 5 Book_Chapter 1-6.indb 5

103

18-12-2023 11:01:03

C o nt e nt s

Numbers Up to 8 Digits ��1 • Understanding Large Numbers • Comparing, Ordering, and Rounding Off Large Numbers

19 22 28

3 Factors and Highest Common

Factor ��35 • Understanding Factors • Highest Common Factor

36 45

ultiples and Least Common M Multiples ��54 • Understanding Multiples • Least Common Multiple

Lines and Angles �� 147

Patterns and Symmetry �� 167

Length and Weight �� 188

Perimeter and Area �� 202

Capacity and Volume �� 215

3-D Shapes on Flat Surfaces �� 230

Time and Temperature �� 247

perations on Numbers Up to O 6 Digits ��18 • Addition and Subtraction of Numbers Up to 6 Digits • Multiplication and Division of Numbers Up to 6 Digits • Choosing The Right Operator

55 58

Fractions ��69

• Understanding Fractions

Operations on Fractions ��86

Introduction to Decimals �� 106

• Addition and Subtraction of Fractions • Multiplication and Division of Fractions • Understanding Decimals • Fractions and Decimals • Types of Decimals • Comparing, Ordering and Rounding off Decimals

87 94

107 116 119

122

8 Operations with Decimals �� 129 • Addition and Subtraction of Decimals • Multiplication of Decimals • Division of Decimals

130 135 141

• Lines and Line Segments • Understanding Angles • 2-D Shapes

• Patterns Around Us • Symmetry and Reflection • Understanding Lengths • Understanding Weights • Perimeter • Area

• Understanding Capacity • Volume • 3-D Shapes • Nets and Views • Maps and Floor Plans • Time • Temperature

148 153 161 168 181 189 195 203 206 216 220 231 234 239 248 254

Money �� 260

• Working with Money • Word Problems in Money

261 265

Data Handling �� 270 • Bar Graphs • Pie Charts • Line Graphs

271 275 279

Answers �� 285

UM24CB_FM.indd 6

22-12-2023 12:36:36

Numbers Up to 8-digits

Let’s Recall

Numbers are used everywhere in our daily life. These numbers are formed using the digits 0 to 9 and are written using commas after every period, starting from the one's period.

Rajesh Pandit 143/3, Sargun Apartments

For example, let us say, the pin code of your area is 201301. This is a 6-digit number. It can be written using commas by representing it in a place value chart. Number

Lakhs

Thousands

Noida, Uttar Pradesh Pincode: 201301

Ones

Lakhs (L) Ten Thousand (TTh) Thousands (Th) Hundreds (H) Tens (T) Ones (O) 2,01,301

periods place

Each of these digits has a place value and a face value. Let us write the face value, place value, expanded form and number name for 201301. 201301

Face Value

Place Value

1 one = 1 × 1

= 1

3 hundreds = 3 × 100

= 300

0 tens = 0 × 10

1 thousand = 1 × 1000

0 ten thousands = 0 × 10000 2 lakhs = 2 × 100000

= 0

= 1000 = 0

= 200000

Expanded form: 2,00,000 + 1000 + 300 + 1

Number name: Two lakh one thousand three hundred one

Letʹs Warm-up Fill in the blanks. 1 2 3 4

The place value of 8 in 8,60,765 is __________________.

he number 4,36,536 can be written in words as: _________________________________ T ____________________________________________________________________________________. The place value of ________ in 4,15,124 and 4,67,890 is the same. 8,76,504 has 6 in the ________ place.

I scored _________ out of 4.

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Understanding Large Numbers Real Life Connect

Dhruv was reading a newspaper. He read news about different states of India that took part in the COVID vaccination drive, and the number of vaccinations done till August 2023. Given below is the data of four states. Delhi

Haryana

Sikkim

Goa

37409161

45546836

1360477

2874477

All About 7-digit and 8-digit Numbers While reading the news, Dhruv got confused and could not read the numbers given in the data. The numbers of vaccinations given were either 7-digit numbers or 8-digit numbers. Sikkim

Goa

Delhi

Haryana

1360477

2874477

37409161

45546836

7-digit Numbers

8-digit Numbers

Place Value, Expanded Form and Face Value 7-digit Numbers Let us help Dhruv in understanding 7-digit numbers! We know that the greatest 6-digit number is 999999. If we add 1 to this number, we will get 1000000. This is the smallest 7-digit number with a new place called the ‘Ten Lakhs’ place. Lakhs period Ten Lakhs (TL)

Thousands period

Ones period

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

+ 1

1 0

new place: Ten lakhs (TL) place

Smallest 7-digit number

7-digit numbers begin with the ten lakhs place in the place value chart (from the left). We saw in the news article that the number of vaccinations in Sikkim was 1360477. Let us try to place this 7-digit number in the place value chart.

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Lakhs

Thousands

periods

Ones

Ten Lakhs (TL)

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hundreds (H)

Tens (T)

Ones (O)

place

Every digit in a number has a fixed position called the place of a digit. The value of the digit depends on its place or position in the number. So, the place value of a digit is the value represented by the digit on the basis of its position in the number. Let us write the face value and place value of 1360477. 1360477

Face Value 7 ones

4 hundreds

7 tens

Place Value

0 thousand

6 ten thousands = 3 lakhs

7×1

4 × 100

400

60000

7 × 10

0 × 1000

6 × 10000

= 3 × 100000

1 ten lakh

= 1 × 1000000

= =

70 0

= 300000

= 1000000

Expanded Form When place values of all the digits are added to form a number, it is known as the expanded form of the number. 1000000 + 300000 + 60000 + 400 + 70 + 7 = 1360477 Expanded Form

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

Standard Form

Face Value The face value of a digit is the value of the digit itself. For example, the face value in the hundreds place is just 4. Reading and Writing 7-digit Numbers

Remember! Place value of a digit = face value of a digit × value of the place.

To read and write 7-digit numbers in figures and words, we group the digits according to periods and put commas after each period, starting from the ones. This helps us read the numbers easily.

Chapter 1 • Numbers Up to 8-digits

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For example, 1360477 can be read in the following way: 13,60,477 Lakhs period

Thousands period

1000000 + 300000

thirteen lakh

Ones period +

60000

sixty thousand

400 + 70 + 7 four hundred seventy-seven

Error Alert! Never write the plural form of ‘periods’ while writing number names. 36,57,648

Thirty-six lakhs fifty-seven thousands six hundred forty-eight

Thirty-six lakh fifty-seven thousand six hundred forty-eight

8-digit Numbers Now, let us help Dhruv in understanding 8-digit numbers! 99,99,999 is the greatest 7-digit number. On adding 1 to it, we get 1,00,00,000. This is the smallest 8-digit number with a new place called the ‘Crores’ place. Crores period Crores (C)

Lakhs period Ten Lakhs (TL)

Lakhs (L)

Thousands period

Ones period

Ten Thousands Thousands Hundreds Tens Ones (TTh) (Th) (H) (T) (O) 9

9 1

Smallest 8-digit number

new place: Crores (C) place 8-digit numbers begin with the crores place in the place value chart (from the left). In the news article, the number of vaccinations in Delhi was 37409161. Now, let us try to place this 8-digit number in the place value chart. Crores

Lakhs

Thousands

Ones

Crores (C) Ten Lakhs Lakhs (L) Ten Thousands Thousands Hundreds Tens Ones (TL) (TTh) (Th) (H) (T) (O) 3

periods place

So, the expanded form of 37409161 is 30000000 + 7000000 + 400000 + 9000 + 100 + 60 + 1 = 37409161 Expanded Form

Standard Form

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From the above table, we can write the place value and face value of each digit of 37409161 in the following way: Let us write the face value and place value of 37409161. 37 409161

Place Value

Face Value 1 one = 1 × 1

= 1

1 hundred = 1 × 100

= 100

6 tens = 6 × 10

= 60

9 thousands = 9 × 1000

0 ten thousand = 0 × 10000 4 lakhs = 4 × 100000

7 ten lakhs = 7 × 1000000

= 9000 = 0

= 400000

= 7000000

3 crores = 3 × 10000000

= 30000000

3,74,09,161 Lakhs period

Crores period

Thousands period

30000000 + 7000000 + 400000 three crore

9000

seventy-four lakh nine thousand

Ones period

100 + 60 + 1 one hundred sixty-one

Think and Tell

Can you now read and write the number of vaccinations taken till August 2023 in Goa and Haryana?

Example 1

Rewrite the numbers using commas in a place value chart. Also, write them in words. 1

54879509 2 6509808

S. No.

Numbers

Crores period

Lakhs period

Thousands period

Ones period

Crores Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones (C) (TL) (L) (TTh) (Th) (H) (T) (O) 1

5,48,79,509

65,09,808

Number Names: 1 Five crore forty-eight lakh seventy-nine thousand five hundred nine 2 Sixty-five lakh nine thousand eight hundred eight Example 2

Write the numbers for the given number names. 1 Seventy-eight lakh, nine thousand, one hundred nine = 78,09,109 2 Nine crore, five lakh, ten thousand, two hundred = 9,05,10,200 Chapter 1 • Numbers Up to 8-digits

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Example 3

Write the expanded form of the number 6,48,70,977. Expanded Form: 6,00,00,000 + 40,00,000 + 8,00,000 + 70,000 + 0 + 900 + 70 + 7

Do It Together

Read the number. Write its number name and expanded form. 6,57,90,284

Number name = ________________________________________________________________________ Expanded form = _______________________________________________________________________

Indian and International Number System We already know that the Indian number system has the ones period, thousands period, lakhs period, and crores period. There is another number system used globally, called the International number system. Millions Ten Millions (TM)

Thousands

Millions (M)

Ones

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

10,000,000 1,000,000

100,000

10,000

1,000

100

3 places

periods place

3 places

Like the Indian number system, we put commas after each period starting from ones. But the same number in the international system will be read differently. 76,012,665 Millions period

Thousands period

Ones period

seventy-six million

twelve thousand

six hundred sixty-five

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

Lakhs

Crores (C)

Thousands

Ones

Ten Lakhs (TL)

Lakhs (L)

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

1,00,000

Ten Millions (TM)

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

Millions

Millions (M)

10,000,000 1,000,000 ×10

×10

Ten Thousands Thousands Hundreds Tens Ones (TTh) (Th) (H) (T) (O) 10,000

1,000

Thousands

100,000

10,000 ×10

So, from the above table, we can say that

100

1,000 ×10

Ones

100 ×10

10 ×10

Indian Number System

1 International Number System

1 ×10

1 lakh = 100 thousands; 10 lakhs = 1 million; 1 crore = 10 millions 6

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Let us represent the number 98710325 on the place value chart. C

TTh

HTh

TTh

Indian Number System International Number System

Indian Number System

International Number System

Standard Form: 9,87,10,325

Standard Form: 98,710,325

Number Name: nine crore eighty-seven lakh ten thousand three hundred twenty-five

Number Name: ninety-eight million seven hundred ten thousand three hundred twenty-five

Expanded form: 9,00,00,000 + 80,00,000 + 7,00,000 + 10,000 + 300 + 20 + 5

Expanded form: 90,000,000 + 8,000,000 + 700,000 + 10,000 + 300 + 20 + 5

Write the numbers in the international number system using commas and as number names.

Example 4

79027348 2 90710946

79,027,348 = Seventy-nine million twenty-seven thousand three hundred forty-eight

90,710,946 = Ninety million seven hundred ten thousand nine hundred forty-six

Write the expanded form of the given numbers.

Example 5

Do It Together

78,200,957 = 70,000,000 + 8,000,000 + 200,000 + 900 + 50 + 7

4,502,456 = 4,000,000 + 500,000 + 2,000 + 400 + 50 + 6

Write numbers or number names, for the following. Also, write their expanded form. 1

Ten million five hundred twenty-nine thousand six hundred five = __ __, __ 29, __ __ __ Expanded form: ________________________________________________________________

65,780,245 = ____________________________________________________________________ Expanded form: ________________________________________________________________

Do It Yourself 1A 1

Write the place value and face value of the underlined digit in the following numbers. a 43,889,385

b 9,32,60,075

c 57,131,060

d 29,543,002

e 61,752,812

2,00,30,543

g 8,00,54,205

h 1,00,81,824

Chapter 1 • Numbers Up to 8-digits

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Write the numbers in the Indian and International number systems. Write their number names and expanded form in both number systems. a 21643332

b 1200621

c 46207219

d 95910158

e 7409230

8656023

g 67890240

h 34565892

Write the numerals for the following number names. a Sixty lakh eight thousand ninety-eight

b Twenty million five hundred sixty-nine

c Four million ninety thousand

d Eight crore one thousand two

Fill in the blanks.

a 10 million = _________ crore

b 1 million = _________ lakh

c 1 crore = _________ thousands

d There are _________ zeroes in 20 million.

e 100 lakhs = _________ millions

How many thousands make:

Do as directed.

a a lakh?

b a million?

5 crores = _________ millions

c a crore?

a Write the greatest 7-digit number that has the smallest odd digit at its hundreds, ten thousands

and lakhs place.

rite the smallest 8-digit number that has the digit 7 at all its odd positions, starting from the ones b W place.

Word Problem 1

In 2019, thirty million one hundred forty-five thousand seven hundred sixty-five people attended the famous, Kumbh Mela. Write the number of people who attended the mela in figures. Also, write its number name in the Indian number system.

Comparing, Ordering, and Rounding Off Large Numbers During COVID, India offered support to 150 affected countries in the form of vaccines, medical equipment, and medicines. Given below is the data of the number of vaccine doses supplied by India to four different countries. Nepal

Bangladesh

Australia

Nigeria

94,99,000

2,80,82,800

3,09,13,200

98,02,000

Comparing Numbers What if we wanted to compare the number of vaccines sent to Nepal and Australia? Let us find out. 8

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We first write the numbers in a place value chart and check the number of digits. C 3

TTh

7 digits 8 digits

Both the numbers have a different number of digits.

Remember!

Since 94,99,000 has 7 digits and 3,09,13,200 has 8 digits, 3,09,13,200 > 94,99,000.

A number with more digits is always greater.

Thus, Australia was donated more vaccine doses.

Now, what if we want to compare two numbers with the same number of digits? Let us consider 4,13,23,456 and 4,13,23,657. Step 1: Write the numbers in a place value chart and check the number of digits.

Both the numbers have the same number of digits, that is, 8 digits.

Step 2: Start comparing the digits from the left till we find different digits. The number with the greater digit is greater. Here, the digits are the same till the thousands place. Now, we compare the digits in the hundreds place. We see that 4 < 6.

TTh

4=4

So, 4,13,23,456 < 4,13,23,657.

Example 6

1=1

3=3

2=2

3=3

4<6

Compare the numbers 6,47,17,389 and 6,47,00,508. C

TTh

6=6

4=4

7=7

1>0

So, 6,47,17,389 > 6,47,00,508 Do It Together

Which is greater: 1,86,01,769 or 1,86,04,766? C

1=1

TTh

8=8

So, ______________ > ______________ Chapter 1 • Numbers Up to 8-digits

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Do It Yourself 1B 1

Which is smaller: 76,24,578 or 87,90,213?

Fill in the blanks with <, >, or =.

a 35,72,123 ____ 35,78,123

b 63,45,789 ____ 63,45,789

c 2,86,73,451 ____ 57,81,290

d 6,24,58,110 ____ 6,24,59,211

e 82,60,154 ____ 89,12,620

Compare the following numbers using <, >, or =.

84,63,758 ____ 7,65,38,453

a 56,53,437 and 97,56,424

b 47,98,560 and 40,87,895

c 4,23,75,449 and 46,67,087

d 48,75,268 and 8,65,43,959

e 98,38,765 and 89,38,453

1,76,54,534 and 98,54,763

Rearrange the digits of the number 5,48,79,802 to form a new number. Compare the two numbers.

Match the numbers so that the number in the second column is 1,00,000 more than the number in the first column. a 99,00,000

10,00,000

b 2,89,52,468

3,00,52,468

c 9,00,000

1,00,00,000

d 2,99,52,468

2,90,52,468

Word Problem 1

Sanket types 25,85,120 words in a week while Mahesh types 24,32,214 words in a week. Who types more words in a week?

Ordering Numbers Remember the number of vaccine doses donated by India to different countries? Let us read them again. Nepal

Bangladesh

Australia

Nigeria

94,99,000

2,80,82,800

3,09,13,200

98,02,000

Can you rearrange them in the ascending order? Let us find out. ing

e sc

de or

smallest to greatest

din

rd e

greatest to smallest

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To arrange the number of vaccine doses in ascending order, follow the given steps. Step 1: Arrange the given numbers in the

place value chart and check the number of digits. We know that the numbers with a greater number of digits are greater.

TTh

7 digits

8 digits

7 digits

From the given table, notice that 7 < 8.

So, 2,80,82,800 and 3,09,13,200 are greater, and 94,99,000 and 98,02,000 are smaller. Step 2: Compare the smaller numbers. Look at the

given table.

From the given table, 94,99,000 < 98,02,000

So, 94,99,000 is the smallest, and 98,02,000 is the next smaller number. Step 3: Compare the greater numbers. Look at the

given table.

From the given table, 2,80,82,800 < 3,09,13,200

So, 2,80,82,800 is the next smaller number, and 3,09,13,200 is the greatest.

TTh

9=9 4<8 C

TTh

2<3

The ascending order is 94,99,000 < 98,02,000 < 2,80,82,800 < 3,09,13,200.

Remember! The predecessor of a number is just before the number. So, it is always less than that number. For example, 100 is the predecessor of 101 and 100 < 101.

Think and Tell

Can you arrange these numbers in descending order?

Forming Numbers Using All the Given Digits Let us understand this using an example. Let us say, we are given the digits 9, 5, 1, 0, 6, 7, and 3. Let us try forming some numbers such that each digit appears exactly once. 5106739

7106539

6107539

5013769

7013569

6013579

5137609

7135609

6135709

Think and Tell

Can you form more such numbers?

Now what if we wanted to form the greatest and the smallest 7-digit numbers using these digits only once?

Chapter 1 • Numbers Up to 8-digits

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To form the greatest number, we will write the digits in descending order:

This becomes the largest 7-digit number that can be formed using the digits exactly once! To form the smallest 7-digit number, we will write the digits in ascending order. But 0 cannot appear in the leftmost place because then this will become a 6-digit number and we are looking for 7-digit numbers. So, 0 can appear in the second position:

This becomes the smallest 7-digit number that can be formed using the digits exactly once! What if repetition of exactly 1 digit is allowed and we want to form 8-digit numbers?

Think and Tell

Why did we choose to repeat the greatest digit?

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

This becomes the greatest 8-digit number that can be formed, while repeating only 1 digit once! To write the smallest number, we repeat the smallest digit, only once. Remember to not put a 0 in the highest place!

1 0 0 3 5 6 7 9

Example 7

This becomes the smallest 8-digit number that can be formed, while repeating only 1 digit once! Descending order Arrange 10,00,000; 38,65,080 and 4,50,50,809 in descending order. 4,50,50,809 4,50,50,809 > 38,65,080 > 10,00,000 is the descending order.

Example 8

38,65,080

Form the greatest and the smallest 8-digit number using the digits 1, 8, 6, 0, 9, 2, 5 and 4. No repetition of digits is allowed.

10,00,000

Greatest number = 98654210; Smallest number = 10245689 Do It Together

Arrange the following numbers in both ascending and descending order. 1,26,56,384, 16,52,349, 86,70,653, 4,00,97,234 Ascending order: ________________________________________________________________________ Descending order: ______________________________________________________________________

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Do It Yourself 1C 1

Arrange the following numbers in ascending order.

a 1,00,36,782; 5,00,00,367; 8,87,21,460; 93,12,820

b 92,56,890; 36,81,910; 6,92,10,350; 8,26,00,031 c 5,00,21,138; 6,04,50,821; 6,50,24,567; 9,45,21,823

Arrange the following numbers in descending order. a 8,26,34,510; 87,92,345; 6,70,81,256; 4,50,00,921

b 42,56,789; 8,01,20,450; 92,11,108; 6,78,20,001 c 5,43,33,867; 9,67,28,891; 7,88,21,134; 5,78,20,010

Write the greatest and the smallest 7-digit numbers using all the digits only once.

Write the greatest and the smallest 8-digit numbers using all the digits but repeating any one digit exactly once.

a 5, 3, 4, 0, 8, 9, 1

a 2, 7, 1, 0, 8, 6, 4

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

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

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

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

Write the greatest 8-digit number and the smallest 7-digit number using: a two different digits

b five different digits

c four different digits

Rounding off Numbers Remember, the number of vaccine doses donated by India to Bangladesh was 2,80,82,800. But what if we wanted to convey this number to a friend? The number 2,80,82,800 is very inconvenient to read and say out aloud. What if we just said that India donated about 2,81,00,00 vaccines to Bangladesh. It still gives a fair idea of about how many vaccines were donated. This is called rounding a number off. While rounding numbers off, terms like “about” and “approximately” are added to convey that the number is close to the exact. We can round off numbers to different places, such as tens, hundreds and thousands. Let us understand how we can do this.

Chapter 1 • Numbers Up to 8-digits

Maths Grade 5 Book_Chapter 1-6.indb 13

Did You Know? The diameter of the Sun is approximately 14,00,000 km.

18-12-2023 11:01:09

Rounding off to the Nearest 10s If the ones digit is less than 5, then the ones digit is replaced by 0. 5 4, 7 0, 8

rounded off to

5 4, 7 0, 8

3<5

If the ones digit is greater than or equal to 5, then the ones digit is replaced by 0 and the tens digit is increased by 1. 2, 6 4, 8 0, 0

rounded off to

2, 6 4, 8 0, 0

7>5

2+1=3

Rounding off to the Nearest 100s

To round off to the nearest 100s, we look at the tens digit. If the tens digit is less than 5, then the ones and tens digits are replaced by 0. 6 4, 3 0, 7

rounded off to

6 4, 3 0, 7

0<5

If the tens digit is greater than or equal to 5, then the ones and tens digits are replaced by 0 and the hundreds digit is increased by 1. 5, 6 4, 3 0, 7

rounded off to

5, 6 4, 3 0, 8

5=5

7+1=8

Rounding off to the Nearest 1000s

To round off to the nearest 1000s, we look at the hundreds digit. If the hundreds digit is less than 5, then the ones, tens, and hundreds digits are replaced by 0. 7 8, 5 1, 4

rounded off to

7 8, 5 1, 0

4<5

If the hundreds digit is greater than or equal to 5, then the ones, tens, and hundreds digits are replaced by 0 and the thousands digit is increased by 1. 2, 8 3, 4 9, 6

rounded off to

2, 8 3, 5 0, 0

6=5

9 + 1 = 10

Think and Tell

How do you think the number of vaccine doses donated by India to Bangladesh was rounded off ? Explain.

Example 9

Round off 3,76,87,519 to the nearest 10s, 100s, 1000s. To the nearest 10s: 3,76,87,519 is rounded off to 3,76,87,520.

To the nearest 100s: 3,76,87,519 is rounded off to 3,76,87,500.

To the nearest 1000s: 3,76,87,519 is rounded off to 3,76,88,000. 14

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Do It Together

Round off 7,39,81,506 to the nearest 10s, 100s, 1000s. To the nearest 10s: 7,39,81,506 is rounded off to ___________.

To the nearest 100s: 7,39,81,506 is rounded off to ____________.

To the nearest 1000s: 7,39,81,506 is rounded off to _____________.

Do It Yourself 1D 1

Round off the following numbers to the nearest tens. a 85,48,749

b 89,05,462

c 6,07,85,888

Round off the following numbers to the nearest hundreds. a 1,25,89,183

b 87,52,368

c 68,67,790

d 77,59,910

Round off the following numbers to the nearest thousands. a 8,97,00,110

b 53,12,069

c 8,21,58,701

d 5,89,89,929

e 5,40,86,566

3,09,24,555

e 5,07,87,345

43,43,551

e 86,75,500

23,74,567

Use the given information to form the greatest 7-digit numbers using the digits 6, 0, 3, 2, 1 and 9. Round off the numbers formed to the nearest thousands. a Repeating the greatest digit only once

d 1,56,48,950

b Repeating the smallest digit only once

An 8-digit number was rounded off to the nearest thousands. The result was 3,23,46,000. Between which two numbers does the original number lie? (Hint: Take a range of 1000 numbers.)

Word Problem 1

The municipal corporation spent ₹65,94,830 on repairing the roads. Rewrite the sentence by rounding off the number to the nearest 1000s.

Points to Remember • • • • • • • •

7-digit number has 7 digits with ten lakhs as its highest place. A An 8-digit number has 8 digits with crores as its highest place. The place value of a digit is the value represented by the digit on the basis of its position in the number. The face value of a digit for any place in the given number is the digit itself. The expanded form of a number is the place values of all the digits. A number with more digits is always greater. The number just before a given number is called its predecessor, and the number just after a given number is called its successor. Numbers can be rounded off to give us an approximate value.

Chapter 1 • Numbers Up to 8-digits

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Math Lab Building Numbers Game Setting: In groups of 2

Materials Required: Flash cards (0 to 9) and coloured pencils. Method: 1

Pick any 8 flash cards containing digits 0 to 9.

Now, arrange these cards in ascending order to form the smallest 8-digit number. Be careful if you choose 0 as one of the numbers.

Next, arrange these cards in descending order to form the greatest 8-digit number.

Make 2 more possible 8-digit numbers using the same digits.

Write their number names and arrange the numbers in descending order. (The team to complete the task first wins the round.)

Repeat steps 1 to 6, two times. (The team to complete the task correctly the maximum number of times wins the game.)

Chapter Checkup 1

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

b c d

63565842

91500084

Sixty million seven hundred fifteen thousand two hundred thirty-nine Eight crore nine lakh fifty thousand two

One million one hundred thousand thirty-nine Twelve lakh fifty-eight thousand forty-three 6,56,52,567 ______ 6,48,90,650 34,57,879 ______ 34,57,879

b d

90,00,518 ______ 90,76,757

13,05,885 ______ 6,74,38,989

Arrange the following numbers in ascending order. a b

42087950

Fill in the blanks using <, >, or =. a

Write the numbers for the following number names: a

3507681

23,56,475; 9,08,04,365; 8,91,63,896; 90,87,687

6,76,12,895; 6,76,87,980; 4,35,46,576; 3,24,35,678

Arrange the following numbers in descending order. a b

4,56,45,768; 5,36,45,787; 2,40,85,167; 43,56,787 80,88,428; 4,90,76,837; 9,09,87,897; 80,68,964

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Form the greatest and the smallest 8-digit numbers using the digits 3, 6, 8, 9, 7, 0, 2, and 5 only once. Also, form the greatest and the smallest 7-digit number using these digits. Arrange the numbers in both ascending order and descending order.

Round off the following to the nearest tens, hundreds, and thousands. a

6,45,87,123

89,09,008

1,08,75,756

24,89,702

List all the numbers that are rounded off to the nearest tens as 16,48,240.

A certain 8-digit number has only fives in the ones period, only sevens in the thousands period, only nines in the lakhs period and only ones in the crores period. Answer the following questions. a b c

Write the number in figures and words using both the Indian and the International number systems. Rearrange the digits of the number to form a new number. Compare the two numbers. Round off the number to the nearest tens, hundreds, or thousands.

10 Sanya wants to solve a 7-digit secret code to a safe. Use the given clues to help Sanya solve the secret code. a b c d e

The digit in the hundreds and ones place is 6.

The digit in the lakhs place is 4 less than the digit in the ones place.

The digit in the ten lakhs and ten thousands place is the smallest odd number. The face value of the digit in the thousands place is 5.

The digit in the tens place is the biggest 1-digit number.

What is the secret code?

Word Problem 1

ina’s mother bought a piece of land and some equipment for ₹1,14,35,860. T Complete the cheque for Tina’s mother by writing the numbers in words in the space provided against ‘Rupees’. BANK OF COUNTRY

0284732492

Greater Noida Branch

Date :

Greater Noida, U.P.

Pay to the order of :

Sheela Yadav

1,14,35,860

Rupees

Authorized Signature :

|: 9027385920 |:

Chapter 1 • Numbers Up to 8-digits

Maths Grade 5 Book_Chapter 1-6.indb 17

89302874

84759

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2 Up to 6 Digits

Operations on Numbers

Let's Recall

Imagine that we have 1150 beads. We find 450 more beads. How many beads do we have now? Th

5 5 0

0 0 0

1 + 1

1 4 6

1150 + 450 = 1600 beads. We have 1600 beads.

This is called addition.

Now, let us say, out of the 1600 beads, we give 200 beads to our friend. How many beads do we have now? Th 1 – 1

H 6 2 4

T 0 0 0

O 0 0 0

1600 – 200 = 1400 So, we have 1400 beads left. This is called subtraction.

Imagine there are 4 children and each one of them has 150 beads. How many beads do they have in total? Let us count. 150 + 150 + 150 + 150 = 600 Or, 4 × 150 = 600 This is called multiplication.

Now what if we say that we have 1000 beads and 4 friends. How many beads will each friend get? 250 4 1000 Let us divide 1000 by 4. –8 20 1000 ÷ 4 = 250 –20 00 This is called division. –0 0 Division is equal sharing.

Letʼs Warm-up

Match the following. 1

180 ÷ 2

5500

7000 – 1500

280

2100 + 250

140 × 2

450

150 × 3

2350 I scored _________ out of 5.

Maths Grade 5 Book_Chapter 1-6.indb 18

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Addition and Subtraction of Numbers Up to 6 Digits Sahil’s father runs a bakery. At the end of every year, they calculate the total sales for the different types of baked goods they sold.

Real Life Connect

Father: We made ₹2,50,678 by selling cakes and ₹1,56,240 by selling cookies. Sahil: Wow! How much did we make last year? Father: Last year we made a total of ₹3,17,500. Sahil wonders whether they have made more or less this year!

Addition and Subtraction Sahil wants to know the total sales they made this year. He also wants to know whether they have made more or less this year than last year. How will Sahil do that? Let us help him out!

Adding Numbers Up to 6 Digits If Sahil wants to find the total sales they made this year, he will have to add the numbers ₹2,50,678 and ₹1,56,240. We have already learnt how two numbers are added. When we put two or more numbers together to find their total, it is called addition. Let us help Sahil calculate the total sales for this year using the following steps.

Add: ₹2,50,678 + ₹1,56,240

Addend

Sum

TTh

So, Sahil’s father made ₹4,06,918 this year, which is more than last year. Example 1

TTh

Add 1,98,794 and 52,250 and 21,000. 1,98,794 + 52,250 + 21,000 = 2,72,044. A company produced 4,56,360 boxes in 2019. The same company produced 3,60,780 boxes in 2018 and 90,995 boxes in 2017. How many boxes have they produced in three years?

Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 19

O carry

Using the steps, we can add the numbers as:

Example 2

+ 2

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Boxes produced in 2019 = 4,56,360

Boxes produced in 2018 = 3,60,780

Boxes produced in 2017 = 90,995 To find the total number of boxes produced, we need to add 4,56,360, 3,60,780 and 90,995.

TTh

Number of poetry books = 26,716

A library has 26,716 poetry books, 43,120 non-fiction books and 80,133 fiction books. How many books are there in the library?

4 +

So, the company produced 9,08,135 boxes in three years. Do It Together

TTh

Number of non-fiction books = 43,120 Number of fiction books = 80,133

Remember!

Total number of books in the library is ____________.

Changing the order of addends doesn’t change the sum.

Subtracting Numbers Up to 6 Digits

Sahil found that they earned ₹4,06,918 this year, which is more than the money they earned last year, which was ₹3,15,500. Now, Sahil wants to know how much more they have earned this year. We know that when we take one number away from another, it is called subtraction. Let us subtract the sales made last year from the sales made this year. Subtract: ₹4,06,918 – ₹3,15,500 ₹4,06,918 – ₹3,15,500 = ₹91,418. Therefore, they earned ₹91,418 more this year than last year. Example 3

–

TTh

Minuend

Subtrahend

Difference

Find the difference of 57,588 and 6765.

Therefore, 57,588 – 6765 = 50,823. A pen company manufactures 9,28,667 pens in total, out of which 58,475 are red pens. Find the total number of pens that are not red.

TTh

TTh 5

–

O After borrowing

We know that we subtract the smaller number from the larger number as shown below:

Example 4

6 0

Th 6 7

H 15 5 7 8

8 6 2

8 5 3

Minuend

Subtrahend

Difference

Maths Grade 5 Book_Chapter 1-6.indb 20

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Total number of pens = 9,28,667 Number of red pens = 58,475 Number of pens that are not red = Total number of pens − Number of red pens Therefore, 9,28,667 – 58,475 = 8,70,192. Do It Together

A store sold 2,85,586 shirts in a year. The sale of shirts for the first month was 4640. How many shirts were sold in the remaining 11 months? Shirts sold in 1 year or 12 months = ___________________

Shirts sold in the first month = ____________________

TTh

Shirts sold in the remaining months = ________________

–

Therefore, the company sold ____________________ shirts in 11 months.

___

Error Alert! Always write the smaller number below the larger number. We can solve 5 – 2 = 3. But we cannot solve 3 – 5.

Do It Yourself 2A 1

Write True (T) or False (F). a When the number is subtracted from itself, the difference is zero. b When 0 is subtracted from a number, the difference is zero. c When the order of the addends is changed, the sum remains the same. d The order of numbers involved in subtraction can be changed.

Add the given numbers. a

+ 3

Subtract the given numbers. a

–

4 8

Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 21

–

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Find the sum of the given numbers. a 8,97,878 and 8689

b 3,44,567, 78,456 and 39,894

c 1,72,744 and 5,56,200

d 5,89,569, 1,24,887 and 56,758

e 5,78,896, 48,798, 67,579 and 13,865

3,86,565, 2,34,567, 56,468 and 46,568

Find the difference of the given numbers. a 87,687 and 5789

b 67,776 and 43,210

c 9,87,609 and 56,000

d 8,76,534 and 87,689

e 5,68,978 and 3,21,098

7,99,098 and 2,67,548

The difference of two numbers is 3,98,460. If the smaller number is 5,05,090, find the greater one.

Word Problems 1

A factory manufactured 59,899 blazers in the year 2020, 78,906 blazers in the year 2021, and 1,34,145 blazers in the year 2022. How many blazers did they manufacture in these three years?

Rashi withdrew ₹1,45,000 from her bank account. She now has only ₹67,000 left in it.

An NGO collected ₹2,89,230 for a charity fund in one year and ₹3,97,500 in another

How much money was in her account before the transfer?

year. They used ₹3,05,700 out of the total amount collected in the two years. How much money are they left with now?

Multiplication and Division of Numbers Up to 6 Digits Real Life Connect

Jay runs a toy store. He also has marbles in all shapes, sizes and colours. Each jar of marbles contains 1225 marbles. Sometimes, the customers want to buy more than one jar. Sometimes, they want to buy only a few marbles from the jar. How do you think Jay calculates the number of marbles a customer is buying?

Multiplication and Division Let us say, a customer wants to buy 2 jars of marbles. Jay will have to multiply 1225 and 2 to find the total number of marbles he will give.

1 2

2 4

Multiplicand

Multiplier

Product

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Now let us say that a customer wants to give marbles from one jar to 5 of his friends.

Divisor

To find out how many marbles each friend will get, he will have to divide 1225 by 5.

245

5 1225 10

22 20 25 25

When we break up a number into equal parts or groups, it is called division.

Quotient

Dividend

Remainder

Multiplying Numbers With 10s, 100s and 1000s A customer wants to buy 10 marble jars. How can he find the total number of marbles? We know that he will have to multiply the number of marbles in one jar with the number of jars he wants. He will have to multiply 1225 by 10. 10 Step 1

Step 2

1225 × 1 = 1225

end of the product.

Multiply the non-zero digits.

So, he will get 12,250 marbles. What if he buys 100 jars?

Put the remaining 0s at the 1225 × 10 = 12,250

Step 1

1225 × 1 = 1225

Multiply the non-zero digits.

Step 2

product.

1225 × 1000 = 12,25,000

Put the remaining 0s at the end of the 1225 × 100 = 1,22,500

So, he will get 1,22,500 marbles.

Anything multiplied by 1 remains the same.

What if he buys 1000 jars?

Step 1

Multiply the non-zero digits.

Remember!

Put the remaining 0s at the end of the product.

Thus, he will get 12,25,000 marbles.

So, when we multiply a number by 10, 100, 1000 and so on, we put as many 0s to the right of the multiplicand as there are 0s in the multiplier to get the product.

Example 5

Multiply 7865 and 30. Step 1

Step 2

7865 × 3

7865 × 30 = 2,35,950

Multiply the non-zero digits. 72 2

81 5

Put the remaining 0s at the end of the product. Therefore, the product of 7865 × 30 = 2,35,950.

Chapter 2 • Operations on Numbers Up to 6 Digits

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Example 6

A book contains about 1244 words on one page. If there are the same number of words on each page, then how many words will there be on 200 pages? Number of words on 1 page = 1244 Number of words on 200 pages = 1244 × 200 = 2,48,800 Therefore, the book has 2,48,800 words in total.

Do It Together

Did You Know? 0 was invented by the great Indian mathematician Aryabhata in the 5th century.

The cost of a scooter is ₹75,250. What would be the cost of 100 such scooters? Cost of 1 scooter = ₹75,250 Cost of 100 scooters = _________ × _________ = ₹_________

Multiplying Numbers Up to 4 Digits Radha wants to buy 125 jars of 1225 marbles each. She wants to know the total number of marbles she bought. Let us find the total number of marbles using multiplication.

TTh

125 = 100 + 20 + 5

1225 × 5

1225 × 20

1225 × 100

We will first expand 125 = 100 + 20 + 5 = 1 hundred + 2 tens + 5 ones

Thus, if Radha buys 125 jars of 1225 marbles each, she will get 1,53,125 marbles.

Mr. Sharma is a businessman. He visited Jay’s store and wants to buy 1121 jars of marbles from his shop. To find the total number of marbles he is buying, we will have to multiply 1225 × 1121. We will first expand 1121 = 1000 + 100 + 20 + 1 = 1 thousand + 1 hundred + 2 tens + 1 one. L

TTh

1121 = 1000 + 100 + 20 + 1

1225 × 1

1225 × 20

1225 × 100

1225 × 1000

So, if Mr Sharma will buy 1211 jars of 1225 marbles each, he will have 13,73,225 marbles.

Remember! Anything multiplied by zero is zero.

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Example 7

Multiply 3479 and 452. L

TTh

Example 8

Expand 452 = 400 + 50 + 2

Step 1: Multiply the ones: 3479 × 2

Step 2: Multiply the tens: 3479 × 50

Step 3: Multiply the hundreds: 3479 × 400 Step 4: Add all three products.

There are 1500 students in a school. The school is planning to take all the students for a trip. Each student has to contribute ₹555 each. What is the total amount collected by the school? We know that: Total amount collected = Total amount paid by 1500 students Since, amount to be paid by 1 student = ₹555 Then, amount to be paid by 1500 students = 1500 × ₹555 = ₹8,32,500 So, the total amount collected by the school is ₹8,32,500.

Do It Together

At an event there is a chair arrangement of 5982 rows and 1313 columns. Find the total number of chairs. To find the total number of chairs, we will have to multiply 5982 and 1313. L

TTh

1313 = ____________ + ____________ + ____________ + ____________

___

5982 × _____________________

___

5982 × _____________________

___

5982 × _____________________

___

5982 × _____________________

___

Therefore, the total number of chairs is __________ × __________ = __________.

Dividing Numbers by 10s, 100s and 1000s Jay, the owner of the toy store, has 40,000 marbles apart from the marbles in the jars. He wants to put them equally into other jars. He can either put them into 10 jars, 100 jars or 1000 jars. If he wants to put them in 10 jars, he will have to divide 40,000 marbles equally into 10 parts. That is, 40,000 ÷ 10.

Chapter 2 • Operations on Numbers Up to 6 Digits

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To divide a number by 10, 1 digit from the right becomes the remainder and the remaining number is the quotient. 40,000 ÷ 10 = 4000; So, he will have to put 4000 marbles in each jar. If he will put them in 100 jars, then he will have to divide them into 100 equal parts. That is, 40,000 ÷ 100. To divide a number by 100, 2 digits from the right form the remainder and the remaining number is the quotient. 40,0000 ÷ 100 = 400 with remainder 00; So, he will have to put 400 marbles in each jar. Similarly, if he will put them in 1000 jars, then he will have to divide them into 1000 equal parts. That is, 40,000 ÷ 1000. To divide a number by 1000, 3 digits from the right form the remainder and the remaining number is the quotient. 40,000 ÷ 1000 = 40 with remainder 000; So, he will have to put 40 marbles in each jar. We can conclude that the same number of digits as the number of 0s in the divisor, from the right of the dividend, becomes the remainder. The remaining digits are the quotient. Example 9

Divide 15,679 by 100. 15,679 ÷ 100 = 156 with remainder 79. Thus, quotient = 156 and remainder = 79.

Do It Together

Divide 67,688 by 1000. 67,688 ÷ 1000 Quotient = _______________ Remainder _______________ Thus, quotient = _______________ and remainder = _______________.

Dividing 5-digit Numbers by 3-digit Numbers

Error Alert! The remainder can never be greater than the divisor. If it is more than the divisor, that means the long division is incomplete or incorrect. 1 12 135 –12 15

11 12 135 –12 15 –12 3

Jay wonders if he could divide all of the 40,000 marbles in 120 smaller jars.

333 120 40000 –360 He can find out by division, using the given steps. 400 –360 Thus, quotient = 333 and the remainder= 40. 400 So, Jay can only put 333 marbles in each jar, and he will be left with 40 marbles. –360 40 26

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Example 10

Divide 76,545 by 115. Divisor = 115 and Dividend = 76,545 665 115 76545 –690 754 –690 645 –575 70 Since 70 is less than 115, the remainder is 70. Thus, the quotient = 665 and the remainder = 70.

Example 11

A room is large enough for 256 people to sit. How many rooms will be required for 125 32,000 people to sit? We know that 256 people can sit in 1 room. So, 32,000 people can sit in 32,000 ÷ 256 rooms. Therefore, the required number of rooms is 125.

Do It Together

The cost of 235 books is ₹56,745. Find the cost of one book. Cost of 235 books = ________________

256 32000 –256 640 –512 1280 –1280 0 235 56745

Cost of one book = ________________ ÷ ________________ Therefore, the cost of one book is ________________.

Do It Yourself 2B 1

Write true or false. a The divisor and remainder in a division sum can be the same. b The result of multiplication is called multiplicand. c When a number is divided by another number, it is called the difference. d Anything multiplied by zero is zero.

Multiply the given numbers by 10, 100 and 1000. a 56,567

b 34,263

c 47,852

d 1,98,454

e 82,587

d 8,95,000

e 9,87,000

Divide the given numbers by 10, 100 and 1000. a 25,000

b 2,45,000

Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 27

c 3,54,000

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Find the product of the given numbers. a 2675 × 23 f

8755 × 1674

b 4561 × 600

g 8784 × 3456

c 6780 × 276

h 7846 × 5678

d 8795 × 895

e 4365 × 4326

9876 × 4788

6784 × 7890

Divide the given numbers. a 67,865 ÷ 500 f

26,486 ÷ 144

b 34,356 ÷ 189 g 77,127 ÷ 567

c 86,576 ÷ 370

h 52,222 ÷ 323

d 45,436 ÷ 284

e 88,755 ÷ 868

34,204 ÷ 274

61,304 ÷ 148

Word Problems 1

Mrs Gupta earns ₹78,562 every month. How much does she earn in 3 years?

A book contains 99,545 words. Each page contains 255 words. How many pages are

Shreya is organising an event. She wants to arrange chairs in rows for the attendees.

there in the book?

Each row will have 2400 chairs and 380 rows. She additionally needs to set up

another 150 rows, each containing 320 chairs. How many chairs does Shreya need in total? 4

The owner of a shop makes a profit of ₹98,000. He decides to keep ₹30,000 for

himself and distribute the rest among 100 staff members. Find the amount each staff member will receive.

Choosing the Right Operator Real Life Connect

On a school trip, there are 6 teachers who accompanied one group out of 2 groups in a class of 54 students. Of the 54 students; 2 groups of 5 students were absent. Ram, a student on the trip wonders what the number of people, including the teachers in each class on the trip is. How can he do that?

Solving Using DMAS Ram tries putting the number and operations in a problem. He comes up with: 6 + 54 ÷ 2 – 2 × 5 Seeing so many operations in a single problem, Ram cannot figure out how to solve it. To solve this problem, he will have to use DMAS. 28

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What is DMAS? When we have two or more operations in a problem, we carry out these operations in a certain order. This order is called D Division ÷ DMAS. Now, let us help Ram figure out how many people are there on the trip.

Multiplication

Addition

Subtraction

–

Using DMAS To help Ram solve the problem: 6 + 54 ÷ 2 – 2 × 5 We always move from left to right. We will follow the following steps of DMAS: Step 1

Step 2

Step 3

Step 4

Simplify division (÷)

Simplify multiplication (×) Simplify addition (+)

Simplify subtraction (–)

= 6 + 54 ÷ 2 – 2 × 5

= 6 + 27 – 2 × 5

= 6 + 27 – 10

= 33 – 10 = 23

= 6 + 27 – 2 × 5

= 6 + 27 – 10

= 33 – 10

Therefore, there are 23 students and teachers in each class on the trip. Example 12

Simplify: 42 ÷ 7 + 3 × 9 – 1 We have: 42 ÷ 7 + 3 × 9 – 1 Step 1

Step 2

Step 3

Step 4

Divide (÷)

Multiply (×)

Add (+)

Subtract (–)

= 42 ÷ 7 + 3 × 9 – 1

=6+3×9–1

= 6 + 27 – 1

= 33 – 1

=6+3×9–1

= 6 + 27 – 1

= 33 – 1

= 32

So, by using DMAS, we get: 42 ÷ 7 + 3 × 9 – 1 = 32. Do It Together

Simplify: 100 – 72 ÷ 8 + 4 × 3 Step 1

Step 2

Step 3

Step 4

Divide (÷)

Multiply (×)

Add (+)

Subtract (–)

= 100 – ___ + 4 × 3

= 100 – ___ + ___

= 100 – ___

= ____

So, by using DMAS, we get: 100 – 72 ÷ 8 + 4 × 3 = ______.

Multi-step Word Problems A teacher has 72 apples. He distributes them equally in 4 bags. Then, 4 students gives the teacher 10 apples each. Finally, all of them together eat 20 apples. Calculate the number of apples left with the teacher. We will first form the problem. Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 29

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The teacher puts the apples equally in 4 bags. That is, 72 ÷ 4. 4 students gives him 10 apples each. So, he has 4 × 10 more apples. That is, 72 ÷ 4 + 4 × 10. Now, all of them ate 20 apples. So, the apples they are left with are: 72 ÷ 4 + 4 × 10 – 20 Let us solve this problem. Step 1

Step 2

Step 3

Step 4

Divide (÷)

Multiply (×)

Add (+)

Subtract (–)

= 72 ÷ 4 + 4 × 10 – 20

= 18 + 4 × 10 – 20

= 18 + 40 – 20

= 58 – 20 = 38

= 18 + 4 × 10 – 20

= 18 + 40 – 20

= 58 – 20

So, by using DMAS, they are left with 38 apples. Example 13

Tom has 2 sets of 10 toy cars. His father gives him 5 more toy cars. He then wants to give 4 cars to his friend. How many cars will he be left with? Cars in sets of 2 = 2 × 10 Father gives him 5 more cars = 2 × 10 + 5

Think and Tell

Gives 4 cars to his friend = 2 × 10 + 5 – 4

Are multiplication and division related?

Number of cars left = ? Let us use DMAS to find out. Step 1

Step 2

Step 3

Step 4

Divide (÷)

Multiply (×)

Add (+)

Subtract (–)

There is no division,

= 2 × 10 + 5 – 4

= 20 + 5 – 4

= 25 – 4 = 21

= 20 + 5 – 4

= 25 – 4

so we can move to the next step.

Therefore, Tom is left with 21 toy cars. Do It Together

Lisa has 50 flowers of 2 colours each. Her mother gives her 8 more flowers. Lisa gives half of the flowers that her mother gave to her friend, Siya. How many flowers does she have now? Initial number of flowers = _________________ Flowers given by mother = _________________ Flowers given to Siya

= _________________

Number of flowers left

= _________________

Think and Tell

If we do not follow DMAS, will we get the same answer?

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Step 1

Step 2

Step 3

Step 4

Divide (÷)

Multiply (×)

Add (+)

Subtract (–)

Therefore, Lisa is left with ____ flowers.

Do It Yourself 2C 1

Tick () the correct answer.

a 100 × 10 – 100 + 2000 ÷ 100 = ____ i

979

920

iii 780

iv 29

÷, ×, +, –

iii ÷ , ×, –, +

iv ×, ÷, +, –

iii 0

iv 64

b The order of DMAS is: i c

+, ×, – , ÷

8 × 8 – 8 = ____ i

d 100 ÷ 10 + 10 × 10 = ____ i

200

110

iii 50

iv 1

iii 23

iv 7

e 63 ÷ 9 + 12 × 4 – 5 = ____

Fill in the blanks. a 9 + 6 – 4 × 2 = ______

b 17 + 8 × 3 = ______

c 28 ÷ 7 + 8 × 5 = ______

d 55 ÷ 11 + 7 = ______

e 82 × 3 + 20 = ______

g 16 + 8 ÷ 2 – 1 × 5= ______

h 25 – 4 × 12 ÷ 4 + 3 = ______

20 + 50 ÷ 2 – 5 = ______

Write true (T) or false (F). a 5 × 4 + 12 ÷ 3 = 24 b In DMAS, we first perform addition/subtraction and then division/multiplication. c In DMAS, the last step is subtraction. d 36 ÷ 6 – 3 × 2 = 2

Simplify. a 40 + 60 ÷ 20 × 16 – 4

b 80 + 20 × 20 – 10 ÷ 5

c 279 + 321 ÷ 3 – 57 × 2

d 724 – 150 ÷ 50 × 4 + 58

e 676 + 835 × 45 ÷ 15 – 10

g 15 × 20 ÷ 10 – 5 + 15

h 39 – 4 × 12 ÷ 3

Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 31

120 – 12 + 36 ÷ 3 × 5

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Word Problems 1

Alex has saved ₹120. He wants to buy a toy for ₹28 and a book. The book is half of the

Anish had 50 stickers. He gave 3 stickers to each of his 4 friends. He then bought 10

Sahil has 15 pencils. He wants to distribute them among his 3 friends. All three

price of the toy. How much money will he have left after buying the toy and the book? more stickers. How many stickers does Anish have now?

friends already have 2 pencils. If he gives each friend the same number of pencils, find the total number of pencils each friend has.

Out of 350 students in a school, 25 students did not come on Monday. The rest of the students brought ₹230 each for donation. How much money did the students bring?

Points to Remember • When two or more numbers are added, they are called addends and their result is called the sum. • The minuend is the bigger number from which the subtrahend, the smaller number is subtracted. Their result is called the difference. • The number to be multiplied is called the multiplicand and the number by which it is multiplied is called the multiplier. Their result is called the product. • When two numbers are divided, the number that is getting divided is called the dividend and the number that divides it is called the divisor. The number that we get as a result is called the quotient and the value that is left after the division is called the remainder. • When we multiply a number by 10, 100, 1000 and so on, we put as many 0s to the right of the multiplicand as there are 0s in the multiplier. • When we divide by 10, 100, 1000 and so on, the same number of digits as the number of 0s in the divisor, from the right of the dividend, becomes the remainder. The remaining digits are the quotient. • When more than two operations are given together in a problem, we use DMAS.

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Math Lab Setting: In pairs

Materials Needed: 3 dice, a pencil and paper. Method: 1

Divide the students into pairs.

Each pair should have a pencil and paper.

One student from each pair would roll the 3 dice.

From the three numbers obtained, one student from each pair will form the largest and

Note down the number obtained on a sheet of paper.

Keep repeating the above process, and each time add the results obtained after

The student who gets the sum of 6,00,000 or more first, wins.

the smallest number. The other students will multiply the two numbers.

multiplication to find the sum.

Chapter Checkup 1

Find the sum of the given numbers. a d

5,65,466 and 3,54,537

8,64,826 and 96,537

9,87,654 and 45,774

2,53,621, 12,365 and 36,586

54,676 and 34,575

2,67,990 and 47,536

8,68,636 and 65,365

9,75,843 and 4646

9,54,863 and 8,45,622

325

896

4546

6457

7656

9876

6789 and 432

8566 and 1564

8765 and 321

4321 and 2981

74,562

8646 and 200

1256 and 3251

Divide the given numbers by 10, 100 and 1000. a

Find the product of the given numbers. a

6,54,321 and 45,216

Multiply the given numbers by 10, 100 and 1000. a

6,73,778 and 5,67,433

Subtract the given numbers. a

56,789 and 23,456

2100

4086

3235

51,200

65,200

Divide the given numbers using long division. a d

42,155 and 125 33,424 and 234

b e

Chapter 2 • Operations on Numbers Up to 6 Digits

Maths Grade 5 Book_Chapter 1-6.indb 33

54,767 and 231 88,757 and 768

c f

56,785 and 314 99,866 and 982 33

18-12-2023 11:01:34

Simplify. a d

25 + 3 × 4

34 + 2 × 5 – 9 ÷ 9

b e

18 ÷ 2 × 3

63 – 4 + 8 ÷ 2 × 30

12 × 6 – 2 + 18 ÷ 3

The product of two numbers is 25,290. If one number is 562, find the other number.

Find the difference of the largest 6-digit number and the smallest 6-digit number.

10 What number must be subtracted from the sum of 5,00,000 and 3,00,000 to make it equal to their difference?

Word Problems 1

I n a town there are 4,10,900 men, 3,05,987 women, and 1,78,678 children. How many people are there in the town?

I sha is planning a road trip that covers a total distance of 7500 km. She plans to drive 150 km each day. In how many days will Isha complete the trip?

garden is divided into 60 equal sections. Each section has 1280 flowers. 30 of A these sections are replanted with the same number of new flowers, and the rest remain the same. How many new flowers are there in the garden now?

eha’s annual income is ₹98,780. She spends ₹50,000 and saves the rest. How N much money will she save in 10 years?

5 6

aya has 80 marbles. She gives 5 marbles each to 6 of her friends. Then, she M finds 15 more marbles. How many marbles does Maya have now? here are 5565 mangoes. The number of oranges is twice the number of T mangoes. How many oranges will each box contain if Riya keeps an equal number of oranges in 5 boxes?

7 8

n auditorium has a capacity of 64,070 people. If one row can seat 430 people, A how many rows are there in the auditorium? hreya had ₹45,000 in her bank account. She got her salary and the amount in S her bank account became thrice of what it was before. She then spent ₹14,500 paying her rent and groceries. How much money was left in her bank account?

Maths Grade 5 Book_Chapter 1-6.indb 34

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Factors and Highest Common Factor

Let’s Recall

We know about multiplication facts of numbers. Let us look at multiplication facts of the number 16. Imagine, we have 16 tiles that we need to arrange in rows and columns. We can do it in the following manner:

4 × 4 = 16

16 × 1 = 16

2 × 8 = 16 8 × 2 = 16 1 × 16 = 16

The different numbers of rows and columns represent the numbers that can divide 16 exactly without leaving a remainder. Thus, all the numbers, including 1, 2, 4, 8, and 16 shown in the above representation divide 16 exactly, without leaving any remainder. So, we can say that 16 is divisible by 1, 2, 4, 8, and 16.

Let's Warm-up

Match the numbers with their factors. Column A

Column B

1, 5, 11

1, 3, 5

1, 23

1, 2, 4

1, 3, 9

I scored ____________ out of 5.

Maths Grade 5 Book_Chapter 1-6.indb 35

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Understanding Factors Real Life Connect

For a group dance event, 12 dancers were to enter the stage together. The dance teacher made all the dancers stand in a single line, but all of them could not be accommodated on the stage.

So, she decided to make 2 lines of 6 dancers each.

In this way, all the dancers could still not fit on the stage. Then, she made 3 lines of 4 dancers each. Now, all the dancers could be accommodated on the stage.

So, here we saw three different ways to make 12 dancers stand in lines having equal rows and columns. 1 × 12

2×6

3×4

Factors of a Number 1, 12, 2, 6, 3, and 4 are all the numbers we multiplied to get 12. Therefore, we can say that 1, 2, 3, 4, 6 and 12 are all factors of 12. The numbers that are multiplied to get a product are called its factors.

Factors

Product

Look at the factors of 12 once more. 36

Maths Grade 5 Book_Chapter 1-6.indb 36

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• 1 is its smallest factor. 1 is the smallest factor of every number. • 1 2 is the greatest factor of 12. Every number is the greatest factor of itself. • E very factor of 12 is less than or equal to 12. The factors of a number are equal to or less than the number.

Remember! When studying factors of a number, we consider factors that are whole numbers. Also, since division by zero is not possible, 0 cannot be a factor of any number.

We know that multiplication and division are opposite operations. Thus, we can also define factors in terms of division. The factor of a number is a number that divides the given number evenly or exactly, leaving no remainder. In our example, the numbers 1, 2, 3, 4, 6, and 12 are all factors of 12 because each of these numbers divides 12 exactly without leaving any remainder.

Finding Factors We can find the factors of a number by recalling the multiplication facts of numbers. We begin with the multiplication table of 1 and gradually move on to higher tables to see where the given number appears. Finding Factors Using Multiplication Let us try to find the factors of 18 using multiplication. Multiplication table of 1 gives 1 × 18 = 18. Multiplication table of 2 gives 2 × 9 = 18. Multiplication table of 3 gives 3 × 6 = 18. Multiplication table of 4 does not give 18 as a product.

Remember! When finding factors by multiplication, always: • S tart with the multiplication table of 1. • S top when any factor starts repeating.

Multiplication table of 5 does not give 18 as a product. Multiplication table of 6 gives 6 × 3 = 18. However, 6 × 3 = 18 is the same as 3 × 6 = 18, which is already covered. Thus, we will stop at the multiplication table of 6. So, the factors of 18 are 1, 2, 3, 6, 9 and 18. Finding Factors Using Division To find the factors of a number, we can also look for numbers that divide the given number exactly, without leaving any remainder. Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 37

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Let us find the factors of 36 using the division method. Number

Divisors

Quotient Remainder

Factors

1 and 36 are factors of 36.

2 and 18 are factors of 36.

3 and 12 are factors of 36.

4 and 9 are factors of 36.

5 and 7 are NOT factors of 36.

6 is a factor of 36.

7 and 5 are NOT factors of 36.

8 and 4 are NOT factors of 36.

STOP! 9 and 4 are already covered above.

When factors are repeated, no further division takes place. So, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Example 1

Find the factors of 20 using multiplication. Multiplication table of 1 gives 1 × 20 = 20. Multiplication table of 2 gives 2 × 10 = 20. Multiplication table of 3 does not give 20 as a product. Multiplication table of 4 gives 4 × 5 = 20. Multiplication table of 5 gives 5 × 4 = 20. However, 5 × 4 = 20 is the same as 4 × 5 = 20, which is already covered above. Thus, we will stop at the multiplication table of 5. So, the factors of 20 are 1, 2, 4, 5, 10 and 20.

Example 2

Find the factors of 30 using the division method. Number

Divisors

Quotient Remainder

Factors

1 and 30 are factors of 30.

2 and 15 are factors of 30.

3 and 10 are factors of 30.

4 and 7 are NOT factors of 30.

5 and 6 are factors of 30.

STOP! 6 and 5 are already covered above.

When factors are repeated, no further division takes place. So, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

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Do It Together

Find the factors of 24 using the division method. Number

Divisors

Quotient

Remainder

Factors

1 and 24 are factors of 24.

___

___ and ___ are factors of 24.

___

___ and ___ are factors of 24.

___

___ and ___ are factors of 24.

5 and 4 are NOT factors of 24.

___

STOP! 6 and 4 are already covered above.

So, the factors of 24 are __________________________________________________________________.

Prime and Composite Numbers Numbers which have only two factors, namely 1 and the number itself are called prime numbers. Numbers having more than two factors are called composite numbers. Example 3

Is 17 a prime number? 17 can only be divided by 1 or 17, which are its only factors. So, 17 is a prime number.

Example 4

Is 16 a prime number? 16 can be divided by 1, 2, 4, 8, and 16, which are its factors.

Remember! • 0 and 1 are neither prime nor composite. • 2 is the lowest and the only even prime number.

So, 16 is not a prime number. 16 is a composite number. Do It Together

Find the factors of the given numbers. Write if the number is prime or composite. Number

Factors

Number of Factors

Prime or Composite

1, 2

Prime

1, 3

1, 2, 4

Composite

5 6

2 1, 2, 3, 6

9 10

1, 3, 9 4

Chapter 3 • Factors and Highest Common Factor

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Do It Yourself 3A 1

Find the factors of the numbers using the multiplication method. a 55

b 14

d 34

e 37

d 81

e 63

e 55

Find the factors of the numbers using the division method. a 15

c 48

b 72

c 41

Sort the given numbers as prime or composite numbers. a 56

b 73

c 81

d 27

g 72

h 11

How many prime numbers are there between 10 and 20? List them.

Write if the given statement is true or false. Give a reason. If 10 is a factor of a number, 2 is also the factor of that number.

Word Problem 1

An event manager is getting chairs arranged for a stage show. He wants to put equal numbers of chairs in rows and columns with no chair left over. What are the different ways in which he can arrange the chairs if he has 189 chairs?

Concept of Divisibility The rules of divisibility will help you find the numbers that divide other numbers without leaving any remainder.

Divisibility by 2, 5 and 10 The divisibility of a number by 2, 5 and 10 can be checked by observing the last digit of the number. Look at the table below: A number is divisible by:

If the last digit is:

0, 2, 4, 6, 8

0, 5

So, a number ending in 0, 2, 4, 6 or 8 is always divisible by 2. A number ending in 0 or 5 is always divisible by 5.

Error Alert! Although 0 itself fulfils all the three conditions above, it is an exception to the rule. 0 is neither prime nor composite, - hence, we say that 0 is divisible by any number.

A number ending in 0 is always divisible by 10. A number is divisible by 2, 5 and 10 if it is divisible by 10. 40

Maths Grade 5 Book_Chapter 1-6.indb 40

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Example 5

Is 64 divisible by 2?

Example 6

Is 45 divisible by 5?

64 ends with the digit 4.

45 ends with the digit 5.

So, 64 is divisible by 2.

So, 45 is divisible by 5.

Put a tick () if the number is divisible and a cross if it is not.

Do It Together

Divisible by

Number

12 15 20 25 45 50

Divisibility by 3 and 9 To test the divisibility of a number by 3 or 9, we add the digits of the number. If the sum is divisible by 3 or 9, we know that the number itself is divisible by 3 or 9 respectively. A number is divisible by:

If the sum of the digits is divisible by:

A number is divisible by both 3 and 9 if it is divisible by 9. Example 7

Do It Together

Is 54 divisible by 3?

Example 8

Is 452 divisible by 9?

5+4=9

4 + 5 + 2 = 11

9 is divisible by 3.

11 is not divisible by 9.

So, 54 is divisible by 3.

So, 452 is not divisible by 9.

Put a tick () if the number is divisible and a cross if it is not. Number

Divisible by 3

28 32 42 29

Think and Tell A number is divisible by 10. Is it divisible by 2 and 5?

33 63 Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 41

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Do It Yourself 3B 1

Circle the numbers that are divisible by 2. a 11

e 160

e 300

b 75

c 37

d 190

g 63

h 25

c 92

d 60

e 130

c 56

d 118

e 919

200

101

Circle the numbers that are divisible by 3. b 63

Circle the numbers that are divisible by 9. a 36

d 49

a 51

a 72

c 38

Circle the numbers that are divisible by both 5 and 10. f

b 24

b 45

A secret passage has a number lock. Given below are the clues to the correct number to open the lock. Which of the given number shows the secret code? a 660

b 530

c 129

•

The number should be divisible by 5.

•

On adding the digits of the number, the sum is divisible by 3.

•

The number has 0 as the last digit.

d 365

Word Problem 1

Era has 324 blueberry cupcakes and 135 chocolate cupcakes. Can she put them in

identical groups of 9 cupcakes without having any cupcakes left over? Give a reason for your answer.

Prime Factorisation A prime factor of a number is a factor that is also a prime number. A composite number can be expressed as the product of prime factors. This is called prime factorisation. A factor tree can be used to find the prime factors of a number. Let us find the factors of 120 using a factor tree. Step 1

Write the number at the top of the factor tree and draw two branches below. Here, we write 120 at the top of the tree with two branches below.

120

Maths Grade 5 Book_Chapter 1-6.indb 42

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Step 2

Fill in the branches with a factor pair of the number above.

120

The number 120 has many factor pairs. As the number ends in a 0, it is easy to choose 12 × 10. So, we have:

Step 3

Continue until each branch ends in a prime number.

120

Both the numbers 12 and 10 are not prime numbers so two branches are extended from each number and filled in with their factors.

Here, we have highlighted the numbers 3, 5, and 2 as they are prime numbers. So, no more branching for these numbers.

However, 4 is not a prime number. So, we need to continue the branches for this number.

12 4 2

120

10 5

We now have each branch ending in a prime number, so this factor tree is now complete.

Step 4

Write the prime factorisation of the number. 120 = 2 × 2 × 2 × 3 × 5

Multiply all the prime numbers found by the factor tree method.

Many different factor trees may be constructed for a number, but the prime factors found will always be the same.

Let us construct factor trees for 24. 24

2 2

6 2

Prime factorisation of 24 = 2 × 2 × 2 × 3 The number can then be written as the product of prime factors as shown above.

Remember! Prime factorisation can be checked by multiplying all the factors. The product should be equal to the given number.

Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 43

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Prime factors can also be found using repeated division. In this method, we start dividing the given number by the smallest possible prime number and continue dividing by prime numbers until we reach 1. Let us factorise 81 by the division method. 3 3 3

Remember!

There is only one unique product of prime factors for any number. For example, prime factors of 40 = 2 × 2 × 2 × 5. There is no other possible set of prime numbers that can be multiplied to make 40.

9 3 1

So, the prime factorisation of 81 = 3 × 3 × 3 × 3. Example 9

Draw a factor tree for the number 36 and then express the number as a product of its prime factors.

Example 10

Use repeated division to find the prime factors of 84. Express the number as a product of its prime factors. 2 84 2 42

3 21 7

2 3

7 1

So, 84 = 2 × 2 × 3 × 7.

So, 36 = 2 × 2 × 3 × 3. Do It Together

Complete the factor tree for the number 64 and then express the number as a product of its prime factors.

64 16

So, 64 = _____ × _____ × _____ × _____ × _____ × _____.

4 8

Do It Yourself 3C 1

Which of the following represents the prime factorisation of 80? a 2×2×4×5

b 2×2×2×2×5

c 2×2×2×3×5

d 2×3×3×4×5

Which of the following represents the prime factorisation of 56? a 2×4×7

b 2×2×2×7

c 2×3×3×7

d 3×7×7

Maths Grade 5 Book_Chapter 1-6.indb 44

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Which of the following has 2 × 2 × 3 × 3 × 5 × 7 as its prime factorisation? a 1206

c 1260

d 1360

Find the prime factorisation of the following using the factor tree method. a 8

b 15

c 20

d 24

g 63

h 72

b 1222

e 33

112

Find the prime factorisation of the following using the division method. a 12

b 16

c 22

d 30

g 51

h 60

e 44

100

148

Word Problem Samantha has 132 flowers that she wants to arrange in vases. She wants to place an

equal number of flowers in each vase. The number of vases that she uses is the largest prime factor of 132. How many vases did she use?

Highest Common Factor Real Life Connect

Mrs. Gupta has 18 students in her class today and her fellow teacher Mrs. Mehra has 27 students. The teachers want to combine the classes for a group activity. The activity requires that the groups be of the same size. Each group should have the same number combination of students from each class. The teachers want to maximise the number of students per group formed.

Finding the HCF The teachers want to divide their students into groups. So, the factors of the total number of children can represent the group size and number of groups. Total number of children in the class = Number of groups × Students per group Mrs. Gupta’s class

Mrs. Mehra’s class

Group Size

Number of Groups

Group Size

Number of Groups

Possible group size = 27, 9, 3 and 1.

Possible group size = 18, 9, 6, 3, 2 and 1. Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 45

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Common Factors Notice that all the possible group sizes are factors of the class sizes. Now, let us list the possible group sizes that are equal in the two classes. We do this in the following manner: So, 1, 3 and 9 are possible equal group sizes for the two classes. These are the common factors of the two numbers. We can say that 1, 3 and 9 are common factors of 18 and 27.

Factors of 18 2 6

Factors of 27 1 3 9

A common factor is a number that can evenly divide a set of two or more numbers without leaving any remainder.

Remember! 1 is the common factor of every two or more given numbers.

Remember! When two or more numbers have the same factor, that factor is called a common factor.

How can we find the common factors of any two numbers? We list the factors of each number and then identify the common factors among them. Let us try to find the common factors of 16 and 40. Factors of 16 = 1 , 2 , 4 , 8 and 16 Factors of 40 = 1 , 2 , 4 , 5, 8 , 10, 20 and 40 Common Factors of 16 and 40 = 1 , 2 , 4 and 8 Example 11

Find the common factors of 15 and 20. Factors of 15 are 1, 3, 5, and 15. Factors of 20 are 1, 2, 4, 5, 10 and 20. Identify the common factors of the two numbers. The common factors of 15 and 20 are 1 and 5.

Example 12

Find the common factors of 12 and 16. Factors of 12 are 1, 2, 3, 4, 6, and 12. Factors of 16 are 1, 2, 4, 8, and 16. The common factors of 12 and 16 are 1, 2, and 4.

Maths Grade 5 Book_Chapter 1-6.indb 46

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Find the common factors of 10 and 14.

Do It Together

Factors of 10: __________________________________. Factors of 14: __________________________________. The common factors of 10 and 14 are __________________________________.

Factor Method The following are the possible group sizes that can be formed in Mrs. Gupta’s and Mrs. Mehra’s classes: Mrs. Gupta’s class of 18: 18, 9, 6, 3, 2 and 1. Mrs. Mehra’s class of 27: 27, 9, 3 and 1 The common factors represent the equal number of group sizes that can be formed. These are 1, 3 and 9. For the activity, the teachers wanted to maximise the number of students per group. The biggest group size that can be achieved is 9! So, 9 is the Highest Common Factor, or simply, the HCF. 9 is the HCF of 18 and 27. Let us see how we can find the HCF of any two numbers, say 12 and 18. We proceed using the following steps: The highest common factor (HCF) of 12 and 18 is 6.

Factors of 12 = 1 , 2 , 3 , 4 , 6 and 12 Factors of 18 = 1 , 2 , 3 , 6 , 9 and 18. Common Factors of 12 and 18 = 1 , 2 , 3 and 6 .

The largest of the common factors is the HCF of the 2 numbers. Example 13

Find the HCF of 56 and 70. Factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. Factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. The common factors of 56 and 70 are 1, 2, 7, and 14.

Remember! The HCF of given numbers cannot be bigger than any one of the given numbers.

The highest common factor (HCF) of 56 and 70 is 14. Example 14

Mike has 16 blue marbles and 8 white ones. If he wants to place them in identical groups without any marbles left, what is the greatest number of groups Mike can make?

Step 1

What do we know? Number of blue marbles: 16 Number of white marbles: 8

Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 47

Step 2

What do we need to find? Greatest number of groups Mike can make

18-12-2023 11:01:44

Step 3 Solve to find the answer. As we need to find the greatest number of groups, we will find the HCF. Factors of 16 = 1, 2, 4, 8, 16

Factors of 8 = 1, 2, 4, 8

Common factors of 16 and 8 = 1, 2, 4, 8. So, the Highest Common Factor (HCF) = 8.

Hence, the greatest number of groups that Mike can make is 8. Do It Together

Find the HCF of 18 and 24. Factors of 18 are 1, 2, 3, 6, 9, and 18. Factors of 24 are _____________________________. The common factors of 18 and 24 are _____________________________. The highest common factor (HCF) of 18 and 24 is ____.

Prime Factorisation Method We have learnt about prime factorisation earlier in this chapter. Now, we have to use the method of prime factorisation to find the HCF of two or more numbers. Follow the steps given below to find the HCF of two or more numbers using the prime factorisation method. Let us take 24 and 32 as an example. Step 1

Find the prime factorisation of both numbers. We use the prime factorisation method. 24 = 2 × 2 × 2 × 3

32 = 2 × 2 × 2 × 2 × 2

Remember! The HCF of two or more numbers is the product of their common prime factors.

Step 2

Find the combination of prime factors that appear in both numbers.

Note that the factors in blue appear in both 24 and 32. These together form the HCF. 24 = 2 × 2 × 2 × 3 32 = 2 × 2 × 2 × 2 × 2

Step 3

Multiply the combination of common factors to get the HCF of the numbers. HCF = 2 × 2 × 2 = 8. Example 15

Find the HCF of 12, 18 and 24. 12 = 2 × 2 × 3 18 = 2 × 3 × 3

24 = 2 × 2 × 2 × 3

The common factors of 12, 18, and 24 are 2 and 3. Hence, HCF = 2 × 3 = 6. 48

Maths Grade 5 Book_Chapter 1-6.indb 48

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Find the HCF of 16 and 48 using the prime factorisation method.

Do It Together

Factors of 16 = 2 × 2 × 2 × 2. Factors of 48 = ____________________. The common factors of 16 and 48 are ____________________. Hence, HCF = ____________________.

Long Division Method In this method, we divide the greater number by the smaller number. The remainder of this division becomes the new divisor, and the previous divisor becomes the new dividend. We continue this process until we get 0 as the remainder. The last divisor obtained is the HCF of the given numbers. Let us find the HCF of 18 and 48 using the long division method. Step 1

Divide the bigger number (48) by the smaller number (18).

Step 2

The remainder (12) becomes the new divisor. The previous divisor (18) becomes the new dividend.

Step 3

Repeat till we get the remainder as 0. The last divisor is the HCF.

Remainder 12 becomes the new divisor. Remainder is 0. So, the last divisor 6 is the HCF. Example 16

2 18 4 8 –3 6 1 12 18 –1 2 2 6 12 –1 2 0

Divisor 18 becomes the new dividend.

Find the HCF of 20 and 24 by the division method. Dividing 24 by 20, we get 4 as the remainder. This remainder becomes the new divisor and 20 becomes the new dividend. We continue this process till the remainder becomes 0, as shown below. The final divisor that leaves no remainder is the HCF. Thus, the HCF of 20 and 24 is 4.

Chapter 3 • Factors and Highest Common Factor

Maths Grade 5 Book_Chapter 1-6.indb 49

20 2 4 1 –2 0 4 20 5 –2 0 0

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Example 17

Find the HCF of 56 and 84 by the long division method. Dividing 84 by 56, we get 28 as the remainder. This remainder becomes the new divisor and 56 becomes the new dividend. We continue this process until the remainder becomes 0, as shown. Since the last divisor is 28, the HCF of 56 and 84 is 28.

Do It Together

Find the HCF of 52 and 68.

56 8 4 1 –5 6 28 56 2 –5 6 0 52 6 8 1

Since the last divisor is ____, the HCF of 52 and 68 is ____.

Do It Yourself 3D 1

Find the common factors for the 2 numbers. a 13, 36

b 30, 45

c 24, 72

d 14, 24

e 21, 16

18, 27

g 42, 22

h 28, 35

50, 45

30, 54

Find the HCF by finding all the factors. a 16, 24

b 25, 45

c 36, 72

d 12, 18

e 27, 81

40, 50

g 15, 90

h 28, 42

27, 47

33, 55

Find the HCF using the prime factorisation method. a 12, 15

b 15, 30

c 32, 40

d 45, 60

e 36, 54

26, 65

g 48, 60

h 75, 90

52, 78

24, 40, 56

Find the HCF of the following numbers by using the long division method. a 16, 64

b 18, 24

e 90, 135

20, 120

81, 108

28, 35

c 150, 225

d 96, 120

g 14, 84

h 216, 256

Suresh has containers with a capacity of 180 and 162 litres of milk. He can fill a bucket repeatedly to pour milk into the two containers. What is the capacity of the largest bucket that can be used to fill the two containers completely?

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Word Problems 1

The length, breadth and height of a box are 75 cm, 85 cm, and 95 cm respectively. Find the length of the longest tape which can measure the three dimensions of the box exactly.

Three pieces of fabric 42 m, 49 m, and 63 m long have to be divided into curtains of

Three drums contain 36 litres, 45 litres and 72 litres of oil. Find the capacity of the largest

the same length. What is the greatest possible length of each curtain?

container that can measure the content of each drum an exact number of times.

Points to Remember • A number that has only two distinct factors i.e., 1 and the number itself, is called a prime number. • A number which has more than two factors is called a composite number. • 1 is neither a prime nor a composite number. • If a number is expressed in the form of the product of prime numbers, then this form is called the prime factorisation of the given number. • The highest common factor of two or more numbers given, is the greatest factor that divides all the given numbers exactly without leaving any remainder.

Math Lab Prime Numbers From 1 to 100 Setting: In groups of 4 Materials Required: number chart for numbers 1 to 100, colour pencils Method: 1

Colour the number 1 on the number chart as 1 is neither prime nor composite.

Circle the number 2 and put a blue dot for all the numbers divisible by 2.

Circle the number 3 and put a red dot for all the numbers divisible by 3.

Circle the number 5 and put a green dot for all the numbers divisible by 5.

Find the next number that is neither circled nor has a dot. Circle that number and

List the numbers that have dots of all the colours.

put a different colour for all the numbers that are divisible by this new number.

Chapter 3 • Factors and Highest Common Factor

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All the circled numbers are prime numbers, and the numbers that have dots are

Write the number of prime numbers between 1 and 100.

Write the number of composite numbers between 1 and 100.

composite numbers.

Chapter Checkup 1

Find the factors of the numbers using the multiplication method. a 20

e 36 e 61

g 57

h 73

Use the divisibility rules to check if the given numbers are divisible by 2, 3, 5, 9, 10. b 84

c 93

d 450

e 700

Find the prime factorisation of the following using the factor tree method. b 65

c 102

d 112

e 140

Use the repeated division method to find the prime factorisation of the numbers. b 21

c 128

d 164

e 91

Find the common factors of each pair of numbers. b 75, 125

c 33, 55

d 120, 156

c 48, 120

d 150, 225

c 60, 225

d 105, 180

Find the HCF by finding all the factors. a 16, 60

d 80 d 46

a 25, 45

c 54 c 38

a 75

b 28 b 31

a 88

e 42

a 12

a 35

d 24

Sort the numbers as prime or composite numbers. f

c 16

Find the factors of the numbers using the division method. a 56

b 12

b 25, 65

Find the HCF using the prime factorisation method. a 34, 38

b 34, 51

10 Find the HCF of the following numbers by using the long division method. a 36, 63

b 119, 187

c 45, 89

d 136, 170

11 Write a 3-digit number such that both 420 and the number have 2 as a common factor. 12 Two tanks contain 250 litres and 425 litres of water respectively. What will be the maximum capacity of

a bucket that can measure water in the two tanks exactly?

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13 Kirti is making identical balloon bunches for a party. She has 24 white balloons and 16 orange

balloons. She wants each arrangement to have the same number of balloons of each colour. What is the greatest number of bunches that she can make if every balloon is used?

Word Problems 1

Jose is making a game board that is 66 inches by 24 inches using square tiles

Neeta wants to distribute refreshments at a party. She has 240 cupcakes and

only. What is the largest square tile he can use and how many tiles will he need? 160 sandwiches. She wants to distribute the food items among her classmates

equally in packets. What is the maximum number of packets she can make and what will be the contents of each? 3

Anita baked 30 oatmeal cookies and 48 chocolate chip cookies to package in

plastic containers for her friends at school. She wants to divide the cookies into identical containers so that each container has the same number of each kind of cookie. If she wants each container to have the greatest number of cookies possible, how many plastic containers does she need?

Chapter 3 • Factors and Highest Common Factor

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Multiples and Least

Common Multiples

Let's Recall We know how to skip count numbers. Let us look at skip counting by the number 6.

Skip counting by 6

Result

0+6

If you look closely, the second column of the table resembles the multiplication table of 6.

6+6

12 + 6

18 + 6

Skip counting by 7 gives the following result.

24 + 6

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …

30 + 6

36 + 6

42 + 6

48 + 6

54 + 6

When we skip count by 6, we add 6 at every step.

This is the same as the multiplication table for 7. Similarly, we can skip count by any number to get the multiplication table for that number.

Let's Warm-up

Match the following. Column A

Column B

Multiplication table of 8

60, 96, 108

Multiplication table of 9

40, 70, 80

Multiplication table of 10

24, 48, 64

Multiplication table of 12

36, 54, 63

I scored _________ out of 4.

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Understanding Multiples Real Life Connect

Ajay is very fond of music. He attends the music class on every fifth day of the month. The rest of the days are devoted to practising what he has already learnt in the class. Ajay wants to mark every fifth day on the calendar so that he does not forget to attend his music classes. So, Ajay skip counts by 5 to make a list of the days on which he will attend his music class in the month of August. Music Classes:

5th August

10th August

15th August

Finding Multiples Here, the list prepared by Ajay gives the first six multiples of 5. When any number is multiplied by 1, 2, 3, 4, …, we get the multiples of that number.

20th August

25th August

30th August

Think and Tell

Do all the numbers when Ajay attends the music classes appear in the multiplication table of 5? What do we call such numbers?

For example, we get the multiples of 6 by multiplying 6 with 1, 2, 3, 4, … and so on. 6 × 1 = 6, 6 × 2 = 12, 6 × 3 = 18, 6 × 4 = 24, … Hence, multiples of 6 are 6, 12, 18, 24, … We get the multiples of 7 by multiplying 7 with 1, 2, 3, 4, … and so on. 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, … Hence, multiples of 7 are 7, 14, 21, 28, … Look at the multiplication sentence given below. 7 and 2 are the factors of 14. 14 is the multiple of 7 and 2. Multiple of 7 7 × 2

= 14

Remember! Multiple of 2

Multiple of a number = Number × Any number

Factors of 14 This is the same as recalling the multiplication tables of those numbers. For example, multiples of 9 are the numbers in the 9 times table. 9×1= 9×2= 9×3= 9×4= 9×5=

9 18 27 36 45

Chapter 4 • Multiples and Least Common Multiples

Maths Grade 5 Book_Chapter 1-6.indb 55

9×6= 9×7= 9×8= 9×9= 9 × 10 =

54 63 72 81 90 55

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Multiples of 15 are the numbers that are in the 15 times table. 15 × 1 = 15

15 × 6 =

15 × 3 = 45

15 × 8 = 120

15 × 2 = 30

15 × 7 = 105

15 × 4 = 60

15 × 9 = 135

15 × 5 = 75

15 × 10 = 150

We can also find the multiples of a number by skip counting on the number line. When we skip count by a number, we get the multiples of that number. +2

10 11 12 13 14 15

The numbers 2, 4, 6, 8, 10, 12 and 14 are the first seven multiples of 2.

Another way to create a list of multiples of a number is to start at the number and add it repeatedly to the sum. The repeated addition method is one of the simplest methods to find the multiples of any given number. For example, the multiples of 25 can be found by: Using the Multiplication Tables

Using Repeated Addition

25 × 2 = 50

25 + 25 = 50

25 × 4 = 100

25 + 25 + 25 + 25 = 100

25 × 1 = 25 25 × 3 = 75

25 × 5 = 125

25 + 0 = 25

25 + 25 + 25 = 75

25 + 25 + 25 + 25 + 25 = 125

Here, 25, 50, 75, 100 and 125 are a few multiples of 25. Hence, it is possible to find the multiples of any number by repeated addition of that number or by recalling the times table of that number. Look at the multiples of 25 once more. 25 is a multiple of itself.

A number is a multiple of itself.

25 is a multiple of 1.

Every number is a multiple of 1.

There are countless multiples of 25.

There is no end to the multiples that you can find for a number.

Each multiple of 25 is either greater than or equal to 25.

Every multiple of a number is greater than or equal to the number itself.

You can also check if a number is a multiple of a given number by using division. If the remainder is zero, then the bigger number (dividend) is a multiple of the smaller number (divisor). 56

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For example:

1 7 9 − 7 2 − 2 0

3 1 1 1 0

On dividing 91 by 7, we get 0 as the remainder. So, 91 is a multiple of 7. Example 1

Show the first eight multiples of 8, on a number line. The first eight multiples of 8 can be obtained on skip counting by 8 on a number line. +8 0

+8 8

+8 16

+8 24

+8 32

+8 40

+8 48

So, 8, 16, 24, 32, 40, 48, 56 and 64 are the first eight multiples of 8. Example 2

+8 56

What is the 9th multiple of 7? The 9th multiple of 7 is equal to 7 × 9 = 63.

Example 3

Is 78 a multiple of 3? On dividing 78 by 3, we get no remainder. So, 78 is a multiple of 3.

Do It Together

Find the first five multiples of 4 using the multiplication tables. Show them on a number line. The first five multiples of 4 can be obtained by multiplying ______________________. So, we have, 4×1=4

4 × 2 = __

4 × 3 = __

4 × 4 = __

4 × 5 = __

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

So, 4, __, __, __, and __ are the first five multiples of 4.

Do It Yourself 4A 1

Write the first 5 multiples of the given numbers. a 8

b 11

Is 7209 a multiple of 9? Justify.

Find the 8th multiple of 15.

c 21

Chapter 4 • Multiples and Least Common Multiples

Maths Grade 5 Book_Chapter 1-6.indb 57

d 25

e 50

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Find the first four multiples of 9 and check whether they are odd or even.

Select any number with 0 in its ones place and write its first ten multiples. State whether the multiples

Write the numbers.

are odd or even.

a Multiples of 8 that are less than 32

b Multiples of 15 that are less than 100

c Multiples of 19 that lie between 57 and 152

d Multiples of 21 that lie between 105 and 210

Word Problem 1

Narendra attends drawing classes which are scheduled once every seven days,

starting from the 7th of August. He marks all the dates on a calendar. Write the dates he will mark for the classes that will be scheduled in the month of August.

Least Common Multiple Real Life Connect

Apart from the music class, Ajay also attends an Art class every fourth day of the month of August. Ajay wants to mark every fourth day on the calendar so that he does not forget to attend his art classes. So, Ajay skip counts by 4 and completes the list of the days on which he will attend his music and art classes in the month of August. Music Classes:

5th August

10th August

15th August

20th August

25th August

30th August

Art Classes:

4th August

8th August

12th August

16th August

20th August

24th August

28th August

Ajay is looking at the lists to see if there is any day on which he would be attending his music and art classes together. To his surprise, the number 20 is common and appears on both the lists. So, 20th of August is the only day in August when Ajay will be attending both the classes.

Common Multiples When a number is a multiple of two or more numbers, it is called a common multiple of those numbers. Let us list the multiples of 5 and 6. 58

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Multiples of 5:

…

Multiples of 6:

…

30 is a common multiple of 5 and 6.

60 is a common multiple of 5 and 6.

There are some multiples that are found on both the lists: 30 and 60. 30 and 60 are known as common multiples of 5 and 6. If the pattern is extended, you will find that 90 and 120 are also common multiples of 5 and 6. Common multiples of 5 and 6 are 30, 60, 90, 120, and so on. Out of these numbers, 30 is the lowest common multiple or LCM of 5 and 6. LCM by Common Multiples The smallest number among the common multiples is called the lowest common multiple or LCM. It is also the smallest number which can be divided by each of the given numbers. Let us understand how to find the LCM of 2 numbers using common multiples. Example 4

Find the LCM of 3 and 5 by finding common multiples. 15 is a common multiple of 3 and 5.

30 is a common multiple of 3 and 5.

Multiples of 3:

…

Multiples of 5:

…

15 is the lowest of all common multiples.

So, 15 is the LCM of 3 and 5.

Common multiples of 3 and 5 are 15, 30 and so on. 15 is the lowest among the common multiples, so it is the LCM of 3 and 5. LCM of 3 and 5 = 15. Example 5

Find the lowest common multiple of 6, 4 and 8. Multiples of 6:

…

Multiples of 4:

…

Multiples of 8:

…

Common multiples of 6, 4 and 8 = 24, 48, … 24 is the lowest among all the common multiples, so it is the LCM of 6, 4 and 8. Lowest Common Multiple or LCM of 6, 4 and 8 = 24 Do It Together

Write the multiples of 12 and 16. Colour the common multiples. Find the LCM. Multiples of 12: Multiples of 16: Chapter 4 • Multiples and Least Common Multiples

Maths Grade 5 Book_Chapter 1-6.indb 59

… … 59

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Common multiples of 12 and 16 are __________, __________ and so on. The lowest common multiple of 12 and 16 is __________.

Do It Yourself 4B 1

Which of the following is a common multiple of 25, 75 and 50? a 75

b 100

e 15 and 12

c 6 and 15

d 10 and 15

b 15 and 20 f

c 9 and 27

9 and 63

g 5 and 55

d 8 and 10

h 7 and 49

Find the common multiples of the 3 given numbers. Write their LCM. a 6, 12 and 18

b 9 and 12

Find the common multiples of the 2 given numbers. Write their LCM. a 12 and 15

d 150

Find two common multiples of the given numbers. a 5 and 10

c 125

b 13, 26 and 39

c 11, 22 and 33

d 6, 9 and 18

Which of these will have the same lowest common multiple as 4 and 15? a 4 and 20

b 6, 25

c 5 and 12

d 8 and 30

Word Problem 1

A bell at the primary school rings after 30 minutes while another bell rings at the

secondary school after 40 minutes. At what time will the two bells ring together next?

LCM by Prime Factorisation Method In this method, we first find the prime factorisation of each number. We then multiply each factor the maximum number of times it occurs in any given number. For example, to find the LCM of 60 and 45, follow the steps given below: Step 1 Write the prime factorisation of the numbers. 2

60 = 2 × 2 × 3 × 5

15 1

45 = 3 × 3 × 5

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Step 2 Then multiply each factor the greatest number of times it occurs in the prime factorisation of any given number.

We take 2 two times, 3 two times, and 5 one time and multiply to get the LCM.

So, LCM of 45 and 60 = 2 × 2 × 3 × 3 × 5 = 180 Example 6

Find the lowest common multiple of 84 and 90. 84 = 2 × 2 × 3 × 7

90 = 2 × 3 × 3 × 5

Now, we take 2 two times, 3 two times, 7 one time, and 5 one time and multiply to get the LCM. So, LCM of 84 and 90 = 2 × 2 × 3 × 3 × 5 × 7 = 1,260 Example 7

Find the LCM of 36, 48 and 56 by the prime factorisation method. 48 = 2 × 2 × 2 × 2 × 3

36 = 2 × 2 × 3 × 3

56 = 2 × 2 × 2 × 7

LCM of 36, 48 and 56 = 3 × 3 × 2 × 2 × 2 × 2 × 7 = 1,008 Do It Together

Find the LCM of 48 and 72 by prime factorisation. 2 48

2 72

2 24

2 12 48 = ____________

72 = ____________

LCM of 48 and 72 = ___

Do It Yourself 4C 1

Find the prime factorisation of the numbers. a 35 and 75

b 44 and 88

c 48 and 12

d 25 and 115

Rohan writes the prime factorisation of the numbers 12, 24 and 56 as shown below. 12 = 2 × 2 × 3

24 = 2 × 2 × 2 × 3

56 = 2 × 2 × 2 × 7

He says that the LCM of the numbers is 24. Is he right? Why or why not? 3

These numbers have already been factorised for you. Find the LCM of the pairs given below. 8=2×2×2

25 = 5 × 5

16 = 2 × 2 × 2 × 2

10 = 2 × 5

a 8, 10

b 8, 16

c 16, 10

d 25, 10

Chapter 4 • Multiples and Least Common Multiples

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Find the LCM of 2 numbers using the prime factorisation method. a 12, 15

b 15, 20

e 36, 45 i

10, 18

63, 105

c 18, 24

g 16, 24

d 25, 35

h 18, 20

Find the LCM of 3 numbers using the prime factorisation method. a 18, 40 and 45 e 28, 56 and 84

b 72, 96 and 108 f

36, 54 and 81

c 48, 56 and 70

d 30, 60 and 90

Word Problem 1

At a school carnival, every 4th student entering gets a candy and every 10th student gets a chocolate. Which student was the first to get a candy and a chocolate? Find the answer using the prime factorisation method.

LCM by Short Division Method In this method, • keep dividing the given numbers by common prime numbers until the quotient is 1. • Then multiply all the prime factors to get the LCM. For example, let us calculate the LCM of 25 and 45 using the short division method. Step 1: Arrange the numbers in a line to find the LCM.

25, 45

Step 2: Start dividing the numbers by the smallest possible prime number, which in this

5, 9

case is 5. Write the quotients in the next row.

1, 9

Step 3: Continue dividing until we get 1 as a quotient for all the numbers. Keep the

1, 3

numbers that are not divisible by the selected prime number as they are in the next row.

1, 1

Step 4: Multiply all the prime factors to get the LCM. LCM = 5 × 5 × 3 × 3 = 225

So, the LCM of 25 and 45 is 225. Example 8

Find the LCM of 6, 12 and 15 by the short division method. 3

6, 12, 15

1, 2, 5

2 5

2, 4, 5 1, 1, 5 1, 1, 1

LCM = 3 × 2 × 2 × 5 = 60 So, the LCM of 6, 12, and 15 is 60.

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Example 9

Find the LCM of 12, 16 and 20 by the short division method. 2 12, 16, 20 2

6, 8, 10

3, 2, 5

3, 4, 5

So, the LCM of 12, 16, and 20 is 240.

3, 1, 5

Do It Together

LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240

1, 1, 5 1, 1, 1

Find the LCM of 24 and 15 by the short division method. 3 24, 15 2

8, 5

LCM = _ ________________________________ ________________________________ So, the LCM of 24 and 15 is ________.

Do It Yourself 4D 1

Find the LCM of 2 numbers using the short division method. a 25, 30 e 60, 75 i

63, 105

c 42, 70

9, 15

g 21, 24

Find the LCM of 3 numbers using the short division method. a 28, 42, 56

e 102, 136, 170

b 32, 48

b 75, 100, 150 f

c 96, 144, 192

21, 63, 105

d 12, 48

h 30, 45

d 90, 135, 180

Rohit finds the common prime factors of 42 and 56 using short division as shown below. 2 2 7

42, 56 21, 28 3, 4

He says that the LCM of 42 and 56 is 28. Is he right? Why or why not? 4

Akash finds the common prime factors of 12, 20 and 32 using the short division method. He says that

Ajay finds the common prime factors of 16, 24, 36 and 54 using the short division method. He says

the LCM of 12, 20 and 32 is 480. Is he right? Why or why not?

that the LCM of 16, 24, 36 and 54 is 72. Is he right? Why or why not?

Chapter 4 • Multiples and Least Common Multiples

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Word Problem 1

Anna and Sylvia are cycling around a park. Anna takes 8 minutes to complete a round and Sylvia takes 12 minutes. After how much time will they meet if both start at the same time and move in the same direction?

Word Problems When to use LCM?

When to use HCF?

When we need to find the smallest number that is a multiple of all the numbers from a group of numbers.

When we need to find the largest number that divides evenly into all the numbers from a group of numbers.

Let us look at some examples to understand this better. Example of using LCM

A store is distributing freebies to its loyal customers. Every third customer receives a free keychain, and every fourth customer receives a free pen. Which customer will be the first to receive both a keychain and a pen? List the multiples of 3 and 4. Then highlight the common multiples. Multiples of 3:

Multiples of 4:

12 is a common multiple of 3 and 4.

24 is a common multiple of 3 and 4.

The Least Common Multiple (LCM) of 3 and 4 is 12. The first customer to get both a keychain and a pen is the 12th customer. Example of using HCF The fifth-grade class is conducting an activity. There are 32 girls and 40 boys who want to participate. Each team must have the same number of girls and the same number of boys. What is the greatest number of teams that can be formed? List the factors of 32 and 40. Then write the common factors. The class can divide 32 girls into 1, 2, 4, 8, 16 and 32 teams. Factors of 32:

Factors of 40:

The class can divide 40 boys into 1, 2, 4, 5, 8, 10, 20, and 40 teams.

The common factors of 32 and 40 are 1, 2, 4, and 8. 64

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The HCF of 32 and 40 is 8. So, the greatest number of teams that can be formed is 8. Example 10

In a morning walk, three people step out together. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps? We need to find the minimum distance they have walked so that they are together again. Now, the distance walked by them will be greater than 80 cm, 85 cm, and 90 cm. So, we need to find the LCM in this case.

LCM = 2 × 5 × 2 × 2 × 2 × 3 × 3 × 17 = 12,240 cm

Therefore, the minimum distance walked so that they are together again = 12,240 cm. Example 11

2 5 2 2 2 3 3 17

80, 85, 90 40, 85, 45 8, 17, 9 4, 17, 9 2, 17, 9 1, 17, 9 1, 17, 3 1, 17, 1 1, 1, 1

Three pieces of cloth 84 cm, 98 cm, and 126 cm long need to be divided into table mats of the same length. What is the greatest possible length of each mat? Here the pieces of cloth need to be divided into table mats of the greatest possible length. So, the length of each table mat would be less than the lengths of the given pieces of cloth. So, we need to compute the HCF in this case. Required length = HCF of 84 cm, 98 cm, and 126 cm 84 = 2 × 2 × 3 × 7

98 = 2 × 7 × 7

126 = 2 × 3 × 3 × 7

HCF = 2 × 7 = 14. Therefore, the required length of each table mat is 14 cm. Do It Together

What is the smallest possible length that can be measured exactly by the scales of lengths 3 cm, 5 cm and 10 cm? We are asked to find the smallest possible length that can be measured by each of the scales of lengths 3 cm, 5 cm and 10 cm. So, we need to find the LCM. 5

3, 5, 10

LCM of 3, 5 and 10 = ______________ So, the smallest possible length that can be measured by each of the given scales is ___ cm. Chapter 4 • Multiples and Least Common Multiples

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Do It Yourself 4E 1

If the students of a class can be arranged in rows of 6, 8, 12 or 16, such that no student is left out. What is the least possible number of students in the class?

Four bells toll at intervals of 8, 14, 18 and 84 minutes. The bells toll together at 12 noon. When will they toll together again?

An electronic device beeps after every 60 seconds. Another device beeps after every 62 seconds. They beep together at 10:00 a.m. What is the next time they will beep together?

After every fifth visit to a restaurant, you receive a free beverage. After every tenth visit you receive a free appetizer. a If you visit the restaurant 100 times, on which visits will you receive both a free beverage and a free appetizer? b At which visit will you first receive a free beverage and a free appetizer?

During a promotional event, a sporting goods store gave a free T-shirt to every 8th customer and a free water bottle to every 12th customer. Which customer was the first to get a free T-shirt and a free water bottle?

Word Problems 1

A milkman has 75 litres of milk in one can and 45 litres in another. What is the

maximum capacity of a container which can measure the milk in either container an exact number of times?

Three sets of English, Mathematics and Science books of the same thickness, containing 336, 240 and 96 books respectively, need to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. What is the total number of stacks?

There are 12 boys and 18 girls in Mrs. Mehra’s maths class. Each activity group must have the same number of boys and the same number of girls. What is the greatest number of groups Mrs. Mehra can make if every student must be in a group?

What is the smallest length of a room into which an exact number of carpets of length 12 metres and 9 metres can fit?

Points to Remember • The Lowest Common Multiple (LCM) of two or more numbers is the smallest among their common multiples. If one of the two numbers given is a multiple of the other, the greater number is the • LCM of the numbers given. 66

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• To determine the LCM of numbers, we first find their prime factorisation. We then multiply each factor the greatest number of times it occurs in the prime factorisation of any number given.

Math Lab Locate the LCM Setting: In groups of 3 Materials Required: Crayons of three colours – blue, orange, and green, and a notebook. Method: 1 The students will make a grid of 10 rows and 10 columns in their notebooks. 2 The students will write the numbers from 1 to 100 in them. 3 The teacher will ask the students to find the LCM of three numbers, say, 3, 6 and 8. 4 The students will highlight 3 and its multiples using blue crayons, 6 and its multiples

using orange crayons and 8 and its multiples using green crayons. The students can be asked to shade one-third of the cell in each colour.

5 The teacher will tell the students that the numbers which are shaded with all 3 colours

are the common multiples of 3, 6 and 8. The teacher will then ask the students to locate the smallest common multiple of 3, 6, and 8 on the grid.

Chapter Checkup 1

Colour the multiples of 9 in the number chart shown below. 3

Which of the following is the smallest number that is divisible by 9, 12 and 15? a

What will be the 9th multiple of the numbers given? a

360

120

180

Ajit picked a number and multiplied it by 7. Which of the following numbers cannot be the result of this multiplication? a

103

105

Chapter 4 • Multiples and Least Common Multiples

Maths Grade 5 Book_Chapter 1-6.indb 67

847

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Select the common multiples of 30 and 45 from the following options. 30 = 30, 60, 90, 120, 150, 180, 210, 240, …

45 = 45, 90, 135, 180, 225, …

90, 180

Only 90

30, 45

finding the LCM of A and B. a

A 2×3×3

B 2×5×7

2×3×5

3×3×5

2×3×3×7

2 × 3 × 11

3×3×5×7

2 × 3 × 5 × 11

2×3×3×5

2×2×3×5

LCM of A and B

Find the LCM by the short division method. d g

63 and 105

121, 132 and 330 125, 75 and 275

b e h

93, 62 and 120

16, 40, and 56

45, 18 and 63

112, 140 and 168 12, 20 and 32 15, 36 and 40

State whether each of the following statements is true or false. a b c d

In the following table, the prime factorisation of two numbers A and B is given. Complete the table by

Only 180

The LCM of 3 and 9 is 9.

The LCM of any two numbers is always greater than their HCF. The least number which is divisible by both 45 and 135 is 135. The LCM of 5, 15 and 45 is 90.

Find the LCM of the given numbers using prime factorisation. a e

15, 12

8, 10, 12

b f

35, 14

6, 10, 16

18, 27

5, 9, 15

10 Three bells ring at intervals of 20 minutes, 30 minutes, and 45 minutes respectively. After how much time will the bells ring together?

Word Problems 1 2

atish is making a board game that is 16 inches by 24 inches. He wants to use S square tiles. What are the dimensions of the largest tile he can use?

akash noticed that the number of questions given for homework is divisible by A both 3 and 13. What is the smallest possible number of questions that could have been given? teacher asks the students to name any three numbers that have 75 as a A multiple. Rahul says 15, Sahil says 25, and Ajay says 12. One of the answers is wrong. Can you identify the wrong answer?

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Fractions

Let's Recall We know that a fraction is defined as a part of a whole. The cake represents a whole. It has been cut into equal slices. So, each slice is a “fraction” of the whole cake. An orange has been cut into 2 equal parts. 1 Each equal part represents a half  . 2 An apple has been cut into 3 equal parts. 1 Each slice represents a third   of the apple. 3 A pear has been cut into 4 equal parts. 1 Each slice represents a quarter   of the pear.  4 In each fraction, the number at the top is called the numerator and the number at the bottom is called the denominator.

1 4

Numerator Denominator

Let's Warm-up

Match the following. 1 2 3 4 5

1 one-half 3 1 two-fourths 2 1 one-fourth 4 1 one-third 8 2 one-eighth 4

I scored _________ out of 5.

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Understanding Fractions Real Life Connect

Richa and Amit are in their art and craft class. Richa: Amit! I have a colourful square origami sheet with me. Amit: I have one too! Let us draw lines on it and see the patterns that come out of it. Richa and Amit drew lines on their sheets in the following way.

Richa’s Sheet

Amit’s Sheet

Reviewing Fractions Look at the pattern on the origami sheet carefully. What do you see? 1 8 1 8

1 8

1 8 1 8

We see 8 triangles on Richa’s sheet. Each triangle is exactly equal in size! What part of the square is each triangle? Since the square sheet is 1 of the square divided into 8 equal parts, one part is 1 out of 8 or 8 sheet. 1 4

What about Amit’s sheet? Here, 4 equal triangles divide the square. 1 So, each part is 1 out of 4 or of the square sheet. 4

1 4 1 4

1 4

Types of Fractions Proper Fractions

Improper Fractions

A fraction whose numerator is less than the denominator is called a proper fraction.

A fraction whose numerator is equal to or greater than the denominator is called an improper fraction.

3 4 5 6 11 , , , , are proper fractions. 4 8 9 9 19

4 5 9 9 19 are improper fractions. , , , , 4 4 5 6 11

3 4

4 8

4 4

5 4

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Like Fractions

Unlike Fractions

Fractions that have the same denominator are called like fractions.

Fractions that have different denominators are called unlike fractions.

2 3 4 5 6 , , , , are like fractions. 7 7 7 7 7

The figures below show the shapes divided into 7 equal parts. Figure A has 2 parts shaded and Figure B has 3 parts shaded. 2 3 So, and are like fractions. 7 7 2 7

3 7

4 6 2 6 3 are unlike fractions. , , , , 5 7 9 11 13 Figure A is divided into 5 equal parts and Figure B is divided into 7 equal parts. 4 6 So, and are unlike fractions. 5 7

4 5

6 7

Mixed Fractions (Mixed Numbers)

Unit Fractions

Fractions which are a combination of a whole number and a proper fraction are called mixed fractions or mixed numbers.

A fraction whose numerator is equal to 1 is called a unit fraction. 1 1 1 1 1 are unit , , , , 2 3 4 5 14 fractions.

Fractional part 1 4

Whole number part 2

Whole number

1 4

1 3

1 2

Proper fraction

4 1 and 2 are mixed numbers. 11 2

4 11

1 2

Conversion of Mixed Numbers to Improper Fractions An improper fraction shows the same number of parts as a mixed number. Both show a value greater than 1. Chapter 5 • Fractions

Maths Grade 5 Book_Chapter 1-6.indb 71

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Here we see each square divided into 4 equal parts. Improper Fraction

Mixed Number

Parts each shape is

Wholes shaded = 2

divided into = 4

Parts shaded = 3 out of 4 3 2 4

Parts shaded = 11 11 4

Convert 4 1

4 into an improper fraction. 5

S plit the mixed number into a whole number and proper fraction. 4

rite the whole W number part as a proper fraction.

4 4 =4+ 5 5

ultiply the numerator M and denominator by the denominator of the fractional part.

4 4 + 1 5

4× 5 4 + 1×5 5

Add the numerators to get the improper fraction.

24 20 4 20 + 4 + = = 5 5 5 5

Quick way: Step 2: Add the result obtained in

Step 3: Write the result obtained in

step 1 to the numerator.

Add 20 + 4

Step 1: Multiply the denominator with the whole number part.

step 2 over the denominator.

Multiply 4×5

4 = 5

(4 × 5) + 4 = 5

24 5

Remember!

o convert mixed numbers to improper ( Denominator × Whole Number ) + Numerator T Denominator fractions, use:

Conversion of Improper Fractions to Mixed Numbers Write

Error Alert!

24 as a mixed number. 5 Divide the

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

numerator by the denominator.

The divisor is

the denominator of the fraction.

The quotient

Whole number part

4 5 24 24 = – 20 5 4 Denominator

becomes the whole number part.

4 5

Numerator

The remainder becomes the numerator of the fraction.

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Example 1

Convert the following mixed numbers into improper fractions. 1 5 3 = 5 × 5 + 3 = 25 + 3 =

Example 2

39 32 + 7 4× 8 + 7 7 = = = 8 8 8 8

Convert the following improper fractions into mixed numbers. 1

8 5 42 – 40 2

42 5

Thus, Do It Together

28 3

Convert. 1

42 2 =8 . 5 5

54 7

Thus,

5 54 =7 . 7 7

7 into an improper fraction. 9

×8+ = 9

59 = 7

7 8 = 9

+7 = 9

7 7 54 49 5

59 into a mixed number. 7

Do It Yourself 5A 1

Are the following like fractions or unlike fractions? a

4 1 and 7 4

5 4 and 8 8

1 1 and 4 5

3 5 and 7 7

6 7 and 18 18

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

4 6 1 58 5 17 12 9 7 12 26 8 4 , , 4 , ,5 , , , ,6 , , , 7 ,15 5 2 7 5 8 11 15 3 9 29 17 9 7 3

Convert the given fractions into mixed numbers. a f

14 3

63 4

58 7

145 4

c h

27 5

165 6

45 6

71 9

543 3

111 7

2 3

Convert the given mixed numbers into improper fractions. 1 2

4 5

Chapter 5 • Fractions

Maths Grade 5 Book_Chapter 1-6.indb 73

4 7

7 8

1 9

14 17

15 19

3 11

2 21 73

18-12-2023 11:01:58

Match the mixed numbers with the improper fractions of the same value. a

1 4 6

314 15 4 c 20 7

144 7

d e

14 15

45 3 264 6

44 25 6

Word Problems 1

87 chocolates are distributed among 5 friends. How many chocolates does each

Sanchita gives a puzzle to her sister. Read the clues and write the fraction that

friend get? Express your answer as a mixed number.

Sanchita is talking about.

a The fraction has a prime denominator and is an improper fraction. b The mixed number 7

2 shows the same fraction. 6

Equivalent Fractions Remember Richa and Amit had drawn lines on their origami sheets. Now, Richa and Amit cut parts of their origami sheets in the given way. The white part shows the sheet that is removed.

Richa

Equivalent Fractions Using Multiplication

Amit

Notice that Amit and Richa have the same size of paper cutouts! 2 Richa has cut 2 out of 8 parts. So, she now has of the entire sheet. 8 1 Amit has cut 1 out of 4 parts. So, he now has of the entire sheet. 4 74

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Since both Amit and Richa have the same size of paper, we can say These are called equivalent fractions.

1 2 = . 4 8

Finding Equivalent Fractions We can multiply the numerator and the denominator by the same number to find any equivalent fraction. ×2 1 4

2 8

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

×2 2 1 1× 2 = = 8 4 4× 2

2 1 and are equivalent fractions. 8 4

Let us see another example of equivalent fractions. ×3 2 4

6 12

×3

2 4

6 12

Write three equivalent fractions for

Example 3

6 3 3×2 = = 14 7 7×2 Do It Together

9 3 3×3 = = 21 7 7×3

Find three equivalent fractions for 4× 2 4 = = 9×2 9

3 . 7

3 3 × 4 12 = = 7 7 × 4 28

4 . 9

4 4×3 = 9 9×

4 4× = 9 9×4

Equivalent Fractions Using Division We can also divide the numerator and the denominator by the same number to find more equivalent fractions.

Chapter 5 • Fractions

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÷2 4 8

2 4

2 4÷ 2 4 = = 4 8÷2 8

Think and Tell

Can equivalent fractions be like fractions?

÷2

4 8 Fraction in its Simplest Form

2 4

To convert a fraction to its simplest form, divide the numerator and the denominator by the highest common factor. 4 Convert to its simplest form. 8 Step 1

Find the HCF of the numerator and the denominator. HCF of 4 and 8 = 4.

Step 2

Divide the numerator and the denominator by their HCF.

4 4÷ 4 1 = = 8 8÷ 4 2

÷4 4 8

1 2

÷4

4 8

Checking for Equivalent Fractions 4 12 Check whether and are equivalent or not. 5 15 Step 1

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

1 2

4 5

Step 2

Check if the products of both the multiplications are the same or not. If the products are the same, then the fractions are equivalent, else the fractions are not equivalent. 4 × 15 = 60 5 × 12 = 60 Since the products are the same,

12 4 and are equivalent. 15 5

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Another way:

4 12 and are equivalent or not. 5 15 Reduce both the fractions to their lowest form and check if they are equal. Check whether

4 is already in its lowest form. 5

12 12 ÷ 3 4 = = 15 15 ÷ 3 5 Both the fractions are equal, so Example 4

12 4 and are equivalent. 15 5

Write any three equivalent fractions for

90 using division. 120

30 90 90 ÷ 3 90 45 90 ÷ 2 90 ÷ 10 9 90 = = = = = = 40 120 120 ÷ 3 120 120 ÷ 2 60 120 120 ÷ 10 12 Example 5

Express

42 in the simplest form. 91

Factors of 42 = 1, 2, 3, 6, 7, 14, 21; Factors of 91 = 1, 7, 13, 91. HCF of 42 and 91 = 7. Divide the numerator and the denominator by 7. 42 ÷ 7 42 6 = = 91 ÷ 7 91 13

42 6 is the simplest form for . 91 13 48 Express in the simplest form. 96 Factors of 48 = ____________________________________. Thus,

Do It Together

Factors of 96 = ____________________________________. HCF of 48 and 96 = ___.

Thus, the simplest form for

48 is 96

Do It Yourself 5B 1

Find any two equivalent fractions by using multiplication for each of the given fractions. a

4 9

Chapter 5 • Fractions

Maths Grade 5 Book_Chapter 1-6.indb 77

11 17

6 15

12 20

14 28

6 7

18-12-2023 11:02:00

Find any two equivalent fractions by using division for each of the given fractions. a

14 56

32 48

64 80

30 42

44 52

91 119

44 66

24 72

Express the given fractions in their lowest form. a

12 20

24 60

27 63

44 55

Check whether the given fractions are equivalent or not. a

4 12 and 8 16

14 22 and 21 33

8 2 and 10 4

3 6 and 5 10

5 25 and 10 50

2 60 and 3 90

12 36

60 15 = 21

Fill in the boxes to make the given fractions equivalent. a

4 = 5 35

21 3 = 8

1 5

5 = 55 11

Word Problems 1

6 of a pizza. Choose the option which represents the same portion of 8 pizza that Ragini ate if both ate an equal quantity of pizza. Tarun ate

3 5

4 8

12 18

15 20

Rahul was eating cookies. He had a total of 12 cookies. He ate 3 cookies in the

morning and 5 cookies in the evening. What fraction of the cookies was he left with? Express your answer in the simplest form.

Comparing and Ordering Fractions Richa and Amit had cut out different parts of their origami sheet.

Richa Amit Although, the sizes of both their sheets were the same, are each of the single triangles in their sheets the same size? Let us see! 78

Maths Grade 5 Book_Chapter 1-6.indb 78

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Comparing Fractions Single triangle in Richa’s sheet = Fraction of the sheet =

1 8

Single triangle in Amit’s sheet = Fraction of the sheet =

1 2 = 4 8

Richa’s sheet

To compare like fractions, compare the numerators.

Amit’s sheet

Greater the numerator, greater the fraction. 2 1 2 > 1. Thus, > . 8 8 We can also see above that the shaded portion in Amit’s sheet is bigger than that in Richa’s sheet. For unlike fractions with the same numerator, we compare the denominators. The fraction with the greater denominator is smaller. Compare

2 2 and . 8 7

7 < 8. Thus,

2 2 > . 7 8

But, what if both the numerators and the denominators are different? Method 1: Make the fractions like, using the LCM Method. Compare

2 1 and . 6 4

1 Step 1: Find the LCM of the Step 2: Find the equivalent fractions of 4 denominators. 2 and such that their denominators are 12. 6 The LCM of 4 and 6 is 12.

So,

3 4 . < 12 12

1 2 < . 4 6

3 3 1 × = 4 3 12

3<4

Thus,

LCM of 4 and 6 =12 3

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

4 2 2 × = 6 2 12

Method 2: Cross-multiplication Method Compare

1 2 and . 5 7

Chapter 5 • Fractions

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Step 1 Cross multiply the denominators with the numerators.

7 < 10. Thus, Example 6

1 × 7 = 7 and 5 × 2 = 10

1 2 < . 5 7

Which fraction is smaller?

6 10 or 8 15

LCM of 8 and 15 = 120 6 × 15 6 90 = = 8 × 15 120 8

90 80 > . 120 120

10 10 × 8 80 = = 15 15 × 8 120

10 is the smaller fraction. 15 Which fraction is bigger? 4 5 or 7 8 Hence,

Example 7

Apply the cross-multiplication method. 4 × 8 = 32

5 × 7 = 35 32 < 35 Thus, Do It Together

4 7

4 5 is bigger than . 7 8

Compare using >, <, =.

3 2 11 6 Apply the cross-multiplication method. 3 × 6 = 18

2 × 11 = __ 18 Thus,

3 11

2 . 6

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Ordering Fractions Arrange

1 3 2 4 , , , in ascending order. 5 7 5 7

Step 1

Step 2

Convert all the fractions into like fractions.

Since the denominators are the same, compare the

LCM of 5 and 7 is 35.

Example 8

1 11 77 7 = ××× = 5 55 77 35

5 15 3 3 = × = 5 35 7 7

22 77 14 2 = 55×××77 = 35 5

4 44 55 20 = ××× = 7 77 55 35

numerators and order the fractions. 7 < 14 < 15 < 20 7 14 15 20 So, . < < < 35 35 35 35 Thus,

1 2 3 4 < < < . 5 5 7 7

Arrange the fractions in descending order. 3, 9 , 4 , 1 4 13 26 2 LCM of 2, 4, 13 and 26 = 52. 9 9 4 36 = × = 13 13 4 52

3 3 13 39 = × = 4 4 13 52

4 4 2 8 = × = 26 26 2 52

1 1 26 26 = × = 2 2 26 52

39 > 36 > 26 > 8 So,

39 36 26 8 . > > > 52 52 52 52

Thus, Do It Together

3 9 1 4 . > > > 4 13 2 26

Arrange the following fractions in ascending order. 5 3 2 3 , , , 6 4 3 2 LCM of 2, 3, 4, 6 = ____________________ 5 5 2 = × = 6 6 2 3 3 = × 4 4

2 2 = × 3 3

3 3 = × 2 2

Thus, the ascending order is ____________________.

Chapter 5 • Fractions

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Do It Yourself 5C 1

Compare the following fractions using >, <, =. a

14 23

17 23

4 5

6 9

3 4

7 9

12 15

14 16

11 13

2 3

1 6

4 20

Put a tick () for the statements which are correct and a cross () for the statements which are wrong. a

4 2 > 5 3

6 5 = 12 10

4 = 8 2 7 15

2 > 4 9 17

7 15 < 8 17

14 > 12 12 11

Arrange the following in ascending order. a

2 1 3 , , 4 6 7

3 5 5 1 7 , , , , 13 7 12 4 12

7 8 4 5 , , , 8 15 8 12

2 1 5 1 , , , 3 5 6 2

1 3 5 1 2 , , , , 4 4 7 2 5

1 2 1 5 5 , , , , 3 5 2 6 8

4 Arrange the given fractions in descending order. a

3 1 3 1 , , , 8 4 5 6

3 8 1 1 , , , 8 9 3 2

4 3 4 2 1 , , , , 5 15 11 3 6

1 1 1 1 , , , 11 14 9 6

1 3 2 3 , , , 2 5 3 4

3 2 5 3 , , , 4 3 25 8

1 1 17 23 , , , 2 5 8 12

14 18 17 5 , , , 5 6 5 7

5 Circle the largest fraction. a

5 8 1 4 , , , 8 13 8 8

45 17 16 2 , , , 15 19 6 3

Word Problems 6 11 hours while Reshma studied for hours. Who studied for 13 23

Shashank studied for

Rakesh, Roshan, Swati, and Prerna were assigned a task each. They completed the

a longer duration?

task in 4 hours, 7 hours, 1 hour and 2 hours, respectively. Arrange the time taken 3 6 8 by each of them in ascending order.

Maths Grade 5 Book_Chapter 1-6.indb 82

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Points to Remember • A fraction indicates one or more parts of a whole. • 2 or more fractions having the same denominators are called like fractions. • 2 or more fractions having different denominators are called unlike fractions. • Fractions with 1 as the numerator are called unit fractions. • A fraction whose numerator is less than the denominator is called a proper fraction. A fraction whose numerator is equal to or greater than the denominator is called an • improper fraction. Mixed fractions or mixed numbers are fractions which are a combination of a whole • number part and fractional part. A fraction obtained by multiplying or dividing the numerator or denominator by the • same non-zero number is called an equivalent fraction of the given fraction. A fraction is in its lowest or simplest form when the HCF of its numerator and • denominator is 1. When we compare fractions with the same denominator, then the fraction with the • greater numerator is greater. To compare fractions with different numerators and different denominators, find • the LCM of the denominators to convert the fractions into like fractions, and then compare the fractions.

Math Lab Aim: To understand how to form mixed numbers and convert them into improper fractions. Materials Required: Three dice, paper, and pencil. Setting: In groups of 3 Method: 1

Roll the three dice and note down the numbers.

The biggest number becomes the denominator; the smallest number becomes the numerator, and the second biggest number becomes the whole number.

(If all the three dice have the same number, the turn will pass to the next player.)

(If two dice have the same number, then the numerator and the denominator will definitely have 3 4

different digits but the whole number could be the same as the numerator or denominator.) Form the mixed number and convert it into an improper fraction and write down the improper fraction.

The person who gets the largest fraction 3 times in a row is the winner.

Chapter 5 • Fractions

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Chapter Checkup 1

Convert the given mixed numbers into improper fractions. 7 2 1 a 7 b 5 c 12 8 3 2

3 8

2 Convert the given improper fractions into mixed numbers. a

5 8

16 24

2 6

11 14

4 9

24 42

108 132

75 195

42 56

75 125

66 84

72 136

5 15 and 8 24

4 28 and 7 42

1 46 and 3 92

1 57 and 6 3 9

18 2 and 24 3

14 1 and 5 2

34 4 and 4 5

24 and 8 3

1 2 2 1 , , , 4 6 5 2

9 1 1 5 , , , 11 2 8 6

3 5 5 3 , , , 8 6 3 9

7 2 1 5 , , , 10 7 3 8

11 1 3 4 , , , 12 2 5 7

4 17 5 7 5 , ,4 ,5 9 3 19 9

Arrange the following fractions in ascending order. a

87 19

Circle () the smallest fraction and make a square () over the largest fraction. a

Compare the fractions and put the correct symbol >, <, =. a

145 6

Check whether the given pairs of fractions are equivalent or not. a

Write the following fractions in the simplest form. a

41 5

Find any three equivalent fractions using division. a

Find any three equivalent fractions using multiplication. a

18 4

5 1 4 1 , , , 12 2 5 6

1 3 3 1 , , , 5 5 8 2

1 2 6 2 , , , 2 3 7 5

Arrange the following fractions in descending order. a

12 10 8 14 , , , 5 8 3 5

1 2 3 4 , , , 2 3 4 5

4 17 19 2 , , , 3 5 17 3

Maths Grade 5 Book_Chapter 1-6.indb 84

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Word Problems 1

Emma baked some cakes for a family gathering. She cut each of the cakes into

12 equal slices. During the party, 47 of those slices were eaten. What is the mixed number representation of the portion of the cake that was eaten?

Ryan and Sarah are sharing stamps. Ryan took 2 of the stamps, while Sarah took 5 3 of the stamps. Who took a greater fraction of the stamps? 8 1 3 Three friends, Alex, Bailey, and Casey are sharing a bag of marbles. Alex took 4 3 3 of the marbles, Bailey took of the marbles, and Casey took of the marbles. 8 12 Who took the least fraction of the marbles and who took the greatest fraction? 2

Four friends, Sara, Mia, Tina, and Sancy are dividing a cake into equal parts.

1 1 Sara ate 3 of the cake, Mia ate of the cake, Tina ate of the cake and 12 18 10 3 Sancy ate of the cake. Arrange them in order from the least to greatest 18 fraction of the cake eaten.

Chapter 5 • Fractions

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Operations on Fractions

Let's Recall

Fractions are parts of a whole. Look at the chocolate bar. The chocolate bar itself is a ‘whole’. Now, what if we divide the chocolate bar into 5 equal parts? Then each part represents 1. 5

1 th of the chocolate – can be added to make one 5 1 whole. That is, if we add five times, we get 1. 5

All the parts:

1 5

All equal parts add together to make a complete whole.

Let's Warm-up

1 5 1 1 1 1 1 + + + + =1 5 5 5 5 5

Fill in the blanks. 1 2 halves 1 together make a _______________. 2

A quarter 1 divides a whole into _______________ equal parts. 4 1 1 1 3 + + = _______________ 3 3 3 4 Adding 1 _______________ times gives one whole. 7 1 5 shows 1 part out of _______________ equal parts from a whole. 6 2

I scored _________ out of 5.

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Addition and Subtraction of Fractions Real Life Connect

Shagun celebrated her birthday. Her mother made pizza and apple pie for the party. Shagun: I will eat 2 slices of the apple pie. Abhishek: I will eat 3 slices of the apple pie. All her other friends also had the apple pie.

Addition of Fractions There are 8 slices in one apple pie. Shagun ate 2 of the 8 apple pie.

3 of the 8 apple pie.

Abhishek ate

What part of the apple pie did they both eat altogether?

2 3 2+3 5 + = = 8 8 8 8

Adding Unlike Fractions What if the denominators of the two fractions being added, are not the same? We first make the denominators the same, that is, make them like fractions. We can then easily add the fractions.

Remember! Sum of like fractions = Sum of numerators Common denominator

Let us add the unlike fractions 2 and 1 . 5 15 We first find the LCM of the denominators of the two fractions. Step 1 Find the LCM of the denominators. LCM of 5 and 15 = 15

Step 2 Find equivalent fractions of the two fractions, so that the denominators of both become the LCM. 2 = 2 × 3 = 6 5 5 3 15

1 = 1 ×1= 1 15 15 1 15

Chapter 6 • Operations on Fractions

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Step 3

2 5

Add the numerators once the denominators

become the same and keep the denominator the same.

LCM of 5 and 15 = 15

6 + 1 =6+1= 7 15 15 15 15

6 15

Step 4

Reduce the fraction to its simplest form if required. 7 is already in its simplest form. 15 Thus,

Example 1

6 15

1 15

7 15

2+ 1 = 7 5 15 15

Add 6 and 5 . 9 12 LCM of 9 and 12 = 36

Example 2

6 6 4 24 5 5 3 15 = × = = × = 9 9 4 36 12 12 3 36

24 15 24 + 15 39 = = + 36 36 36 36

39 (1 × 36) + 3 1 × 36 3 3 = = + = 1 36 36 36 36 36 6 5 3 1 = 1 = 1 Thus, + 9 12 36 12 Do It Together

1 15

Add 2 and 5 . 7 9 2 5 2 9 5 7 18 35 + = × + × = + 7 9 7 9 9 7 63 63 18 35 18 + 35 53 += = 63 63 63 63 2 5 53 Thus, + = 7 9 63

2 9 and . 7 14 LCM of 7 and 14 = ______.

Add

2 2 = × 7 7

9 9 = × 14 14

2 9 + = + = 7 14 14 14 14

Adding Mixed Numbers Remember Shagun, who had a birthday party?

3 5 Her friends ate 2 whole apple pies and of an apple pie and 3 of apple pie was left 6 6 over. How much apple pie was there in total?

Maths Grade 5 Book_Chapter 1-6.indb 88

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Method 1: Adding whole number with whole number and fraction with fraction. Adding Mixed Numbers with Like Fractions Add 2

5 3 and 3 . 6 6

2+3=5 5 3 5+3 8 += = 6 6 6 6

Add the whole number with the whole number. Add the fractional part with the fractional part. Convert the fraction to a mixed number.

8 2 =1 6 6

Add the results of step 1 and step 3.

5+1

Write the fraction in its simplest form.

2 2 = 6 6 6

2 1 5 1 = 6 . Thus, 2 and 3 3 = 6 . 6 3 6 3 6

Method 2: Convert the mixed numbers to improper fractions and then add. 1 1 What if the total apple pie eaten was 2 and the apple pie left over was 3 , then how 3 2 much apple pie was there in total? Adding Mixed Numbers with Unlike Fractions Add 2

1 1 and 3 . 3 2

Convert the mixed numbers to improper fractions. 1 = 2 3

(2 × 3= ) + 1 6= +1 7 3

1 = 3 2

(3 × 2= ) + 1 6= +1 7 2

7 7  7 2   7 3  14 21 14 + 21 35 (Add the improper fractions.) + =  × + ×  = + = = 3 2 3 2 2 3 6 6 6 6 35 = 6

(5 × 6 ) + 5= 5 × 6 + 5= 5 5 6

Thus, 2

Example 3

Add 3 1 = 3 3

1 1 5 +3 = 5 3 2 6

1 1 and 5 . 3 4

(3 × 3= ) + 1 9= + 1 10 3

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 89

1 = 5 4

4) + 1 20 + 1 21 (5 × = = 4

18-12-2023 11:02:12

10 21  10 4   21 3  40 63 103 + =  × + × = + = (Add the improper fractions.) 3 4  3 4   4 3  12 12 12 103 = 12

(8 × 12) + 7= 8 × 12 + 7= 8 7 12

Thus, 3 Do It Together

1 1 7 8 +5 = 3 4 12

Add 3 2 and 3 1. 6 4 On converting to improper fractions, we get: 3

2 1 2 1 3 = 3 +3 = = 4 6 4 6

Do It Yourself 6A 1

Circle the pair of fractions in each problem which add together to make 1 . a

1 2 3 5 , , , 4 4 4 4

1 5 1 2 , , , 8 8 2 4

2 15 15 12 , , , 5 10 25 35

14 and 15 22 11

2 and 3 2 4 7

2 and 1 2 3 9

1 c 4, 1 and 3

1 , 2 and 6 2 4 5

3 8 12 4 , , , 7 21 21 14

Add the unlike fractions and write the answer in the simplest form. a

5 and 4 9 10

4 and 3 5 15

1 and 18 9 21

63 and 11 14 7

7 2 and 18 3

Add the mixed numbers and write the answer in the simplest form. a

1 and 3 3 5 5

11 and 5 14 13 13

15 and 11 7 34 17

1 and 13 17 3 18

21 44 and 21 , 4 5 20

2 1 1 4 , 5 and 2 7 28 5

Find the sum of: a

1 5 , and 11 7 7 14

1 2 , 4 and 5 1 3 9 18

Answer the questions. a What is the sum of

5 45 and 3 ? 7 14

b What is the sum of

15 7 and 5 ? 21 30

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Word Problems 1

Rakesh has two ropes of length 3

A basket has 1

Zaid completes three tasks in

1 5 m and 1 m. What is the total length of rope? 3 6

2 2 kg of mangoes. Rajiv puts 2 kg more mangoes in it. What is the 5 5 total weight of the mangoes in the basket? 11 74 1 minutes, 5 minutes and minutes. What is 7 15 2 the total time taken by Zaid to complete all the tasks?

Subtraction of Fractions Remember, Shagun and Abhishek had eaten How much apple pie was left?

3 left 8

5 of an apple pie. 8

A whole is represented as 1. 5 3 8 5 8−5 = − = = Apple pie left = 1 – 8 8 8 8 8 3 Thus, of the apple pie was left. 8

5 eaten 8

Subtracting Unlike Fractions

1 3 of an apple pie was left. Shagun ate of the 3 4 leftover apple pie. How much apple pie was left? Make the fractions like using the LCM method. Subtract

1 3 from . 3 4

9 12

1 1 4 4 = × = 3 3 4 12

Subtract the like fractions.

5 3 1 – = 4 3 12

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 91

4 12 −

4 12 5 12

9−4 9 4 5 − = = 12 12 12 12

Thus,

LCM of 3 and 4 = 12 9 12

LCM of 3 and 4 = 12

3 3 3 9 = × = 4 4 3 12

1 3

3 4

Remember! Difference of like fractions =

Difference of numerators Common denominator

18-12-2023 11:02:14

Find 7 – 1. 8 4

Example 4

Do It Together

Convert the fractions into like fractions. LCM of 8 and 4 = 8 7 7 1 7 = × = 8 8 1 8

1 1 2 2 = × = 4 4 2 8

7 2 7−2 5 − = = 8 8 8 8 Thus,

5 7 1 – = . 8 4 8

Find the difference: 7 – 1 16 14 LCM of 16 and 14 = ______ 7 7 = × 16 16

7 1 – = 16 14

–

Thus,

1 1 = × 14 14

7 1 − = 16 14

Subtracting Mixed Numbers Subtracting Mixed Numbers with Like Fractions 2 1 If the total apple pie available was 5 and the portion eaten was 2 , then how much 3 3 apple pie would be left? Subtract 2

1 2 from 5 . 3 3

2 (2 × 3) + 2 6 + 2 8 = = = 2 3 3 3 3 So,

1 (5 × 3) + 1 15 + 1 16 = = = 5 3 3 3 3

16 8 16 − 8 8 −= = 3 3 3 3

8 (2 × 3) + 2 2 × 3 2 2 = = += 2 3 3 3 3 3 Thus, 5

1 2 2 –2 =2 3 3 3

Subtracting Unlike Fractions from Mixed Numbers

1 2 If the total apple pie available was 4 and the portion eaten was 2 , then how much 3 5 apple pie would be left? Subtract 2

1 2 from 4 . 3 5

1 (2 × 3) + 1 6 + 1 7 = = = 2 3 3 3 3

2 (4 × 5) + 2 20 + 2 22 = = = 4 5 5 5 5

22 7  22 3   7 5  66 35 66 − 35 31 − × − ×  = – = = = 5 3  5 3   3 5  15 15 15 15 92

Maths Grade 5 Book_Chapter 1-6.indb 92

18-12-2023 11:02:16

31 (2 × 15) + 1 2 × 15 1 1 = = + = 2 15 15 15 15 15 2 1 1 2 Thus, 4 − 2 = 5 3 15 Example 5

Subtract 24 from 31 5 5 Convert the mixed numbers into improper fractions.

4 (2 × 5) + 4 10 + 4 14 1 (3 × 5) + 1 15 + 1 16 = = = 2 = = = 3 5 5 5 5 5 5 5 5 So,

16 14 2 − = 5 5 5

Thus, 3 Do It Together

Find 5

1 4 2 −2 = 5 5 5

4 2 −3 9 3

2 4 (5 × 9) + 4 __ + __ __ 3 = = = ______________ = 5 3 9 9 9 9 5

4 2 −3 = 9 3

Do It Yourself 6B 1

Find the difference. a

5 2 − 7 4

9 1 − 13 2

17 5 − 12 6

9 3 − 12 9

14 11 − 11 22

7 3 − 13 7

2 11 − 3 24

Subtract. a

1 4 − 5 5

5 14 − 12 6

3 7 − 4 10

2 17 − 3 6

1 14 − 4 12

Subtract the mixed numbers and write the answer in its simplest form. a

1 4 −2 8 9

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 93

1 7 −4 4 8

6 1 − 2 9 3

2 1 − 15 7 14 93

18-12-2023 11:02:17

Simplify. a

2 5 3 + − 3 6 9

2 1 2 +2 −1 3 4 4

10 − 3

1 3 +5 5 5

1 1 5 +3 −1 3 6 12

Answer the questions. a What is 4

1 7 less than 5 ? 3 8

b What must be added to 3 c Subtract the sum of 2

3 1 to get 5 ? 4 3

1 1 1 1 and 3 from the sum of 5 and 2 . 3 4 4 3

d Subtract the difference of 2

1 1 1 1 and 1 from the difference of 5 and 3 . 3 3 2 2

Word Problems 2 1 hours on Monday. He studied 1 hours less on Tuesday 3 4 than on Monday. How many hours did Navneet study on Tuesday?

Navneet studied for 4

A drum has 50

3 1 kg of wheat. Sanjay uses 10 kg of wheat. What quantity of wheat 4 2 is left in the drum? 1 Raghuveer purchased 4 types of fruit weighing 7 3 kg. He purchased 1 kg of 2 4 2 1 apples, 2 kg of pears, 1 kg of oranges and some litchis. What was the weight of 3 4 the litchis that he purchased?

Multiplication and Division of Fractions Real Life Connect

Mithun, a shopkeeper, sells dairy products like milk, curd, and paneer. 1 He packs milk in litre packets. 4

Multiplication of Fractions Mithun packed 8 packets of

1 litre of milk each. What quantity of milk did he pack? 4

Maths Grade 5 Book_Chapter 1-6.indb 94

18-12-2023 11:02:18

Multiplying Fractions and Whole Numbers To show this using fraction strips, we can say that there are 8 groups of What will we get when 8 one-fourths are put together?

1 4

1 . 4

1 4

2 wholes We can also find the answer using equal grouping or repeated addition of like 1 fractions . 4 1 1 8 groups of is the same as 8 × . 4 4 1 1 1 1 2 1 1 1 8 1 + + + + + + + = = or 2. 4 4 4 4 4 4 4 4 1 4 Another way to multiply the whole number 8 by a fraction Step 1

Step 2

To multiply a whole number with a fraction,

Convert the improper fraction into a mixed number.

fraction. The denominator remains the same.

8 =2 4

8×

Thus, the total quantity of milk packed is 2 litres.

multiply the whole number by the numerator of the

Example 6

1 is given below. 4

8×1 8 1 = = 4 4 4

Multiply 5 and 5× 30 = 7

6 . 7

Do It Together

6 5×6 30 = = 7 7 7

Multiply 5 by 21. 9 5 5 21 = × 21 = × 9 9

2 4× 7 2 ( 4 × 7) + = 2 += 4

Thus, 5 ×

6 2 4 = 7 7

Chapter 6 • Operations on Fractions

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18-12-2023 11:02:19

Multiplying Two Fractions Multiply

2 3 and 5 7

Let us multiply 2 fractions using pictures. 1 To show 2, divide the rectangle vertically 5 into 5 equal parts. Shade 2 out of 5 parts.

3 D raw crosses in 3 out of 7 parts, horizontally. × × ×

× × ×

6 35

2 5 2 Then, to show 3 , divide the rectangle horizontally 7 into 7 equal parts.

× × ×

4 C ount the number of equal parts which have both the shading and the cross (×). 6 out of 35 parts have both the green colour and the cross (×).

We can also multiply two fractions by multiplying the numerator with the numerator and the denominator with the denominator. 6 2 2×3 3 × = = 35 5 5×7 7

2 6 3 × = 5 35 7 When multiplying two fractions, we can also cancel out the common factors in the numerator of one fraction and the denominator of the second fraction. Thus,

16 33 × 81 52 Step 1

Remember! Fraction of a fraction: ‘of’ means multiplication when multiplying fractions.

Find the common factors between 16 and 52. Find the common factors between 33 and 81.

Common factor between 16 and 52 = 4. Common factor between 33 and 81 = 3.

Step 2 Divide both 16 and 52 by 4. Divide both 33 and 81 by 3. 4 11 16 33 = 16 ÷ 4 × 33 ÷ 3 = × × 81 52 81 ÷ 3 52 ÷ 4 27 13 96

Maths Grade 5 Book_Chapter 1-6.indb 96

18-12-2023 11:02:19

Step 3 Multiply the numerators of the 2 fractions. Multiply the denominators of the 2 fractions. 4 × 11

27 × 13

Example 7

44 . 351

Multiply 2 by 7. 3 8 2 × 7 2 7 14 ×= = 3 8 3 × 8 24

Example 8

4 7 7 4 of = × 9 2 2 9 4 7 4÷2 7 2 7 14 × = × = × = 9 2 9 2÷2 9 1 9

So,

14 7 = 24 12 Thus,

Find 4 of 7. 9 2 In this statement, ʹof ʼ means to multiply.

2 7 7 × = 3 8 12

14 = 9 Thus,

Do It Together

Multiply

5 1×9 5 (1 × 9 ) + = 5 + = 1 9

4 7 5 × = 1 9 2 9

5 4 by using fraction strips and cancelling out the common factors. 8 7

4 5 × = 7 8

Do It Yourself 6C 1

Multiply the whole numbers and fractions. a

6 13

15 ×9 17

2×

1 4

7×

Multiply the fractions using fraction strips. a

5×

5 2 × 6 5

1 3 × 4 3

8 7

12 × 12 15

7 5 × 8 6

Multiply the fractions and write the answer in the simplest form. a

7 8 × 10 5

15 3 × 9 20

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 97

13 4 × 8 26

48 4 × 6 12

12 ×

157 12

14 21

14 ×

2 5 × 7 7

24 ×

11 6 97

18-12-2023 11:02:20

4 Simplify. a

1 3 5 × × 3 2 9

5 7 × × 14 4 8

1 3 × × 18 2 9

5 8 72 × × 7 9 2

5 Answer the questions. a What is

4 of 27? 9

16 of b How much is

c What is

3 of 2 hours? 4

2 of 1 week? d How many days are

18 ? 4 7

Word Problems 1

Mahesh reads 2 pages of a book. The total number of pages in the book is 100. How 5 many pages has he read? 3 of a cake. The total weight of the cake was 2 kg. What weight of cake 4 did Sushen eat?

2 Sushen ate 3

3 of the fuel is used up to travel from one 5 place to another. What quantity of fuel is left in the fuel tank? 1 1 Shefali has 2000 marbles of different colours. of the marbles are red, of them 2 4 are blue and rest of the marbles are green. How many green marbles does she have? A car has 50 litres of petrol in its fuel tank.

Division of Fractions

1 of the cake. 2 1 How many cake boxes are needed? We need to find 2 ÷ . 2 Lalit has to pack 2 cakes in boxes. Each box can hold

Let us find out!

Dividing a Whole Number by a Fraction To find the number of boxes packed by Lalit, first let us understand the concept of reciprocal. Reciprocal of a Number The reciprocal of a number or a fraction is obtained by interchanging the numerator and the denominator of the fraction. 98

Maths Grade 5 Book_Chapter 1-6.indb 98

18-12-2023 11:02:21

What is the reciprocal of 5? 5 5= 1 1 Reciprocal of 5 = 5

What is the reciprocal of Reciprocal of

1 5 = or 5. 5 1

1 ? 5

Two number are said to be reciprocals of each other when their product is 1. Thus, reciprocals are the multiplicative inverse of each other. Consider, 8 and

1 . 8

Think and Tell

8×1 1 8 1 8 8× = × = = =1 8 1 8 1×8 8 Thus, 8 and

What is the reciprocal of 1?

1 are reciprocals of each other. 8

Now, let us find the number of boxes packed by Lalit. There are 2 boxes. Each box can hold So, we need to find 2 ÷

1 of the cake. 2

1 or how many halves will fit into 2 wholes. 2

Let us see using figures. 2÷

1 means how many halves will fit into 2 wholes. 2

4 halves

2 ÷ 1 2

1 2

Let us see the other way in which we can divide a whole number by a fraction. 1 Write the whole number as a fraction.

3 Write the reciprocal of the fraction.

2 ÷ 1 = 2 × 2 = 2×2 = 4 = 4 2 1 1 1×1 1 2 Reverse the ‘÷’ symbol to ‘×’.

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 99

4 Multiply the fractions to get the answer.

18-12-2023 11:02:21

Keep, change and flip method 2

1 2

Keep

Change

Flip

2 1

Thus, 4 boxes of cake were packed by Lalit. Example 9

Divide 5 by 1. 5 5

Keep

Change

Flip

5 1

Thus, 5 ÷ Do It Together

1 5

Error Alert! Always remember to flip the second fraction while dividing fractions.

1 = 25 5

5÷

5 5 5 = × 4 1 4

5÷

5 5 4 = × 4 1 5

Divide 7 by 14 using fraction strips and the keep, change, and flip method. 4 7÷

14 7 14 7 = ÷ = 4 1 4 1

Dividing a Fraction by a Whole Number Divide

1 by 4 . 2

1 2

1 divided into 4 equal parts 2

1 8

Let us understand this in another way.

100

Maths Grade 5 Book_Chapter 1-6.indb 100

18-12-2023 11:02:22

1 Write the whole number as a fraction.

3 Write the reciprocal of the fraction.

1÷4=1÷4=1×1=1×1=1 2 2 1 2 4 2×4 8 4 Multiply the fractions to get the answer.

2 Reverse the ‘÷’ symbol to ‘×’.

Example 10

Divide 1 by 6. 3

Do It Together

Divide 4 by 2. 8

4 4 2 4 1 4×1 4 1 ÷2= ÷ = × = = = 8 8 1 8 2 8 × 2 16 4

1 divided into 3 6 equal parts

1 3 Thus,

Example 11

Thus,

1 18

4 1 ÷2= 8 4

1 1 ÷6= 3 18

Divide 3 by 9 using fraction strips and using reciprocals. 4 3 ÷9= 4

Dividing a Fraction by a Fraction Divide

1 1 by . 2 6

Method 1 1 2

1 2

So,

Method 2 ÷

1 6

= 1 6

1 1 ÷ = 3 2 6

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 101

1 6

1 1 fits in 2 6 three times.

1 2 Keep 1 2

1 6

Change

Flip

6 1

101

18-12-2023 11:02:22

Example 12

Divide 1 by 1. 3 6 1 3

Example 13

1 6

1 3 Thus, Do It Together

4 2 4 7 4 × 7 28 ÷ = × = = =2 7 7 7 2 7 × 2 14

1 6 1 6

= 1 6

Thus,

1 1 fits in 6 3 two times.

1 1 ÷ = 2 3 6

Divide 4 by 2. 7 7

4 2 ÷ = 2 7 7

Divide 3 by 1 using fraction strips and using reciprocals. 4 8 3 1 ______ ÷ = 4 8

Do It Yourself 6D 1

Find the reciprocal of the given fractions.

12 5 6 5 c d e 14 9 19 4 Divide the whole numbers by the fractions and reduce to the simplest form. a

36 3 3 5 4 b 4÷ c 14 ÷ d 64 ÷ e 5÷ f 12 6 4 4 5 Divide the fractions by the whole number and express the answer in its simplest form. a

2 3

18 ÷

18 19 b c 23 ÷ 23 d 4 ÷ 12 e 12 ÷ 24 ÷2 ÷ 54 6 5 4 9 54 4 Divide the fraction by a fraction and express the answer in its simplest form. a

4 45 ÷ 6 14

32 12 ÷ 15 25

5 Answer the given questions. a What is 45 divided by c What is

15 ? 7

12 18 divided by ? 18 12

3 27 ÷ 5 42

49 18 ÷ 12 14

b What is

23 19 ÷ 8 69

17 divided by 17? 9

d What is 54 divided by the sum of

3 7 12 ÷

6 5

11 ÷ 14 36

1 2 ÷ 10 5

12 8 and ? 8 12

102

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Word Problems 3 kg each. How many such 4

Ravi has 21 kg of rice. He packs the rice in packets of

57 How many bottles of capacity 3 L can be filled from a container of capacity L? 4 4

packets does he pack?

Points to Remember • To add or subtract like fractions, simply add or subtract the numerators and keep the denominators the same. To add or subtract unlike fractions, first convert the unlike fractions into like fractions • and then add or subtract the fractions. To add or subtract mixed numbers, convert the mixed numbers into improper • fractions and then perform addition or subtraction. • To multiply 2 fractions, find the product of their numerators and the product of their denominators. • The reciprocal of a number is that number which when multiplied by the original number gives 1 as the answer. • To divide a given fraction by another fraction, multiply the first fraction with the reciprocal of the second fraction.

Math Lab Fraction Bingo Aim: To understand fractions and add fractions.

Materials Required: Fraction Bingo cards, markers, or coloured pencils Method: 1

Create the fraction bingo cards and create the list of fractions that are to be called out.

Each student gets a fraction bingo card and a coloured pencil or marker.

Call out fractions from the prepared list.

The students put a tick or cross over the fraction that is called out.

Once a row, column, or diagonal of the fraction bingo is crossed out, the student adds

all the fractions in that line. If the sum of all the fractions is more than 3 wholes, then that student is the winner.

After the winner is declared, students can check in their fraction bingo which row has the sum of more than 3 wholes.

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 103

103

18-12-2023 11:02:24

Chapter Checkup 1

Add the given unlike fractions. a

5 17 + 9 18

6 21 + 17 34

1 14 + 3 6

15 26 + 6 4

18 64 + 5 15

2 Add the mixed numbers and express the answer in its simplest form. a

7 5 +3 17 17

2 4 +5 3 6

12 3 − 5 4

14 3 − 16 8

7 1 − 18 3

1 7 −2 2 8

4 1 −2 9 3

5 7 −5 6 8

5×

4 7

6×

5 9

15 ×

8×

2 7

2 3 × 9 8

3 4 × 5 16

1 4 + 10 5 10

17 5 − 24 8

1 2 +3 7 14

47 5 − 54 18

2 1 −1 3 2

14 5

Find the multiplicative inverse of the given fractions. 4 5

18 17

24 1 − 36 2

2 1 +1 9 3

2 4 −7 5 5

7 15 −3 8 24

5÷

2 5

12 ÷

3 4

3 9

3×

6 12 × 12 18

22 33

11 ×

4 27 × 9 36

78 14

1 5

4 3

123 121

11 ÷3 5

5 ÷ 18 9

12 3 ÷ 17 4

15 5 ÷ 24 6

Divide and find the quotient. a

Multiply to find the product. Also, write the answer in its simplest form.

21 18 + 6 8

Find the difference of the mixed numbers and write the answer in its simplest form. a

1 2 + 12 5 5

Subtract the unlike fractions and express the answer in its simplest form. a

Write if true or false. a

The reciprocal of every fraction is a proper fraction.

0 ÷ any fraction = 0.

The reciprocal of 0 is 0.

The order of multiplicand and multiplier does not alter the product while multiplying fractions.

The multiplicative inverse of any fraction is always greater than 1.

Fill in the blanks. a b c d

1 is _______. 2 1 There are _______ quarters in 14 . 2 1 Dividing by is the same as multiplying by _______. 3 _______ is the reciprocal of itself. The multiplicative inverse of 2

104

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10 Answer the following questions. a

Which is more: the difference of 2

Which is less: the product of 5 and

1 1 1 1 and 8 or the sum of 2 and 4 ? 8 4 4 3

1 4 4 or the sum of 1 and 2 ? 4 5 5 1 2 What is the difference of the product of 4 and and the product of and 3? 5 5

Word Problems Raghav jumped 34 m in the long jump competition while his friend Utkarsh 9 jumped 7 m further than him. How far did Utkarsh jump? 8 1 3 1 2 A man travelled 4 km by bicycle, 2 km on foot and 10 km by car. What is the 7 4 2 total distance covered by him? 1

Subhankar's height is 1471 cm. His friend Subhangi is 111 cm shorter than him. 2 3 What is the height of Subhangi?

3 of a parking lot is full when there are 66 cars in it. How many cars can be 4 parked inside the parking lot?

14 cans can hold 1441 litres of water. What is the capacity of each can? 2 2 6 Radhika purchased 20 m of cloth. She used 11 m of cloth for the curtains and 5 35 m of cloth for a bed sheet. How much cloth does she have left? 6 5

Chapter 6 • Operations on Fractions

Maths Grade 5 Book_Chapter 1-6.indb 105

105

18-12-2023 11:02:27

Introduction to

7 Decimals Let's Recall Fractions are part of a whole.

Rajesh has an apple pie which is divided into 10 equal parts. He eats 1 part out of it.

1 10 All the equal parts join together to form a whole. The fraction of pie eaten by him =

1 1 1 1 1 1 1 1 1 1 + + + + + + + + + =1 10 10 10 10 10 10 10 10 10 10 If a whole is divided into 10 equal parts, then each part is equal to 1 . 10

If a whole is divided into 100 equal parts, then each part is equal to 1 . 100

1 whole

Letʼs Warm-up

Fill in the blanks. 1 1 1 1 + + = __________. 10 10 10 2

Each equal part in a hundred grid is equal to __________.

When a whole is divided into 10 equal parts then each part is equal to __________.

1 1 1 1 1 + + + + = __________. 5 5 5 5 5

The fraction for 3 out of 10 = __________.

I scored _________ out of 5.

Maths Grade 5 Book_Chapter 7-12.indb 106

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Understanding Decimals Real Life Connect

Alfred the carpenter goes to the market to purchase nails and screws. The weight written on the packet of nails was 2.5 kg.

Tenths Alfred thinks 2.5 is not a whole number. This number lies between 2 and 3. What are these numbers called?

Decimal point

The number that uses a decimal point followed by digits is called a decimal number. Decimals have a whole number part and a fractional (or decimal) part, which is separated by a decimal point.

Whole number part

Decimal part

Reading and Writing Tenths When 1 whole is divided into 10 equal parts, then each part is called one-tenth. 1 Fractional representation = 10 Decimal representation = 0.1 Let us see some more tenths. Each whole is divided into 10 equal parts

Shaded part = 0.2

Shaded part = 0.5

Shaded part = 0.9

Step 1

Step 2

Step 3

Step 4

Read the digit to the left

Say ‘and’ for the decimal

Read the digit to the

Say the place name of

of the decimal point as a point. whole number. 0

zero

Chapter 7 • Introduction to Decimals

Maths Grade 5 Book_Chapter 7-12.indb 107

right of the decimal point. 6

six

the last digit.

tenths

107

18-12-2023 11:15:22

Let us see how to read 0.6. It is written as zero and six tenths. zero

and

six tenths

0.6 zero Example 1

six

Write the decimal numbers in words. 1

0.7 2 0.8 3 0.4

Zero and seven tenths Do It Together

point

Zero point eight

Zero and four tenths

Write in figures. 1

Zero point seven = 0.7

Zero and one tenth = __________ 4 Zero point nine = __________

Zero point two = __________

Combining Whole Numbers and Tenths Write the decimal for the shaded part.

Remember! 1 1 Whole number part

0.4 1.4

Decimal point

Example 2

The decimal part after the decimal point is always less than 1 whole.

4 10

Decimal part

Write the decimal number for the coloured part.

2 wholes

2 2.2

2 10 0.2

Thus, the coloured part represents 2.2. 108

Maths Grade 5 Book_Chapter 7-12.indb 108

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Do It Together

Write the decimal number for the coloured part.

____________________

10 __________

Thus, the decimal number for the coloured part is __________.

Expanded Form of Decimals up to Tenths As we move towards the right-hand side of the decimal point, the value of the digit decreases 10 times. Let us write the expanded form of decimals up to tenths. Write 14.6 and 45.7 in the place value table and in their expanded form. Tens (10)

Ones (1)

Decimal point

14.6

45.7

14.6

Tens (value of 1 is 10)

Tenth (value of 6 is

Tenths 1 10 6 7

6 or 0.6) 10

Ones (value of 4 is 4)

Expanded form: 10 + 4 +

6 10

14.6 = 10 + 4 + 0.6

Expanded form of 45.7 = 40 + 5 + 0.7 Example 3

Write the decimals 15.7 and 23.9 in the place-value table. Also, write these in words and in expanded form. 15.7

Tens

Ones

Decimal point

Tenths

23.9

15.7 is written as fifteen and seven tenths.

7 Or 15.7 = 10 + 5 + 0.7 10 23.9 is written as twenty-three and nine tenths. 9 Or 23.9 = 20 + 3 + 0.9 23.9 = 20 + 3 + 10 15.7 = 10 + 5 +

Chapter 7 • Introduction to Decimals

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109

18-12-2023 11:15:23

Do It Together

Shade to show the decimal. Write it in expanded form.

3.4 = __________

Do It Yourself 7A 1

Shade to show the decimals and then write in words.

1.6 2

Write the decimal number for the coloured part. Write the decimals in words.

Write the words as decimal numbers. a Fifty-two point one

b Four hundred thirteen and two tenths

c One hundred and five tenths

d Eight hundred five point seven

Write the expanded form of the given decimal numbers. a 45.1

2.3

b 17.9

Write in decimal form. 5 1 a b 4+ 10 10

c 143.5

c 3+

8 10

d 548.1

e 789.4

9 10

e 78 +

5 10

985.3

80 + 6 +

8 10

Word Problems 1

Raj measured his height but forgot to put a decimal. His height is 1255 cm. Where

Rashmi ate 3 out of 10 slices of a pizza. What fraction of the pizza did she eat? Also,

should he put the decimal point so that he writes his correct height in metres? write your answer in decimal form.

110

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Hundredths When 1 is divided into 100 equal parts then each part is equal to one-hundredth. 1 Fractional representation = 100 Decimal representation = 0.01 One-hundredth is one-tenth further divided into 10 equal parts. Let us see a few more hundredths.

Shaded part =

8 100

Shaded part = 100

Reading and Writing Hundredths Let us learn how to read decimal numbers up to hundredths. fifteen

and

fifty-four hundredths

15.54 fifteen

point

five four 1 1 Whole number part

0.32 1.32

Decimal point

32 100

Decimal part

Write the given decimal numbers in words.

Example 4

Do It Together

3.14 = three and fourteen hundredths

154.87 = one hundred fifty-four point eight seven

6.51 = six and fifty-one hundredths

Write two hundred fifty-three and forty-five hundredths in figures. Two hundred fifty-three and forty-five hundredths = ________________

Chapter 7 • Introduction to Decimals

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111

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Expanded Form of Decimals up to Hundredths Write 53.21 in expanded form.

53.21

Tens

Ones

Decimal point

Tenths

Hundredths

5 tens

3 ones

2 tenths

1 hundredths

53.21 = 50 + 3 + 0.2 + 0.01

Expanded form: 50 + 3 + Example 5

Write 34.18 in expanded form.

Number

Tens (10)

Ones (1)

Decimal point

3 tens

Expanded form: 30 + 4 + Do It Together

2 + 1 10 100

1 + 8 10 100

Tenths 1 10 1

Hundredths 1 100 8

4 ones

1 tenths

8 hundredths

34.18 = 30 + 4 + 0.1 + 0.08

Shade the grid for 1.34 and write the number in the place-value chart and in its expanded form.

Tens

Ones

Decimal point

Tenths

Hundredths

1.34

Expanded form - ___________________________________________________________________________

Do It Yourself 7B 1

Write the decimal number for the coloured part. a

112

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Colour the grid for the decimal numbers.

1.72 3 4

0.54

Write the decimal numbers in words. a 42.14

b 87.08

c 81.87

d 178.64

Write as decimal numbers.

a Seventy-two point three three

b 79.06

c 41.37

Write in decimal form. 4 a 100 4 d 500 + 3 + 100

457.18

d Three hundred sixty-one point zero four

Write the expanded form for the decimal numbers. a 45.13

b Four hundred seventeen and two hundredths

c Twenty-six and forty-seven hundredths

e 874.09

d 486.72

2 100 3 e 70 + 100

b 14 +

e 879.34

c 43 +

4 8 + 10 100

800 + 60 + 7 +

976.07

1 7 + 10 100

Word Problem 1

A cyclist travelled 23.48 km in a day. Express the decimal number in its expanded form.

Thousandths

One-thousandth

When 1 is divided into 1000 equal parts, then each part is equal to one-thousandth. 1 Fractional representation = 1000 Decimal representation = 0.001 One-thousandth is one-hundredth further divided into 10 equal parts.

Reading and Writing Thousandths Let us read decimal numbers in thousandths. Write 14.179 in the place-value chart. Number

Tens 1

Chapter 7 • Introduction to Decimals

Maths Grade 5 Book_Chapter 7-12.indb 113

Ones 4

Decimal point .

Tenths 1

Hundredths 7

Thousandths 9

113

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fourteen

one hundred seventynine thousandths

and

14.179 fourteen Example 6

Do It Together

one seven nine

point

Write the decimal numbers in words. 1

4.153 = Four and one hundred fifty-three thousandths.

108.018 = One hundred eight and zero one eight.

Write the words in figures. 1

Forty-one and five hundred thirty-one thousandths = __________

One hundred twelve point three four three = __________

Twenty and two thousandths = __________

Expanded Form of Decimals up to Thousandths Write 81.078 in expanded form. Number

Tens

Ones

Decimal point .

Tenths 0

81.078

Tens (value of 8 is 80)

Hundredths

Thousandths

Thousands (value of 8 is 0.008) Hundredths (value of 7 is 0.07)

Ones (value of 1 is 1)

81.078 = 80 + 1 + Example 7

Tenths (value of 0 is 0)

7 8 + 100 1000

81.078 = 80 + 1 + 0.07 + 0.008

Write 76.103 in expanded form.

Number

Tens (10)

Ones (1)

Decimal point

7 tens

76.103 = 70 + 6 +

6 ones

1 + 3 10 1000

Tenths 1 10

Hundredths 1 100

Thousandths 1 1000

1 tenths

0 hundredths

3 thousandths

76.103 = 70 + 6 + 0.1 + 0.003

114

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Do It Together

Write the decimal for the expanded form. 7 tens + 5 ones + 9 hundredths + 2 thousandths = ____________________ 7 2 600 + 20 + 2 + = ____________________ 1000 5 1 3 5 tens + 2 ones + 5 hundredths + 1 thousandths = 50 + 2 + + = _______________ 100 1000 1

Equivalent Decimals Equivalent decimals are decimal numbers which have the same value. They are also called equal decimals.

four tenths 0.4

forty hundredths 0.40

seven tenths

seventy hundredths

4 10

40 = 4 100 10

7 10

70 100

0.7

Examples of equivalent decimals: 0.5 = 0.50 = 0.500 = 0.5000

0.70

1.6 = 1.60 = 1.600 = 1.6000

If we place zeroes before the end of the decimal, then it changes the value of the number, and thus the decimal numbers are not equivalent. Example: 0.5 and 0.05 are not equivalent decimals.

Remember! five hundredths

five tenths 0.5

0.05 5 100

5 10

Example 8

When we place zeroes on the right-hand side of a decimal number, its value remains the same.

Write any three equivalent decimals for 14.3. 14.3 = 14.30 = 14.300 = 14.3000

Do It Together

Shade the grids to find the equivalent fraction for 0.8

0.8 = __________

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Do It Yourself 7C 1

Write the decimal number in words. a 25.127

b 42.872

c 16.012

d three hundred two and five thousandths

b 54.642

c 78.602

Write in decimal form. 6 2 5 + a b 14 + 1000 100 1000 8 3 7 + d 106 + e 500 + 6 + 1000 10 1000

d 111.001

e 486.145

154.031

1 6 + 10 1000 9 8 2 + + f 400 + 80 + 9 + 10 100 1000

c 10 + 6 +

Shade the tenths grid for the decimal. Shade the hundredths grid to show its equivalent decimal.

0.3 6

765.004

Write the expanded form for the decimal numbers. a 15.062

b fifty-two and four hundred two thousandths

c twelve and forty-one thousandths

e 965.045

Write as decimal numbers. a thirty-one point three five two

d 174.201

0.4

Write any two equivalent decimals for the decimal numbers. a 45.6

b 187.2

c 87.02

d 963.14

e 12.701

189.221

Word Problems 1

A recipe calls for 1.269 kg of rice. How do we write 1.269 in words?

A person travels 3.659 km. Write 3.659 in expanded form.

Fractions and Decimals Real Life Connect

1 kg of cocoa 5 powder. She has a weighing balance with her which has readings in the decimal form.

Rashmi has to bake a cake. She requires

How can she weigh the cocoa powder? 116

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Conversion Between Fractions and Decimals Rashmi thinks of converting the fraction into its decimal form to weigh the cocoa powder. Let us learn about this!

Converting Fractions to Decimals

Rashmi now converts 1 into a decimal number. 5 Step 1: Multiply the denominator and the numerator so that we get a power of 10 in the denominator.

Step 2: Write a fraction with the denominator as 10, 100 or 1000

1 1×2 2 = = = 0.2 5 5 × 2 10 So, the weighing scale will show 0.2 kg of cocoa powder since 1 is 5 the same as 0.2 kg.

Step 3: Insert a decimal point before the number of places equal to the number of zeroes in the denominator from the rightmost side. Example 9

Convert the given fractions into decimals. 1 = 1 × 2 = 2 = 0.2 5 5 × 2 10 2 3 = 3 × 4 = 12 = 0.12 25 25 × 4 100 3 3 = 3 × 125 = 375 = 0.375 8 8 × 125 1000 1

Do It Together

Think and Tell Why do we only convert fractions into fractions with a denominator of 10, 100 or 1000 before converting them into decimals?

Convert the given fractions into decimals. 1

8 2 20

8 = 8×5 = 20 20 × 5

6 8

6× 6= 8 8×

Converting Decimals to Fractions Rashmi also wanted 0.25 kg of sugar for the cake. She wanted to know how she could represent this number as a fraction. Let us see! Step 1: Write the decimal with a denominator of 1.

0.25 =

Step 2: Convert the denominator into multiples of 10, 100, or 1000 to eliminate the decimal point.

0.25 0.25 × 100 25 25 ÷ 25 1 = = = = 1 1 × 100 100 100 ÷ 25 4

Step 3: Express the fraction in its simplest form.

Thus, Rashmi used 1 kg of sugar for the cake. 4 Chapter 7 • Introduction to Decimals

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Example 10

Convert the decimals into fractions. 1 2

Do It Together

4.2 = 4.2 = 4.2 × 10 = 42 = 21 1 × 10 10 5 1

0.174 = 0.174 = 0.174 × 1000 = 174 = 87 1 × 1000 1000 500 1

Convert the decimals into fractions. 1

14.50 2 1.75 14.5 = 1.75 = 14.50 = 14.5 = 1

Do It Yourself 7D 1

Identify the number by which you need to multiply the denominator of the fraction to convert it to a decimal number. 1 2 a b 4 5

11 20

Convert the fractions into decimal numbers. 9 8 4 a b c 10 25 5

14 25

7 8

11 40

17 50

6 8

38 80

e 2.302

32.120

Rewrite the decimals in their fractional form. a 1.3

b 14.8

c 12.54

d 176.25

Write as fractions and decimals. a Two and one tenth

b Five thousandths

c Two and fifty thousandths

d Thirty-five hundredths

e Five hundred twenty-four thousandths

Insert >, < or = in the blanks. 3 a 12.2 5 7 d 0.375 8

b 1

15.2

e 5.5

1 4

Three and one hundred twenty thousandths c 1

0.125 5 8

Word Problems 1

Ravi was eating a cake. He ate 0.375 of the cake. What fraction of the cake did he eat?

A contractor completed

5 5 of the work in 7 days. What is in decimal form? 8 8

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Types of Decimals Real Life Connect

Sanjana and her friends were measuring their heights. Sanjana is 1.32 m tall, Sana is 1.2 m tall, Reshma is 1.26 m tall and Muskan is 1.4 m tall.

Like Decimals Sanjana wondered why her height and Reshma’s height look similar while Sana’s height and Muskan’s height look similar. Let us see how they are similar! 1.32 and 1.26 have 2 digits after the decimal point. 1.2 and 1.4 have 1 digit after the decimal point.

1.32

1.26

Decimal numbers which have the same number of digits after the decimal point are called like decimals.

Same number of digits after the decimal point. Example 11

Thus, 1.32 and 1.26 are like decimals.

Sort the groups of like decimals out of the following decimal numbers. 1.4, 5.64, 5.5, 48.14, 38.6, 147.47 Group 1 (With 1 digit after the decimal point) 1.4, 5.5, 38.6

Group 2 (With 2 digits after the decimal point) 5.64, 48.14, 147.47

Tens Ones Decimal Tenths point 1.4

5.64

5.5

48.14

147.47

38.6 Do It Together

Hundreds Tens Ones Decimal Tenths Hundredths point

Check whether 51.54 and 187.37 are like decimals or not. Number of digits after the decimal point in 51.54 = 2 Number of digits after the decimal point in 187.37 = _______ So, the numbers of digits after the decimal point (are/are not) equal. Thus, 51.54 and 187.37 (are/are not) like decimals.

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Unlike Decimals Decimal numbers which have a different number of digits after the decimal point are called unlike decimals. Example: 1.32 and 1.2 are unlike decimals. So, Sanjana was able to understand why she thought her height was looking similar to Reshma’s but different from Sana and Muskan’s. Example 12

Which option shows unlike decimals? (a) 14.2, 54.8, 23.6

(b) 71.25, 89.35, 41.51

Option (a) and (b) are groups of like decimals. The numbers in (c) are unlike decimals since 13.147 has 3 decimal places and 879.78 has 2 decimal places. Do It Together

Check whether 3.145 and 54.14 are unlike decimals or not. Number of digits after the decimal point in 3.145 = 3 Number of digits after the decimal point in 54.14 = _______ So, the numbers of digits after the decimal point (are/are not) equal. Thus, 3.145 and 54.14 (are/are not) unlike decimals.

Converting Unlike Decimals into Like Decimals Remember Sanjana who measured her height and the height of her friends? She wanted the heights of all of them to look similar. Let us see how this can be done! Consider 1.3 and 1.46 1.3

Add 1 zero

Unlike decimals can be converted into like decimals by finding the equivalent fraction.

1.46 1.46

1.30

Same number of digits after the decimal point

Put zeroes at the end of the decimal number such that the number of digits after the decimal point is the same.

For example, 1.2 = 1.20 and 1.4 = 1.40 Thus, the heights of all the friends can be written as: Name

Height

Sanjana 1.32 m

Sana

1.20 m

Reshma 1.26 m

Muskan 1.40 m

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Convert 14.5, 74.543 and 15.64 into a set of like decimals.

Example 13

Convert each of the decimal numbers into equivalent decimals having 3 decimal places by adding 0s at the end. 14.5 = 14.500

74.543 = 74.543

15.64 = 15.640

Thus, 14.500, 74.543 and 15.640 are a set of like decimals. Do It Together

Convert 189.4745 and 1.2 into like decimals. 189.4745 =

1.2 =

Thus, _______ and _______ are a set of like decimals.

Do It Yourself 7E 1

Which of the following are a group of like decimals? a 1.2, 5.4, 8.9, 6.54, 1.3

b 2.23, 4.26, 4.89, 4.2, 6.584

c 7.89, 7.2, 64.594, 45.2, 56.5

d 81.564, 78.512, 453.125, 486.154, 86.15

Tick () if the decimals are like and cross out () if the decimals are unlike. a 4.2 and 6.25

b 17.23 and 691.56

c 11.3 and 17.8

d 5.157 and 64.581

e 10.001 and 12.690

1.111 and 1.11

Identify the number of zeroes to be added to the decimal numbers to convert them into like decimals. a 1.45, 1.6

b 81.566 and 12.2

c 17.98 and 14.221

d 11.001 and 11

e 8.21 and 3.560

1.101 and 967.1

Convert the unlike decimals into like decimals. a 13.15, 1.2

b 3.48, 1.2

c 4.8, 1.526

d 1.4, 47.584

e 53.23, 17.164

Convert the fractions into like decimals. a

12, 1 20 5

14, 5 40 10

5, 3 8 4

1.002, 348.1

4, 5 5 20

1, 1 2 8

Word Problem 1

A company’s profit in crores for the year 2021 is `23.2, for 2022 is `26.987, and for 2023

is `24.54. Convert all the decimals into like decimals.

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Comparing, Ordering and Rounding off Decimals Real Life Connect

Bhawna volunteered at the sports event at her school. There was a long jump competition and Bhawna recorded the jumps of various participants. Suhani jumped 3.2 m, Sobhita jumped 3.15 m, Kiran jumped 3.18 m and Mridula jumped 3.7 m.

Comparing and Ordering Decimals Bhawna wanted to compare the lengths of the jumps of the players. Let us see how we could do this!

Comparing Like Decimals Let us compare the jumps of Suhani and Mridula.

3.2

3.7

The coloured part in 3.2 is less than the coloured part in 3.7 Since 3.2 < 3.7, Mridula jumped farther than Suhani. Let us learn a quicker way to compare decimal numbers. Compare 3.15 and 3.18

3.15

DECIMAL POINT 3=3 1=1 5<8

3.18

If the whole numbers are the same, compare the tenths. If the tenths are the same, compare the hundredths. Follow the same process till you find unequal digits.

3.15 < 3.18 Example 14

Compare the whole numbers. The number with a larger whole number will be bigger.

Which of the given numbers is smaller—16.47 or 16.42? 16.47

16.42

Tens

Ones

Decimal point

Tenths

Hundredths

Look at the digit in the hundredths place: 7 > 2. So, 16.42 < 16.47. 16.42 is smaller. 122

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Do It Together

Which is smaller—174.54 or 174.05? Hundreds

174.54

Tens

174.05

Ones

Decimal point .

Tenths

Hundredths

_______________________________________

Comparing Unlike Decimals

Error Alert!

Remember Bhawna? She now wants to compare the jumps of all the players and announce the winner.

If the number of digits in a number is more, then the number is not necessarily bigger.

Let us see how she finds the winner!

23.645 > 23.65

23.645 < 23.65

Let us compare 3.2 m, 3.15 m, 3.18 m and 3.7 m. To compare unlike fractions, make all the decimals like decimals. Then, compare the like decimals. 3.20

3.2

3.15

3.18

3.7

3.70

3.70 is the biggest number. Thus, Mridula jumped the farthest and is the winner. Do It Together

Which is greater—145.14 or 145.4? The digits up to the ones are the same for both numbers. Compare the tenths and then the hundredths. Hundreds

Tens

Ones

Decimal point

145.14

145.4

145.14

__________

145.4

Tenths

Hundredths

___________

Ordering Fractions The distances jumped by 4 players are: Suhani– 3.2 m, Sobhita– 3.15 m, Kiran– 3.18 m and Mridula– 3.7 m. Arrange the distances in order, from the longest jump to the shortest. Convert the given fractions into like decimals. 3.2

3.20

3.15

3.18

Compare the decimals and arrange them.

3.7

3.70

Ones

Decimal point

Tenths

Hundredths

3.20

3.15

3.18

3.70

3.70 > 3.20 > 3.18 > 3.15 Chapter 7 • Introduction to Decimals

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Example 15

Arrange 12.14, 12.4, 12.04, 12.41 in ascending order. Make the decimals like: 12.14

12.14

12.4

12.40

12.04

12.41

So, 12.04 < 12.14 < 12.40 < 12.41. Do It Together

Arrange 15.47, 15.4, 15.744 and 15.3 in descending order. Make the decimals like: 15.47

15.470

15.4

Tens

15.470

15.400

Ones

15.744

Decimal point .

15.400

15.744

Tenths

15.3

Hundredths

15.300

Thousandths

15.744

15.300

Descending order: _________________________________________________________________________.

Do It Yourself 7F 1

Colour and compare using >, < or =. b

0.5 2

e 186.6 or 186.5

e 47.654 or 47.650

b 4.3 or 4.5 f

294.98 or 294.864

b 89.14 or 89.4

c 15.64 or 15.67

d 87.654 or 87.65

c 81.174 or 81.714

d 853.68 or 853.6

Arrange the decimal numbers in ascending order. a 14.14, 14.1, 14.01, 14.101

c 184.2, 184.23, 184.1, 184.112

0.89

Identify the decimal number which has the smaller value. a 15.64 or 15.46

0.9

Find the greater decimal number. a 4.2 or 4.15

0.6

b 84.56, 84.5, 84.6, 84.55, 84.65 d 64.23, 54, 64.32, 64.22, 64.33

Put the given decimal numbers in order, from the largest to the smallest. a 7.4, 7.5, 7.44, 7.54

c 921.4, 921.26, 921.35, 921.41

b 48.6, 48.36, 48.63, 48.66

d 185.5, 185.55, 185.45, 185.501

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Word Problems 1

A bottle contains 0.75 litres of water while another bottle has 0.9 litres of water. Which

The temperature for Monday was 34.2°C, for Tuesday was 34.5°C, for Wednesday was

In a racing competition, three athletes finish the race in 24.54 minutes, 24.68 minutes

water bottle has more water in it?

33.7°C, and for Thursday was 35.2°C. Arrange the temperatures in ascending order. and 24.36 minutes, respectively. Arrange the numbers in descending order.

Rounding off Decimals In a 100 m race event, Rupali completed the race in 21.3 seconds and was the winner. Rupali’s coach wanted to know approximately how many seconds she took to complete the race.

Rounding off to the Nearest Whole Number 20

Mark the decimal number on the number line.

Check which whole number is closer to the decimal number.

20.3

20.3 is closer to 20 and farther away from 21.

Thus, 20.3 rounds off to 20 when rounded off to the nearest whole number. Let us understand the other way for rounding decimal numbers off to the nearest whole number. Check the digit in the tenths place. The digit in the tenths place = 3 20.3

Tens 2

Ones 0

Decimal point .

Tenths 3

If the digit in the tenths place is less than 5, then round down; or else, round up. Here, 3 < 5, so we will round down. 20.3 ≈ 20. Example 16

Round 14.7 to the nearest whole number. 14.7 lies between 14 and 15. Digit in the tenths place = 7

7 is greater than 5

So, 14.7 will round up. (when 5 or more, round up, and when 4 or less, round down) Thus, 14.7 rounds up to 15 when rounded to the nearest whole number.

Chapter 7 • Introduction to Decimals

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Do It Together

Round off 263.5 to the nearest whole number. Mark 263.5 on the number line. 263

264

263.5 lies between 263 and 264. 263.5 rounds to __________. Thus, 263.5 rounds to __________ when rounded to the nearest whole number.

Do It Yourself 7G 1

Write if true or false.

a If the digit in the tenths place is 5, then the number rounds up to the nearest whole number.

b If the digit in the ones place is 4 or less, the number always rounds down to the lower whole

number.

c 12.2 rounds up to 13 when rounded to the nearest whole number.

d 99.9 rounds up to 100 when rounded to the nearest whole number.

Round off the numbers to the nearest whole number. a 2.6

b 7.9

c 48.2

d 15.1

e 71.3

156.4

Answer the questions.

a What is 31.9 rounded off to the nearest whole number?

b What is 14.5 rounded off to the nearest whole number?

c What is 123.4 rounded off to the nearest whole number?

d What is 543.1 rounded off to the nearest whole number?

Which of these, when rounded off to the nearest whole number, gives 23 - 22.9 or 21.2? Give a reason.

Round off the numbers to the nearest whole number first and put the correct symbol <, >, =. a 45.2 __________ 45.6

d 17.3 __________ 17.6

b 14.1 __________ 14.2

e 197.1 __________ 198.6

c 87.6 __________ 88.2

Word Problems 1

A city received 15.7 inches of rainfall in a year. What is 15.7 rounded off to the nearest

A student scored 88.5 marks in a test. What is 88.5 rounded off to the nearest whole

whole number? number?

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Points to Remember •

Decimals are numbers between whole numbers.

•

The decimal part in a number is always less than a whole.

•

Decimals are fractions with denominators of 10, 100 or 1000.

•

Adding a 0 at the end of a decimal number does not change its value.

• If the number of decimal digits after the decimal point is the same, then the decimal numbers are called like decimals; or else, they are called unlike decimals.

Math Lab Aim: To understand the visual relationship between decimal numbers and whole numbers. Setting: In groups of 2 Materials required: Printed worksheet with a grid (blank or light outlines), coloured pencils or markers

Method: 1 Draw a square grid with 5 rows and 5 columns. Write 15 decimal numbers in the squares (e.g., 0.5, 0.75, 1.2, 2.25, etc.). leaving some cells blank. Colour all squares with decimal numbers less than 0.5 in one colour.

2 Instruct the students to use the coloured pencils or markers to colour the squares based on certain criteria. For example: a

Colour all squares with whole numbers in a different colour.

Leave squares with blank or non-decimal labels uncoloured.

Chapter Checkup 1

Shade the grid and write the expanded form of the decimals. a

1.2

1.48

814.36

Write the decimal number in words. a

15.2

71.65

Chapter 7 • Introduction to Decimals

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152.1

176.254

176.801 127

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Write as decimal numbers. a c e

Ninety-five point two two three

11.1

189.3

16.54

1 8

4 5

490.02

12 20

17.684

36 40

843.033

4 16

12 15

12.6

52.2

120.5

20.8

10.125

25.315

12.3 and 12.4

222.22 and 222.02

b e

14.5 and 14.55

187.540 and 187.504

c f

999.9

102.3 and 102.25 3.003 and 3.033

Put the given decimal numbers in order, from smallest to largest. a c

Thirty-one and five thousandths

Four hundred twenty and eleven thousandths

Circle the smaller decimal number and put a square on the greater decimal number. a

Sixty-three and four tenths

Identify the number by which the decimal has to be multiplied and then convert it into a fraction. a

One hundred three and five hundredths

Convert the fractions into decimal numbers. a

Place the numbers in the place value chart and write the expanded form of the decimal numbers. a

Forty point one zero one

14.23, 14.2, 14.3, 14.25

235.64, 235.66, 235.6, 235.666

87.64, 87.6, 87.5, 87.7

888.88, 888.888, 888.8, 888.08

19.4, 19.44, 19.54, 19.501

Arrange the numbers in descending order. a c

1.3, 1.31, 1.2, 1.33

555.5, 555.55, 555.05, 555.555

748.01, 748.101, 748.11, 748.1

10 Round off the decimal numbers to the nearest whole number. a

1.3

14.8

99.6

100.2

106.7

Word Problems 1

Consider the time taken by different riders in a bicycle race. Who is the winner? Raj– 14.3 minutes Rekha– 15.2 minutes Utkarsh– 13.92 minutes Ali– 13.99 minutes r. Goyal completes a task in 40.6 minutes. What is 40.6 rounded off to the M nearest whole number?

Shalu purchased a book for ₹156.25. Write 156.25 in its expanded form.

Mr. Jadeja spent $12.99, Sarah spent $13.01, Alice spent $12.9 and Jacob spent $13.1. Arrange the amounts spent in ascending order.

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Operations with Decimals

Let's Recall 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

Sana and her friends measured their heights and represented them in decimal form. Sana is 1.51 m tall, Rakhi is 1.45 m tall, Anjali is 1.34 m tall and Meenakshi is 1.5 m tall. The increasing order of the heights of the 4 friends is 1.34 m < 1.45 m < 1.5 m < 1.51 m

Thus, Sana is the tallest, followed by Meenakshi, then Rakhi, while Anjali is the shortest among them.

Letʼs Warm-up

Fill in the blanks using >, < or =. 1

1.14 __________ 1.41

1.4 __________ 1.400

3.15 __________ 3.5

2.33 __________ 2.3

1.54 __________ 1.45

Did You Know? As of 2021, the population of India is 140.76 crores.

I scored _________ out of 5.

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Addition and Subtraction of Decimals Real Life Connect

Ana and her friend Rita went to a café. Ana: I need a glass of lime soda. Rita: I need a cup of soup. Cashier: Sure! You need to pay $1.2 for the lime soda and $1.5 for the soup. Ana made the total payment at the cafe.

Addition of Decimals To find the total amount paid by Ana at the café, add 1.2 and 1.5. Step 1

Step 2

Represent 1.2 and 1.5 visually.

Add the wholes with the wholes and tenths with the tenths.

1.2

1 whole

2 tenths + 5 tenths = 7 tenths

1.5

Thus, the total cost of the lime soda and soup is $2.7.

0.2 + 0.5

1.2 + 1.5 = 2.7

Let us learn another way to add decimals. Ones 1

Write the digits in columns and align the decimal points and the digits.

. Tenths

Add the numbers. Use regrouping if required.

Put the decimal point in the answer at the same place as the numbers above it.

Let us add some more tenths and learn how to add hundredths.

0.4 + 0.3 = 0.7

0.5 + 0.7 = 1.2

Ones .

Tenths

Ones .

Tenths

4 7

0 1

. . .

7 2

0.35 + 0.25 = 0.60 O . 0

+ 0 0

. 3

. 2

. 6

0.67 + 0.66 = 1.33

+ 0

0 1

h 7 6 3

130

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Add 12.45 and 1.3 using the column method. T

+ 1 Example 1

Decimal numbers with the same number of decimal places are called like decimals.

Find the sum of the given numbers visually. 1

0.5 + 0.3

0.6 + 0.4

0.5 + 0.3 = 0.8 Ones 0 + 0 0 Example 2

Remember!

. . . .

Tenths 5 3 8

0.44 + 0.22

0.6 + 0.4 = 1.0 Ones .

Tenths

. .

0.58 + 0.42

0.44 + 0.22 = 0.66 O

+ 0

0.58 + 0.42 = 1.00

4 6

0 1

h 8 2 0

Nikhil requires 3.5 kg of rice and 2.25 kg of sugar for his home. What is the total weight of rice and sugar that Nikhil requires? Weight of rice = 3.5 kg

Weight of sugar = 2.25 kg

Total weight = weight of rice + weight of sugar = 3.5 kg + 2.25 kg Thus, Nikhil requires 5.75 kg of rice and sugar. Do It Together

Add the numbers visually and also show the column method. 1

0.8 + 0.6

0.8 + 0.6 = __________

O 3 + 2 5

. . . .

t 5 2 7

h 0 5 5

0.14 + 0.93

0.14 + 0.93 = __________

Do It Yourself 8A 1

Add the tenths visually and find the sum. a 0.1 + 0.2

Chapter 8 • Operations with Decimals

Maths Grade 5 Book_Chapter 7-12.indb 131

b 0.2 + 0.5

131

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Add the hundredths visually and find the sum. a 0.15 + 0.25

b 0.25 + 0.38

Add the numbers using the column method. a 12.3 + 23.6

b 12.5 + 87.74

d 53.41 + 64.472

c 63.47 + 15.72

e 823.1 + 3.718

89.04 + 963.478

Match the following. a 3.5 + 154.2

53.9

b 41.56 + 12.34

88.58

c 31.23 + 57.35

51.003

d 45.913 + 5.09

157.7

Compare using >, < or =. a 45.2 + 32.9

79.1

c 62.1 + 81.009

b 14.5 + 31.47

143.009

d 61.81 + 17.568

45.97 79.378

Word Problems 1

Rani went to the market. She bought a matchbox for ₹0.50, a bar of soap for ₹8.75,

A special cookie is made by mixing three ingredients of 10 grams, 12.82 grams and

In a relay race, Rashi covers 1.548 km, Prerna covers 2.328 km and Navya covers

Read the table showing the weight of 4 students. Answer the questions below.

and a pen for ₹6.5. How much money did she spend in total?

8.72 grams, respectively. What is the weight of the cookie?

1.986 km. What is the total distance (in km) covered by the three girls?

Name

Rakesh

John

Ismail

Shahid

Weight (in kg)

34.53

43.5

36.35

40.235

a What is the total weight of Rakesh and Ismail? b What is the total weight of John and Shahid?

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Subtraction of Decimals

Ana paid $3 for the lime soda and soup which cost her $2.7. How much money will she get back?

−

Ones

Tenths

Write the digits in the column method and align the decimal points and the digits. Remember to convert the decimals to like decimals if required. Subtract. Use regrouping if required. Put the decimal point in the answer at the same place as the numbers above it.

Let us see how to subtract visually.

0.7 – 0.3 = 0.4

1.5 – 0.8 = 0.7

Ones .

Tenths

Ones .

Tenths

−

. .

−

0.46 – 0.25 = 0.21 O

−

0 0

. . .

4 2

0.6 – 0.06 = 0.54 O

−

6 0 5

h 0 6 4

Subtract 16.84 from 17.2 using the column method. T O . 6

1 7

0 0

− 1 6

Example 3

2 0 8 4 3 6

Find the difference of the numbers visually. 1

0.8 – 0.3 O

– 0

0 0

0.8 – 0.3 = 0.5

Chapter 8 • Operations with Decimals

Maths Grade 5 Book_Chapter 7-12.indb 133

0.84 – 0.36

– 0

8 3 4

h 4 6 8

0.84 – 0.36 = 0.48

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Example 4

When Sushil was born, his weight was 3.75 kg. After 1 year, his weight increased to 8.5 kg. By how much did Sushil’s weight increase in 1 year? Sushil’s weight when he was born = 3.75 kg

O 8 – 3 4

Sushil’s weight after 1 year = 8.5 kg

Increase in weight = 8.5 kg – 3.75 kg Thus, Sushil’s weight increased by 4.75 kg in 1 year. Do It Together

. t h 14 1 . 5 0 . 7 5 . 7 5

Subtract the numbers visually and also show the column method, in your notebooks. 1

1.4 – 0.3

1.54 – 0.87

1.4 – 0.3 = __________

1.54 – 0.87 = __________

Do It Yourself 8B 1

Subtract visually and find the difference. a 1 – 0.3

b 1.23 – 0.46

2 Subtract using the column method. a 5.6 – 2.4

b 12.5 – 11.92

c 113.09 – 93.989

d 523.2 – 514.26

e 513.654 – 471.817

932.45 – 821.647

3 Match the following. a 45.3 – 25.6

550.584

b 51.48 – 11.68

678.877

c 896.4 – 345.816

19.7

d 789.987 – 111.11

39.8

4 Compare using >, < or =. a 47.8 – 36.6 d 256.71 – 59.357

11.2

b 96.4 – 51.84

196.353

e 512.2 – 461.718

44.46 51.482

c 125.6 – 84.934 f

827.601 – 739.798

35.666 87.802

134

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5 Answer the given questions. a What is the difference of 12.45 and 9.496? b What is 15.84 less than 93.457?

Word Problems 1 Ramesh is a milkman. He has a milk and water mixture of 48.6 L. The mixture has 41.2 L of pure milk. What quantity of water is there in the mixture?

2 Iyer is 1.65 m tall and his brother is 0.18 m shorter than him. What is his brother’s height? 3 The distance between A and B is 1.2 km, and the distance between B and C is 2.35 km. If the total distance between A and D is 7 km, what is the distance between C and D? 7 km 1.2 km A

2.35 km B

4 Rashmi had ₹5555.5 in her wallet. She bought a saree for ₹555.5 and an umbrella

for ₹55.5. She paid ₹5.5 to the auto driver for dropping her home. What was the total amount of money she had left?

Multiplication of Decimals Real Life Connect

The next day, Ana and her friends went to a restaurant. They ordered 10 apple pies. The cost of each pie was $0.5.

Multiplying Decimals The pies looked delicious. Ana wanted to know the total cost of the pies. How could they find the total cost? Let us learn!

Multiplying Decimals by 10, 100, 1000 … Cost of each pie = $0.5 Total number of pies = 10 Total cost of 10 pies = 10 × $0.5 = $5

Chapter 8 • Operations with Decimals

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Multiplication by 10

When we multiply any decimal number by 10, we move the decimal point 1 place to the right.

10 × 0.614 = 6.14

10 × 6.14 = 61.4

10 × 61.4 = 614

Multiplication by 100

When we multiply any decimal number by 100, we move the decimal point 2 places to the right.

100 × 0.157 = 15.7

100 × 1.57 = 157

100 × 15.7 = 1570

When we multiply any decimal number by 1000, we move the decimal point 3 places to the right.

1000 × 0.356 = 356

Multiplication by 1000

0 . 6 1 4

0 . 1 5 7

6 . 1 4

6 1 . 4

1 . 5 7

1 5 . 7 0 (Add 1 zero)

0 . 3 5 6

Thus, we move as many decimal places to the right as there are 0s in the multiplier (10, 100 or 1000).

1000 × 3.56 = 3560

1000 × 35.6 = 35600

(Add 1 zero)

(Add 2 zeroes)

3 . 5 6 0

3 5 . 6 0 0

Remember! 614. is written as 614 as we do not put a decimal point at the end of the number that has no decimal value.

Error Alert! Adding 0s at the end of the decimal number does not change the value. Move the decimal point right while multiplying by 10, 100 or 1000.

Example 5

12.54 × 100 = 1254

Solve. 1

Do It Together

12.54 × 100 = 12.5400

4.52 × 10

23.48 × 100

ove the decimal M point 1 place to the right.

ove the decimal M point 2 places to the right.

ove the decimal M point 3 places to the right.

4.52 × 10 = 45.2

23.48 × 100 = 2348

1.413 × 1000 = 1413

32.4 × 1000 = 32400

1.413 × 1000

32.4 × 1000

Fill in the blanks. 1

1.415 × 10 = __________

2.547 × __________ = 254.7

1.01 × 1000 = __________

256.47 × __________ = 25647

136

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Multiplying Whole Numbers and Decimals We can multiply a decimal number with a whole number. Let us multiply 4 by 0.4. 4 × 0.4 = 0.4 + 0.4 + 0.4 + 0.4

0.4 + 0.4 + 0.4 + 0.4 = 4 × 0.4 = 1.6

The above numbers can also be multiplied using the column method with the following steps.

0.6

Ignore the decimal point in the decimal number and multiply the whole numbers. 4 × 4 = 16

Count the total number of digits after the decimal point in both the numbers.

× 1

4 4 6

There is 1 digit after the decimal point.

Place the decimal point in the product so as to obtain as many decimal places as there are in the decimal number. Example 6

Multiply 0.6 and 3 visually.

0 1

. .

4 4 6

0 decimal places 1 decimal place (0 + 1) = 1 decimal place

1 0.8 Thus, 3 × 0.6 = 1 + 0.8 = 1.8 Example 7

Multiply: 506 × 1.06 Ignore the decimal point in the decimal number and multiply the whole numbers. 506 × 106 Count the total number of digits after the decimal point in both numbers. There are 2 digits after the decimal point.

5 5

× 3 0 0 3

5 1 0 0 6 6

0 0 3 0 0 3

6 6 6 0 0 6

Put the decimal point in the product so that it has 2 decimal places. 506 × 1.06 = 536.36 Do It Together

Multiply 0.7 and 4 visually and then show column multiplication.

0.7 × 4 = ___________________ Chapter 8 • Operations with Decimals

Maths Grade 5 Book_Chapter 7-12.indb 137

137

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Multiplying Two Decimals We can also multiply a decimal number with another decimal number. Let us multiply 1.5 by 2.6. Ignore the decimal point in the decimal numbers and multiply the whole numbers.

× +

15 × 26 = 390 Count the total number of digits after the decimal point in both numbers.

1 2 3

Sum of the decimal places in the given decimal numbers = 1 + 1 = 2

1 decimal place 1 decimal place (1 + 1) = 2 decimal places

What is the product of the smallest 4-digit number and 45.623?

1.5 × 2.6 = 3.90 Multiply. 1

1.4 × 3.66 × 1 3 5

3 4 6 1

6 1 6 6 2

6 4 4 0 4

15.46 × 36.1

1.4 × 3.66 = 5.124 Example 9

5 6 9

9 6 5

4 5

1 × 1 2 3 8

5 3 5 7 8 1

4 6 4 6 0 0

6 1 6 0 0 6

15.46 × 36.1 = 558.106

The cost of 1 kg of apples is ₹126.5. What is the cost of 2.5 kg of apples? Cost of 1 kg of apples = ₹126.5 Weight of apples required = 2.5 kg Total cost of 2.5 kg of apples = 2.5 × ₹126.5

Thus, the cost of 2.5 kg of apples is ₹316.25. Do It Together

5 6 0 0 0

Think and Tell

Put the decimal point in the product so that it has 2 decimal places. Example 8

. . .

3 3

1 2 9 0 9

2 3

1 × 6 5 1

2 3 3 6

6 2 2 0 2

5 5 5 0 5

What is the product of 23.5 and 45.12? 4

Thus, 23.5 × 45.12 = _____________ 138

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Do It Yourself 8C 1

Multiply the decimal number by the whole number visually. a 0.2 × 4

b 0.4 × 5

2 1

3 1

4 1

Multiply. a 45.6 × 10

b 15.478 × 100

e 8 × 5.45

12 × 14.945

c 63.157 × 1000

d 4 × 2.5

g 1.6 × 1.58

h 12.3 × 35.69

Compare using >, < or =. a 5.15 × 10

515

b 83.482 × 1000

d 715.15 × 8

5720.12

e 2.13 × 31.4

83482 6.5882

c 6 × 1.658

10.948

1.5 × 634.71

94.2065

Fill in the blanks. a 12.15 × ______ = 121.5

b 15.26 × ______ = 15260

c 61.235 × ______ = 612.35

d 14.1 × ______ = 14100

e 6 × 1.645 = ______

1.1 × 47.503 = ______

Matthew was solving a puzzle based on decimal numbers. The first step is to think of a number and then find six times of that number. If he thinks of 4.6, what is the final answer?

Word Problems 1

If the cost of 1 kg of oranges is ₹41.23, what will be the cost of 5 kg of oranges?

There are 24 workers at a factory. 11 workers are given ₹252.54 each and the rest of

The cost of 1 kg of rice and 1 kg of pulses is ₹75.5. If the cost of 1 kg of rice is ₹43.75,

them are given ₹364.52 each. How much money is given to the workers in total? what is the cost of 5 kg of pulses?

Chapter 8 • Operations with Decimals

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Currencies From Different Countries Anaʼs father travelled all over the world for business. Whenever he visited a country, he exchanged Indian currency for the currency of that country. Let us learn about the currencies of different countries along with their value in Indian rupees. Consider the chart which shows how many Indian rupees we can get when we exchange the money of different countries. Country

Money

Changed into Indian Rupees (₹)

Nepal

Rupee (NPR)

0.62

U.A.E

Dirham

South Korea

Won (₩)

Sri Lanka

0.062

Rupee (LKR)

0.25

22.51

England

Pound (£)

104.06

Hong Kong

Dollar (HKD)

10.54

China

Yuan (¥)

11.51

U.S.A

Dollar ($)

South Africa

82.67

Rand (R)

Germany

4.43

Euro (€)

89.19

(This is the rate on 25-08-2023.) How many Indian rupees are there in 5 Nepalese rupees? (1 NPR = ₹0.62) 1 NPR = ₹0.62

5 NPR = 5 × ₹0.62

× 3

So, 5 × 0.62 = 3.10 Thus, 5 NPR = ₹3.1 Example 10

How many Indian rupees are there in $6? ($1 = ₹82.67) $1 = ₹82.67 $6 = 6 × ₹82.67

× 4

Thus, $6 = ₹496.02 Do It Together

× 4

6 1

2 5 0 7 6 2 7 6 2

Read the exchange rate table given above. Complete the table. Currency

Changed into Indian Rupees (₹)

How many?

Changed into Indian Currency

Won

0.062

500

500 × 0.062 =

Euro

89.19

Dirham

22.51

140

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Do It Yourself 8D 1

How many Indian rupees are the same as 1000 Euros? (1 Euro = ₹89.19)

2 1

Babita’s father brought a present for her from China. It cost 55 yuan. How much would it cost

3 1

Shweshitha went to England on a holiday. She wanted 500 pounds. How much Indian money did she

4 1

Suresh’s father is working in U.A.E. He gets 800 dirham as a salary. Saurav’s father who is working in

in Indian rupees (1 yuan = ₹11.51)?

have to give to get 500 pounds? (1 pound = ₹104.06)

Sri Lanka gets 2200 SriLankan rupees. Who gets more Indian rupees as salary and by how much? (1 dirham = ₹22.51 and 1 LKR = 0.25)

5 1

Asmitha has some pounds and rands. She has 180 pounds and 54 rands. How much money does she have in Indian rupees (1 pound = ₹104.06 and 1 rand = ₹4.43)?

Word Problem 1

Anjali is fond of collecting different currency notes. She has 2 pounds, 5 rands, 6 US

dollars, 5 Euro and 15 yuan. How much money does Anjali have in Indian rupees? (1

pound = ₹104.06, 1 rand = ₹4.43, 1 US dollar = ₹82.67, 1 Euro = ₹89.19, 1 yuan = ₹11.51)

Division of Decimals Real Life Connect

Ana and her friends had a great outing. Now, they have to pay the bill. They spent a total of $92.5.

Dividing Decimals Ana wanted to know what is the share of each person if there were 10 people in total. How could she find out? Let us learn!

Dividing Decimals by 10, 100, 1000, … Total amount spent = $92.5, Total number of people = 10 Share for each friend = $92.5 ÷ 10 = $9.25 Let us divide 987 by 10, 100 and 1000. Division by 10

When we divide any decimal number by 10, move the decimal point 1 place to the left. 987 ÷ 10 = 98.7 9 8 7 . 0

Chapter 8 • Operations with Decimals

Maths Grade 5 Book_Chapter 7-12.indb 141

Division by 100

Division by 1000

987 ÷ 100 = 9.87 9 8 7 . 0

987 ÷ 1000 = 0.987 9 8 7 . 0

When we divide any decimal When we divide any decimal number by 100, move the decimal number by 1000, move the point 2 places to the left. decimal point 3 places to the left.

141

18-12-2023 11:15:42

Thus, when we divide by 10, 100 or 1000, we move as many decimal places to the left as there are 0s in the divisor. Example 11

Divide. 1

453.2 ÷ 10 = 45.32

3 471.12 ÷ 1000 = 0.47112 Do It Together

53 ÷ 100 = 0.53

4 65.001 ÷ 1000 = 0.065001

Find the quotient. 1

15 ÷ 10

346.4 ÷ 100

12.3 ÷ 1000

15 ÷ 10 = 1.5 1 5 . 0

Remember! Adding a 0 before any number does not change its value.

Dividing Decimals by Whole Numbers Divide 0.6 by 3. 6 tenths are put in 3 equal parts which is equal to 2 tenths in each part. 0.6 � 3 = 0.2 Steps

Step 1: Place the

decimal point directly above the decimal point in the dividend.

Step 2: Divide the Step 3: Divide the ones.

tenths.

Step 4: Divide the hundredths.

When the dividend is less than the divisor (1.45 ÷ 5) . 5 1.45

0. 5 1.45

0.2 5 1.45 – 10 45

0.29 5 1.45 – 10 45 – 45 0

When the dividend is more than the divisor (6.48 ÷ 4) 1. 4 6.48

1. 4 6.48 – 4 24

1.6 4 6.48 – 4 24 – 24 08

1.62 4 6.48 – 4 24 – 24 08 – 8 0

142

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When there is a remainder while dividing (4.5 ÷ 2) . 2 4.5

Example 12

2. 2 4.5 – 4 05

Divide 3.44 by 2.

2.2 2 4.5 – 4 05 – 4 1

Example 13

1.72 2 3.44 – 2 14 – 14 04 – 4 0

Add an extra zero at the hundredths place to complete the division. Adding a zero to the dividend on the right side does not change the value of the number. 2.25 2 4.50 – 4 05 – 4 10 – 10 0

Find the quotient. 5.62 ÷ 4 1.405 4 5.620 – 4 16 – 16 02 – 00 20 – 20 0

Thus, 3.44 ÷ 2 = 1.72

Thus, 5.62 ÷ 4 = 1.405 Do It Together

Divide 5.25 by 5.

. 5 5.25

Thus, 5.25 ÷ 5 = _______

Do It Yourself 8E 1

Divide the given decimal numbers visually. a 0.8 ÷ 4

Chapter 8 • Operations with Decimals

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143

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b 1.6 ÷ 8

Fill in the blanks. a 15.6 ÷ 10 = _________

d 513.47 ÷ 100 = _________ g 23.1 ÷ 1000 = _________

b 154.64 ÷ 10 = _________ e 1.2 ÷ 100 = _________ h 2 ÷ 10 = _________

c 51.23 ÷ 100 = _________

32.561 ÷ 1000 = _________

11.236 ÷ 1000 = _________

Find the quotient. a 120.6 ÷ 6

b 100.43 ÷ 11

4 Divide until the remainder is zero. a 252.3 ÷ 6

b 105.1 ÷ 5

c 272.22 ÷ 6

d 20.45 ÷ 5

e 37.905 ÷ 7

292.05 ÷ 9

c 2.67 ÷ 5

d 18.9 ÷ 4

e 82.56 ÷ 5

236.85 ÷ 6

5 Compare using >, < or =. a 13.25 ÷ 5

2.65

b 33.04 ÷ 4

8.24

c 202.08 ÷ 8

25.28

d 150.1 ÷ 5

30.02

e 332.16 ÷ 8

41.22

378.18 ÷ 9

42.04

6 What number should 154.265 be divided by so that the resulting answer is 1.54265?

Word Problems 1

The total weight of 9 cartons is 37.8 kg. Find the weight of one carton.

There are 70.728 kg of rice in a basket. The rice is put into 12 smaller baskets. What

How many pounds should Hari take with him to England, if the total expenses for

Six 1 litre bottles of oil cost $37.92. What is the cost of a litre of oil?

is the weight of the rice in each basket?

the trip add up to ₹1,04,000? (Given: 1£ = ₹104 approximately)

Points to Remember • To add or subtract decimal numbers, always remember to convert into like decimals first and align the decimal points one below the other.

• On multiplying a decimal number by 10, 100 or 1000, the decimal point moves to the right by 1 place, 2 places or 3 places, respectively. • When multiplying a decimal with a whole number or a decimal number, remove the decimal point and multiply the whole numbers. After the multiplication, place the decimal point at the correct place.

• On dividing a decimal number by 10, 100 or 1000, the decimal point moves to the left by 1 place, 2 places or 3 places, respectively. 144

Maths Grade 5 Book_Chapter 7-12.indb 144

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Math Lab Aim: To add, subtract, multiply and divide decimals. Setting: In pairs Materials Required: Pack of cards (20 cards) and an empty box. Method:

he teacher writes the decimal numbers and whole numbers like 10, 100, 1000 on 1 T different cards.

2 Mix all the cards and give 5 cards to each student. Put the rest of the cards in the empty box. 3 The first student shows one card and the second student shows 1 card. a b

I f both cards are decimal cards, then the first student will add the decimals and the second will subtract.

I f one card is a decimal and the other a whole number, the first student will multiply and the second will divide.

4 Now, it’s the turn of the second student to ask for a card.

5 Once all the cards have been used, the students can pick up more cards from the box. The student whose answer is correct the greatest number of times wins the game.

Chapter Checkup 1

Add or subtract the numbers visually. a

0.23 + 0.47

1.4 – 0.62

2 Add or subtract the numbers using the column method. a d

64.51 – 12.718

534.115 – 15.36

18.963 + 14.517

649.41 – 534.489

6 × 0.2

5 × 0.6

Fill in the blanks. a d

1.14 + 45.2

Find the product of the numbers visually. a

45.61 + 17.513

45.12 × _______ = 451.2

2.002 × _______ = 20.02

184.351 × _______ = 184351 81.36 × _______ = 81360

c f

2.517 × _______ = 251.7 1.001 × _______ = 1001

Find the product of the numbers. a

12 × 1.54

18 × 3.251

Chapter 8 • Operations with Decimals

Maths Grade 5 Book_Chapter 7-12.indb 145

9 × 32.14

1.2 × 21.36

41.5 × 12.45

120.2 × 12.06 145

18-12-2023 11:15:44

Fill in the blanks. a d

8143.12 ÷ ______ = 8.14312

14.2 ÷ 1000 = _______

123.321 ÷ ______ = 1.23321

12.145 + 18.415 + 2.51

41.54 + 56.81 – 12.379

15.47 + 81.415 – 41.555

31.23 + 17.28 – 11.111

15.145 + 81.1 – 45.64

71.64 + 16.87 – 63.998

110.4 ÷ 6

98.80 ÷ 8

270.18 ÷ 9

139.17 ÷ 3

165.55 ÷ 7

133.562 ÷ 11

Match the following. a

1.51 + 23.72

24.85

27.63 – 2.78

22.8852

4 × 6.17

24.68

3.26 × 7.02

25.23

Divide to find the quotient. a

1.21 ÷ _______ = 0.00121

531.14 ÷ _______ = 5.3114

Divide to find the quotient. a

Simplify. a

47.01 ÷ 10 = _______

56.8 ÷ 5

22.49 ÷ 2

189.2 ÷ 8

313.5 ÷ 6

356.1 ÷ 5

535.47 ÷ 6

Read the exchange rate chart of different countries given in the section above. Answer the questions. a c e

What is 6000 won in Indian rupees?

Convert 850 rand into Indian rupees.

What is 50,000 NPR in Indian rupees?

b d

What is €150 in ₹?

What is 100.5 dirham in ₹?

Word Problems 1 2

J ason bought an item for ₹45.8. He gave the shopkeeper ₹50. How much change will he get back from the shopkeeper? Gorillas sleep for an average of 84.7 hours a week. How long do they sleep in a day?

im has to travel a distance of 8.75 km. He has already covered a distance of 4.79 km. T How many more kilometres does he have to travel to reach his destination? 4 Raj wants to buy a DVD player for $47.85, a DVD holder for $21.36 and a stereo for $22.01. If he has $90, how much more money does he require? 3

eorge’s friend comes from the USA. He exchanges $168 for Indian currency. If G he has to give $1 for every $20 as a tax for the exchange, what amount of Indian currency does he have? ($1 = ₹89.12)

car runs for 4 hours every day and consumes 6.5 L of oil each hour. If the total oil A consumption is 133.9 L, for how many days does the car run in total?

J ohn’s weight is 0.89 times that of Harry’s and Harry’s weight is 0.56 times that of Jacob’s. If Jacob weighs 51.25 kg, what is John’s weight?

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Lines and Angles

Let's Recall Lines can be seen everywhere! Lines can be sleeping, standing or slanting.

Sleeping line

Standing Line

Slanting Line

For example, We see a slanting line in the pen. We see a standing and sleeping line in the book.

Letʼs Warm-up

Look at the figures. Write the type of line (Sleeping, Standing, Slanting) these things show.

_________________

I scored _________ out of 5.

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Lines and Line Segments Real Life Connect

Vandana is sitting on the window seat of the train while travelling to New Delhi with her father. Vandana: Look, Papa! There are so many railway tracks. Some are straight, and some are crossing each other. Papa: Yes, Vandana! These tracks help different trains to reach different cities.

Types of Lines Lines can be seen everywhere, whether on railway tracks, roads, or airport runways. Lines provide direction and control. Let us learn more about lines.

Point A point is a fixed location that cannot be moved. It can be shown using a dot. It has no length, breadth or thickness.

Dot New Delhi

A p X

Example 1

Which of these are points?

L S

L, M and N are points. Example 2

Mark 4 points inside the given circle and name them A, B, C and D. 4 points can be marked and named inside the circle as A

B D C 148

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Do It Together

Count the number of points inside and outside the given square. Also, write their names. E

F C

Number of points inside the square = 2

Points inside the square = ____, G

Number of points outside the square = ____

oints outside the square = ____, P ________________________

Line Segment A line segment is the shortest distance between two points. It has a definite length. The edges of a book can be shown as line segments. We can measure the length of a line segment with the help of a ruler.

Shortest Distance

Line segment

A 0

B 1

L 11

The line segment is named by its end points. The above line segment can be named as AB or BA. The length of the line segment can be given as 6 cm. Example 3

Which of the following represents a line segment?

Sky

Window bars

A dot on the paper

The window bars represent line segments. Do It Together

Below are some line segments. Name the line segments and write their measurements. M 0

N 1

P 11

Q 1

Name of the line segment = ___ or ___

Name of the line segment = PQ or QP

Length = 10 cm

Length = _______ cm

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Line A line is a collection of points extended endlessly in both the directions along a straight path. For example, the railway tracks seem to be a line. A line has no beginning and no end. It has no end points. A line can be represented by a two-headed arrow showing that it can be extended in both the directions. Q

A line is named by using two points on the line. The above line can be named as QP or PQ . Lines are further divided into different types. Pair of straight lines that never meet.

Two lines that cross each other or will cross each other on extension.

ABCD

Intersecting Lines C

A D

A vertical line standing exactly above a horizontal line.

AB and CD are parallel.

Parallel Lines

Perpendicular Lines

B Example 4

Arrow heads at both ends.

AB and CD are a set of intersecting lines. Point X is the point of intersection.

AB and CD are perpendicular lines.

The symbol of a perpendicular line is ⊥.

Which of the given figures shows parallel lines?

A plus sign

A pair of scissors

A ladder

The ladder shows parallel lines. 150

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Do It Together

Write the names of the given lines. P

Intersecting lines

Y X

_______________

Ray A ray is a part of a line which extends indefinitely in one direction. It has only one end point. The beam of light coming out of a torch is an example of ray. It is represented by a single-headed arrow ( ). We name a ray beginning with its starting point and any other point on it. A

AB is a ray with A as the starting point and B as a point on the ray. Example 5

Which of the given figures shows a ray?

A needle

Headlight of a car

Tip of a pencil

The traffic light represents a ray. Do It Together

Tick () the figure that represents a ray and cross () the rest.

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Do It Yourself 9A 1

Match the given figures with their names. a

Line

Intersecting lines

Parallel lines

Ray

Write if true or false. a A line segment AB can be denoted as AB.

____________________

b A ray CD can be denoted as CD.

____________________

c A line PQ can be denoted as PQ.

____________________

d Two parallel lines are denoted by the symbol ||.

____________________

Select the word that best describes the given figures. Line segment

A bridge 4

Curve

Ray

Point

Tip of a nail

Sunlight

b A line segment MN

c A line PQ

d A ray ST

See the figure and answer the given questions.

Read and draw. a A point M

Sides of a railway track

Parallel lines

e Two intersecting lines UV and WX with O as the point of intersection

a Name all the points and lines in the figure.

b Write the name of a line segment.

c How many rays are there in the figure? Write their names.

C D

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Word Problem 1

Disha went to a museum. She saw a pair of swords hanging on the wall. What kind of lines does the pair of swords show?

Understanding Angles Real Life Connect

Soumya and her friend Rohan are learning to read a wall clock. Soumya: Look at the clock, Rohan. The hands of the clock are making an L shape. Rohan: Yes! It’s 3 o’clock. Soumya: You are right! Rohan: I also saw a V shape made by the hands of the clock this morning. Both the friends talked about the different shapes that they observed on the clock.

Types of Angles We also see different shapes made by the hands of the clock. How can the same hands make different shapes? This happens with the help of different angles. Two rays with a common starting point form an angle.

Did You Know? Angles can be seen everywhere! Every object around you is made by placing its parts at different angles.

Shown below are two rays, OA and OB with a common starting point O. O is the vertex of the angle.

OA and OB are the arms of the angle

An arc is drawn to show an angle. O

An angle is named using the symbol ∠. The angle can be named as ∠AOB or ∠BOA or ∠O.

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Right Angle Look at the angles that the edges of the window or the sides of a blackboard make.

Angles that look like an L shape are called right angles. Let us look at some right angles.

Example 6

Which of the given figures represents a right angle? 1

As a right angle makes an L shape, figure 2 represents a right angle. Do It Together

Draw more right angles on the clocks.

Draw the other arm of the angle to make right angles on points Q and P. P

Straight Angle The angle formed on a straight line is called a straight angle.

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Example 7

Which of the given figures represents a straight angle? 1

As a straight angle is an angle formed on a straight line, figure 3 represents a straight angle. Do It Together

Draw a straight angle on the given clock. One is made for you.

raw 2 points: G and H. Draw straight angles on each point. One should be D vertical and one should be horizontal.

H G

Acute Angle Angles that measure less than a right angle are called acute angles. Given below are some acute angles.

Example 8

Which of the given figures represents an acute angle? 1

Acute angles are angles that measure less than a right angle. Figure 3 shows an acute angle. Do It Together

Draw an acute angle on the given clock. One is made for you.

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Obtuse Angle Angles that are more than a right angle are called obtuse angles. Example 9

Which of the given figures represents an obtuse angle? 1

Obtuse angles are more than right angles. Figure 1 represents an obtuse angle. Do It Together

Draw an obtuse angle on the given clock. One is made for you.

Draw the other arm of the angle to make obtuse angles on points X and Y. Y

Do It Yourself 9B 1

Label the arms and vertex of the given angle.

Name the angles of the given figures. P

Q 3

Y M

Look at the figure with ∠1, ∠2, ∠3 and ∠4 marked.

Which of the angles in the figure are obtuse angles?

N 1 3

2 4

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Write the type of angle made by the hour and minute hands of the given clocks. Write acute, obtuse,

Label the given figure. Mark and write the number of right angles in the

straight or right angles.

given figure.

Word Problem 1

Rishi’s school starts at 7:30 a.m. and ends at 1:30 p.m. What types of angles do the hands of the clock make at the school’s start and end time?

Measuring and Drawing Angles We have learnt about angles and their types. Let us now learn how to measure and draw angles.

Measuring Angles Angles are measured in degrees. A degree is denoted by °. A complete turn around a point is divided into 360 parts. Each part is denoted by 1°. Angles can be measured with the help of a protractor. A protractor is semicircular in shape; and

Each line on the protractor measures 1°.

therefore, it can measure angles up to 180°.

A protractor has two sets of measurements called scales. The scale inside is the inner scale and the scale outside is the outer scale.

This is the centre of the protractor. This is the baseline.

Let us learn to use a protractor to measure angles. 1

Place the centre point of the protractor on the vertex of the angle.

djust the protractor (without shifting the centre from the vertex) so that one arm of A the angle is along the baseline.

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Example 10

ook at the scale where the baseline arm points to 0° L (inner scale in this case).

ead the measurement of this angle where the other R arm crosses the scale.

The measure of ∠ AOB = 50°

Measure ∠ LMN using a protractor. L

Do It Together

M N N The measure of ∠ LMN = 60°

Read the protractor and write the correct measure for the given angles. A

∠ABC = ___________

Error Alert! Always look at the scale where the baseline arm points to 0°. A

∠ AOB = 130°

Remember! The length of the arms of an angle does not affect the measure of the angle.

∠ AOB = 50°

Drawing Angles We can draw an angle of any measurement using the inner or the outer scale of a protractor by following some steps. Let us draw an 80° angle using both inner and outer scales. 158

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Using the inner scale of the protractor Using the outer scale of the protractor Step 1

Step 1

Draw a ray of any name such that it points to

Draw a ray of any name such that it points to 0° of the

0° of the inner scale. Here we are drawing a ray with the name OC. O

outer scale. Here, we are drawing a ray with the name OC.

Step 2

Place the centre of the protractor on the vertex

Place the centre of the protractor on the vertex O and

O and 0° on the arm OC. Mark a point A at 80°.

0° on the arm OC. Mark a point A at 80°. A

Step 3

Remove the protractor and draw a ray from O to Remove the protractor and draw a ray from O to A A using a ruler. Thus, ∠AOC = 80°. A

using a ruler. Thus, ∠AOC = 80°.

80°

80° C

Draw a 105° angle using the inner scale of a protractor.

Example 11

Do It Together

105°

Draw the given angles using the protractor. 1

30° using the outer scale Chapter 9 • Lines and Angles

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95° using the outer scale

45° using the inner scale

Do It Yourself 9C 1

Name the following angles in two different ways. a

A L

Measure each angle using a protractor. a

P N

O 3

c ∠ DOA

e ∠ DOB

b ∠ AOE

d ∠ COB f

F A

∠ FOB

D C B

Draw the given angles using the inner scale of a protractor. a 50°

Use a protractor to measure the angles. a ∠ AOF

b 100°

c 85°

d 120°

e 145°

160°

Draw a line segment PQ equal to 6 cm. At P, draw ∠QPR = 95° using the outer scale.

Word Problem 1

Priya is going to her friend’s house. She goes straight from her house, and then takes a right turn. What is the angle made by her at the turning point?

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2-D Shapes Real Life Connect

Kavya, Richa and Rishi are playing a game. Richa is giving hints and Kavya and Rishi are making shapes. Richa: It is a closed figure. It has 5 sides. The shape has 5 angles. Kavya and Rishi made the given shapes. 1

Kavya’s shape

Rishi’s shape

All of them wondered how the same hints led to different shapes!

Triangles, Quadrilaterals and Polygons Kavya and Rishi made different shapes using the same hints. Do you know why? This is because they placed the sides of the shape at different angles. Let us learn more about shapes having different numbers of sides and angles. Triangles

A triangle has 3 sides and 3 angles.

The sides can be named as AB, BC and CA.

The symbol ∆ is used to show the triangle: ∆ABC.

C Quadrilaterals

A quadrilateral has 4 sides and 4 angles. A quadrilateral is named

using the names of its vertices.

Let us see some quadrilaterals. All the quadrilaterals have 4 sides, 4 angles and 4 vertices.

Square

Rectangle

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Rhombus

The name of the given quadrilateral can be ABCD, BCDA, CDAB and so on. The name can begin with any vertex but it needs to move in either a clockwise or anticlockwise manner.

Think and Tell

Can we make more shapes with 4 sides and 4 angles?

Parallelogram

Trapezium

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Polygons: A polygon is a closed figure that has 3 or more straight sides.

Think and Tell

Polygons can be categorised on the basis of the number of sides and angles.

polygons too?

Are triangles and quadrilaterals,

Given below are some polygons along with their names, number of sides and number of angles.

5 Sides

Example 12

6 Sides

7 Sides

9 Sides

8 Sides

5 Angles

6 Angles

7 Angles

8 Angles

9 Angles

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Which of the following is not a polygon? 1

We know that polygons are closed figures with three or more straight sides, hence figure 3 is not a polygon. Do It Together

Write the number of sides and angles for each of the shapes.

Number of sides - 8

Number of sides - ____

Number of sides - 4

Number of angles - ____

Number of angles - 4

Number of angles -____

Do It Yourself 9D 1

Name the sides of the given triangle.

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Write the names of each of the polygons. a

Match the shapes with the number of angles they have. a

three

five

nine

Which of the following is not a quadrilateral? a

seven

Write if true or false. a A polygon is a closed figure that has two or more straight sides. b An octagon has eight angles. c A pentagon has five sides. d A rhombus has five sides.

Tick () the polygons and cross () the non-polygons. a

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Word Problem 1

Answer the riddle: I am a closed shape. I have twice the number of sides and angles

than a triangle has. Who am I?

Points to Remember •

A point is a fixed location that cannot be moved. It is represented by a dot.

•

A line segment is the shortest distance between two points. It has a definite length.

• A line is a collection of points extended endlessly in both directions along a straight path. •

A ray is a part of a line which extends indefinitely in one direction.

•

Two rays with a common starting point form an angle.

• A polygon is a closed figure that has 3 or more straight sides. Each polygon has as many angles as there are sides in it.

Math Lab Let’s Find Out the Angles! Setting: In groups of 4 Materials Required: Paper cutouts of polygons, protractor, pen and paper. Method: Distribute the polygons among the groups. The teams measure the angles made by different sides of the polygons. The teams record the observations on their paper, along with the names of the polygons. The team that records the angles of all the polygons first, wins!

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Chapter Checkup 1

Which word best describes the figure? a

Q 2

Look at the figure and write if true or false. a

AC is a line segment.

PQ and RS are sets of parallel lines.

CS is a line.

U P

Write the names of the angles made by the given figures.

Look at the given figure and name the angles as right, straight, acute or obtuse. a b c d e

∠ AOC

∠ AOD ∠ AOF

∠ BOF

∠ BOC

B A

Write the measure. a

Measure the given angles using a protractor. a

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Draw the given angles using the outer scale of a protractor. a

65°

75°

140°

155°

180°

Write the names of the polygons. Also, give the number of sides and angles they have. a

115°

Tick () the polygons that have one or more right angles. a

10 Draw a line segment MN equal to 10 cm. At N, draw ∠MNO = 125° using the inner scale.

Word Problems 1

Rohan is going to the park with his family. On the way, they stop at a zebra crossing. What kind of lines does the zebra crossing show?

ujata made a shape that had half the number of S sides and angles that a decagon has. What shape has been made by Sujata?

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Patterns and Symmetry

Let's Recall We can see patterns all around us.

1212121 In Nature

In Fabric

In Shapes

In Numbers

Let us now see some patterns and try to expand them!

We can see that the pattern is in the order of circle, triangle, circle, triangle. So, our unit will be a circle, then a triangle and so on.

Now, look at this pattern. Similarly, here we can extend the pattern as:

Letʼs Warm-up

Draw 2 more shapes to extend the given patterns. 1 2 3 4

I scored _________ out of 4.

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Patterns Around Us Real Life Connect

The teacher took Rani and her classmates to visit an art museum. Teacher: Do you notice the patterns on the paintings on the walls? Rani: What is a pattern? Teacher: A pattern is a sequence of repeating objects, shapes or numbers. Rani: So, does that mean these circles and semicircles make a pattern? Teacher: Yes, Rani! Patterns exist all around us.

Extending and Creating Patterns Rani notices that the frames of the paintings are placed in the form of a pattern. They are of so many shapes, colours and sizes.

Looking at the wall she wonders, if there had to be more frames, what would the next unit in the pattern be? Let us learn how we can extend or continue a pattern.

Repeating Patterns When the same sequence keeps repeating one after the other, we call it a repeating pattern. Let us learn how to extend a pattern made with shapes.

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Step 1 First, we have to find the shapes that are repeating in the pattern to help us find the rule.

Step 2

Unit of Repeat

The shapes that are repeating are triangle, circle, circle and

rectangle. Together these are called the unit of repeat or element.

Step 3

Find the rule in the given pattern. The rule here is ABBCABBC.

Therefore, if we need to add more shapes, the next shape would be: triangle, circle, circle, rectangle, triangle, circle, circle and so on. Example 1

Colour to extend the repeating pattern.

Identify the unit of repeat in the pattern.

Unit of Repeat

The colours are in the following order: Yellow, Red and Blue. So, the rule is ABC. On extending, our pattern will be:

A Example 2

Create a pattern using the given figures with the rule – ABCCBABCCB.

The pattern can be:

Think and Tell

Can you think of some examples of a repeating

Do It Together

Colour to extend the repeating patterns.

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pattern that you see every day?

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Rotating Patterns Look at the picture of the triangles below. Sometimes the shapes that are repeating are turned around at every step.

Observe the dot on each triangle. Patterns where a shape changes its direction with each step are called rotating patterns. 1 When the triangle makes a quarter or turn, it rotates at 90° or a right 4 angle.

90° 180°

1 When the triangle makes a half or turn, it rotates at 180° or two right 2 angles.

3 When the triangle makes three quarters or turns, it rotates at 270° or 4 three right angles.

270° 360°

When the triangle makes a complete turn, it rotates at 360° or four right angles.

A unit in a rotating pattern can turn either clockwise or anti-clockwise.

Clockwise Direction

Example 3

What will be next?

Anti-clockwise Direction

_______ _______

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The required pattern will be:

Example 4

_______ _______ Create a pattern by turning the given figure 60° at every step.

Error Alert! We can see that the rule of this pattern is turning 60° at every next step.

Always check if a rotating pattern is clockwise or anti-clockwise.

The required pattern will be:

Do It Together

The given shapes are rotating clockwise. Fill in the missing shapes. Shape

Turn 90°

Turn 180°

Turn 270°

Turn 360°

Do It Yourself 10A 1 Which of these do not show a pattern? a

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Observe and extend the pattern. _____

_____

Identify the rule for the given rotating pattern. Also draw the next three units of it.

____________ 4

_____

____________

Identify and write the rule for the rotating patterns. Is there any pattern which is not following any rule? If yes, give your answer with a reason. a

Use the given shapes to draw rotating patterns. The pattern should rotate clockwise at right angles.

Word Problem 1

Rakesh has made a rotating pattern where each shape rotates at half turns. Which of the given patterns has been made by Rakesh? a

Growing and Tiling Patterns Rani notices that the art gallery is filled with a lot of different patterns. She comes across some more paintings and wonders what kind of patterns they make. Let us have a look at some paintings Rani has come across.

Growing and Reducing Patterns

Here, we have triangles forming a pattern.

On the other hand, in this painting, the triangles are forming a tree-like pattern.

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We see that both the purple and the black triangles keep increasing by one unit.

A triangle is reduced at every next step until there is only one triangle left.

The increasing unit is:

The decreasing unit is: This is a reducing pattern.

This is a growing pattern. Example 5

Extend the pattern by choosing the correct toy train that will come next. ___________________________________ 1

We can see that at every next step of the repeating sequence, one block is added. So, this is a growing pattern. Our repeating unit is: Do It Together

. So, the correct option is 3.

Identify and write whether the following are growing or reducing patterns. Pattern

What will come next?

Type of pattern

Tiling Patterns Now, Rani is well aware of what patterns are. As she is walking through the halls of the art gallery, she notices a pattern on the floor. Rani: Wow! Do these tiles on the floor make a pattern too? Teacher: Yes, Rani. This type of pattern is called tiling. What is tiling? The pattern formed by the repetition of a single unit or shape over and over again without leaving any gap is called a tiling pattern. When this pattern is repeated, it fills the area with no gaps and no overlaps. Chapter 10 • Patterns and Symmetry

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Did You Know? The structure of a beehive is also an example of a tiling pattern.

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This is also called tessellation. Types of tessellation Rectangular tessellation

Triangular tessellation

Square tessellation

The arrangements of bricks in a wall.

Triangles arranged in a tiling pattern.

Pink and white squares arranged in tiling.

Look at the given images. Are these tiling patterns?

Think and Tell

Does a tangram also have a tiling pattern?

No. Even though the units are of the same size, these are not tiling patterns because there are gaps between the units. Example 6

Complete the tiling pattern.

Do It Together

Draw and colour to complete the pattern. A small part of it is done for you.

Do It Yourself 10B 1

What would be the next unit in the pattern and why?

_____________ 174

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Draw the pictures that come next in each growing or reducing pattern. a

Which of these is not a tiling pattern? Give your answer with a reason. a

Colour to complete the tile.

Which tile does not belong in the tiling pattern? a

Word Problem 1

Ankit bought these tiles for his bathroom. Which of these shows a tiling pattern made by the tiles bought by Ankit? a

Number Patterns After returning from the art gallery, the teacher tells the students that patterns exist not only on the walls, frames and the floor as they saw but they exist everywhere, even in letters and numbers.

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Now look at some patterns. Pattern 1: The numbers are the same, although the order of the numbers is reversed. 25 + 80 +10

Pattern 2: One more number is getting added each time.

10 + 80 + 25

37 +8

Some numbers are exactly the same even when we read them backwards. These are called palindrome numbers. One example is 363. Let us see how to get special numbers or palindromes. Take a number, say

Now interchange the digits

Add the two numbers together,

28 + 82 = 110

Is this a special number? No! Let’s carry on. Write the digits of 110 back to front

011

Add the numbers 110 and 011

121

121 is a special number or a palindrome.

Coding and Decoding Patterns Different methods are used for coding and decoding of passwords Letters and numbers are used in different ways for coding or decoding. Here are the letters of the English alphabet in their order. Let the corresponding letters and numbers represent each other. A

We use a particular code pattern to express a word in English. We can write a word, say CAB, in the code language. We can also decode the code for the word ?Imagine. C

Let us now try decode a new code. But how can we do that? 1I2 3L4I5K6E 7M8A9T10H11S. 176

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First remove all the numbers from the given code. 1I2 3L4I5K6E 7M8A9T10H11S. 1I2 3L4I5K6E 7M8A9T10H11S. So, the code says, I LIKE MATHS. Interesting, right? Now, let’s solve some more examples. Example 7

Decode the given code. 1P2A3T4T5E6R7N8S 9A10N11D 12S13Y14M15M16E17T18R19Y. 1P2A3T4T5E6R7N8S 9A10N11D 12S13Y14M15M16E17T18R19Y. Removing all these numbers we get: PATTERNS

A N D

SYMMETRY

Therefore, the code is patterns and symmetry. Do It Together

In a certain language, the letters are coded as follows: A

Decode the word: CNGGREA.

We can see that: Therefore, the given word is __________________.

Patterns in Numbers We have learnt about many types of patterns. We can apply the same pattern rules to numbers as well. Look at the sequence: 1, 1, 2, 4, 7, 11, 16, 22, 29. If we observe the numbers, the rule being followed is +0, +1, +2, +3, ... and so on, hence making a pattern.

Did You Know?

In a Fibonacci series, a number is the sum of the two preceding terms. 0, 1, 1, 2, 3, 5, 8, 13, 21…

We can represent some numbers in the form of an equilateral triangle arranged in a sequence. These numbers are called triangular numbers.

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Numbers like 1, 3, 6, 10 and 15 are some examples.

1 dot

3 dots

6 dots +3

10 dots +4

Triangular Numbers

15 dots +5

When we multiply a number by itself, we call it a square number. Some square numbers are 1, 4, 9 and 16. We can also represent these numbers in the shape of a square.

1×1=1

2×2=4

3×3=9

4 × 4 = 16

Magic Triangle

Look at the picture. This is called a magic triangle. Here, we have written numbers from 1 to 6 in such a way that the sum of each side is 9. 1+5+3=9

3+4+2=9

5 3

1+6+2=9

Magic Squares In a magic square, the sum of the numbers in each row, column and diagonal is always the same. Rows:

Columns:

Diagonals:

2 + 6 + 10 = 18

4 + 6 + 8 = 18

5 + 6 + 7 = 18

9 + 4 + 5 = 18

7 + 8 + 3 = 18

9 + 2 + 7 = 18

5 + 10 + 3 = 18

Here, the sum of each row, column and diagonal is 18. Similarly, we can make magic squares of different numbers as well. This is Pascal’s Triangle. If we observe properly, it begins with 1 on the top and with 1s running down the two sides of the triangle. Each number lies between and below them, and its value is the sum of the two numbers above it.

6 2

4 9

10 3

9 + 6 + 3 = 18

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This triangle has many amazing patterns.

Think and Tell

Check the horizontal sum of each row.

Can you see more patterns in

They are: 1, 2, 4, 8, 16, 32.

Pascal’s triangle?

We can see that they are also in a pattern. Every number is double the previous number.

2 numbers

Patterns with Odd Numbers

1+3=2×2

When we add a certain number of odd numbers, their sum is equal to the product of that certain number with itself.

1+3+5=3×3 1+3+5+7=4×4 1+3+5+7+9=5×5

We can go on for as many numbers as required. Let us see some more patterns. Let us take a number, say 10. Multiply the number with 1, 2, 3, 4,… at every next step. Also, add 2 at each step.

Example 8

10 ⨯ 1 10 ⨯ 2 10 ⨯ 3 10 ⨯ 4 10 ⨯ 5 10 ⨯ 6 10 ⨯ 7

+ + + + + + +

2 2 2 2 2 2 2

= = = = = = =

12 22 32 42 52 62 72

1 + 3 + 5 + 7 + 9 + 11 = 6 × 6 1 + 3 + 5 + 7 + 9 + 11 + 13 = 7 × 7

Can we see a pattern? Yes. The numbers 12, 22, 32, 42, 52, 62, 72 are in a pattern. The terms increase by 10 at every step.

Observe the pattern and write the next two lines of number patterns. 25 + 11 = 36 36 + 13 = 49 49 + 15 = 64 We can see that the pattern of the given terms is: 5 × 5 + 11 + 0 = 36; 6 × 6 + 11 + 2 = 49; 7 × 7 + 11 + 4 = 64

Example 9

Without actual addition, find the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 is the sum of the first 8 odd numbers. The sum of a certain number of odd numbers is equal to the product of that number with itself. So, 8 × 8 = 64. Therefore, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

So, the next two terms will be: 8 × 8 + 11 + 6 = 81 9 × 9 + 11 + 8 = 100 Chapter 10 • Patterns and Symmetry

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Do It Together

The stars are forming squares—4 stars, 9 stars, 16 stars and so on. Draw stars and complete the pattern.

1 + 3 = ____

3 + 6 = ____

6 + 10 = ____

____ + ____ = ____

Do It Yourself 10C 1

Look at the numbers and write the rule describing each pattern. a 15, 19, 23, 27, 31, 35

b 6, 8, 10, 12, 14, 16, 18

d 1, 6, 11, 16, 21, 26

e 1, 1, 2, 3, 5, 8, 13

Use the given code to find the numbers. 1

a HUEHU

c 1, 4, 9, 16, 25, 36, 49

b IOQCB

c CEHBU

d WYBQO

e HUYIOQ

Shreya helps her mother water her plants every weekend. One weekend there were 1 red and 1 yellow flowers. The next weekend there were 2 red and 2 yellow flowers. The next weekend, there were 3 red and 3 yellow flowers. How many flowers will be there after the fifth weekend?

A pattern starts with 60 and decreases first by 1, then by 2, then 3 and so on. Write the numbers up to

Create a magic triangle with a sum of each side that is equal to 12.

the 10th term.

Word Problems 1

A bakery is receiving a lot of orders. They got 12 orders in January, 24 in February and

A swimming pool has small chlorine tablets added every day to keep it clean. On

36 in March. If the same pattern continues, how many orders will they get in July?

Monday, 1 tablet is added. On Tuesday, 3 tablets are added. On Wednesday, 7 tablets

are added. On Thursday, 15 tablets are added. How many tablets are added on Friday?

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Symmetry and Reflection Real Life Connect

Kush was at a pottery shop. He noticed there were a lot of different pots. He looked at a pot and realised no matter from which angle he looked at the pot, it looked exactly the same. How is that possible? Let’s find out!

Symmetry We can divide or fold a shape, letter or a number into two identical halves along a straight line. This is called symmetry. This straight line that divides a shape or figure into two halves that are identical and match perfectly when they overlap is called the line of symmetry. We can say that one of these two parts is the mirror image or the reflection of the other. These two halves are said to be symmetrical. Let us see some types of symmetry. Vertical Symmetry

A line of symmetry that runs down vertically in a shape or figure.

Horizontal Symmetry

Diagonal Symmetry

A line of symmetry that runs across horizontally in a shape or figure.

A line of symmetry that runs across diagonally in a shape or figure.

Drawing Lines of Symmetry Let us see symmetry and lines of symmetry in some shapes. Have a look at these shapes:

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How many lines of symmetry can these shapes have?

Rectangle - 2

Equilateral Triangle - 3

Square - 4

Circle - Infinite

Notice how these shapes, when divided, give 2 identically equal parts. Now, if you observe properly, many letters have symmetry too.

1 line of symmetry

2 lines of symmetry

B A C M D E T U W K H I O X

Error Alert! Every figure need not always be symmetrical when divided into 2 parts.

Meanwhile, letters like F, G, J and L have no line of symmetry. Now, observe these pictures. Are the dotted lines drawn lines of symmetry for this tree?

The two halves obtained in both the pictures, when compared, look very different. Therefore, there is no symmetry. This is called asymmetry.

Think and Tell

Can you list some symmetrical and asymmetrical objects around you?

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Example 10

Draw all possible lines of symmetry.

In a square, we have 4 lines of symmetry.

Example 11

Draw the other half of the image.

The complete image is: Do It Together

Draw the other identical halves of the given figures.

Do It Yourself 10D 1

Match the figures with their lines of symmetry. a

Asymmetrical

1 horizontal line of symmetry c

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Many lines of symmetry

2 lines of symmetry

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2 Draw the lines of symmetry for the given figures. a

3 Draw the lines of symmetry (if any) for the given letters. a

4 Write true or false.

a When two halves look identical, they are called asymmetrical.

b The letters F, G, J and L have two lines of symmetry. c The number 8 has 2 lines of symmetry.

d The number 3 has a horizontal line of symmetry.

5 Complete the missing half of each of the shapes.

Word Problem 1

Ayan is an architect who creates layouts of houses. He was constructing 2 flats facing each other that are identical to

each other. He lost the layout of one of the flats. Can you help Ayan by colouring the grid to show the flat?

Points to Remember • When the same sequence repeats over and over again, it forms a repeating pattern. • The unit of repeat is that part of the pattern that gets repeated each time. When the figure repeatedly rotates, it is called a rotating pattern. • A pattern where there’s an increase or decrease by a certain unit, is called a growing • or reducing pattern, respectively. The pattern formed by the repetition of a single unit or shape over and over again • without any gaps is called tiling.

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• When a figure is divided into two identical halves along a straight line, we say symmetry exists. This straight line is said to be the line of symmetry. •

Figures or shapes that have no lines of symmetry are said to be asymmetrical.

Math Lab Materials Required: A piece of thread, paint and a sheet of paper. Method:

1 Dip the thread in the paint. 2 Take the thread out of the paint and put it on the sheet of paper.

3 Fold the sheet of paper in half with the thread in between.

4 Press and hold the sheet with one hand. 5 Remove the thread slowly with the other hand. 6 You will obtain a symmetrical pattern.

Chapter Checkup 1

Complete the tiling patterns by drawing 4 more shapes. a

2 Tick () the figures that are symmetrical. a

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What will come next in the given patterns? Draw two figures for each. a

Generate number patterns with 5 terms following the given rules. a c

Start with 5 and add 11.

Start with 1 and add the square of itself.

Start with 10, multiply with 2 and add 1.

Start with 15 and add 3 and 5 alternately.

Examine the number pattern below. 331, 316, 301, 286, 271, ……

Write the next three numbers in the pattern. Also, write the rule followed. Fill in the missing numbers. a

15 25 35

37 39 42 46 51

90 89 87 84 80

Draw all the possible lines of symmetry for the given figures.

Draw the other halves of the figures.

In a certain language, the letters are coded as follows. E   1

L   2

M   4

N   8

27 81 243

O   16

Find the code for the word LEMON. 186

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10 Compare the patterns.

2, 4, 8, 16, … and 3, 6, 12, 24, … . How are these patterns similar and how are they different?

Word Problems 1

teacher gave his class these digits: 1, 2, 5, 7, 8, 9. He asked them to make the A smallest number possible. What would that number be?

ily is drawing flowers. The first flower has 5 petals. The next flower has 8, then L 11, then 14 and so on. How many petals will the next three flowers have?

bee is building a honeycomb. It starts with 2 cells, then 4, 8, 16, 32, 64, …. What A will be the next three numbers in this pattern? Also write the pattern rule.

mily is hiking in the mountains. On the first day, she takes 100 steps. On the E second day she takes 120 steps, then 150 on the third day, 190 on the fourth day and so on. How many steps will she take on the sixth day?

ihir gets pocket money every month. He saved ₹20 in January, ₹40 in February, M ₹80 in March, ₹160 in April and so on. How much money did he save in August?

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11 Length and Weight Let's Recall We use “length” to describe how long or short something is. For example, take this laptop and table. The laptop is shorter in length than the table. We measure length in different units. We use millimetres (mm), centimetres (cm), metres (m) and kilometres (km). What if we had a pencil that is 8 cm long? Since 1 cm = 10 mm, the length of the pencil can also be written as 80 mm (= 8 × 10). We use “weight” to describe how heavy or light something is. Is the laptop heavier or the table? The table is made of solid wood and so it is heavier! Hence, the weight of the table is more. Like length, different units, such as milligrams (mg), grams (g) and kilograms (kg) are used to measure weights of objects. Let us say that we have a pumpkin that weighs 2 kg. Since 1 kg = 1000 g, the weight of the pumpkin can also be written as 2000 g (= 2 × 1000).

Letʼs Warm-up Fill in the blanks. 1

The length of your mathematics textbook is __________ (Use a ruler to measure).

The length of your table in the classroom is __________ (Use a ruler to measure).

If a child weighs 5000 g, then his weight in kg will be __________.

I f the weight of a 10-rupee coin is about 8 g, then the weight of this coin in mg will be __________.

If a newborn baby’s weight is 2 kg, then __________ is its weight in g.

I scored _________ out of 5.

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Understanding Lengths Real Life Connect

Sana wants to get a pair of trousers made for herself. So, she goes to the market to buy cloth for it. Sana: 1 m 85 cm of cloth, please. Shopkeeper: Could you please tell me the length, in metres?

Sana is confused about the metres of cloth she should buy.

Estimating Length Sana thinks of buying the minimum length of cloth that will fit her required length of cloth. So, she chooses to buy 2 m of cloth. This is exactly what estimating length is! Estimation helps us to find out the approximate measure of things. It gives us a fair idea about how long or tall (length) an object is just by looking at it. It also gives us an approximate measure of how far (distance) a place is. For example, look at the given objects.

The width of a grain of rice is about 6 mm.

Did You Know? The height of Mount Everest is about 8848 m.

The distance from the floor to The thickness of a notepad is about 1 cm. the door handle is about 1 m.

10 km Home The distance between Shalu’s home and school is about 10 km.

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School

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Example 1

The rope in the figure is 8 cm long. What can be the best estimate of the length of the battery? Estimated length of the battery = Half the length of the rope.

8 cm

So, the length of the battery is about 4 cm. Do It Together

Circle the best estimate. 1 2 3 4

Width of a finger

1 mm 10 mm

10 cm

10 m

Height of a car

16 mm

16 cm

160 cm

Length of a pen

14 cm 140 cm

Length of your shoe

160 mm

12 m 12 cm

14 mm 12 mm

14 m

1200 mm

Measuring Lengths Now, Sana takes the cloth to the tailor for stitching. She wants front pockets of length 4 cm on her trousers. Let us see how we can measure the length of the pockets. We use a centimetre (cm) ruler to measure the lengths of small objects. On this ruler, there are both small lines and long lines. The shortest distance between two small lines is 1 millimetre (mm). The numbers below the longer lines show the measurement in cm. 1 millimetre

Front

1 centimetre

10 millimetres

When measuring the length of an object, we always place the object along the ‘0’ mark of the ruler. Then, read the marking on the ruler where the object ends. For example, look at the pocket. Here, the pocket ends at the 4th longer line of the ruler. So, the width of the pocket = 4 cm.

What if the tailor made the pocket without measuring it? For example, look at the pocket. Let us see how we can measure its width. Here, the pocket is beyond the 4th longer line but shorter than the 5th longer line. In this case, we will count the number of shorter lines after 4 cm. There are 5 shorter lines after 4 cm. Hence, the pocket is 4 cm 5 mm = 4.5 cm long.

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Example 2

A pencil is placed along the ruler. What is the length of the pencil in cm? The pencil is longer than 7 cm but shorter than 8 cm. There are 5 mm lines after 7 cm. Hence, the pencil is 7 cm 5 mm = 7.5 cm long.

Do It Together

Find the length of the toffee. The toffee is longer than __________ but shorter than __________ cm. There are __________ mm lines after 3 cm. Hence, the toffee is __________ cm __________ mm = __________ cm long.

Units of Length Recall that Sana bought 2 m of cloth from the market to make a pair of trousers. How would you measure this length in metres? Let us see how we can do this. This is a metre ruler. On this ruler, there are small lines and long lines. The shortest distance between two small lines is 1 centimetre (cm). So, from the ruler given below, we see that 1 m = 100 cm. 1 cm 1 metre

We know that the standard unit of length is metre (m). We have also learnt about the units given below. Kilometre (km) more than a metre metre (m) standard unit of length centimetre (cm) millimetre (mm)

Chapter 11 • Length and Weight

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less than a metre

Remember! The units of measuring length, such as millimetres (mm) and centimetres (cm) are used to measure smaller lengths whereas metres (m) and kilometres (km) are used to measure longer lengths.

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Apart from these units, we also have a few more units for measuring length. Let us see how they are connected to each other in a place-value chart. Divide by 10 for every step as you move left ÷ 10

Thousands kilometre (km)

÷ 10

Hundreds hectometre

÷ 10

Tens decametre

Ones metre (m) Base Unit

Bigger Units × 10

× 10

÷ 10

tenths decimetre

÷ 10

hundredths thousandths centimetre (cm) millimetre (mm)

Smaller Units

× 10

Multiply by 10 for every step as you move right

× 10

Let us convert the length of cloth (2 m) bought by Sana to cm. 1 m = 100 cm. So, 2 m = 2 × 100 m = 200 cm Convert 40 cm to m. 100 cm = 1 m. So, 1 cm =

1 m 100

1 × 40 m = 0.4 m 100 What if Sana had bought 1 m 85 cm of cloth? Let us see how you would write it in metres. 40 cm =

1 m = 100 cm 1 m 85 cm = 1 m + Example 3

85 cm = 1.85 m. 100

Convert 2356 dm to m.

Example 4

10 dm = 1 m 1 m 1 dm = 10 1 2356 × 2356 m = = 235.6 m 2356 dm = 10 10 Do It Together Fill in the blanks to convert 5 m 230 mm to m. ______ mm = 1 m 1 m 1 mm = 5 m 230 mm = 5 m + 230 ×

Convert 6.547 hm to m. 1 hm = 100 m 6.547 hm = 6.547 × 100 m = 654.7 m

Error Alert! 1

2 m 8 cm = 28 cm

2 m 8 cm = 208 cm

= 5 m + ______ m = ______ m

Word Problems on Length Recall that Sana bought 2 m of cloth for making a pair of trousers. After some time, she also thought of buying 1 m 55 cm of cloth for making a shirt. Let us see what length of cloth she has altogether.

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Cloth bought by Sana for making a pair of trousers = 2 m Cloth required by Sana for making a shirt = 1 m 55 cm = 1.55 m Total cloth required by Sana = 2 m + 1.55 m

Think and Tell

So, the total cloth required by Sana is 3.55 m.

How do you think 1 m 55 cm is converted into 1.55 m?

Samantha buys two rolls of tape. One is 9.56 m long and the other is 6.80 m long.

Example 5

What is the total length of the two rolls?

What is the difference of the lengths of the two rolls?

Length of the first roll of tape = 9.56 m; Length of the second roll of tape = 6.80 m 1

otal length of the two rolls of tape T = 9.56 m + 6.80 m

So, Samantha buys 16.36 m in total.

Example 6

Do It Together

ifference in the length of the two D rolls = 9.56 m – 6.80 m So, the difference of the two rolls of tape is 2.76 m.

–

A 34 m long rope is divided into 10 equal pieces. What is the length of each small piece? 3.4 Length of a rope = 34 m 1 0 3 4. 0 Number of equal pieces of rope needed = 10 - 3 0 4 0 Length of each piece of rope = 34 m ÷ 10 - 4 0 0 Therefore, the length of each piece is 3.4 m. If 5 m 80 cm of cloth is needed to make a saree, then how much cloth will be needed for 5 such sarees? Length of cloth needed to make a saree = 5 m 80 cm = 5.80 m Number of sarees required = 5 Total length of cloth needed for making 5 sarees = __________ m × __________ So, _____________________ m of cloth is required for making 5 sarees.

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Do It Yourself 11A 1 Esimate the length of these items. Measure the lengths. Find the difference. a

Pencil

Eraser

2 Measure the lengths of the objects. a

3 Convert the given measurements. a 45 mm into cm

d 1056 m into km

d Book Sharpener

b 6892 dm into m e 547 mm into m

c 7.698 hm into km f 2.034 hm into m

Complete the table. S. No. a b c

Full form

In the bigger unit

In the smaller unit

16 m 80 mm

16.080 m

_______________ mm

45 m 10 cm

_______________

_______________ m 280.5 m

4510 cm

2805 dm

Find the measurement of the pencil. Express your answer in km.

Word Problems 1 Shagun travelled 2 km 578 m by bicycle, 21 km 870 m by bus and 1 km 346 m on foot. What is the total distance that Shagun travelled?

2 Yamini bought a pair of jeans which was 75 cm 6 mm in length. A part of it got stuck around a screw in the almirah and was torn. Now, the length of the jeans is reduced to 72 cm 3 mm. What length of the jeans was torn?

3 If 3 m 586 cm of cloth is required for one skirt, then how much cloth is needed for 6 such skirts?

4 If the length of a door is 2 m 1 dm and that of a wall is 3 m 2 dm, then what is the total length in dm?

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Understanding Weights Malti went to the market to buy sugar for preparing desserts required for a dinner get-together.

Real Life Connect

Malti: I want 1500 g of sugar, please.

Shopkeeper: I have only packets of 1 kg sugar.

Malti was confused about how many packets of sugar she should buy.

Estimating Weights Malti thought of buying the minimum weight of sugar that would fit her requirement. So, she chose to buy two 1 kg packets of sugar. This is exactly what estimating weight is! We guess the weight of different objects either by looking at them or by holding them. For example, look at the given objects.

Example 7

About a miligram

About a gram

About a kilogram

grain of sugar

small paper clip

Book

Which of these objects measures about a gram?

Leaf The weight of the leaf is about a gram. Do It Together

Ball

Shoes

Circle the best estimate. 1 2 3 4

A drop of medicine

1 mg

10 mg

10 kg

10 g

A photo album

600 mg

600 g

60 g

600 kg

A 2-rupee coin

A tube of toothpaste

4 mg

12 kg

14 kg

120 g

14 g

120 mg

140 mg

12 g

Units of Weight This is a weighing balance. Starting from 100 g, read the weights on the balance, clockwise. What do you notice? Here, 1 kg = 1000 g! Chapter 11 • Length and Weight

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We know that the standard unit of weight is kilogram (kg) and gram (g). We have also learnt about the units given below. kilograms (kg)

more than a gram

grams (g)

standard unit of weight

milligrams (mg)

less than a gram

Apart from these units, we also have a few more units of measuring mass (weight). Let us see how they are connected to each other in a place-value chart. ÷ 10

÷ 10

Thousands

kilogram (kg)

Hundreds

hectogram

decagram

× 10

÷ 10

Ones

decigram

Base Unit

× 10

÷ 10

Tenths

gram (g)

Bigger Units Value increases by 10 times × 10

The units of measuring weight, such as milligrams (mg), grams (g), and kilograms (kg) are used to measure light as well as heavy objects. Divide by 10 for every step as you move left

÷ 10

Tens

Remember!

Hundredths Thousandths centigram

milligram (mg)

Smaller Units Value decreases by 10 × 10

Multiply by 10 for every step as you move right

÷ 10

× 10

Convert 2 kg, the quantity of sugar that Malti bought, into g. 1 kg = 1000 g. So, 2 kg = 2 × 1000 = 2000 g Convert 400 mg to g.

Error Alert!

1 g 1000 400 mg = 1 × 400 g = 400 = 0.4 g 1000 1000 1000 mg = 1 g. So, 1 mg =

2 g 3 mg = 23 mg

2 g 3 mg = 2003 mg

What if she bought 1500 g of sugar? Let us see how we would write it in kilograms. Quantity of sugar purchased by Malti = 1500 g Since 1000 g = 1 kg, 1500 g = 1500 = 1.500 kg 1000 Example 8

Convert 7695 dg to g.

Example 9

Convert 3.476 hg to g.

10 dg = 1 g

1 hg = 100 g

1 dg =

3.476 hg = 3.476 × 100 g = 347.6 g

1 g 10 1 7695 × 7695 g = = 769.5 g 7695 dg = 10 10 196

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Do It Together

Fill in the blanks to convert 5 g 230 cg to cg. 1 g = ___________ cg 5 g 230 cg = 5 × ___________ cg + 230 cg = ___________ cg

Word Problems on Weights Recall that the sugar bought by Malti weighs 2 kg. Later, she also bought rice that weighed 2 kg 106 g. Let us see which food item weighed less and by how much. Weight of sugar = 2 kg = 2.000 kg

Think and Tell

Weight of rice = 2 kg 106 g = 2.106 kg

How do you think 2 kg 106 g is converted

Here, 2 < 2.106.

into 2.106 kg?

So, the sugar weighs less than the rice bought. To find out by how much the sugar weighs less, we will subtract the two weights.

Therefore, the sugar weighs less than the rice by 0.106 kg or 106 g. Example 10

Total weight of vegetables = 8 kg 250 g = 8.250 kg

carry

Total weight of potatoes and ladiesʼ fingers = 1.250 kg + 3.750 kg = 5 kg. Now, weight of onions = 8.250 kg – 5 kg

Therefore, Samaira bought 3.250 kg or 3 kg 250 g of onions.

1 . 2

5 . 0

3 . 7

A shopkeeper purchased 6 bags of salt, each weighing 2 kg 789 g. What is the total weight of the salt that he bought? Weight of 1 bag of salt = 2 kg 789 g = 2.789 kg

Weight of 6 bags of salt = 2.789 kg × 6 Therefore, the total weight of salt bought by the shopkeeper is 16.734 kg. Do It Together

Samaira bought 1 kg 250 g of ladiesʼ fingers, 3 kg 750 g of potatoes and some onions. If she bought 8 kg 250 g of vegetables in total, then find the weight of the onions that she bought. Weight of ladiesʼ fingers = 1 kg 250 g = 1.250 kg; Weight of potatoes = 3 kg 750 g = 3.750 kg

Example 11

× 1

9 6

The weight of a suitcase is 9 kg 200 g. Two books weighing 840 g each are removed from it. What is the weight of the suitcase now? Chapter 11 • Length and Weight

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Total weight of books removed from the suitcase = _________________________ Weight of the suitcase after removing the books = _________________________

Do It Yourself 11B 1

Circle the best estimate of the weight of the objects. 5000 kg

5000 g

Frog 2

85 g

30 g

Baseball Cap

350 g

5 kg 200 kg

100 g

70 kg

Dresser 500 kg

Iron

15 kg 2 kg

Convert the measurements. a 79 g into kg

b 4677 kg into g

c 6876 cg into g

d 1655 dg into g

e 975 g into mg

390 g 45 dag into mg

Convert the weights into bigger units and smaller units. a 6 kg 10 g

b 16 g 80 mg

c 547.6 g

d 3 g 8 cg

e 87 kg 6 dag

12 hg 42 g

By the time a kitten is about 4 months old, it weighs about 1688 g. What is its weight in kg?

An egg has a mass of 40 g. How many eggs can be bought in 1 kg?

Word Problems 1

Siya bought 2 kg 450 g of apples, 1 kg 547 g of guavas and 2 kg 136 g of pears. What is

The weight of 2 chairs is 16 kg 400 g. If the weight of one chair is 10 kg 300 g, then

An egg has a mass of about 65 g. What is the mass of 3 dozen eggs?

A truck can carry 8 cartons of packed food. If they weigh 32 kg 448 g in total, then

A man carried 4 soap bars weighing 258 g each and 3 detergent cakes weighing 352 g

the total weight of fruits Siya bought?

what is the weight of the other chair?

what is the weight of each carton? each.

a What is the total weight he carried?

b If 1 soap bar and 1 detergent cake was used, then what quantity of soap bars

and detergent cakes was left with him in total?

Three dg per kg of baking soda is used in cakes. If a baker prepares 17 cakes of 1 kg in a day, then what quantity of baking soda is used in total?

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Points to Remember • Estimation is a fair idea of how far (distance) something is, and how long or tall (length) an object is just by looking at the object. • To measure lengths of small objects, we use a centimetre (cm) ruler. The shortest distance between two small lines is 1 millimetre (mm). 10 such millimetres make 1 centimetre (cm). • To measure the weight of lighter objects, we use grams (g). To measure the weight of heavier objects, we use kilograms (kg). 1 kg = 1000 g.

Math Lab Setting: In groups of 5

Measure and Build Playland!

Material required: Building blocks, clay, cardboard, craft sticks, a pair of scissors, measuring tape or a ruler, objects of different lengths and weight, pencils, erasers and an A4-size sheet of paper Method: All 5 members of each group must follow these steps. 1

Create a rough draft of your playland on a sheet of paper.

Create different dummy objects of your choice like ramps, towers, slides, children, etc.

Estimate the weights of these objects by holding them.

According to the lengths and weights of the dummy objects, assemble and build a playland

using clay and building blocks, and measure the lengths of these objects.

of your choice on the cardboard. Make sure to decorate it beautifully so that it looks similar to the rough draft you initially made.

Each group presents their playland to the class, explaining their design choices and

showcasing how they integrated the measured objects. After presentations, allow time for students to explore and play in each other’s playlands.

Chapter Checkup 1

Guess the best units of length (m or cm) and weight (kg or g) for the given objects. a

Measure the objects. a

Chapter 11 • Length and Weight

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James has some cotton candy which is 3 cm longer than the cotton candy shown below. How long is

How much longer is the red straw than the blue straw?

Convert the lengths.

James’ cotton candy?

8 m into km

4 hm 35 m into hm

1232 m into mm

897 m into dam

4 g 64 cg into g

5487 g into mg

43 kg 7 dag into kg

Convert the weights. a

5 kg into g

A baby koala is called a joey. A young joey weighs about 0.38 kg. How much is that in g?

A candle weighs 125 g. How much is it in mg?

What is Meenakshi’s weight? Observe the clues and answer. a b

Meenakshi’s weight is half of her grandmother’s weight.

If you add the smallest 2-digit number to 28 less than the smallest 3-digit number, you will get the grandmother’s weight.

If you have two blank strips of paper, one 4 cm long and the other 8 cm long, then how many strips of

each would you need to measure lines of the following lengths: a

12 cm

20 cm

24 cm

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Word Problems 1

Rimi has 12 stamps that are each 3 cm long. a What is the total length of all the stamps? b Will all 12 stamps fit into a single row across a stamp album that is 24 cm across?

A lift in a building is allowed to carry up to 260 kg in weight. The people below want to enter the lift. Their weights are mentioned. Amit: 85 kg

Priya: 70 kg

Priyanshi: 58 kg

Raju: 80 kg

Can they enter the lift together? Why? 3

To bake one cake, a baker needs the following items: 200 g of flour, 3 eggs, 75 g of butter, 100 g of sugar, some milk How many kilograms of flour, butter and sugar would be needed to bake 100 cakes?

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12 Perimeter and Area Let's Recall Perimeter is the total distance around the boundary of a closed shape. It is calculated by adding the measures of all the sides of a closed shape. As it is the sum of lengths, it is measured in the same units as length— in mm, cm, m, or km. cm

Look at the given shape. We simply add all the side lengths to find the perimeter!

5.8 cm

Perimeter = 5 cm + 4 cm + 5.8 cm + 7 cm + 5.8 cm = 27.6 cm Area is the total space occupied by the closed shape. But how do we find the area of a figure?

5.8 cm

7 cm

We draw on a marked grid as shown and find the squares that fall inside it. If each square in the grid is 1 square cm, the area of the figure becomes the number of squares it occupies. Now, if we have a figure drawn as shown, we can count the number of squares the figure occupies. That number of squares is the area of the figure in square cm.

Start Here

End Here

Number of unit squares = 18 Area = 18 sq. cm

Letʼs Warm-up

Look at the figures drawn on a 1 cm square grid. Fill in the blanks. P

Perimeter ___________

___________

Area

I scored _________ out of 4.

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Perimeter Real Life Connect

Students of 5B are decorating their classroom for Independence Day! They want to stick a ribbon around the class blackboard and the bulletin board. How will they do this?

Perimeter of Squares and Rectangles The design that the students have thought of has the ribbon on all sides of the board and the bulletin board. That is exactly what perimeter is! So, if the students can find the perimeter of the blackboard and the bulletin board, they will be able to find the length of ribbon needed.

Remember! Perimeter is the total distance around the boundary of a closed shape.

Also, note that the board is a rectangle in shape. Its opposite sides are equal!

The bulletin board, on the other hand, is a square. All its sides are equal. If the students find the perimeter of the blackboard (rectangle) and the bulletin board (square) they will know the length of the ribbon they need.

Perimeter of a Rectangle If we know the perimeter of any rectangle, we can find the perimeter of the blackboard too. Let us take a rectangle with length = l cm and breadth = b cm, as shown. The perimeter is the sum of all sides = l + b + l + b

l b

b l

So, the Perimeter of a rectangle = 2l + 2b The teacher and students together measured the size of the blackboard. They found the length of the blackboard as 100 cm and breadth as 80 cm.

Chapter 12 • Perimeter and Area

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Perimeter of a rectangle = 2l + 2b

100 cm 80 cm

So, perimeter of the blackboard = 2 × 100 + 2 × 80 = 200 + 160 = 360 cm The students will need 360 cm of ribbon to decorate the blackboard. Example 1

Find the perimeter of the window. Here, length (l) = 38 cm and breadth (b) = 28 cm Perimeter of a rectangle = 2l + 2b = 2 × 38 + 2 × 28

= 76 + 56 cm = 132 cm

28 cm

38 cm

So, the perimeter of the window is 132 cm. Do It Together

Find the perimeter of the given pillow.

60 cm

Here, length (l ) = 60 cm and breadth (b) = _____ cm

20 cm

Perimeter of a rectangle = 2l + 2b The perimeter of the pillow = 2 × _____ + 2 × _____ = 120 + _____ cm = _____ cm

Perimeter of a Square We know that the bulletin board is square in shape. To cover the boundary of the bulletin board with the decorative ribbon, we need to find the perimeter of the square. Now, if we know the perimeter of a square, we can find the perimeter of the bulletin board as well. Let us consider a square with side length s.

Perimeter is the sum of all the sides of a figure.

s s

So, the perimeter of the square = s + s + s + s = 4s Perimeter of a square = 4 × s Now, the teacher and students together measured the sides of the bulletin board as 40 cm each.

40 cm

Since the perimeter of a square = 4 × s and here s = 40 cm, we get the perimeter of the bulletin board = 4 × 40 = 160 cm. So, the students need ribbon of length 160 cm to decorate the bulletin board.

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Example 2

Find the perimeter of a square of side 10 m. Here, s = 10 m Since the perimeter of a square = 4 × s, perimeter of the square = 4 × 10 m. Perimeter of the square = 40 m

Do It Together

45 cm

Find the perimeter of the chessboard. Here, side (s) = ________ cm. Perimeter of a square = _____________________ So, the perimeter of the chessboard is ________ cm.

Do It Yourself 12A 1

Find the perimeter of the rectangles of the following measurements. a l = 12 cm, b = 15 cm

c l = 18 m, b = 20 m

Find the perimeter of the squares of the following measurements. a s = 32 m

b l = 55 m, b = 85 m

b s = 53 cm

c s = 27 m

d s = 78 cm

Find the perimeter of the given objects. a

20 cm

10 cm

30 cm 20 cm

20 cm

50 cm

100 cm 4

Find the perimeter of each option to complete the tables. a

l (in m)

110

b (in m)

Perimeter of Rectangle (in m) b

s (in mm) Perimeter of Square (in mm)

Find the missing sides. a Rectangle with length = 32 mm, breadth = ?, Perimeter = 112 mm b Square with Perimeter = 148 mm, side = ?

Chapter 12 • Perimeter and Area

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Word Problems 1

A farmer needs to fence his square-shaped garden. The distance from one corner to the

Seeta and Geeta are playing with a thread. They first form a square with a side length

next corner is 45 m. How much wire would he require to fence the entire garden?

of 10 cm with no length of thread remaining. They then use the same thread to form a rectangle. They have the length of the rectangle as 12 cm. What would the breadth of the rectangle be?

Area Real Life Connect

Have you ever noticed that some walls in your house are big and some are small? When we want to paint the walls, do you know which one needs more paint? The wall with more space on it needs more paint. We can use the word �area� to talk about how big a wall is. It's like measuring how much space there is to paint!

Area of Squares, Rectangles and Triangles Now let us look at different shapes of walls and their areas.

Area of a Square What if we have a square of side 3 m each? Imagine a square of side 3 m on a square grid as shown below. We know that the area of the square is the number of 1 m squares that it encloses. The area of each small square in the grid is equal to 1 sq. m. Count all the small squares inside the big square to find the area of the wall. Area of the square = Area of 9 small squares = 9 sq. m We can say that the area of a square = side × side or Area of a Square = s × s

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Think and Tell

How many rows are there? How many small squares are there in each row? Write a multiplication sentence to find the total number of squares. What do you notice? 1m

Units of Area Area is always measured in square units (sq. units). The method of finding the area also remains the same. If the size of the object is in metres, the area then will be in sq. m. If the side of an object is in centimetres, the area in that case will be in sq. cm.

1m 1m

1 cm

A bed occupies a large part of the room and is measured in metres. 1 cm Therefore, the area it occupies will also be measured in sq. m. Compare this with an eraser which is only a few centimetres long. Its area is measured in sq. cm.

Did You Know? The smallest country in the world is Vatican City. It surrounded by the city of Rome. It has a total area of about 0.44 sq. km.

Example 3

Find the area of a square of side 5 cm.

Example 4

Here, s = 5 cm

Here, s = 16 cm

Area of a square = s × s

Area of a square = s × s The area of the cushion = 16 cm × 16 cm = 256 sq. cm

= 5 cm × 5 cm = 25 sq. cm Do It Together

Find the area of a cushion of side 16 cm.

Find the area of the photo frame. Here, s = 8 cm The area of the photo frame = ____ cm × ____ cm

8 cm

Area of a square = s × s The area of the photo frame = ____ sq. cm.

Area of a Rectangle Let us consider a rectangle of length 5 cm and breadth 3 cm.

Chapter 12 • Perimeter and Area

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Let us again place this shape on a square grid of 1 cm squares. Let us count the small squares along a row and a column of the rectangle.

= 5 × 3 = 15 small squares

row 1 column

Area of the rectangle = number of squares in a row × number of squares in a column

1 cm

2 3

Do you see any relation between the area of the rectangle and its sides? Yes! The area of the rectangle = 5 cm × 3 cm = 15 sq. cm We can say that the area of a rectangle = length × breadth or, Area of a rectangle = l × b Area of Compound Shapes Compound shapes are formed with more than one basic shape.

5 cm 3 cm

For example, this is a compound shape formed with two rectangles A and B, as shown in the given figure.

7 cm

Now, let us find the area of the given compound shape.

12 cm

We will follow these steps: Step 1

Step 2

Identify the shapes that come together to form the

Find the missing lengths.

compound shape. In this case, it is A and B as shown. 5 cm A

5 cm 3 cm

3 cm

7 cm

7 cm B

12 cm

7 – 3 = 4 cm

Step 3

Step 4

Find the area of rectangle A.

Find the area of rectangle B.

Area of rectangle A = l × b = 5 × 3 sq. cm = 15 sq. cm

Area of rectangle B = l × b = 12 × 4 sq. cm = 48 sq. cm

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Step 5 Find the area of the whole shape. Area of the whole shape = area of rectangle A + area of rectangle B

= 15 + 48 sq. cm = 63 sq. cm

Error Alert! Always calculate the area before comparing the sizes of two shapes. A shape can “appear” to be bigger and yet have a smaller area.

Example 5

Find the area of a rectangle of length 15 cm and breadth 20 cm. Here, l = 15 cm and b = 20 cm Since the area of a rectangle = l × b, the area of the rectangle = 15 × 20. Area of the rectangle = 300 sq. cm

Example 6

Find the area of the given shape. 4 cm

4 cm 3 cm 5 cm

A 3 cm 6 cm

B 2 cm

3 cm

5 cm

9 cm

3 cm 6 cm

15 cm

2 cm

First split the shape into three rectangles A, B, and C and find the missing lengths. The area of rectangle A = l × b = 4 × 3 sq. cm = 12 sq. cm The area of rectangle B = l × b = 9 × 3 sq. cm = 27 sq. cm The area of rectangle C = l × b = 15 × 2 sq. cm = 30 sq. cm The area of the whole shape = area of rectangle A + area of rectangle B + area of rectangle C Do It Together

= (12 + 27 + 30) sq. cm = 69 sq. cm 3 cm

Find the area of the shape.

The area of rectangle B = _______________________ The area of the whole shape = _______________________.

4 cm

The area of rectangle A = 3 × 2 = ______ sq. cm

2 cm B 12 cm

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Area of a Triangle Look at the rectangular wall. It has been divided into two right-angled triangles of two different colours. The length of the wall is 10 m and the breath is 5 m as shown. How do we find the area of the triangular part of the wall? Note that the two triangles are exactly the same size. So, both will have an equal area! 10 m

10 m 5m

10 m

Area of Right triangle 1 + Area of Right triangle 2 = Area of Rectangle Area of Right triangle 1 = Area of Right triangle 2 So, we can conclude that the area of each triangle = Area of the rectangle = l × b = 10 × 5 = 50 sq. m. So, the area of each triangle =

1 of the area of the rectangle 2

1 × 50 = 25 sq. m. 2

What if we want to find the area of a triangle that is not a right-angled triangle? How would we find the area? Let us use the 1 cm square grid again! Step 1

Step 2

Step 3

Draw the triangle on the square

Draw a rectangle around the

Split the triangle into two.

grid. Find the base of the triangle.

Here, the base of the triangle covers

triangle, as shown below.

rectangles. The two triangles

6 squares. So, the base is 6 cm.

1 cm

Here, 1 and 2 become the two within them are both right-

1 cm

angled triangles.

1 cm

6 cm

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Step 4

Step 5

Step 6

the part of the triangle within it.

Area of triangle within rectangle 1 +

Find the area of rectangle 1 and Length of rectangle 1 = 2 cm

Breadth of rectangle 1 = 6 cm Area of rectangle 1 = 6 × 2 = 12 sq. cm

So, the area of triangle 1 = half of 12 = 6 sq. cm Example 7

Find the area of rectangle 2 and Length of rectangle 2 = 4 cm

Breadth of rectangle 2 = 6 cm

Area of the whole triangle =

Area of triangle within rectangle 2 = 6 sq. cm + 12 sq. cm = 18 sq. cm

Area of rectangle 2 = 6 × 4 = 24 sq. cm

So, area of triangle 2 = half of 24 = 12 sq. cm

What is the area of the red triangle? 1 unit

1 unit

1 2

To find the area of this triangle, draw a rectangle around the triangle and split the triangle into two parts. Area of triangle 1 = half of area of rectangle 1 = half of 6 = 3 sq. units

Did You Know? The ancient Egyptians used the concepts of perimeter

Area of triangle 2 = half of area of rectangle 2 = half of 6 = 3 sq. units

and area to measure and plan fields for agriculture.

Area of the whole triangle = 3 sq. units + 3 sq. units = 6 sq. units Do It Together

Find the area of the given triangle.

To find the area of this triangle, draw a rectangle around the triangle and split the triangle into two parts. Area of rectangle 1 = ____________; Area of triangle 1 = ____________ Area of rectangle 2 = ____________; Area of triangle 2 = ____________ Area of the whole triangle = ____________. Chapter 12 • Perimeter and Area

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Do It Yourself 12B 1

Which unit would you prefer to find the area of the following? a board

b your book

e your city

lake

h a farmer’s field

b l = 47 m, b = 36 m

c l = 58 m, b = 70 m

b s = 67 cm

c s = 24 m

d s = 40 cm

Find the area of the given compound shapes. 3 cm

10 cm

8 cm

5 cm

3 cm

9 cm 4 cm

3 cm

12 cm 6

d l = 88 cm, b = 62 cm

Take each side of a square on the squared paper as 1 unit. Find the area of the triangles. a

g table

Find the area of the squares of the given measurements. a s = 56 m

d your school

Find the area of rectangles of the given measurements. a l = 15 cm, b = 22 cm

c classroom

8 cm

3 cm

6 cm

Use squared paper to show rectangles or squares with these measurements. Find the measurement that is not given for the following shapes. a Perimeter = 36 cm, Area = ?

b Area = 24 sq. cm, Perimeter = ?

Word Problems 1

Advita wants a new carpet for her room. If her room measures 12 m by 12 m, then how

The perimeter of a rectangular field is 80 m. If the breadth of the field is 20 m, find the

much carpet does she need to do to cover her entire floor? area of the field.

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Points to Remember • Perimeter of a rectangle = 2l + 2b and Perimeter of a square = 4 × s. • Area of a rectangle = l × b and area of a square = s × s. Area is always measured in square units. • To find the area of compound shapes, we split the shapes into squares or rectangles. •

Math Lab Build a Paper House!

Setting: In groups of 3

Materials Required: Origami sheets, a pair of scissors, glue or tape, rulers, markers or coloured pencils, templates of squares, rectangles and triangles Method: All 3 members of each group must follow these steps. 1 Collect templates of different shapes, such as squares, rectangles, and squares from your teacher. These templates will serve as the bases for your paper houses.

2 Choose a shape from the templates and cut it out of the origami sheets.

3 Decorate the paper shape to represent different parts of a house, such as walls, roof, doors and windows.

4 Calculate the perimeter and area of each part of your paper house using the respective formulas.

5 Once the decoration and calculations are done, fold and assemble the paper shapes to create your house. You may use glue or tape to secure the different parts of the house together.

Chapter Checkup 1

Find the perimeter and area of the following rectangles. a

l = 27 m, b = 21 m

l = 49 m, b = 33 m

l = 70 cm, b = 57 cm

s = 45 m

s = 80 cm

Find the perimeter and area of the following squares. a

l = 14 cm, b = 18 cm

s = 23 cm

s = 32 m

Complete the given table. S. No.

Shape

Rectangle

24 cm

Square

–

Rectangle

Chapter 12 • Perimeter and Area

Maths Grade 5 Book_Chapter 7-12.indb 213

Length/side

34 m

Breadth

Perimeter

Area 960 sq. cm

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Which rectangle has the largest area and the smallest perimeter? 8 cm 5 cm

4 cm

12 cm

11 m

4 cm 3 cm 8m

1 cm

7 cm 5 cm

5 cm

A triangle whose sides are equal and a square of side 10 cm were joined as shown in the figure. The total perimeter of the new shape formed is a

70 cm

40 cm

60 cm

50 cm

This shape is made from five squares that are all the same size. If the area of the figure is 80 sq. cm, then its perimeter is a

Find the area of the following triangles. a

6 cm

5 cm

3 cm

6 cm

Find the area of the following compound figures. a

5 cm

7 cm

11 cm

6 cm

52 cm

40 cm

48 cm

36 cm

Three identical rectangular tiles are placed as shown. If the perimeter of each tile is 22 cm, find the perimeter of the shape formed.

10 If the length and width of a rectangle double, the area of the rectangle also becomes double. Is this statement true? Explain your answer.

Word Problems 1

Find the total area of 8 square wooden panels if each measures 25 cm.

A floor is 15 m long and 12 m wide. A square carpet of sides 13 m is laid on the floor. Find the area of the floor that is not carpeted.

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13 Capacity and Volume Let's Recall Height of the vessels

We use ‘capacity’ to measure how much liquid a container can hold. It depends on the shape and size of the container. For example, look at these two containers. The first container is narrower than the second but both of them have the same height. So, the capacity of the first container is less than the second.

We measure capacity by filling bigger containers using smaller containers. For example, look at the bucket and mugs. = It takes 20 mugs of water to fill the bucket. So, the capacity of the bucket is the same as 20 mugs. We use different units to measure capacity such as litres (L) and millilitres (mL). To measure larger quantities we use litres and to measure smaller quantities, we use millilitres. Suppose we have a 3 L water bottle. How do we write its capacity in millilitres? Since 1 L = 1000 mL, the capacity of the bottle in millilitres (mL) will be (3 × 1000) = 3000 mL.

Letʹs Warm-up Fill in the blanks. 1

Between a jug and a glass, __________ has more capacity.

If 8 glasses of juice fill a jug, then __________ is the capacity of the jug.

our tin cans of paint fill a bucket of paint. If the capacity of 1 tin can is 2 L, then F __________ is the capacity of the bucket.

5 L = __________ mL

8 L = __________ mL I scored _________ out of 5.

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Understanding Capacity Real Life Connect

Samidha went to a shop to buy a bottle of water with her mother. Samidha: Mummy, I want a 2 L bottle. Which bottle should I buy? Mother: The small bottle is full of water. Its capacity is 1 L. Now, guess how many of the small bottles will fill this big bottle. Let us help her out!

Estimating Capacity We guess the capacity of any container/packet either by looking at it or by holding it. This is exactly what estimating capacity is. For example, look at these objects:

The bottle of water holds about 1 L of water.

The medicine dropper holds about 2 mL of medicine.

Therefore, we can estimate that the capacity of the small bottle is half a litre since its size looks half of the big bottle. Example 1

Which of these objects can have a capacity of about a litre?

The capacity of the kettle can be about a litre. Do It Together

Circle the best estimate. 1

apacity of a bottle of C ghee

1 mL

10 mL

10 L

apacity of a watering C can

Capacity of a bathtub

15 mL 1500 mL 15 L 150 L 16 L 160 mL 16 mL 160 L

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Measuring Capacity Things like bottles, buckets and jars are various types of containers. But, can all these containers be filled with the same amount of water? Have you ever wondered how much liquid these containers can hold? This is where capacity measurement comes into play. Let us learn about measuring the capacity of a container, and the units that we use for it. We can measure the amount of liquid using different measuring containers. The figure below shows a set of measuring cups of capacity 1 L, 500 mL and 300 mL.

Think and Tell

How do we know how much water each cup holds?

Look at the amount of water in the first cup. The first cup contains 900 mL of water, while the second and third cups contain 100 mL and 200 mL of water, respectively. Example 2

Look at the syringe. What is the amount of the liquid in it? The syringe contains only 6 mL of liquid.

Example 3

Look at the measuring cups. Find the amount of water in each of them?

1000 mL

800

900

700

600

500

400

300

400

300

200

300

200

100

600

500

400

900 700

600

500

Cup B contains 400 mL of water.

Do It Together

1000 mL

900

Cup A contains 700 mL of water. Cup C contains 50 mL of water.

1000 mL

200

100

What is the amount of juice in the jug? The smaller markings on the jug stand for 50 mL. The jug contains ______ mL of liquid.

Units of Capacity We know that litre (L) is the standard unit of capacity. It is used to measure containers with big capacities. Similarly, we use millilitres (mL) for measuring containers with smaller capacities.

Remember! Millilitre (mL) is less than a litre (L).

Apart from these units, we also have a few more units of measurement of capacity. Let us see how they are connected to each other. Chapter 13 • Capacity and Volume

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Now, look at the conversion chart. ×10 Kilolitre

ltip

×10

Hectolitre

×10

Decalitre

÷10

Div

ide

ste

Millilitre

÷10

do wn

×10

Centilitre

÷10

×10

Decilitre

÷10

ste

×10

Litre

÷10

by 1

ly b y1

÷10

For converting bigger units to smaller units, multiply by 10 for every step as we move towards the right. As Samidha bought 2 L of milk, let us convert it to mL. 1 L = 1000 mL

2 L = 2 × 1000 = 2000 mL

Similarly, for converting smaller units to bigger units, divide by 10 for every step as we move towards the left. Let us say Samidha bought 2500 mL. It can be converted to L as: 1000 mL = 1 L 2500 mL = Example 4

1 mL =

1 L 1000

2500 1 × 2500 L = = 2.5 L 1000 1000

Convert 6548 dL to L.

Example 5

10 dL = 1 L 1 dL =

1 L 10

6548 dL = Do It Together

1 6548 × 6548 L = = 654.8 L 10 10

Fill in the blanks to convert 5 L 250 cL to L. 1 L = ______ cL 5 L 250 cL = 5 × _____ cL + 250 cL = ______ cL

Convert 2.342 hL to L. 1 hL = 100 L 2.342 hL = 2.342 × 100 L = 234.2 L

Error Alert! Multiply the number of litres by 1000 and then add millilitres to convert into millilitres. 2 L 8 mL = 28 mL

2 L 8 mL = 2008 mL

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Do It Yourself 13A 1

Circle the best estimate of the given objects. a Teaspoon

250 L

50 mL

5 mL

250 mL

b Pan

250 mL

2 mL

800 L

Convert the given measurements. a 658 mL into L

b 8437 L into mL

d 5054 cL into L

e 821 L into daL

S. No.

Full form

8 L 60 mL

32 L 89 cL

In bigger units

136 L 80 dL into mL

In smaller units 8060 mL

32.89 L 4012.3 L

90 kL 6 daL

40,123 dL

90.06 kL

Ridhi pours 3050 mL of liquid into each of the containers. She says container C has the least capacity. Is she correct? Express the quantity of liquid in decilitres (dL). a

Complete the given table.

c 2567 dL into L

Are there enough bowls to fill a jar of 2700 mL? Use the information to find your answer. Given: =

600 mL

Word Problems 1

A tap drips water at the rate of 150 mL in 2 hours. How many litres of water would

A bottle holds 1050 mL of tea. How many tea cups of 150 mL each can be filled from it?

drip from it in 10 hours?

Chapter 13 • Capacity and Volume

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Volume Real Life Connect

Adil and Mitali want to distribute sweets to their relatives on Diwali. They had planned to buy a tasty sweet called milk cake. So, they go to the market, buy milk cake and get it packed into boxes. Adil: Please pack these sweets in a box. Shopkeeper: I have different sizes of boxes. Which one do you want? Both of them wanted to buy a bigger box of sweets.

Volume of Solids Using Unit Cubes

Box 1

Box 2

When we want to compare the sizes of two solid objects, we do so by measuring the amount of space they occupy. This is called the volume of the solid object. Now, what if we want to know which box can hold a greater number of sweets? To measure the number of sweets in a box, the shopkeeper would need to take each milk cake one by one and start filling them in the box. Box 1

The shopkeeper puts one layer of 6 milk cakes in Box 1. 1 layer Then, the shopkeeper puts one more layer of 6 milk cakes in the box. There was no more space left in the box.

2 layers So, Box 1 can hold 12 (= 6 × 2) milk cakes.

Box 2

The shopkeeper puts one layer of 16 milk cakes in Box 2. 1 layer The shopkeeper could put in two more layers of 16 milk cakes. Only then the box was full.

3 layers So, Box 2 can hold 48 (= 16 × 3) milk cakes.

So, Box 2 holds a larger number of sweets. Therefore, we can say that Box 2 has a greater volume than Box 1. Here, we can say that the volume is the amount of space a solid object occupies or the amount of space enclosed within a container. 220

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Now, let us say we have an open box, as shown. What will be the volume of this box if we have to use unit cubes to fill the box? Let us see how we can do this. A unit can be a millimetre, a centimetre, or a metre. So, to find the volume of a box, we can use cubes of three kinds: 1 cm

1 mm

1 cm

1 mm This is a millimetre cube (mm cube). It is about the size of a grain of sugar cube.

1 cm This is a centimetre cube (cm cube). It is about the size of a dice.

1m 1m

This is a metre cube (m cube). It is about the size of a very large cubical carton.

An ‘mm’ cube is used to measure the volume of very small objects. An ‘m’ cube is used to measure the volume of large objects. Let us see how we can find the volume of the box by filling it with ‘cm’ cubes. Step 1 Fill the base of the box with one layer of ‘cm’ cubes. 4 rows of 5 cubes each make up a layer in the box. So, number of cubes in 1 layer = 4 × 5 = 20 cubes

Step 2 Fill the box with as many layers of cubes as required. The box takes 3 such layers. So, total number of cubes in the box = 20 × 3 = 60 cubes

Since the box contains 60 cubes, the volume of the

box is 60 cubic centimetres (cu. cm) as each side of the small cube is 1 cm. If 1 cm = 1 unit, then the volume of the box is 60 cu. units. Therefore, the unit of volume is cubic units (or cu. units),

Example 6

where units can be ‘mm’, ‘cm’ or ‘m’.

Did You Know?

Which object has the greater volume?

The volume of an adult

bottle of jam Chapter 13 • Capacity and Volume

Maths Grade 5 Book_Chapter 13-17.indb 221

elephant is about 6 cu. m.

shoe box 221

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The shoe box occupies more space than the bottle of jam. So, the shoe box has a greater volume than the bottle of jam. Example 7

Find the volume of the given solids. Do they both have the same volume?

A Solid A has 10 unit cubes.

B Solid B has 10 unit cubes

So, the volume of solid A is 10 cu. units.

So, the volume of solid B is 10 cu. units.

Yes, the solids are different in shape and size but have the same volume. Do It Together

Find the volume of the solid. Layer 1 (blue) has _____ unit cubes. Layer 2 (yellow) has _____ unit cubes. Total number of unit cubes = _____ Volume of the given solid = ______ cu. units

Volume of Solids Using the Formula Remember Adil and Mitali who went to buy milk cake? The shopkeeper had two more boxes of sweets, A and B, as shown below. Each box was of a different shape and size.

br 5 c ea m dt h

cm 12 gth n e l

5 br cm ea dt h

Box A

5 cm height

8 cm height

m 5 c th g len

Box B

Now, let us say this time Adil and Mitali require cube-shaped pethas. Let us see the volume of pethas these boxes will hold. Let’s say 1 petha = 1 unit cube.

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Box A

Box B

Length = 12; Breadth = 5

Length = 5; Breadth = 5

Thus, number of pethas (unit cubes) in

1 layer = 12 × 5 = 60

1 layer = 5 × 5 = 25

Height = 8

Height = 5

So, number of pethas (unit cubes) in box A = 60 × 8 = 480 pethas

So, number of pethas (unit cubes) in box B = 25 × 5 = 125 pethas. Thus, volume of box B = 125 cu. units

Thus, volume of box A = 480 cu. units Therefore, volume = number of unit cubes in a layer × number of layers =l×b×h Example 8

What is the volume of the container if it is to be completely packed with unit cubes? Length = 4 cm, breadth = 3 cm So, the number of unit cubes that can be put along the length = 4 5 cm

and the number of unit cubes that can be put along the breadth = 3 Thus, the number of unit cubes in 1 layer = 4 × 3 = 12 Since height = 5 cm, the number of unit cubes in 5 such layers = 12 × 5 = 60

4 cm

So, the volume of the container = 60 cu. cm Chapter 13 • Capacity and Volume

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Do It Together

Look at the box. If the box is completely filled with unit cubes, what is the volume of the box? 3 cm

Number of unit cubes = ________________

Word Problems on Volume and Capacity

5 cm

Sneha bought two containers A and B for packing food.

2c m

So, volume of the box = _____ × _____ × h = _____ × _____ × 3 = _____ cu. cm

What is the volume of the two containers?

10 cm

12 cm 16 cm

10 cm

Container A

10 cm

Container B

For container A To find the volume of container A, we need to multiply its length, breadth and height. So, volume of the container = l × w × h = 16 × 12 × 10 = 192 × 10 = 1920 cu. cm

For container B To find the volume of container B, we need to multiply its length, breadth and height.

Think and Tell

So, volume of the container = l × w × h = 10 × 10 × 10 = 1000 cu. cm Example 9

The two containers are of different shapes and sizes. Will the volume of water in the 2 containers be the same?

John filled a truck with 20 L 126 mL of petrol, a car with 13 L 679 mL of petrol, and a bus with the rest. If the total quantity of petrol used to fill all three vehicles is 50 L 342 mL, then find the quantity of petrol used to fill the bus. Quantity of petrol in the truck = 20 L 126 mL = 20.126 L Quantity of petrol in the car = 13 L 679 mL = 13.679 L Total quantity of petrol = 20.126 L + 13.679 L

L mL 1 1 20 . 1 26 13 . 6 7 9 33 . 8 0 5

Think and Tell

How do you think 20 L 126 mL and 13 L 679 mL is converted into 20.126 L and 13.679 L, respectively?

So, the total quantity of petrol in the truck and car is 33.805 L. 224

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Total quantity of petrol used to fill three vehicles = 50 L 342 mL = 50.342 L Quantity of petrol filled in the bus = 50 L 342 mL – 33 L 805 mL = 50.342 L – 33.805 L

L –

So, the quantity of petrol in the bus is 16.537 mL (or 16 L 537 mL). Example 10

4 5 3 1

9 10 13 0. 3 3. 8 6. 5

A rectangular container measures 15 cm by 10 cm by 8 cm. It is completely filled with sugar cubes. What volume of the container is filled with sugar cubes?

3 4 0 3

12 2 5 7

8 cm

Length = 15 cm, breadth = 10 cm Volume of the container = l × w × h = 15 × 10 × 8 = 150 × 8 = 1200 cu. cm Do It Together

15 cm

10 cm

Ketan wants to rent two of the rooms in his house. He uses the store room that measures 20 m × 12 m × 10 m for storing his household things in cubical cartons with sides of length 1 m. If the room is completely filled with the cartons, then what is the volume of the room? Number of cartons that are placed along the length = _____ Number of cartons that are placed along the breadth = _____ Number of cartons in 1 layer of the room = ____ × ____ = ____ Number of cartons in 10 such layers = ______ × 10 = ______ So, volume of the room = ______ cu. m

Do It Yourself 13B 1

Tick () the object that occupies more space.

Find the volume of each solid. Circle the solid with the greater volume. a

Chapter 13 • Capacity and Volume

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A big box that measures 10 cm by 15 cm by 6 cm is packed with smaller boxes. What is the volume of

A fish tank is emptied to fill it with cubical boxes. If the fish tank is 30 cm by 22 cm by 10 cm, then find

Rinku’s lunch box measures 16 cm by 8 cm by 3 cm, whereas Rita’s lunch box measures 10 cm by 7 cm

the bigger box? its volume.

by 3 cm. Whose lunch box is bigger in size?

Word Problems 1

Rhea wants to fill an almirah that measures 3 m by 2 m by 1 m with cartons of old

A container is filled with 16 L of juice. The juice is poured equally into some bottles. If

Srishti has a jug filled with juice. She pours it into 2 glasses of different sizes. One

clothes. What is the volume of the almirah?

each bottle can hold 4000 mL of juice, how many such bottles are needed?

glass has 350 mL juice and the other has 890 mL juice. If the jug contained 1.5 L of juice, how much juice was left in the jug?

Points to Remember • While estimating the capacity of different objects, we either guess their capacity by looking at them or by holding them.

• Volume is the amount of space an object occupies or the space enclosed within a container. The greatest amount of liquid a container can hold is the capacity of the container.

•

The volume of a solid object is the amount of space it occupies.

• Volume of a cuboid/cube (using unit cubes) = number of unit cubes along the length × number of unit cubes along the breadth × number of layers

Math Lab Cube Art Gallery Setting: In groups of 5 Materials Required: paper cubes, cardboard, glue/tape, a pair of scissors, a ruler, markers

or crayons or coloured pencils, decorative materials like stickers, glitter, etc. of your choice

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Method: All 5 members of each group must follow these steps. 1 Create a unit cube using a sheet of paper. You may use the given design and make

copies of it to make multiple unit cubes. Colour and highlight using glitter or stickers.

2 Use these unit cubes to build different kinds of structures on a base made of cardboard. You may stack, arrange, and combine the cubes to bring your designs to life.

3 Once the structures are built, glue or tape the cubes together so that they do not fall apart. 4 After completing your cube art structures, measure the dimensions (length, breadth, and height) of your creations and find the volume of each of them.

5 The whole class now creates a gallery of all the artworks. The one who creates a structure with the highest volume wins a reward of the teacher’s choice.

Chapter Checkup 1 Tick () the best estimate. a

Glue in a small bottle

50 L

2000 L

50 mL

Paint in a bucket

50 L

50 mL

20 L

1 mL

Ketchup in a packet

500 mL

20 mL

50 L

Shampoo in a bottle

250 mL

250 L

Water in a pool

20 L

500 mL

5000 L

The container holds a volume of 750 mL of water. Estimate the capacity of the container.

Read and write the amount of coloured water in each of the jugs.

600 mL 4

Convert the measurements. a d

980 mL into L

3456 cL into L

Chapter 13 • Capacity and Volume

Maths Grade 5 Book_Chapter 13-17.indb 227

b e

6869 L into mL 243 L into daL

c f

9796 dL into L

907 L 56 dL into mL

227

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20 mL

150 mL 120 mL 90 mL 60 mL 30 mL

90 mL

10 mL 125 mL

150 mL 120 mL 90 mL 60 mL 30 mL

50 mL 40 mL 30 mL 20 mL 10 mL

Find the volume of the figures. a

50 mL 40 mL 30 mL 20 mL 10 mL

Find the length, breadth, and height of the solids by counting the number of cubes. a

100 mL 80 mL 60 mL 40 mL 20 mL

20 mL

Set B

100 mL 80 mL 60 mL 40 mL 20 mL

50 mL 40 mL 30 mL 20 mL 10 mL

50 mL

Set A

50 mL 40 mL 30 mL 20 mL 10 mL

40 mL

Colour the beakers up to the amount of water written with each beaker.

60 mL

Compare the volumes of the solids.

A 9

The containers are partly filled with unit cubes. Find the volume of each container. a

10 A carton that measures 50 cm by 30 cm by 25 cm is packed with small cubical boxes. What is the volume of the carton?

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Word Problems 1

Juhi drinks 235 mL of juice from a pack of 1 L of juice. How much juice is left in

Ashish had two flasks. One flask can hold 850 mL of water, and the other flask

Seema’s bucket can hold six and a half litres of water. The water tank holds four

5 cups of water can fill a bottle and 5 cups of water can fill a bowl. 5 bottles of

the pack?

can hold 1 L 250 mL of water. How much water can the flasks hold together? times as much water. So, how much water does the tank hold?

water can fill a jug while 3 jugs and 7 bowls of water can fill a pail. If each cup holds 90 mL of water, what is the capacity of the pail?

Chapter 13 • Capacity and Volume

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3-D Shapes on Flat Surfaces

Let's Recall Two sisters, Isha and Misha, are enjoying playing on the seesaw in the park. They see some shapes but they can't remember their names.

Cuboid Cone

Cylinder Cube

Sphere

Let's Warm-up

Fill in the blanks with the correct names of the shapes. 1

The

is in the shape of a __________ (cube/cuboid). is in a __________ (cylindrical/cubical) shape. is in the shape of a __________ (cuboid/cone). is in the shape of a __________ (sphere/cone).

I scored _________ out of 4.

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3-D Shapes Real Life Connect

Isha and Misha are at the market. Both of them decide to eat something. Let us see what they have bought.

I have bought a sandwich.

Isha: I see that my can of juice is long and round.

Misha, I have bought a can of juice.

Misha: I see that my sandwich is triangular. They both wondered at the shapes of the things that they had bought.

The objects that we see in our daily lives are solid shapes. We can feel, touch, see and hold these objects. We call them three-dimensional (3D) shapes. The faces of 2-D shapes are joined together to form 3-D shapes. Solid shapes like a cube

, cuboid

, cylinder

, cone

, and a sphere

have faces, edges and corners.

Features of 3¯D Shapes

Vertex Edge

• The flat or curved side of a solid shape is called a face. • The line where two faces meet is an edge.

Face

• The point where two edges meet is a corner or vertex. The plural of vertex is vertices. Let us have a look at some objects, their shapes and the number of faces, edges, and vertices in each of them. Cube: 6 flat faces, 8 vertices, 12 edges

Cylinder: 2 flat faces, 1 curved face, 2 curved edges, no vertices Flat face Curved edges

Cuboid: 6 flat faces, 8 vertices, 12 edges

Chapter 14 • 3-D Shapes on Flat Surfaces

Maths Grade 5 Book_Chapter 13-17.indb 231

Curved face

Cone: 1 flat face, 1 curved face, 1 curved edge, 1 vertex Curved face Curved edge

Sphere: 1 curved face, no edges, no vertices

Rectangular pyramid 5 flat faces, 8 edges, 5 vertices

Vertex Flat face

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Let us read about two types of solid shapes: prisms and pyramids. Prisms

Pyramids

A prism is a 3-D solid shape with two identical flat

A pyramid is a 3-D solid shape with a flat base and

a cuboid are also called prisms.

meet at a point or vertex.

bases and rectangular lateral surfaces. A cube and Square Prism

Triangular Prism

6 flat faces 8 vertices 12 edges

Square Bases Example 1

three or more sides in the shape of triangles that Rectangular Pyramid

5 flat faces 6 vertices 9 edges

Triangular Bases

Triangular Pyramid

5 flat faces 5 vertices 8 edges

Rectangular Base

4 flat faces 4 vertices 6 edges

Triangular Base

Which of the given objects looks like a cone? 1

Among the above objects, object 3 looks like a cone. Example 2

Do It Together

Write the number of edges for the given shapes. 1

Sphere: 0 edges

Square pyramid: 8 edges

Triangular prism: 9 edges

Look at the objects and complete the table. 3-D Shape

Name

Cone

Cube

Faces

6 flat

Edges Vertices

12 1

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Drawing 3¯D Shapes

We can draw 3-D shapes on a square grid with the help of 2-D shapes. Let us draw the picture of a cube on a square grid.

Draw 2 squares as shown in the figure. Example 3

Draw a cylinder with the help of rectangles on a square grid.

Draw a rectangle as shown below.

Do It Together

Join the corners of the squares to make a cube.

Join the vertices of the rectangles using curved lines to form a cylinder.

Draw a cuboid on the square grid.

Do It Yourself 14A 1 Circle the shape that is a a Cube

b Cuboid

c Cone

d Cylinder

2 Write the number of vertices, edges and faces of a cone, sphere and a cuboid. 3 Write the number of faces and types of faces in a square prism and a cylinder. 4 Draw a cone on a square grid. 5 How is a prism different from a pyramid? How are they similar? Chapter 14 • 3-D Shapes on Flat Surfaces

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Word Problem 1 Rohan and Simi have pitched a tent each, for themselves, as shown in these

pictures. Name the shapes of the tents. What is similar and different in their tents?

Rohan's tent

Simi's tent

Nets and Views Real Life Connect

Isha and Misha went to the canteen and purchased two doughnuts. The shopkeeper took a paper cut-out and quickly packed the doughnuts in these beautiful boxes. The sisters wondered how a simple paper cut-out turned into a box.

Nets of 3-D Shapes

After having the doughnuts, the sisters unfolded the box. They removed the extra flaps so that the cut-out looked like this:

Remember!

Hence, we can say that a net is a 2-D figure that can be folded to form a 3-D shape.

A shape can have more than one net.

Let us look at the nets of a few more 3-D shapes.

Shape Cuboid

Cylinder

Cone

Triangular prism

Triangular pyramid

Net

Example 4

Which of these is not the net of a cube? 1

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Net 1 can be folded as: 1

Hence, it is a cube. 2

Net 2 can be folded as:

Did You Know? Drawings of bridges, structures, homes, etc. are created using nets.

As two faces are overlapping, it is not the net of a cube. Example 5

Select the correct net of the given shape. 1

The shape has a pentagon-base with 5 triangular faces. Hence, 3 is the correct net of the shape. Do It Together

Name the 3-D shapes formed by each net.

___________________

Cuboid

___________________

Do It Yourself 14B 1 Identify the net of the given shape. a

Chapter 14 • 3-D Shapes on Flat Surfaces

Maths Grade 5 Book_Chapter 13-17.indb 235

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2 Look at the net and identify the object it belongs to. a

3 Match the net with the object that has the shape formed by the net. b

Pyramid

Candle

Ice cube

Ice cream cone

4 Draw the net of the given shapes. a

5 Draw the net of a hexagonal prism.

Word Problem 1 Rishi and Megha made the net of

Rishi’s drawing

Megha’s drawing

a square-based pyramid. Who made the net correctly? Explain your answer with reasons.

Views of Cube Structures Isha and Misha saw their father make a structure with small wooden cubes, as shown in this image. He then asked them to look at the structure from each side and draw the 2-D shapes that they see. Misha who was standing in the front of the cube structure drew the front view. Isha was standing on the side so drew the side view. Which is the third view?

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Misha’s Drawing

Isha’s Drawing

Front view

Side view

Which view is this?

The third view is the top view. Example 6

Identify the top, front and side view of the cube structure.

Top view Example 7

Front view

Side view

Which is the top view of the shape? 1

1 is the side view of the shape. It is also the front view. But, 2 is the top view of the shape. Do It Together

Colour the square grids to make the top, front and side view of the object.

Front view

Top view

Side view

Do It Yourself 14C 1 If you look at this object from the front, what will you see? a

Front view

Chapter 14 • 3-D Shapes on Flat Surfaces

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2 Draw the top view of the shape.

3 Draw the views of these figures. a Right-side view of

b Top view of

c Front view of

d Top view of

4 This table is made of many cubes joined together. Draw and label three different views of the table.

5 M eena drew three different views on square grids. Identify which cube structure is like the views made by her.

Front View

Top View

Right-side View

Cube structures: a

Word Problem 1 M ehar has made a shape using her building blocks. Count and write the number of building blocks she used to make the shape. Also draw the front, side and top view of the shape.

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Maps and Floor Plans Real Life Connect

Isha and Misha are at the park. Isha wants to buy a gift for their mother from the nearby mall. Isha: Let us go to the mall, Misha. Misha: But, I don’t know the way to the mall. Isha: Don’t worry, I have a map to go there.

Reading Maps We already know that a map is a drawing of all or parts of a particular place. Its purpose is to show where things are. Maps show rivers, forests, buildings, and roads in the form of symbols. Misha shows the map to her sister.

School

Fire Station

Post Office

Pond

House

Mall

Park

Library

She tells her sister that they need to step out of the park, move left and take the first right. On going straight they will see the mall on the left. We can also see directions on a map. The 4 directions on a map are East, West, North and South. North Can we show a distance of 1 km or more on a piece of paper?

When we show a big area on a piece of paper, we have to reduce the area to fit the dimensions of the paper. This reducing of the actual size of a place to fit on a piece of paper is called scaling. Chapter 14 • 3-D Shapes on Flat Surfaces

Maths Grade 5 Book_Chapter 13-17.indb 239

West

East South

239

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Tilak Marg

Man Singh Road 4

4 cm

In this map, the scale is 1 cm = 10 km. Let us find the actual distance between National Stadium and India Gate with the help of the map.

Rajpath

Pandara road 3 cm

Distance between National Stadium and India Gate on the map = 3 cm

E S

Kasturba Gandhi Marg

India Gate

2 cm

Sher Shah Marg National Stadium Scale: = 1 cm = 10 km

Scale on the map = 1 cm = 10 km Actual distance between National Stadium and India Gate = 3 × 10 km = 30 km. We can also use a square grid to find and compare the areas of different places.

3 cm

This is a map with the directions and scale marked on it.

Error Alert! Do not miss the scale while reading a map in order to find the distance.

Think and Tell

If you had to draw the pictures in a 1 cm grid, what would you 2 change?

1 cm grid

Look at the map and answer the questions. 1

hich places are to the north and the east W of the Little Town neighbourhood?

Little Town Lake Lake St.

uhani is standing close to the Little Town S Lake. In which direction is the post office from her location?

Little Town Neighbourhood

Little Town Park

Rail St.

Post Office

Ans: The post office is towards the south from Suhani's location. Do It Together

Library

Hospital

Ans: The hospital and the library are to the north and the Little Town Park is to the east of the Little Town neighbourhood. 2

Park St.

Cross St.

Post Rd.

N Answer the questions and also find the area of different places given on the cm grid W E Y X map. S

The city park is _____ of the highway.

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Maths Grade 5 Book_Chapter 13-17.indb 240

Highway

Example 8

2 cm grid

City Park

Hotel

Z City Lake

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Post Office

The hotel is _____ east of the highway.

The area of the city park is ________ 12 cm2 on the grid.

The area of City Lake is ___________ on the grid.

The area of the highway is ___________ on the grid.

Post Rd.

N Y

E S

C Highway

A City Park

Hotel

Z City Lake

Do It Yourself 14D North

1 Look at the map and answer the questions. a The hospital is to the _____ of the airport.

Town Y

b The hotel is to the _____ of the bank.

3.5 cm

c The airport is to the _____ of the school.

House

2 On a map, the distance between two cities is 5 cm. Taking the scale as 1 cm = 25 km, what would

be the actual distance between these two cities?

Market School

Airport

Police College Station

West Hospital

6 cm Cafe

3 What is the actual distance between the complex and the college?

3.5 cm

House College Market

2 cm 6 cm

Library

Complex

East

2 cm Bank Library

Complex

Hotel

scale: South 1 cm = 12 km

4 W hat is the length and width of the cupboard, table and desks? Map scale: 1 square = 2 m Key:

Town Y

School

Cupboard Table Desk Chair Door Window

scale: 1 cm = 12 km

5 The distance between Ronnyʼs house and the school is 5 cm on the map. What is the actual distance if the scale is 1 cm = 2.5 km? Cupboard Table Desk

Chapter 14 • 3-D Shapes on Flat Surfaces

Chair Door

241

Window

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Word Problem 1 M ihir goes to the stadium for football practice after school. On the map, his house is 5 cm from the school and the stadium is 2 cm farther from the school. What is the actual distance from his house to the stadium if the scale of the map is 1 cm = 5 km?

Floor Plans and Deep Drawings While going to the shopping mall, the two sisters saw many buildings and houses that were being made. They also saw a sheet of paper in the hand of one of the architects. Curious, they asked the architect what that sheet of paper was. The architect told them that it was the floor plan of a house. A floor plan is the outline of a house. It is like a map or a net that is drawn on a square grid. We use floor plans to make the map of a house when we start to design it.

Remember! Sometimes it is not possible

It is a 2-D shape that illustrates where the windows and to show all the windows and doors of the house will be located, The special way of doors on a deep drawing. drawing a house which is deep, so as to show its length, width and height is called a deep drawing of a house. A deep drawing of a house is a 3-D representation of the map of a house. Look at the drawings of a floor plan and a deep drawing of a house as given below.

Window

Window Window

Door

Window

Floor Plan Example 9

Deep Drawing

Compare the floor map and deep drawing and identify if the drawing is correct. Floor Map

Deep Drawing

Window Window

Door Door Window Window 242

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The door is between two windows and on the left-hand side in the floor map and deep drawing. On comparing, we see that both the floor plan and deep drawing have 2 windows and 1 door placed exactly at the same side. Hence, the deep drawing is correct for the given floor plan. Do It Together

Label and count the number of windows and doors in the floor plan. Does the deep drawing given on the side match the floor plan? Why? We can label the windows and door on the floor map as shown below. Window

Window

Door

Window

There are _______ windows and _______ door in the floor map. Window

Window

__________________________________________________________________________________________ Window Door Window Window

Window

Door

Window

Do It Yourself 14E Door

Window

Door Door

1 Look at the floor mapWindow of the house and label the doors and windows. Window

Window

2 How many window(s) does the front of the Door house Door have in the floor map? Window

Window

Door

Window

Door

Window

Window Door

Window Window

Window

Door

3 Draw the floor planWindow for the given deep Door drawing.

4 Which room will you Door be in if you enter through the back door? Door

Door

Door Door

Window

Bedroom

Back door Kitchen Window

Window

Door

Door Door

Bathroom

Bedroom

Dining Living room Front

Window

5 Which are the window(s) that you cannot show in a deep drawing? Chapter 14 • 3-D Shapes on Flat Surfaces

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Word Problem 1 Rakesh made the floor plan of his house which is in a cuboid shape. He had 4 windows on

each side of his house. How many windows can he not show in the deep drawing of his house?

Points to Remember • All 3-D objects have faces, edges, and/or corners. • All 3-D objects can be seen from 3 different views—top, side view and front view. • 2-D shapes that are folded to make 3-D shapes are called nets. • A floor map of a house shows where the doors and windows are in the house. • A deep drawing of the room shows the length, breadth, and height of the room.

Math Lab Exploring Views and Deep Drawings Setting: In groups of 5

Materials Required: Colourful blocks, sketch pens, a sheet of drawing paper Method:

1 Take out the colourful blocks which you have brought from home and place them on the table.

2 The groups can now build a tower of any shape using those blocks. 3 Now, take a sheet of drawing paper and draw the top, side and front views of the tower. 4 Now draw doors and windows in the blocks to show the deep drawing of the tower. Let the other groups draw the floor plan of this deep drawing.

5 The group that makes the submission first wins.

Chapter Checkup 1 Join the corners of these figures and write the name of the solid. a

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2 Fill in the blanks. a

A pattern that can be cut and folded to make a model of a 3-D shape is called a ___________________.

A special way of drawing a house which is deep to show its length, width and height is called a

___________________ and ___________________ have same number of faces, edges, and vertices.

The 2-D representation of the map of a house is called a ___________________.

___________________.

3 Identify the shapes and write the number of faces, edges and vertices of each shape. Objects a

______________

4 Draw the net of these shapes. a

Fro

Side

5 Draw the top, front and side views of the figures. a

Side

Fro

Front

Side

Front

Side

Front

Side

Fro

Front

6 Match the distance on the map with the actual distance, if the scale of the map is 1 cm = 8 km. Distance on the map

Actual distance nt

4 cm

8 cm

15 cm

32 km

25 cm

256 km

32 cm

64 km

Side

Fro

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Maths Grade 5 Book_Chapter 13-17.indb 245

120 km

Front

Side

200 km

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Sid

7 The floor plan of a library is ready. Draw the doors and windows on the deep drawing of the library. Window Window

Door Door

Door Window

Window

KITCHEN

Window

BEDROOM

LIVING ROOM

Is there any window which you could not show on the deep drawing? Circle them. How many windows were you able to show on the deep drawing?

8 How many doors do you see in the floor plan?

9 Look at the layout of Ruchi’s apartment. How many times bigger is the bathroom than closet 1?

Apartment Floor Plan A

KITCHEN

Bedroom 1

BEDROOM

LIVING ROOM

KITCHEN

BEDROOM

Closet 2

Bedroom 2 C

LIVING ROOM

m roo

th Ba

Storage loset 1

KITCHEN

Hall

Window Window

Window

Window Window

Window

BEDROOM

LIVING ROOM

Living Room

Kitchen

Apartment Floor Plan A om

hro

Closet 2

Bedroom 2

om hro

Bedroom 1 Bat

Scale: 1 cm = 1 m

Storage loset 1

Apartment Floor Plan A

Hall

Bedroom 1 Bat

Hall

Apartment Floor Plan A Scale: 1 cm = 1 m C

Storage loset 1

Closet 2

Living Room and Kitchen house show it to your friends/family. 10 Draw a floor plan of your Bedroom 2

Kitchenom

Scale: 1 cm = 1 m Closet 2

Bedroom 2

en goes from his house to May’s house B Living Room Kitchen and then to Jule’s house. What is the total distance covered by Ben?

Jessleʼs house Patʼs house Carlʼs house

2 cm 3 cm

Scale: 1 cm = 1 m

Benʼs house

Hall

Word ProblemBedroom 1 Bathro

Storage loset 1

Living Room

Tonʼs house Wayneʼs house

Mayʼs house

4 cm

Scale: 1 cm on the paper = 3 km

Danʼs house

Juleʼs house

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Time and Temperature

Let's Recall Time is the measure of each moment in our life. The basic unit of time is the second. We also use units like minutes, hours, days, weeks, months, and years to measure longer periods of time. For example, brushing our teeth takes a few minutes; tying our shoelaces takes a few seconds; we spend many hours at school. We have a school timetable for a week, while the summer holidays last for months and birthdays come once a year. Clocks are used to read time. The hour hand is the shortest hand, which points to the current hour, and the minute hand is the longest hand, which shows the current minute. There is also a thin hand that moves the fastest. It measures the seconds.

Letʼs Warm-up

Match the events with the time they take. 1

Taking a shower

Months

Putting on a jacket

Days

Going for a trip

Minutes

Swimming lesson

Seconds

Rainy season

Hours

I scored _________ out of 5.

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Time Real Life Connect

Kiran and Sahil are in the school cricket team. They are talking about the amount of time they spend in their cricket practice. Kiran: Sahil, how much time do you spend practising cricket? Sahil: I practise for about 90 minutes every day. My brother practises for even longer. He spends almost two and a half hours on the field every day. Kiran: Wow! I just spent about half an hour practising.

Converting Between Units of Time We know that time can be measured in different units. We can also convert the units of time from one to another.

Converting From a Bigger Unit Into a Smaller Unit We can convert units of time from bigger to smaller as shown below. × 60 Hours

× 60 Minutes

Seconds

Kiran and Sahil played cricket for 3 hours. Let us convert this time to minutes. In order to do so, we need to convert hours to minutes. We can do that by multiplying the number of hours with 60 as one hour is always equal to 60 minutes. So, 3 hours = 3 × 60 = 180 minutes

Did You Know? 1 millennium is 1000 years.

Hence, they played for 180 minutes. Example 1

How many minutes are there in 2 hours and 30 minutes? 1 hour = 60 minutes 2 hours and 30 minutes = (2 × 60) + 30 minutes = 120 + 30 = 150 minutes

Example 2

Neethu takes 6 minutes to walk to her school bus from her home. For how many seconds does she walk? Time spent walking = 6 minutes 1 minute = 60 seconds

Think and Tell

How many seconds are there in a day?

6 minutes = 6 × 60 seconds = 360 seconds Neethu walks for 360 seconds. 248

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Do It Together

Raj and his friend Tina were eager to watch the solar eclipse. But it would happen after 4.5 hours. How many minutes would they have to wait for the eclipse? Time remaining before the eclipse = _____ hours. 1 hour = _____ minutes 4.5 hours = 4.5 × _____ = _____ minutes So, there are _____ minutes remaining before the solar eclipse.

Converting From a Smaller Unit Into a Bigger Unit We can convert units of time from smaller to bigger as shown below. hours

minutes ÷ 60

seconds ÷ 60

Kiran practised playing badminton for 75 minutes in the morning. She again practised for 50 minutes in the afternoon. Let us find the total number of hours she practised playing badminton. In order to do so, we must divide the bigger units of time by 60. Thus, the total minutes of practice by Kiran = 75 min + 50 min = 125 minutes Dividing 125 by 60 can be given as: Therefore, Kiran played badminton for 2 hours and 5 minutes. Example 3

2 60 125 120 5

2 hours 5 minutes

How many hours and minutes are in 220 minutes? Total minutes = 220 60 minutes = 1 hour 220 1 × 220 = 220 minutes = 60 60 Quotient = 3

3 60 220 180 40

3 hours 40 minutes

Remainder = 40 So, 220 minutes is equal to 3 hours and 40 minutes. Do It Together

Kiran and Rohit were timing how long their friend Shreya could juggle some tennis balls. Shreya managed to juggle for 250 seconds. How many minutes and seconds is that? Number of seconds for which Shreya juggles = 250 seconds We know that 60 seconds = 1 minute 250 seconds =

× 250 =

250

Thus, Shreya juggled the tennis balls for _____ minutes and _____ seconds. Chapter 15 • Time and Temperature

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250

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Do It Yourself 15A 1 Convert into minutes. a 3 hours

d 3 hours 10 minutes

b 4 hours

c 2 hours 20 minutes

b 8 minutes

c 9 minutes 10 seconds

e 3 hours 50 minutes

4 hours 30 minutes

2 Convert into seconds. a 5 minutes

d 10 minutes 20 seconds

e 12 minutes 40 seconds

15 minutes 50 seconds

3 Fill in the blanks. a 120 minutes = _____ hours c 150 minutes = _____ hours _____ minutes

d 200 minutes = _____ hours _____ minutes

Match the seconds with their conversions. a 240 seconds

5 minutes 50 seconds

b 480 seconds

8 minutes 40 seconds

c 350 seconds

4 minutes

d 440 seconds

9 minutes 10 seconds

e 520 seconds

7 minutes 20 seconds

b 180 minutes = _____ hours

550 seconds

8 minutes

How many seconds are there in 1 hour and 20 minutes?

Word Problems 1 Maya reads a book for 50 minutes in the morning and 45 minutes in the afternoon. For how much time does she read the book?

2 A train takes 8 hours and 30 minutes to reach its destination. How many minutes is the journey?

3 Sarah practised playing the piano for 55 minutes in the morning and 50 minutes in the evening. How much time did she practise in total?

Calculating Duration The students in the school cricket team are of different ages. Sahil is 10 years and 5 months old whereas Manish is 12 years and 3 months old.

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Let us find how much older Manish is than Sahil. Step 1

Step 2

Write the information.

Apply the correct operation.

Sahil Age = 10 years 5 months

We need to find how much older Manish is than Sahil, so we will subtract their ages.

Age of Manish = 12 years 3 months

Step 3 Write the data vertically and perform the operation.

3 + 12 = 15

12 years

3 months

10 years

5 months

1 years

10 months

Error Alert! 1 year = 12 months 4

6 + 10 = 16

5 years - 2 years

6 months 8 months

2 years

5 years - 2 years 2 years

8 months

6 + 12 = 18 6 months 8 months 10 months

Hence, Manish is 1 year 10 months older than Sahil. Example 4

On 20 November, Raj and Tina began counting down the number of days left for their school's annual sports day. The annual sports day is on 15 December. Find out the number of days they need to count down. The annual sports day is on 15 December.

Raj and Tina start the countdown from 20 November. We know November has 30 days.

Number of days left in November from 20 November to 30 November = 10 days

Number of days in December starting from 1 December to 15 December = 15 days So, adding the number of days = 15 + 10 = 25

Therefore, they are counting down 25 days till the sports day. Example 5

Komal went to her dance class for 7 weeks and 4 days. She also joined a painting class for 3 weeks and 5 days. How many weeks and days did she spend in the two classes? Step 1

Step 3

Write the information.

Write the data vertically and perform the operation.

Time spent in dance class = 7 weeks 4 days

Time spent in painting class = 3 weeks 5 days

7 weeks

4 days

+ 3 weeks

5 days

Step 2

10 weeks

9 days

Apply the correct operation.

We need to find the total time spent in both the classes; hence we will add the duration.

Komal spent 11 weeks and 2 days in the two classes. Chapter 15 • Time and Temperature

Maths Grade 5 Book_Chapter 13-17.indb 251

7 + 2 days 1 week

11 weeks 2 days

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Do It Together

Ananya planted a tree on 5 June 2015, and it reached its full height on 10 August 2023. How many years and months did it take for the tree to grow to its full height? Start Year: 2015; End Year: ______

Remember! 7 days = 1 week

Start Month: June (6th month); End Month: ______ (_____ month) Difference in years: ____________________ years Difference in months: ____________________ months The tree took _____ years and _____ months to grow to its full height.

Do It Yourself 15B 1 Find the sum. a 3 weeks 10 days + 6 weeks 5 days

c 2 years 6 months + 1 year 5 months

b 5 weeks 2 days + 25 days d 5 years + 11 months

2 Find the difference. a 7 weeks 20 days – 2 weeks 5 days

c 11 years 7 months – 2 year 4 months

b 10 weeks 10 days – 5 weeks

d 3 years 1 month – 7 months

3 Suhani is 13 years and 5 months old. Her brother Kunal is 3 years 10 months older than she is. How old is Kunal?

4 A new museum opened on 25 October 2020, and closed on 15 February 2022. For how many months and days was the museum open?

5 Manya is 15 years 7 months old. Her friend Shubhi is 8 months younger than she is. Her sister Megha is 3 years 5 months younger than Shubhi. How old is Megha?

Word Problems 1 A school was closed for 1 month 25 days in the summer vacation and for 20 days in the winter vacation. For how many months was the school closed? (Consider 30 days = 1 month) 2 Sam started learning to play the violin on 10 June 2019, and reached an advanced level on 5 December 2023. How many years and months did it take for Sam to become an advanced violinist?

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Calculating Time Sahil and his team practised for 3 hours 45 minutes on Saturday and 4 hours 25 minutes on Sunday. Let us find the total time they practised over the weekend.

3 hours

45 Minutes

+ 4 hours

25 Minutes

7 hours

70 Minutes

60 + 10 Minutes

Time spent practising on Saturday = 3 hours 45 minutes

1 hour

Time spent practising on Sunday = 4 hours 25 minutes

8 hours

We need to find the total time spent on practising over the weekend; hence, we will add the duration. Sahil and his team practised for 8 hours and 10 minutes over the weekend. Example 6

Remember! 60 minutes = 1 hour

Arun practises yoga from 4:45 p.m. to 5:25 p.m. For how long does he practise yoga? Start Time: 4:45 p.m. = 4 hours 45 minutes End Time: 5:25 p.m. = 5 hours 25 minutes Since both times are on the same day, we will subtract the time to find the duration. Hence Arun practises yoga for 40 minutes.

Example 7

10 Minutes

25 + 60 = 85

4 5 hours

25 Minutes

- 4 hours

45 Minutes

4 hours

85 Minutes

- 4 hours

45 Minutes

0 hours

40 Minutes

Mahi started preparing for his match on 23 June. The match is scheduled for 25 days later. On which date was the match scheduled? Starting date of the preparation = 23 June; Preparation time = 25 days Date on which the match was scheduled = ? We can find the match date by counting forward. 23 June to 30 June = 30 - 23 = 7 Days left after the month of June = 25 - 7 = 18 Mahi’s match is on 18 July.

Do It Together

A science experiment began at 2:30 p.m. and lasted for 3 hours and 45 minutes. When did the experiment end? Starting time: 2 hours 30 minutes Time which the experiment lasted = ____ hours ____ minutes We need to find the time at which the experiment ended, hence we will add the data.

2 hours

30 Minutes

+ 3 hours

45 Minutes

The experiment ended at ____ hours ____ minutes = ____ p.m.

Chapter 15 • Time and Temperature

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Do It Yourself 15C 1 Solve. a 5 hours + 55 min

c 7 hours 5 min – 5 hours 30 min

b 4 hours 30 min + 6 hours 20 min d 10 hours 10 min – 45 min

2 Can you tell the time that the clock will show after 3 hours and 10 minutes, if the clock shows 10:30 a.m. now?

3 If a meeting started at 9:30 a.m. and ended at 12:45 p.m., how long did the meeting last in hours and minutes?

4 If it’s 2:40 p.m. now, what time will it be 35 minutes later? 5 A train stops for 40 minutes 25 seconds during the day and for 45 minutes and 40 seconds at night. Calculate the duration of the train stops.

Word Problems 1 An online class started at 10:00 a.m. and ended at 11:30 a.m. How many minutes did the class last?

2 A school event started at 2:00 p.m. and ended at 5:30 p.m. How many hours and minutes did the event last?

3 If a film begins at 7:15 p.m. and finishes at 9:30 p.m., how long is the duration of the film?

4 Raman started driving at 11:20 am. He reached his destination at 4:45 pm. How long did it take him to reach his destination?

Temperature Real Life Connect

Riya and Aarav were spending their summer vacation camping at a hill station. Riya noticed that the nights were colder than the days. She asked Aarav, “Why is it so chilly in the mountains?” Aarav explained that the level of heat or cold was determined by the temperature. It could change based on the object, location and time of day. Curious to learn more, Riya and Aarav decided to find out how hot or cold the objects around them were.

Think and Tell

Can you name the coldest and hottest place you have been to?

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Measuring Temperature Temperature is the measurement of how much heat an object or place has or does not have. It’s a number that tells us how warm or cool something is. We use different units to talk about temperature. We use a tool called a thermometer to measure temperature. The thermometer has a scale that helps us measure the temperature. When it’s hot, the scale goes up, and when it’s cold, the scale goes down. This helps us to know if things are hot or cold. The units we use to measure temperature are 1

Celsius (°C)

Fahrenheit (°F)

However, when we talk or write the temperature of something, we also use the word degrees along with the unit. For example, the temperature shown in the thermometer is 30℃.

Did You Know?

Mercury is a metal that can change its state from liquid to solid and vice versa at different temperatures. It’s used in thermometers because it expands when it gets hot and contracts when it gets cold.

Remember! Freezing point of water = 0℃ Water changes to steam = 100℃

Measuring Body Temperature Clinical thermometers are used to measure body temperature. They have numbers from 35°C to 42°C. The normal body temperature for a healthy body is approximately 37°C.

A thermometer is usually placed under the tongue, in the armpit, or in the ear to get an accurate reading. The mercury in the thermometer rises when the temperature increases or gets hotter. This helps us understand if we have a fever or if our body temperature is normal.

Chapter 15 • Time and Temperature

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Example 8

During her fever, Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature? Normal body temperature = 37°C. Aliya’s body temperature during her fever = 37°C + 2.3°C = 39.3°C

So, Aliya’s body temperature was 39.3°C. Do It Together

Write the readings of the given thermometers.

Temperature = 45°C

Temperature = ____

Measuring Temperature Around Us We can measure the temperature of the air around us, which tells us if it’s a warm or chilly day. We also measure the temperature of objects like water or food to know if they’re hot or cool. Knowing the temperature helps us make choices about what to wear or how to handle different things. The temperature can change with different weather conditions. On a sunny day, the air feels warm, and the thermometer shows a higher temperature. On a cloudy or rainy day, the air feels cooler, and the temperature on the thermometer drops. Weather conditions like rain, wind, and sunlight can all affect the temperature of the air. Temperature Range (°C) Below 0

Very Cold

11–20

Mild

0–10

Cold

21–30

Warm

Above 40

Very Hot

31–40

Example 9

Weather

Hot

Did You Know? Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

In a city, the highest temperature was 37.5°C, and the lowest was 21.2°C. What is the difference between these two temperatures? To find the difference, subtract the lowest temperature from the highest temperature. 37.5°C – 21.2°C = 16.3°C The temperature difference is 16.3°C.

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Do It Together

If the temperature early in the morning was 12°C and warmed up by 8°C due to the sun, what was the temperature during the day? Initial temperature

= 12°C

Increase in temperature = _______ New temperature

= _______

Do It Yourself 15D 1 Choose the correct temperature. a The temperature of a cup of hot coffee is around

i 30°C.

ii 85°C.

b What is the possible temperature of a snowman?

i 0°C

ii 50°C

2 Look at the thermometer and write the temperature.

_________________

3 The temperatures of different cities at different times are as follows. Answer the questions based on the data.

a Which place is the coolest? b Which place is the hottest? c What is the difference between the temperatures of Delhi

and Shimla?

d How many degrees will the temperature need to rise in

Bangalore to reach 25°C?

City

Temperature (°C)

Chennai

44.5

Bangalore Jaipur

Srinagar Delhi

Shimla

21.2 18.3 8.8

41.2 11.5

4 The highest temperature of a city was 25.6°C while the lowest temperature was 18.4°C. What is the difference between the temperatures?

5 Arrange the cities from the hottest to the coldest.

City A = 23.6°C, City B = 29.3°C, City C = 23.9°C, City D = 26.0°C

Chapter 15 • Time and Temperature

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Word Problem 1

On a certain day, the temperature of Jammu and Kashmir was 17.3°C and Kerala was 39.8°C. What is the difference between the temperatures of the two cities?

Points to Remember • To convert hours to minutes and minutes to seconds, multiply them by 60. To convert seconds to minutes and minutes to hours, divide them by 60. • • • •

Calculating duration requires subtracting the starting time from the ending time. Temperature measures heat or cold and is used for weather predictions. Body temperature is around 37°C for a healthy individual.

Different temperature ranges indicate different weather conditions.

Math Lab Time Trek Game Materials:

• Prepare event cards with different scenarios (e.g., “Birthday,” “Sports Day”).

• Prepare time unit cards (years, months, weeks, days, hours, minutes, seconds). • A calendar.

Teams: Divide the class into small teams. Activity Steps:

Each team draws an event card and a time unit card. They need to match the event with the appropriate time unit that represents how long it will take for the event to happen.

Duration Calculation: Teams use the calendar to visualise and calculate the duration.

Starting from the current date, they calculate days, weeks, months, or years to determine the timing of the event.

Presentation: Each team presents their matched event and time unit, explaining their calculation process.

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Chapter Checkup 1

Convert the time into smaller units. a d

2 hours

5 minutes 20 seconds

b e

1 hour 30 minutes

6 minutes 40 seconds

c f

2 hour 20 minutes

6 minutes 55 seconds

Fill in the blanks. a c

90 minutes = _____ hours _____ minutes

160 minutes= _____ hours _____ minutes

b d

115 minutes = _____ hours _____ minutes

280 seconds = _____ minutes _____ seconds

A flight takes 6 hours and 30 minutes to reach its destination. How many minutes does the flight take

Shreya sleeps for 6 hours 15 minutes a day. For how many minutes does she sleep?

A soccer match started at 3:45 p.m. and continued for 1 hour and 15 minutes. When did the match end?

Prashant was 5 years 4 months old when he first went to school. Today he is 12 years 3 months old.

Fill in the blanks.

to reach its destination?

For how long has he been going to school?

If a meeting starts at 11:00 a.m. and lasts for 1 hour and 20 minutes, it will end at _______.

If you start reading a book at 7:45 p.m. and read for 1 hour and 15 minutes, you will stop reading

An online class begins at 2:15 p.m. and runs for 50 minutes. The class will end at _______ p.m.

at _______.

The highest temperature in a city today was 37.5°C, while the lowest was 21.2°C. What was the

Find the finishing date if the starting date is 6 July and the duration is 40 days.

The doctor advised Kunal to take his medicine every 75 minutes. How many times did Kunal take the

temperature difference?

medicine in 12 hours?

Word Problems 1

riya has taken 15 days leave from her office. Her holidays will start on Saturday, P 25 June. On which day and date will the holidays end?

anya started to draw a picture at 1:32 p.m. She completed it at 5:15 p.m. How S much time did she take to draw the picture?

ara is in an air-conditioned room with a room temperature of 32°C. If she S decreases the temperature by 2°C every minute, how many minutes will it take to reach 16°C?

train arrived at Jaipur at 10:45 a.m. It reached Jaipur 1 hour 15 minutes late. A What was the scheduled time of arrival of the train at Jaipur?

Chapter 15 • Time and Temperature

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16 Money Let's Recall In India, money is used in the form of rupees (₹) and paise (p). We use it to buy or sell goods. Let us say we want to buy a pencil that costs ₹5.50. So, we can say, “The cost of the pencil is 5 rupees and 50 paise”, or we can say, “The cost of the pencil is five rupees and fifty paise.” ₹5.50

The number to the left of the decimal point shows the rupee amount.

The decimal point separates the rupee amount from the paise.

The number to the right of the decimal point shows the amount in paise.

How can we convert rupees to paise and paise to rupees? Remember that 1 rupee = 100 paise. Therefore, to convert rupees to paise, we remove ₹ and (.) and write p for paise at the end. For example, ₹7.25 = 7.25 × 100 = 725 p To convert rupees to paise, we count 2 digits from the right, put a dot (.) and write the rupee symbol (₹) before the numbers. Or, we divide the amount by 100 to convert it into rupees. For example, 725 p = 725 ÷ 100 = ₹7.25

Letʹs Warm-up Fill in the blanks.

Amount in Rupees

Amount in Paise

₹3.70

_______________________

412 p

₹6.10

_______________________

305 p I scored _________ out of 4.

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Working With Money Real Life Connect

Mr Gupta owns a stationery store. He buys stationery items from a wholesaler and sells them to the customers at his shop for a relatively higher price.

Think and Tell

What are the different coins and banknotes available in our country?

Ram wants to buy 5 pens from Mr Gupta’s shop. Mr Gupta shows him a pack of 10 pens, which is available for ₹60. What should Ram do?

Unitary Method We often meet situations in which we know the value of a number of units, but we need to find the value of a specific quantity of the item. In this case, Ram knows the price of 10 pens, but he wants to buy only 5 pens. How will he find the price of 5 pens? Here are the steps he needs to follow: Step 1

Step 2

Find the cost of one pen.

Find the cost of 5 pens.

Cost of one unit = Total cost ÷ Number of units We divide the total cost of the pack of pens (₹60) by the number of pens (10). = ₹60 ÷ 10 = ₹6 So, the cost of 1 pen is ₹6.

We know the cost of 1 pen = ₹6

Cost of number of pens = Cost of one pen × number of pens

Multiply the cost of one pen (₹6) with the total number of pens (5). = ₹6 × 5 = ₹30

So, the cost of 5 pens is ₹30.

So, Ram should give Mr Gupta ₹30 for the number of pens he wants. This way of finding the value of a single unit and then using that value to find the value of multiple units is called the unitary method.

Chapter 16 • Money

Maths Grade 5 Book_Chapter 13-17.indb 261

Remember! 1. We divide to find the price of one unit. 2. We multiply to find the total amount of all units.

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Example 1

A bookstore sells 3 books for ₹210. How much would 7 books cost at the same price? Cost of 3 books = ₹210

Cost of 1 book = Total cost ÷ Number of books = ₹210 ÷ 3 = ₹70

Did You Know?

So, the cost of 1 book is ₹70.

Cost of 7 books = Cost of one book × number of books wanted = ₹70 × 7 = ₹490

So, the cost of 7 books is ₹490. Example 2

The Indian currency is called the Indian Rupee. Its short form is “INR”. It has a unique symbol ₹, which was adopted in 2010.

Which of these two will be a better buy? Or

Rani pays ₹40 for 5 balls.

Rani pays ₹56 for 8 balls.

When buying an item, a better buy is the one where we have to pay less money for more items. How should we decide in this case? We will compare the money to be paid in each case. The case where we pay less for 1 ball is a better buy. 1 Rani pays ₹40 for 5 balls. Cost of one ball = Total cost ÷ Number of balls = ₹40 ÷ 5 = ₹8

2 Rani pays ₹56 for 8 balls. Cost of one ball = Total cost ÷ Number of balls = ₹56 ÷ 8 = ₹7

So, ₹56 for 8 balls will be a better buy. Do It Together

Fill in the table using the unitary method. Cost of Multiple Items

Cost of One Item

Cost of Given Items

Cost of 3 pens = ₹60

Cost of one pen = ₹20

Cost of 5 pens = ₹100

Cost of 8 oranges = ₹400

Cost of one orange = ₹ ________

Cost of 12 oranges = ₹600

Cost of 4 notebooks = ₹2000

Cost of one notebook = ₹500

Cost of 3 notebooks = ₹________

Do It Yourself 16A 1

Fill in the blanks. a If the cost of one candy is ₹ __________ , then the cost of 7 candies will be ₹35. b If Lisa received ₹240 as pocket money for 4 weeks, the amount of money she received every week

was ₹__________.

c Riya saved ₹600 in 3 months. She saved ₹________ every month.

d A bakery sells 10 cupcakes for ₹400. They sell one cupcake for ₹________ and 55 cupcakes

for ₹________.

e If the price of a pack of 10 crayons is ₹50, then the cost of 3 crayons is ________.

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Find the cost of 20 candles at ₹15 per dozen. (1 dozen = 12 units)

The cost of 35 envelopes is ₹630. What will be the cost of 57 such envelopes?

A worker gets ₹200 for every 10 chores she completes. How many rupees will she earn if she

1 3 metres of cloth cost ₹54. Find the cost of metre of cloth. 2 4

completes 9 chores at the same rate?

Word Problems 1

A shopkeeper sells 15 kg of sugar for ₹450. Another shopkeeper is selling 20 kg of

Anita has ₹200 and wants to buy some chocolates that cost ₹25 each. How many

10 bags of rice cost ₹1230. If each bag contains 3 kg of rice, find the cost of 1 kg of rice.

sugar for ₹500. From whom is it better to buy sugar?

chocolates can she buy with her money and how much money will she have left?

Profit or Loss As we read above, Ram bought pens for ₹6 each. He now decides to sell the pens he bought to his friends. He sold one pen to Om for ₹10, which is more than the amount he paid for the pen. When the amount of money earned by selling an item is more than the amount of money spent, the seller is said to have gained a profit.

If the amount of money Ram earned is ₹10 and the amount of money he spent is ₹6, then how much profit did he gain?

• Profit earned = Amount earned by selling the item – Amount spent to get the item = ₹10 – ₹6 = ₹4

Remember! Profit is gaining money. So, the money earned by selling is more than the money spent to get the item. Loss is losing money. So, the money spent to get the item is more than the money earned by selling.

Think and Tell

Why do you think Ram sold the ₹6 pen for ₹10?

Now, what if Ram had sold the pen to Om for ₹4? We know that he bought the pen for ₹6. When the amount of money earned is less than the amount of money spent, the seller is said to have incurred or suffered a loss. Chapter 16 • Money

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If the amount of money Ram would have earned is ₹4 and the amount of money he spent to get the pen is ₹6, then how much loss did Ram incur?

• Loss incurred = Amount spent to get the item – Amount earned by selling the item = ₹6 – ₹4 = ₹2

Cost Price and Selling Price The amount of money Ram paid to Mr Gupta for a pen (₹6) is said to be the cost price of the pen. • Cost price (CP) is the amount of money you spend to buy an item. The amount of money Ram earned by selling the pen to Om (₹10) is called the selling price of the pen. • Selling price (SP) is the amount of money you sell an item for. Therefore, we can understand Profit and Loss in terms of CP and SP.

A profit is gained when SP is more than the CP. That is, we earned more money on selling the item than the amount we spent to get it. • Profit = SP – CP

Think and Tell

A loss is incurred when the CP is more than the SP. That is, we spent more money to get the item than the amount we received on selling it. • Loss = CP – SP Example 3

Do you think exchanging ₹5 candy for a ₹10 chocolate bar leads to a profit or loss? Why?

Shreya bought a toy for ₹150 and sold it for ₹100. Did she make a profit or a loss? How much was it? Cost price (CP) of the toy = ₹150

Selling price (SP) of the toy = ₹100

Since, CP > SP, Shreya incurred a loss.

Remember!

Loss = CP – SP = ₹150 – ₹100 = ₹50

SP > CP → Profit CP > SP → Loss

Thus, Shreya incurred a loss of ₹50. Do It Together

Complete the table. Statement Mary bought a necklace for ₹500 and sold it for ₹750. A shopkeeper bought a football for ₹200 and sold it for ₹500.

Yash bought a pair of shoes for ₹800 and sold them to his friend for ₹540.

Cost Price (CP)

Selling Price (SP)

Profit/Loss

Profit gained/ Loss incurred

₹500

₹750

SP > CP Profit

₹750 – ₹500 = ₹250

________

₹500

________

________________

________

₹800 – ₹540 = ₹260

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Do It Yourself 16B 1

Find the missing value. a CP = ₹1200, Profit = ₹85, SP = ?

b SP = ₹200, Profit = ₹55, CP = ?

c Loss = ₹550, SP = ₹7850, CP = ?

d CP = ₹525, Loss = ₹70, SP = ?

Fill in the blanks. a When the cost price is higher than the selling price, then

b When the selling price is

a profit.

than the cost price, then the seller is said to have gained

c The amount of money that a seller spends to pay for an item is called

d A flower vase is sold at ₹725 at a profit of ₹225. Its CP is

A shopkeeper bought 30 buckets for ₹1200. Find the selling price of each bucket, if he wants to make

Khushi bought a dress for ₹2500. She spent ₹435 on accessories to make the dress look more

A man bought 350 books at the rate of ₹70 each. He sold 160 books at the rate of ₹80 and the

a profit of ₹27 on each bucket.

attractive. She then sold the dress to her friend for ₹3500. Find the profit or loss.

remaining at the rate of ₹25 each. Find his profit or loss on the whole transaction.

Word Problems 1

Mita buys 8 sharpeners for ₹40. She sold each sharpener at a profit of ₹2. What is the

A shopkeeper buys 3 kg of oranges for ₹510 and 4 kg of apples for ₹456. He sells the

selling price of each sharpener? What is the total profit?

oranges at ₹32 per kg. At what price (per kg) should he sell the apples so that he is able to recover his loss and break even (no profit - no loss)?

Word Problems in Money Real Life Connect

Seema is the owner of a company that makes and sells toys. It costs her ₹80 to make two toys and they are sold at ₹100 each. Seema wants to find out how much profit they make on one toy and how much profit they make on 70 toys. What should she do?

Chapter 16 • Money

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Profit gained on each toy Total profit gained on each toy

Total profit gained on 70 toys

CP of one toy = Total cost ÷ Number of units

CP of 70 toys = ₹40 × 70 = ₹2800

= ₹80 ÷ 2 = ₹40

SP of 70 toys = ₹100 × 70 = ₹7000

CP = ₹40, SP = ₹100

Total Profit = Total SP – Total CP

Profit = SP – CP

= ₹7000 – ₹2800 = ₹4200

= ₹100 – ₹40 = ₹60

Thus, the total profit gained on selling 70 toys is ₹4200.

Thus, the profit gained on each toy is ₹60.

Did You Know? Remember! When an item is sold at the same price as its cost price there is no profit or loss.

Example 4

Earlier, people used to follow a barter system where goods and services were directly exchanged without using money.

Amy bought two shirts for ₹850 and sold both of them for ₹300 each. Calculate her profit/loss for: 1 One shirt

2 Two shirts

1 For one shirt:

2 For two shirts:

CP of one shirt = Total cost ÷ Number of units

Total CP of two shirts = ₹850

= ₹850 ÷ 2 = ₹425

Total SP of two shirts = ₹300 × 2 = ₹600

CP = ₹425, SP = ₹300

Now, since CP > SP, this means Amy incurred a loss.

Total Loss = Total CP – Total SP

Loss = CP – SP = ₹425 – ₹300 = ₹125

= ₹850 – ₹600 = ₹250

Thus, the loss incurred on one shirt is ₹125. Thus, the total loss incurred on two shirts is ₹250. Do It Together

Vansh bought a box of 20 chocolates for ₹200. He sold all the chocolates to the kids in the neighbourhood for ₹20 each. How much profit/loss did he make on: 1 One chocolate?

2 All chocolates?

Error Alert! Do not jump to the answers directly. Always follow the steps!

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1 One chocolate

2 All chocolates

CP of each chocolate = Total cost ÷ Number of units

Total CP = ₹200

= __________ ÷ 20 = ₹10

Total SP = ________

SP of each chocolate = ________

Since, SP > CP, Vansh gained a profit.

Total Profit = Total SP – Total CP

Profit = SP – CP

=_______–_______

=_____–_____

hus, the profit gained by Vansh is T ________.

Thus, the total profit gained by Vansh is ________.

Do It Yourself 16C 1

Neha bought 9 cups for ₹270 and sold them at ₹50 each. Calculate the profit/loss for each cup.

Janvi bought 27 balls for ₹9 each. She sold them at a shop for a total of ₹250. Calculate the total

The school purchased 150 books for ₹3000. If they sold each book at ₹15, did they make a profit or

Rani bought 7 apples for ₹49 and sold each apple for ₹5. Determine her total profit or loss.

A shopkeeper bought 20 pencil boxes for ₹480. If he sold each pencil box for ₹50, calculate the profit

Ayan bought an old smartphone for ₹5620. He spent ₹530 to repair it. He then sold the smartphone

Aryan bought a new bicycle for ₹7830 and spent ₹170 on transportation. He then sold the bicycle for

Mohit bought 15 toys for ₹990. He then purchased another 15 toys for ₹1080. He also spent ₹50 on

Sahil sold his TV for ₹30,000 at a profit of ₹1563. Find the CP of the TV.

profit/loss.

loss? How much was the profit or loss on each book?

or loss on each pencil box.

to his friend for ₹6150. Did Ayan make a profit or a loss and by how much? ₹8000. Find the profit or loss.

packing all the toys. If he sold all the toys for ₹70 each, calculate the profit or loss.

10 A shopkeeper lost ₹1750 on selling a refrigerator for ₹18500. What was the CP of the refrigerator?

Points to Remember •

We use the unitary method to find the value of a single unit from multiple units. Cost of one unit = Total cost ÷ Number of units

• •

Cost Price (CP): Amount of money you spend to buy an item. Selling Price (SP): Amount of money you sell an item for.

• Profit: When the selling price is higher than the cost price (SP > CP), then the seller gains a profit which is the difference between the SP and CP. Chapter 16 • Money

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• Loss: When the selling price is less than the cost price (SP < CP), then the seller incurs a loss, which is the difference between the CP and SP.

• When the selling price is equal to the cost price (SP = CP), then the seller neither gains a profit nor incurs a loss.

Math Lab Setting: In 2 groups Materials Required: Pencils, notebooks, paper price tags, play money (can be made out of paper)

Instructions: 1 Set up a mini store with items like pens and notebooks on the table with labels showing the cost price of each item.

2 Divide the class into two groups, shopkeepers and customers. 3 Shopkeepers should sell the items by stating a selling price to the customers and the customers will buy items using play money and the shopkeepers will write down the selling prices.

4 Calculate the profit or loss after each transaction and discuss the results.

Chapter Checkup 1

Fill in the blanks. a

____________ is used to find the value of a single unit and then using that value to find the value of

If the cost of one apple is ₹____________, then the cost of 9 apples will be ₹54.

Rama bought a ring for ₹260 and sold it for ₹500. The cost price of the ring is ₹____________.

When the cost price is higher than the selling price, then the seller is said to have ____________.

State true or false. a b c d

multiple units.

Selling price is the amount of money a seller pays to buy an item for his store. ____________ When CP < SP, the seller makes a profit. ____________

Priya bought a bag for ₹350 and sold it for ₹400. She made a profit. ____________ If the cost of 5 pencils is ₹30, then the cost of one pencil will be ₹8. ____________

Which of these will be a better buy? 4 pencils for ₹24

10 pencils for ₹50

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Which chair will cost more: 36 chairs at ₹550 each or 22 chairs for ₹14300?

Calculate the profit or loss. a

CP = ₹167 and SP = ₹185

CP = ₹36 and SP = ₹29

CP = ₹147 and SP = ₹125

Ram is paid ₹6705 for 9 days of work. How much will he get paid in 28 days?

Mr Sharma bought 12 glass bowls for ₹660, but as he was going home, 3 of them broke. He incurred a

3 workers can paint a house in 6 days for ₹3000. If each worker divided the amount equally among

A store bought 7 phones at ₹2100 each. They want to sell them at a total profit of ₹700. What should

loss of ₹63 on selling the remaining glass bowls. What is the selling price of 1 glass bowl? each other, then how much does each worker gets paid every day? be the selling price of each phone?

10 Shreya works at a shopping centre. In her first year, she earned 2 valuable stamps. In her second year

she earned twice as many stamps as in the first year. How many stamps did she collect in these 2 years? If each stamp is worth ₹450 and she sold all the stamps at a profit of ₹50, find the total selling price of all the stamps.

Word Problems 1

Kanika sold 20 bulbs for ₹15 each at a profit of ₹5 on all the bulbs. What was the

Amit purchased two tea sets for ₹780 and ₹675. He sold both of them for ₹1450.

Ansh bought 50 flowers for ₹750. He kept 4 flowers and then sold the rest of the

A retailer bought 30 shirts for ₹200 each and 20 shirts for ₹100 each. He sold all

cost price of each bulb?

Did he make a profit or a loss? How much was it? flowers for ₹12 each. Calculate the profit or loss.

of them for ₹300 each. Calculate the total profit or total loss made by the retailer.

Chapter 16 • Money

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17 Data Handling Let's Recall

For example, the bar graph shows the number of people and their favourite fruit.

40 35 30 Number of People

Bar graphs are the pictorial representation of data (generally grouped), in the form of vertical or horizontal rectangular bars, where the length of bars is proportional to the measure of data.

From the given graph, we can interpret that 35 people like apples, 30 people like oranges, 10 people like bananas and 25 people like kiwi fruit.

25 20 15 10 5 0

an Or

n Ba

Letʼs Warm-up

The number of stamps that Kavya, Jishnu and Rehan have are shown below in the form of a bar graph. Read the graph and answer some questions.

Number of Stamps

Stamp Collection of 3 Students

50 40 30 20 10 0

Kavya

Jishnu

Name

Rehan

Who has the highest number of stamps in their collection?

____________

Who has an equal number of stamps as Jishnu?

____________

Who has half of the total number of stamps?

____________

What is the total number of stamps that the three children have?

____________

How many more stamps does Kavya have than Rehan?

____________ I scored _________ out of 5.

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Bar Graphs Real Life Connect

Rohan conducted a survey to find out the favourite outdoor activities of his classmates. He asked 40 classmates and recorded their answers in the form of a bar graph.

Reading and Drawing Bar Graphs Interpreting Bar Graphs The bar graph shown below was prepared by Rohan. Let us try to read and answer a few questions based on the bar graph.

Remember! When data is represented using vertical rectangular bars they are called vertical bar graphs. graphs

ow many students chose football as their favourite H sport?

The number of students who chose football is 10. 2

What is the favourite sport of the most students?

The longest bar is that of cricket (green line), hence the favourite sport of the most students is cricket. The same data can be shown using horizontal bar graphs.

16 14 12

No. of Students

10 8 6 4 2 0

Football Cricket Cycling Sport

Racing

Think and Tell

Sport

Cycling

Does drawing the bars horizontally make any

Cricket

difference to the data?

Football 0

No. of Students

We saw that bar graphs are used to read the data of a group. But do you know that bar graphs can also be used to compare two data groups? It can be done with the help of double bar graphs. Chapter 17 • Data Handling

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Blue bars show the favourite

sport of boys.

10 9 8 7 6 5 4 3 2 1 0

Red bars show

the favourite sport of girls.

Remember! We read double bar graphs in the same way we read Foootball

Cricket Cycling Boys Girls

Racing

single bar graphs.

Let us read the double bar graph shown above and answer some questions. How many students liked cycling? Number of girls who like cycling = 5; Number of boys who like cycling = 3 So, the total number of students who like cycling = 5 + 3 = 8 students.

Do It Together

The marks of a student in different subjects are shown in the bar graph. Read the graph and answer the questions. 100 1

ow many marks did the student score H in Science? 65

hat is the difference of the marks W scored in French and the marks scored in Economics? 75 – 70 = 5 marks

I n which subject were the highest marks scored? Maths

Marks Scored

Example 1

90 80 70 60 50 40 30 20 10 0

French

English

Maths

Science Economics

Subject

Read this bar graph. It shows the runs scored by two players in different matches. Then, answer the questions. 90 80

Number of Runs

70 60 50 40

What is the score of player 2 in match 4? ___________

hat is the total score of player 1 in match W 2 and 3? ___________

hat is the total score of player 2 W throughout the 5 matches? ___________

Player 1

Player 2

20 10 0

Match 1

Match 2

Match 3

Match 4

Match 5

Matches

Drawing Bar Graphs Rohan now tries to find the favourite subject of his classmates. He first writes the data in tabular form.

Error Alert! Look at the correct bar while answering the question.

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Subject

English

Hindi

Maths

Science

Number of Students

Let us help him draw a bar graph for the data. Step 2: Mark the horizontal line with subjects and

Step 1: Draw a vertical and horizontal line as

shown below. Choose a scale as 1 unit = 2 students.

vertical line with numbers at fixed intervals so that all the readings can be marked on the graph.

Step 3: Draw rectangular bars for all subjects

The same graph can be drawn by interchanging the data on the x and y axis as shown below.

adjacent to each other. Remember the height of the bar will match the number given in the table. 14

Science

No. of Students

12 8 6 4 English

Hindi

Maths Subject

Hindi

Science

6 8 10 12 No. of Students

Read the table showing the favourite fruit of 57 children. Draw a horizontal bar graph using the data. No. of Children

Grapes

Bananas

Apples

Mangoes

Fruit

Apples Mangoes 0

10 15 Number of Children

The number of trees planted by an organisation is given below. Complete the bar graph. Year

Number of Trees Planted

2012

120

2013

150

2014

200

2015

220

2016

300

Chapter 17 • Data Handling

Maths Grade 5 Book_Chapter 13-17.indb 273

350 Number of Trees Planted

Do It Together

Maths

English

2 0

Example 2

Subject

300 250 200 150 100 50 0

2012

2013

2014 2015 Year

2016

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Do It Yourself 17A 1

The bar graph represents the number of

students in different sections of Grade 5 of

50 No. of Students

a school.

Which section has the least and most number of students?

40 30 20 10

Read the bar graph that shows the number of animals in a village. Fill in the blanks.

a The number of cows in the village was b There were 20 more ___________ than

___________ in the village.

V-A

V-B

Class

V-C

V-D

Horse Animal

___________.

Buffalo Cow Goat

c The animal that had the least number

was the ___________.

Number of Animals

The students of Grade 5 conducted a “Green Choices” project where they collected data on the number of plastic bottles recycled over 5 days.

Day 1—20; Day 2—15; Day 3—18; Day 4—25; Day 5—22 Create a horizontal bar graph to represent the data. 4

The fifth-grade students at Green Valley Elementary School conducted a survey to find their favourite book genres. They asked 60 students to choose their favourite genres, and they wanted to create a bar graph to represent the results. Here’s the data they collected: Mystery

Adventure

Fantasy

Science Fiction

Historical Fiction

18 students

15 students

12 students

10 students

5 students

Create a bar graph to represent this data. 5

The number of carnival tickets sold to adults and children on certain days of a week are shown on the bar graph. Read the graph and answer the questions. a What is the difference of the tickets sold to

180

children on Sunday and those sold on Friday? throughout the four days?

c What is the difference of the tickets sold to adults

on Friday and the tickets sold to children on Sunday?

d How many more tickets were sold on Sunday as

compared to Saturday?

Number of People

b What is the total number of tickets sold

160 140 120 100

Adult Child

80 60 40 20 0

Thursday

Friday Saturday Sunday Day of the Week

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Word Problem A survey was conducted in a local forest to count the number of squirrels in it. The data of that survey has been put in this table.

No. of Squirrels

Monday

Tuesday

Wednesday

Thursday

Friday

Create a bar graph to represent the data.

Pie Charts Real Life Connect

Jaya and Shreya wanted to treat their mother on Mother’s Day. So, they decided to prepare a big family-size apple pie for the party. Their mother was happy and overwhelmed by their surprise. She said she’d serve the apple pie and divided it into 16 equal parts.

Reading and Drawing Pie Charts Let us now learn to read and draw pie charts.

Interpreting Pie Charts

Apple pie

Jaya, Shreya and their mother shared the apple pie. The pie chart shows their shares. Let us try to find some information from the chart. 1

Who ate the smallest share of the apple pie? Mother

How many slices of apple pie did Jaya have? 3 Slices of apple pie eaten by Jaya = of 16 = 6 slices. 8 Hence, Jaya ate 6 slices of apple pie.

ow many more slices of apple pie did Shreya eat as H compared to Jaya? Slices of apple pie eaten by Shreya = 1 × 16 = 8 slices. 2

1 of 16 = 2

1 2

Mother

1 8 3 8 Jaya

Shreya

Remember! We multiply the numerator of the fraction with the whole number and divide the product by denominator.

ifference of the number of slices of apple pie eaten by Shreya and Jaya D = 8 – 6 = 2 slices. Chapter 17 • Data Handling

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Example 3

Look at the pie chart. Rhea and her classmates were asked about their favourite subjects.

Favourite Subjects

If 72 students were surveyed, how many students voted for Art? 1 of the students voted for The pie chart shows that 4 Art as their favourite subject.

Hindi

Art

Hindi

English

English Maths EVS

EVS

Art

Maths

1 1 of 72 = × 72 =18 students. 4 4 Do It The pie chart below shows how a family spends its monthly income. Use the chart to Together answer the questions. Hence, number of students who voted for art =

hat is the largest expense of the W family? __________

On comparing the fractions, ______ 3 _____ > > _____ ______ 10 Hence, ______ is the largest expense in the family. 2

1 Others 10

Expenses

Entertainment 1 10

Rent 2 5

Rent Bills Food Entertainment

3 Food 10

1 Bills 10

Others

I f the family’s total monthly income is ₹25,000, how much is spent on rent and bills combined? ____________________________ Total fraction of amount spent on rent.

__________________________________________________________________________________________ I f the family decides to cut their entertainment expenses by half, what fraction will they be spending on entertainment? ____________________________

Representing Data on a Pie Chart Remember Jaya, Shreya and their mother sharing an apple pie? After some days, they prepared some laddoos to distribute among the needy people. Father also helped them. Below is a table showing the number of laddoo boxes filled by each member. Name

Mother

Father

Jaya

Shreya

Number of Boxes Filled

To draw a pie chart, we must know how to represent fractions on a circle. We first find the angle of each data category using the fraction. We know that a circle makes an angle of 360° and a pie chart is a circular graph that is used to represent data in terms of fractions or quantities. To represent the data on a pie chart, we follow the steps. 276

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18-12-2023 11:38:22

Step 1 Find the fraction and angle for each category by multiplying 360° with the fraction of each category. Name

Mother

Father

Jaya

Shreya

Number of Boxes Filled

Fraction of Boxes Filled

2 10

3 10

1 10

4 10

Angle of Each Category

360° ×

2 = 72° 10

360° ×

3 = 108° 10

360° ×

1 = 36° 10

360° ×

4 = 144° 10

Step 2 Draw a circle and its radius. Mark the first angle, keeping the radius as the reference line.

Boxes of Laddoos Packed Jaya

Step 3 Mark the next angle keeping the previous as angle line as the reference. Continue until all the angles are marked.

144° Shreya

36°

Mother 72° 108° Father

Step 4 Colour the categories and write the data inside the pie chart. Example 4

Error Alert!

A travel agency conducted a survey to find Add the angles before marking them on the the most popular vacation destinations chart. The sum must be 360°. among its customers. They surveyed 60 customers and wanted to create a pie chart to represent the results. Here’s the data they collected: Beach Resorts

Cultural Tours

Mountain Getaways

Adventure Expeditions

25 customers

12 customers

15 customers

8 customers

Create a pie chart to represent this data. Vacation Destination

Customers

Fraction of Boxes Filled

Beach Resorts

25 5 = 60 12

360° ×

Mountain Getaways

15 1 = 60 4

360° ×

1 = 90° 4

Cultural Tours

12 1 = 60 5

360° ×

1 = 72° 5

Adventure Expeditions

8 2 = 60 15

360° ×

Chapter 17 • Data Handling

Maths Grade 5 Book_Chapter 13-17.indb 277

Angle of Each Category 5 = 150° 12

2 = 48° 15

277

18-12-2023 11:38:22

Using the above angles for each destination, a pie chart can be created. Popular Vacation Destination

Adventure Expeditions, 8

Beach Resorts, 25

Cultural Tours, 12

Mountain Getaways Cultural Tours Adventure Expeditions

Mountain Getaways, 15 Do It Together

Beach Resorts

A school club organised an event where students participated in different activities. The activities and the number of students who joined each activity are as follows: Activity

No. of Students

Art and Crafts

Gardening

Cooking

Sports

Draw a pie chart to show the above data of students across these activities during the event.

Do It Yourself 17B 1

The pie chart shows the uses of pocket money managed by Riya.

Saving

Expenses

a What activity consumes the largest portion of her pocket

Food

money?

b If she receives ₹100 as pocket money, how much money

Stationery

goes towards savings?

1 of her pocket money on 10 snacks/food, how much money is that approximately?

c If she decides to spend only

Food

walk

The pie chart represents the different modes of transport used

Stationery

Saving

Modes of Transport 1 10

by students to travel to school.

a Which mode of transport is the least preferred? b If there are 200 students in the school, how many students

Rickshaw

Bus

3 10

1 5

travel by bicycle?

c If there are 200 students in the school, how many students

take the rickshaw to school?

Bicycle Bus

Bicycle

2 5 Rickshaw

walk

278

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18-12-2023 11:38:23

Rohan recorded the time he spends on different activities during a day over the weekend. Here is the data he collected:

Playing: 3 hours

Studying: 4 hours

Watching TV: 2 hours

a Find the fractions and angles for the data.

b Draw a pie chart and label the data.

A school cafeteria serves the following items for lunch: Curd rice: 25

Salad: 20

Pulav: 15

Create a pie chart to represent the lunch choices. 5

Reading: 3 hours

Veg soup: 10

Mixed juice: 30

Read the pie chart showing the data for 200 students and

Subject

answer the questions.

a If there are 200 students in total, how many students are

interested in Maths?

Music 9 40

Science

b If 40 students are interested in Science, what fraction of

the total students does this represent?

Maths 1 4

Sports 11 40

c Which subject are the most students interested in?

d If 10 students are interested in EVS, what fraction of

Science

the students are interested in EVS?

Maths

EVS

Sports

Music

Word Problem 1

In a survey of 60 students, they were asked about their favourite sports. The results

are as follows: 15 students like soccer, 12 students prefer basketball, 10 students enjoy swimming, and the rest are into tennis. Create a pie chart to represent this data.

Line Graphs Real Life Connect

Neha and her brother love reading books. They have a small library at their house. They even get books from family and friends. Both the siblings started reading books in the year 2018. They recorded the number of books read by them across the years.

Neha’s father put the data in the form of a line graph. A line graph uses lines to connect individual data points. Study the graph and answer the questions.

Chapter 17 • Data Handling

Maths Grade 5 Book_Chapter 13-17.indb 279

Number of Books

Reading Line Graphs 50 45 40 35 30 25 20 15 10 5 0

2018

2019

2020 Year

2021

2022

279

18-12-2023 11:38:26

In which year did the siblings read the most books? o find the greatest number of books, look at the T peak of the given line graph. Now see the year corresponding to the peak.

Remember! Read the data corresponding to both the x and y axis.

ence, 45 is the greatest number of books read by H the siblings in the year 2020. 2

How many books did the siblings read in 2021? Looking at the corresponding data for 2021, the siblings read 40 books in this year.

How many books did the siblings read in total for these 5 years? o find the number of books read in each year, look at the corresponding data for T each year. ence, the total number of books read across 5 years = 25 + 30 + 45 + 40 + 40 = 180 H books.

Do It Together

Given below is a line graph that shows the annual food grain production (in tons) from 1992 to 1997. Read the graph and fill in the blanks.

Production (in tons)

120

he production of food grain in the year T 1992 was 20 tons.

he highest production of food grain T was in the year ______.

he difference in production between T the year 1995 and the year 1993 was ______________.

he total production across the years = T _________________________________________.

100 80 60 40 20 0

1992

1993

1994

Year

1995

1996

1997

Do It Yourself 17C The graph shows Kavya’s internet usage

across five days. 10 9 8 7 6 5 4 3 2 1 0

The line graph shows the number of persons who visited different cities on a certain day.

Name the cities visited by the most and least number of people.

Mon

Tues

Wed Day

Thurs

Fri

On which days did she spend the highest number of hours on the internet?

Number of People

Number of Hours

900 800 700 600 500 400 300 200 100 0

Chennai

Goa

Pune

City

Kolkata

Delhi

280

Maths Grade 5 Book_Chapter 13-17.indb 280

18-12-2023 11:38:39

The line graph represents the weekly

temperature readings (in °C) in a city for

guitars over the course of six years.

and answer the questions. Temperature in °C

12 10 8 6 4 2 0

Week 1

Week 2

Week 3 Week

Week 4

1000 900 800 700 600 500 400 300 200 100 0

Number of Guitars Sold

five weeks. Study the graph carefully

Rahul drew a line graph to show the sales of

Week 5

a What was the temperature recorded

2016

2017

2018

Year

2019

2020

2021

a In which year did Rahul sell 700 guitars?

during the second week?

b How many guitars were sold in the year

2021?

b In which week was the highest

c In which year was the least number of

temperature recorded?

guitars sold?

c What was the temperature difference

between the first and the last week?

The graph shows the runs scored by a cricketer across various matches. Read the line graph and answer the questions.

a In which match/es did the cricketer hit a

century?

b What is the lowest score of the cricketer? c How many times did the cricketer score

Runs Scored

less than a century?

120 100 80 60 40 20 0

Match 1 Match 2 Match 3 Match 4 Match 5 Match 6 Match 7 Matches

d What is the total number of runs scored

by the cricketer in the initial four matches?

Word Problem 10

Mohit’s father runs a clothing factory. The total sale of clothes from the factory is

shown. Read the graph and answer the questions.

a In which month did the factory sell 8000

items of clothing?

b In which month did the factory sell the

most, and how many, items?

Sales (in thousands)

9 8 7 6 5 4 3 2 1 0

April

May

June Month

July

August

c What is the difference between the sales

of April and July?

Chapter 17 • Data Handling

Maths Grade 5 Book_Chapter 13-17.indb 281

281

18-12-2023 11:38:55

Points to Remember • A bar graph is a way to represent data using bars of different lengths. Each bar represents a category, and the height of the bar corresponds to the quantity or value of that category. A pie chart helps represent data visually and shows how different parts relate to the whole. • •

A line graph uses lines to connect individual data points.

Math Lab Setting: In groups of 5 Materials required: Chart paper or whiteboard, Coloured pencils or markers, Protractor, Ruler, Sheets with different sets of data Method: Each group gets a chance to choose a sheet having different sets of data. The groups start analysing the data and putting the data in the form of bar graphs and pie charts. The group that is the first to put all the data correctly in both types of graphs, wins!

Chapter Checkup 1

An organisation conducted a survey on pets to find the favourite pets in a certain locality. They asked 60 families about their preferences and created a bar graph to represent the results. Look at the data and fill in the blanks. ______ families like cats.

______ families prefer birds and rabbits

The number of families who like dogs

as their pets.

as pets is ______ than the number of families who like fish as pets.

Number of Families

16 14 12 10 8 6 4 2 0

Dogs

Cats

Birds

Pets

Fish

Rabbits

282

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2 The graph shows the number of pages read by boys and girls for five days of a week. a

On which day did boys and girls

Who read more pages on

Wednesday?

How many pages did the boys read

What is the difference in the total

on Tuesday?

Number of Pages

read an equal number of pages?

15 10

Boys Girls

5 0

Monday

Tuesday Wednesday Thursday

Friday

Day

number of pages read by girls and by boys?

3 A school recorded the number of absentees in a certain week. Draw a bar graph for the data. Days

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Number of absentees

125

100

105

125

115

4 There are 1000 books in a library. The pie chart shows the types of books in the library. Read the chart and answer the questions. a

How many poetry books are there in the library?

What is the difference in the number of comics and story

Comic Mystery

Story 1

Poetry

What type of book has the highest number?

Story

Mystery

Others

The data shows the number of hours spent by Kunal on

different activities in a day. Draw a pie chart for the given data.

Poetry 1 5

Activity

School

Homework

Play

Sleep

Others

Number of hours

Read the graph showing the amount

of fruit sold in a week. Answer the questions. a

On which day did the fruit seller sell

How many kilograms did he sell

15 kg of fruit?

35 30 25 20 15

altogether on Saturday and

What is the difference of the sales of

Sunday? c

books?

Fruits Sold (in kg)

Comic 3

Others 1

the initial 4 days and the last 3 days?

Chapter 17 • Data Handling

Maths Grade 5 Book_Chapter 13-17.indb 283

5 Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Day

283

18-12-2023 11:39:05

This graph shows the number of customers visiting a coffee shop

between 4 p.m. to 8 p.m. Read it and answer the questions. a b

At what time did the coffee shop have the least customers?

c d

10 5 0

4:00 p.m.

5:00 p.m.

6:00 p.m.

7:00 p.m.

8:00 p.m.

Chennai

Jaipur

Ooty

Time

temperature of different cities. Answer

coffee shop at 8:00 p.m.?

Read the graph showing the

How many customers were in the

the questions.

Number of Customers

Which is the hottest city?

Which is the coldest city?

What is the temperature in Chennai? Which city has a temperature of

Temperature in °C

20 15 10 5 0

Mumbai

Bangalore

City

10℃?

The data represents the sale of refrigerators by a showroom in the last six months. Draw a horizontal bar graph.

Months

July

August

September

October

November

December

Number of Refrigerators

In a music class, students were asked about their favourite musical instruments. Out of 40 students

surveyed, 12 students liked the guitar, 10 liked the piano, 8 enjoyed playing the drums, and the remaining students liked other instruments. Draw a pie chart to represent the data.

Word Problem 1

Kunal carried out a survey among 50 children

from his locality about their favourite ice cream. He made a pie chart with the survey results. a b c d

Strawberry 6 25

Which ice cream is the most popular?

Mango 4 25 Blueberry 3 25

Which is the least popular ice cream?

How many children like chocolate ice cream? How many children like either vanilla or strawberry ice cream?

Vanilla 5 25

Chocolate 7 25

284

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18-12-2023 11:39:14

A n swe rs Chapter 1

1. 8,00,000 thousand five hundred thirty-six

Let's Warm-up

Do It Yourself . 3,000,000; 3 1A 1. a

2. four lakh thirty-six 3. 4 4. thousands

b. 9,00,00,000; 9 c. 100,000; 1 d. 20,000,000; 2 e. 60,000,000; 6 f. 30,000; 3 g. 0; 0 h. 800; 8 2. a. Indian System: 2,16,43,332; Two Crore Sixteen Lakh Forty-Three Thousand Three Hundred Thirty-Two 2,00,00,000 + 10,00,000 + 6,00,000 + 40,000 + 3000 + 300 + 30 + 2; International System: 21,643,332; Number Name: Twenty-One Million Six Hundred Forty-Three Thousand Three Hundred Thirty-Two; 20,000,000 + 1,000,000 + 600,000 + 40,000 + 3000 + 300 + 30 + 2 b. Indian system: 12,00,621; Twelve Lakh Six Hundred Twenty-One; 10,00,000 + 2,00,000 + 600 + 20 + 1; International System: 1,200,621; One Million Two Hundred Thousand Six Hundred Twenty-One; 1,000,000 + 200,000 + 600 + 20 + 1 c. Indian system: 4,62,07,219; Four Crore Sixty-Two Lakh Seven Thousand Two Hundred Nineteen; 4,00,00,000 + 60,00,000 + 2,00,000 + 7000 + 200 + 10 + 9; International System: 46,207,219; Forty-Six Million Two Hundred Seven Thousand Two Hundred Nineteen; 40,000,000 + 6,000,000 + 200,000 + 7000 + 200 + 10 + 9 d. Indian system: 9,59,10,158; Nine Crore Fifty-Nine Lakh Ten Thousand One Hundred Fifty-Eight; 9,00,00,000 + 50,00,000 + 9,00,000 + 10,000 + 100 + 50 + 8; International System: 95,910,158; Ninety-Five Million Nine Hundred Ten Thousand One Hundred Fifty-Eight; 90,000,000 + 5,000,000 + 900,000 + 10,000 + 100 + 50 + 8 e. Indian system: 74,09,230; Seventy-Four Lakh Nine Thousand Two Hundred Thirty; 70,00,000 + 4,00,000 + 9000 + 200 + 30; International System: 7,409,230; Seven Million Four Hundred Nine Thousand Two Hundred Thirty; 7,000,000 + 400,000 + 9000 + 200 + 30 f. Indian system: 86,56,023; Eighty-Six Lakh Fifty-Six Thousand Twenty-Three; 80,00,000 + 6,00,000 + 50,000 + 6000 + 20 + 3; International System: 8,656,023; Eight Million Six Hundred Fifty-Six Thousand Twenty-Three; 8,000,000 + 600,000 + 50,000 + 6000 + 20 + 3g) g. Indian system: 6,78,90,240; Six Crore Seventy-Eight Lakh Ninety Thousand Two Hundred Forty; 6,00,00,000 + 70,00,000 + 8,00,000 + 90,000 + 200 + 40; International System: 67,890,240; Sixty-Seven Million Eight Hundred Ninety Thousand Two Hundred Forty; 60,000,000 + 7,000,000 + 800,000 + 90,000 + 200 + 40 h. Indian system: 3,45,65,892; Three Crore Forty-Five Lakh Sixty-Five Thousand Eight Hundred Ninety-Two; 3,00,00,000 + 40,00,000 + 5,00,000 + 60,000 + 5000 + 800 + 90 + 2 International System: 34,565,892; Thirty-Four Million Five Hundred Sixty-Five Thousand Eight Hundred Ninety-Two 30,000,000 + 4,000,000 + 500,000 + 60,000 + 5000 + 800 + 90 + 2 3. a. 60,08,098 b. 20,000,569 c. 4,090,000 d. 8,00,01,002 4. a. 1 b. 10

Answers

Maths Grade 5 Book_Chapter 13-17.indb 285

c. 10,000 d. 7 e. 10 f. 50 5. a. 100 b. 1000 c. 10,000 6. a. 91,19,199 b. 170,70,707 Word Problem 1. 30,145, 765; Three crore one lakh forty-five thousand seven hundred sixty-five. 1B 1. 76,24,578 2. a. < b. = c. > d. < e. < f. < 3. a. < b. > c. > d. < e. > f. > 4. Answer may vary. Sample answer. 5,79, 48,280; 5,79,48,280 > 5,48,79,802 5. a. 1,00,00,000 b. 2,90,52,468 c. 10,00,000 d. 3,00,52,468 Word Problem 1. Sanket 1C 1. a. 93,12,820 < 1,00,36,782 < 5,00,00,367 < 8,87,21,460 b. 36,81,910 < 92,56,890 < 6,92,10,350 < 8,26,00,031 c. 5,00,21,138 < 6,04,50,821 < 6,50,24,567 < 9,45,21,823 2. a. 8,26,34,510 > 6,70,81,256 > 4,50,00,921 > 87,92,345 b. 8,01,20,450 > 6,78,20,001 > 92,11,108 > 42,56,789 c. 9,67,28,891 > 7,88,21,134 > 5,78,20,010 > 5,43,33,867 3. a. 98,54,310; 10,34,589 b. 87,65,321; 12,35,678 c. 65,43,210; 10,23,456 4. a. 8,87,64,210; 1,00,24,678 b. 9,98,76,431; 1,13,46,789 c. 9,97,54,320; 2,00,34,579 5. a. 9,99,99,998; 10,00,000 b. 9,99,98,765; 10,00,234 c. 9,99,99,876; 10,00,023 1D 1. a . 85,48,750 b. 89,05,460 c. 6,07,85,890 d. 1,56,48,950 e. 5,40,86,570 f. 3,09,24,560 2. a . 1,25,89,200 b. 87,52,400 c. 68,67,800 d. 77,59,900 e. 5,07,87,300 f. 43,43,600 3. a . 8,97,00,000 b. 53,12,000 c. 8,21,59,000 d. 5,89,90,000 e. 86,76,000 f. 23,75,000 4. Answer may vary. Sample answer. a. 99,63,210; 99,63,000 b. 96,32,100; 96,32,000 5. 3,23,45,500 and 3,23,46,449 Word Problem 1. The municipal corporation spent ₹65,95,000 on repairing the roads.

Chapter Checkup

1. a. Indian system: 35,07,681; Thirty-Five Lakhs Seven Thousand Six Hundred Eighty-One; 30,00,000 + 5,00,000 + 7000 + 600 + 80 + 1; International system: 3,507,681-Three Million Five Hundred Seven Thousand Six Hundred EightyOne; 3,000,000 + 500,000 + 7000 + 600 + 80 + 1 b. Indian system: 4,20,87,950; Four Crores Twenty Lakhs Eighty-Seven Thousand Nine Hundred Fifty; 4,00,00,000 + 20,00,000 + 80,000 + 7000 + 900 + 50 International system: 42,087,950; Forty-Two Million Eighty-Seven Thousand Nine Hundred Fifty; 40,000,000 + 2,000,000 + 80,000 + 7000 + 900 + 50 c. Indian system: 6,35,65,842; Six Crores Thirty-Five Lakhs Sixty-Five Thousand Eight Hundred Forty-Two; 6,00,00,000 + 30,00,000 + 5,00,000 + 60,000 + 5000 + 800 + 40 + 2 International system: 63,565,842; Sixty-Three Million Five Hundred Sixty-Five Thousand Eight Hundred FortyTwo; 60,000,000 + 3,000,000 + 500,000 + 60,000 + 5000 + 800 + 40 + 2 d. Indian system: 9,15,00,084; Nine Crore Fifteen Lakh Eighty-Four; 9,00,00,000 + 10,00,000 + 5,00,000 + 80 + 4 International system: 91,500,084 - Ninety One Million Five Hundred Thousand Eighty-Four; 90,000,000 + 1,000,000 + 500,000 + 80 + 4

285

18-12-2023 11:39:14

A n swe rs 2. a. 60,715,239 b. 8,09,50,002 c. 1,100,039 d. 12,58,043 3. a. > b. < c. = d. < 4. a. 23,56,475 < 90,87,687 < 8,91,63,896 < 9,08,04,365 b. 3,24,35,678 < 4,35,46,576 < 6,76,12,895 < 6,76,87,980 5. a. 5,36,45,787 > 4,56,45,768 > 2,40,85,167 > 43,56,787 b. 9,09, 87, 897 > 4,90,76,837 > 80,88,428 > 80,68,964 6. 9,87,65,320; 2,03,56,789; 98,76,532; 20,35,678 Ascending order: 20,35,678 < 98,76,532 < 2,03,56,789 < 98,765,320; Descending order: 9,87,65,320 > 2,03,56,789 > 98,76,532 > 20,35,678 7. a. 6,45,87,120; 6,45,87,100; 6,45,87,000 b. 89,09,010; 89,09,000; 89,09,000 c. 1,08,75,760; 1,08,75,800; 1,08,76,000 d. 24,89,700; 24,89,700; 24,90,000 8. 16,48,235; 16,48,236; 16,48,237; 16,48,238; 16,48,239; 16,48,240; 16,48,241; 16,48,242; 16,48,243; 16,48,244 9. a. Indian number system: 1,99,77,555 One Crore Ninety-Nine Lakh Seventy-Seven Thousand Five Hundred Fifty-Five; International Number system: 19,977,555 Nineteen Million Nine Hundred SeventySeven Thousand Five Hundred Fifty-five b. Answer may vary. Sample answer: 1,77,55,995 Hence, 1,99,77,555 > 1,77,55,995 c. 1,99,77,560; 1,99,77,600; 1,99,78,000 10. 1 2,15,696 Word Problem 1. One Crore Fourteen Lakh Thirty-Five Thousand Eight Hundred Sixty

Chapter 2 Let's Warm-up 5. 450

1. 90

Do It Yourself 2A 1. a. True

2. 5500

3. 2350

4. 280

b. False c. True d. False 2. a. 89,285 b. 2,64,859 c. 15,54,313 3. a. 91,660 b. 71,092 c. 9,21,018 4. a. 9,06,567 b. 4,62,917 c. 7,28,944 d. 7,71,214 e. 7,09,138 f. 7,24,168 5. a. 81,898 b. 24,566 c. 9,31,609 d. 7,88,845 e. 2,47,880 f. 5,31,550 6. 9 ,03,550 Word Problems 1. 2,72,950 blazers 2. ₹2,12,000 3. ₹3,81,030 2B 1. a. False b. False c. False d. True 2. a. 5,65,670; 56,56,700; 5,65,67,000 b. 3,42,630; 34,26,300; 3,42,63,000 c. 4,78,520; 47,85,200; 4,78,52,000 d. 19,84,540; 1,98,45,400; 19,84,54,000 e. 8,25,870; 82,58,700; 8,25,87,000 b. 24,500; 2450; 245 3. a. 2500; 250; 25 c. 35,400; 3540; 354 d. 89,500; 8, 8950; 895 e. 98,700; 9870; 987 4. a. 61,525 b. 27,36,600 c. 18,71,280 d. 78,71,525 e. 1,88,82,990 f. 1,46,55,870 g. 3,03,57,504 h. 4,45,49,588 i. 4,72,86,288 j. 5,35,25,760 5. a. Quotient = 135; Remainder = 365 b. Quotient = 181; Remainder = 147 c. Quotient = 233; Remainder = 366 d. Quotient = 159; Remainder = 280

e. Quotient = 102; Remainder = 219 f. Quotient = 183; Remainder = 134 g. Quotient = 136; Remainder = 15 h. Quotient = 161; Remainder = 219 i. Quotient = 124; Remainder = 228 j. Quotient = 414; Remainder = 32 Word Problems 1. ₹28,28,232 2. 391 pages 3. 9,60,000 chairs 4. ₹680 2C 1. a. (ii) b. (ii) c. (i) d. (ii) e. (ii) 2. a. 7 b. 41 c. 44 d. 12 e. 266 f. 40 g. 15 h. 16 3. a. True b. False c. True d. False 4. a. 84 b. 478 c. 272 d. 770 e. 3171 f. 168 g. 40 h. 23 Word Problems 1. ₹78 2. 48 stickers 3. 7 pencils 4. ₹74,750 Chapter Checkup 1. a. 80,245 b. 6,99,537 c. 10,33,428 d. 12,41,211 e. 9,20,003 f. 3,02,572 2. a. 20,101 b. 7,68,289 c. 2,20,454 d. 8,03,271 e. 9,71,197 f. 1,09,241 3. a. 3250; 32,500; 3,25,000 b. 8960; 89,600; 8,96,000 c. 45,460; 4,54,600; 45,46,000 d. 64,570; 6,45,700; 64,57,000 e. 76,560; 7,65,600; 76,56,000 f. 98,760; 9,87,600; 98,76,000 4. a. 29,32,848 b. 28,13,565 c. 17,29,200 d. 1,33,97,224 e. 1,28,80,901 f. 40,83,256 5. a. Quotient = 210; Quotient = 21; Quotient =2, Remainder = 100 b. Quotient = 408, Remainder = 6; Quotient = 40, Remainder = 86; Quotient = 4, Remainder = 86 c. Quotient = 323, Remainder = 5; Quotient = 32, Remainder = 35; Quotient = 3, Remainder = 235 d. Quotient = 5120; Quotient = 512; Quotient = 51, Remainder = 200 e. Quotient = 6520; Quotient = 652; Quotient = 65, Remainder = 200 f. Quotient = 7456, Remainder = 2; Quotient = 745, Remainder = 62; Quotient = 74, Remainder = 562 6. a. Quotient = 337; Remainder = 30 b. Quotient = 237; Remainder = 20 c. Quotient = 180; Remainder = 265 d. Quotient = 142; Remainder = 196 e. Quotient = 115; Remainder = 437 f. Quotient = 101; Remainder = 684 7. a. 37 b. 27 c. 76 d. 43 e. 179 8. 45 9. 8, 99, 999 10. 6,00,000 Word Problems 1. 8,95,565 people 2. 50 days 3. 38,400 flowers 4. ₹4,87,800 5. 65 marbles 6. 2226 oranges in each box 7. 149 rows 8. ₹1,20,500

Chapter 3

1. 1, 2, 4 5. 1, 5, 11

Let's Warm-up 4. 1, 23

2. 1, 3, 9

3. 1, 3, 5

Do It Yourself 3A 1. a. 1, 5, 11 and 55

b. 1, 2, 7 and 14 c. 1, 2 , 3, 4, 6, 8, 12, 16, 24 and 48 d. 1, 2, 17 and 34 e. 1 and 37 2. a. 1, 3, 5 and 15 b. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 c. 1 and 41 d. 1, 3, 9, 27 and 81 e. 1, 3, 7, 9, 21 and 63 3. Composite numbers: a, c, d, e, g, i Prime numbers: b, f, h, j 4 . 4 prime numbers - 11, 13, 17 and 19 5. True. For example, 2 and 10 both are factors of 20, 2 and 10 both are factors of 30, etc.

286

Maths Grade 5 Book_Chapter 13-17.indb 286

18-12-2023 11:39:15

Word Problem Rows

Columns

Total chairs

189

3B 1. b, c, e

2. d, e, f, i 3. a, b, d 4. a, b 5. 660 Word Problem 1. Yes, because both 324 and 135 are divisible by 9. 3C 1. b 2. b 3. c 4. a. 8 = 2 × 2 × 2 b. 15 = 3 × 5 c. 20 = 2 × 2 × 5 d. 24 = 2 × 2 × 2 × 3 e. 33 = 3 × 11 f. 60 = 2 × 2 × 3 × 5 g. 63 = 3 × 3 × 7 h. 72 = 2 × 2 × 2 × 3 × 3 i. 90 = 2 × 3 × 3 × 5 j. 112 = 2 × 2 × 2 × 2 × 7 5. a. 12 = 2 × 2 × 3 b. 16 = 2 × 2 × 2 × 2 c. 22 = 2 × 11 d. 30 = 2 × 3 × 5 e. 44 = 2 × 2 × 11 f. 45 = 3 × 3 × 5 g. 51 = 3 × 17 h. 60 = 2 × 2 × 3 × 5 i. 100 = 2 × 2 × 5 × 5 j. 148 = 2 × 2 × 37 Word Problem 1. 11 3D 1. a. 1 b. 1, 3, 5, 15 c. 1, 2, 3, 4, 6, 8, 12, 24 d. 1, 2 e. 1 f. 1, 3 , 9 g. 1, 2 h. 1, 7 i. 1, 5 j. 1, 2, 3, 6 2. a. 8 b. 5 c. 36 d. 6 e. 27 f. 10 g. 15 h. 14 i. 1 j. 11 3. a. 3 b. 15 c. 8 d. 15 e. 18 f. 13 g. 12 h. 15 i. 26 j. 8 4. a. 16 b. 6 c. 75 d. 24 e. 45 f. 20 g. 14 h. 8 i. 27 j. 7 5. 18 litres 2. 7 m 3. 9 litre Word Problems 1. 5 cm Chapter Checkup 1. a. 1, 2, 4, 5, 10 and 20 b. 1, 2, 3, 4, 6 and 12 c. 1, 2, 4, 8 and 16 d. 1, 2, 3, 4, 6, 8, 12 and 24 e. 1, 2, 3, 6, 7, 14, 21 and 42 2. a. 1, 2, 4, 7, 8, 14, 28 and 56 b. 1, 2, 4, 7, 14 and 28 c. 1, 2, 3, 6, 9, 18, 27 and 54 d. 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80 e. 1, 2, 3, 4, 6, 9, 12, 18 and 36 3. C omposite number: a, c, d, f, g, j; Prime number: b, e, h, i 4. a. divisible by 5 b. divisible by 2 and 3 c. divisible by 3 d. divisible by 2, 3, 5, 9 and 10 e. divisible by 2, 5 and 10 5. a. 88 = 2 × 2 × 2 × 11 b. 65 = 5 × 13 c. 102 = 2 × 3 × 17 d. 112 = 2 × 2 × 2 × 2 × 7 e. 140 = 2 × 2 × 5 × 7 6. a. 75 = 3 × 5 × 5 b. 21 = 3 × 7 c. 128 = 2 × 2 × 2 × 2 × 2 × 2×2 d. 164 = 2 × 2 × 41 e. 91 = 7 × 13 7. a. 1,5 b. 1, 5, 25 c. 1, 11 d. 1, 2, 3, 4, 6, 12 8. a. 4 b. 5 c. 24 d. 75 9. a. 2 b. 17 c. 15 d. 15 10. a. 9 b. 17 c. 1 d. 34 11. Answers may vary. Sample answers: 120 12. 25 litres 13. 8 balloons Word Problems 1. 6 inches 2. 80; 3 cupcakes; 2 sandwiches 3. 6 containers

Chapter 4 Let's Warm-up

1. 24,48,64

4. 60,96,108

2. 36,54,63

3. 40,70,80

c. 21, 42, 63, 84, 105 d. 25, 50, 75, 100, 125 e. 50, 100, 150, 200, 250 2. Yes 3. 120 4. 9, 18, 27 and 36, Odd multiples – 9, 27, Even multiples – 18, 36 5. Answers may vary. Sample answers: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100; Even.

Maths Grade 5 Book_Chapter 13-17.indb 287

Chapter 5

1. one-third 5. two-fourths

2. one-half

Let's Warm-up 4. one-eighth

3. one-fourth

Do It Yourself 5A 1. a. Unlike fractions

b. Like fractions c. Unlike d. Like fractions e. Like fractions

fractions

2. Proper fractions:

4 12 12 , , . 5 15 29

Improper fractions:

1 7

6 58 17 9 26 , , , , . 2 5 11 3 17

5 7 8 4 , 6 , 7 , 15 8 9 9 7 3 2 c. 5 d. 7 6 5 3 h. 27 i. 181 6

Mixed numbers: 4 , 5 3. a. 4 f. 15

3 4

4. a.

Do It Yourself 4A 1. a. 8, 16, 24, 32, 40 b. 11, 22, 33, 44, 55

Answers

6. a. 8, 16, 24 b. 15, 30, 45, 60, 75, 90 c. 76, 95, 114, 133 d. 126, 147, 168, 189 Word Problem 1. August 7, 14, 21, 28. 4B 1. d 2. Answers may vary. Sample answers: a. 10,20 b. 36,72 c. 30, 60 d. 30,60 3. a. 60 b. 60 c. 27 d. 40 e. 60 f. 63 g. 55 h. 49 4. a. 36 b. 78 c. 66 d. 18 5. c Word Problem 1. 120 minutes 4C 1. a. 35 = 5 × 7; 75 = 3 × 5 × 5 b. 44 = 2 × 2 × 11; 88 = 2 × 2 × 2 × 11 c. 48 = 2 × 2 × 2 × 2 × 3; 12 = 2 × 2 × 3 d. 25 = 5 × 5; 115 = 5 × 23 2. No, since the LCM = 168 3. a. 40 b. 16 c. 80 d. 50 4. a. 60 b. 60 c. 72 d. 175 e. 180 f. 315 g. 48 h. 180 i. 90 5. a. 360 b. 864 c. 1680 d. 180 e. 168 f. 324 Word Problem 1. 20th student. 4D 1. a. 150 b. 96 c. 210 d. 48 e. 300 f. 45 g. 168 h. 90 i. 315 2. a. 168 b. 300 c. 576 d. 540 e. 2040 f. 315 3. No, Rohit’s calculation is incorrect, as he didn’t consider all the prime factors. 4. Yes, Akash is correct as he has considered all the prime factors. 5. No, Ajay is not correct as he has not considered all the prime factors. Word Problems 1. 24 minutes 4E 1. 48 students 2. 8:24 p.m. 3. 10:31 a.m. 4. a. 10, 20,30,40,50,60,70,80,90 and 100 b. 10 5. 24th customer Word Problems 1. 15 litres 2. 3360 stacks 3. 6 groups 4. 36 metres Chapter Checkup 1. 9, 18 and 27 2. a. 63 b. 81 c. 99 d. 127 e. 144 3. d 4. a 5. b 6. a. 630 b. 90 c. 1386 d. 6930 e. 180 7. a. 315 b. 3720 c. 1680 d. 7260 e. 560 f. 480 g. 4125 h. 630 i. 360 8. a. True b. True c. True d. False 9. a. 60 b. 70 c. 54 d. 45 e. 120 f. 240 10. 3 hours Word Problems 1. 8 inches 2. 39 3. Ajay

109 5

5. a.

2 3

9 2

b. 8

1 4 18 b. 7

2 7

g. 36

119 8

25 6

b. 20

55 9 116 h. 17

14 15

35 3 243 i. 19

144 7

e. 7 j. 15 e.

135 11

d. 15

8 9

6 7

485 21

e. 44

22 2 2. 3 5 5B 1. Answers may vary. Sample answers:

Word Problems 1. 17

8 12 , 18 27

22 33 , 34 51

12 18 , 30 45

24 36 , 40 60

287

18-12-2023 11:39:15

28 42 , 56 84

12 18 , 14 21

2. Answers may vary. Sample answers:

6 3 , 10 5 15 10 , e. 21 14 a.

7 2 , 28 8

4 5

22 11 , 26 13

3 7

Word Problems 1. d.

16 8 , 24 12

32 16 , 40 20

13 2 1 e. f. 17 3 3 4. a. Not equivalent b. Equivalent c. Equivalent d. Equivalent e. Not equivalent f. Equivalent 5. a. 28 b. 70 c. 2 d. 84 e. 56 f. 25

3. a.

2 5

5C 1. a. < b. 

b. > c. 

1 3

c. > d. 

1 3 2 < < 6 7 4 5 4 8 7 < < < c. 12 8 15 8

3. a.

d. < e. 

e. < f. 

f. <

2. a. 

2 2. a. 4 4

103 c. 8

1 11 1 b. 8 c. 24 d. 4 5 19 6 3. Answers may vary. Sample answers:

10 15 20 4 6 8 b. , , , , 16 24 32 12 18 24 8 12 16 d. , , 18 27 36 4. Answers may vary. Sample answers: a.

1 14 < 2 5

b. 6

4 34 < 5 4

51 d. 8

22 33 44 , , 28 42 56

2. four

3. 1

1 2 , 2 4

3 12 , 7 21

b. 1

d. 6

4 5

d. 11 b. 5

b. 22

1 18

e. 12

199 210

6B 1. a. 3

1 57 = 3 9

1 3

3. a. 1 4. a.

11 18

f. 11

b. 3

1 12

c. 4

3. a. b. 5

3 25

5 12

b. 2

1 4 b.

5 12

16 17

1 4

b. 15

5 16

35 48

10 91

3 4

13 14

7 12

1 12

1 3

7 34

4 kg 5

d. 4

e. 3 d.

1 12

1 20

5 12

31 3 or 2 14 14

d. 1 2. 39

e. 12

2. 3

d. 6

c. 2

7 hours 12

b. 7 b.

49 72

5 9

5. a. 6

5 24 3 b. 8 2 c. 12 5

2 15 , 5 25

c. 7

1 m 6

5. six

43 21 or 1 22 22

1 10

73 140

17 22

5 26

4. seven

d. 5

b. 15

4 13

5 18

9 14

1 28

Word Problems 1. 5

Word Problems 1. 2

2. a.

c. 4

1 minutes 210

14 2 2. a. 5 5 f. 2 6 1 4. a. 1 6

6C 1. a. 2

3 13

4. a. 1

7 10

3. 12

19 1 or 1 18 18

85 1 or 6 14 14

5. a. 1 13 24

c. Not equivalent.

24 =8 3

61 63

f. 23

54 36 18 12 8 4 8 4 2 b. c. , , , , , , 66 44 22 21 14 7 12 6 3 25 15 5 d. , , 65 39 13 11 9 3 3 5. a. b. c. d. 14 17 4 5

7. a. 2

17 18

3. a. 5

6. a. Equivalent. b. Not equivalent. d. Not equivalent.

1. whole

4 4

45 17 17 5 5. a. b. c. d. Word Problems 1. Reshma 15 8 5 8 2 4 7 2. hours = hours < hours < 1 hour 3 6 8

17 b. 3

Chapter 6

2. a.

1 1 3 3 < < < 18 12 18 10

6A 1. a. 1 , 3

3 2 3 5 f. > > > 4 3 8 25

15 Chapter Checkup 1. a. 2

Greatest: Bailey

Let's Warm-up Do It Yourself

3 1 5 7 5 < < < < 13 4 12 12 7

1 1 2 5 d. < < < 5 2 3 6 1 2 1 5 3 1 2 1 5 5 e. f. < < < < < < < < 4 5 2 7 4 3 5 2 8 6 3 3 1 1 8 1 3 1 4. a. b. > > > > > > 5 8 4 6 9 2 8 3 1 1 1 1 4 2 4 3 1 c. d. > > > > > > > 6 9 11 14 5 3 11 15 6 3 2 3 1 e. > > > 4 3 5 2

1 5 3 5 2 7 1 1 ; b. ; c. ; d. ; 8 6 9 3 7 10 4 2 1 5 1 4 1 3 1 3 9. a. b. < < < < < < 6 12 2 5 5 8 2 5 2 1 2 6 1 4 3 11 c. d. < < < < < < 5 2 3 7 2 7 5 12 14 8 12 10 4 3 2 1 10. a. b. > > > > > > 5 3 5 8 5 4 3 2 17 4 19 2 7 17 4 5 c. d. 5 > > > > >5 >4 5 3 17 3 9 3 9 19 11 Word Problems 1. 3 2. Ryan 3. Least: Alex and Casey; 12 8. a.

1 2

3 kg 4

d. 8 d.

10 49

2 3

1 4

d. 2

c. 3

d. 22

3. 2 e. 9

3 5

e. 157

1 kg 3 f. 9

1 3

f. 44

6 7

288

Maths Grade 5 Book_Chapter 13-17.indb 288

18-12-2023 11:39:17

5. a. 12

c. 1

b. 8

1 hours 2

3 kg 2. 2

Word Problems 1. 40 pages

6D 1. a. 3 2

2. a. 22

1 2

f. 10 e.

1 3

4. a.

e. 10

67 152

1 5. a. 21 b. 9

1 c. 2 4

Word Problems 1. 28

2. 19

Chapter Checkup 1. a. 1 f. 5

3 4

e. 5

2. a. 17

2 7

f. 5

16 e. 27

6. a.

f. 2

5 4

7. a. 12

8. a. False 9. a.

2 5

4. 88

13 20

1 b. 2 9

1 2

i. e.

c. False

3. a. 52.1

b. 413.2 c. 100.5 d. 805.7 1 9 4. a. 40 + 5 + or 40 + 5 + 0.1 b. 10 + 7 + or 10 + 7 + 0.9 10 10

3 5

c. 42

d. 1

1 3

121 123

16 17

e. False

11 km 28

1 cm 6

9 litre 28

6. 4

3 10

23 m 30

2. 5.

3. 136

3 10

1 or hundredths 100

e. 700 + 80 + 9 +

5. a. 0.5

1 12

1 d. 6

1 3

13 15

f. 900 + 80 + 5 +

23 c. 1 24

3 4

d. True

1 18

e. 7

4 5

2. 17

4. 1

5 162

d. 9

b. Product of 5 and

23 m 72

1 or tenths 10

14 15

5 or 100 + 40 + 3 + 0.5 10

1 or 500 + 40 + 8 + 0.1 10

4 or 700 + 80 + 9 + 0.4 10

3 or 900 + 80 + 5 + 0.3 10

b. 4.1

Word Problems 1. 125.5

7B 1. a. 1.25

b. 3.61

c. 3.8 2.

d. 0.9

e. 78.5

f. 86.8

3 ; 0.3 10

;

3 4

d. 1

1 1 and 4 3 4

d. 14 c.

3 20

Maths Grade 5 Book_Chapter 13-17.indb 289

1 3

1 b. 3 3

11 15

Answers

8 2 or 2 3 3

Let's Warm-up 3.

c. 8

Chapter 7

4 9

2. a. 0.5 = Zero and five tenths or zero point five b. 4.0 = Four and zero tenths or four point zero

d. 500 + 40 + 8 +

12 17

b. 16

5. 10

1 27

Two and three-tenths.

c. 100 + 40 + 3 +

b. 7

d. 5

Word Problems 1. 4

, One and six-tenths;

7 3

f. e. 10

1 5

1 4

14 78

10. a. Sum of 2

1 3

b. 4

33 34

c. 3

19 6

12 d. 24 13

b. 58

d. 21

1 12

b. True

4. 500

28 135

17 18

1 2

14 12

6 5. a. 2 7

2 7

1 5

7A 1.

3. 20 L

5 4. a. 8

1 f. 1 4

1 3

1 2

3. a. 1

1 f. 6

3 e. 8 5

e. 7

5 9

3 5

1 2

b. 1

11 504

c. 11

19 216

3. a.

19 108

4 5

b. 5

1 108

d. 3

9 5

Do It Yourself

d. 2 days

2 5

3. a. forty-two and fourteen-hundredths or forty-two point one four b. eighty-seven and eight hundredths or eighty-seven point zero eight c. eighty-one and eighty-seven hundredths or eighty-one point eight seven d. one hundred seventy-eight point six four or one hundred seventy-eight and sixty-four hundredths e. eight hundred seventy-four point zero nine or eight hundred seventy-four and nine hundredths f. four hundred fifty-seven point one eight or four hundred fifty-seven and eighteen hundredths 4. a. 72.33 b. 417.02 c. 26.47 d. 361.04 5. a. 40 + 5 +

1 3 or 40 + 5 + 0.1 + 0.03 + 10 100

b. 70 + 9 +

6 or 70 + 9 + 0.06 100

c. 40 + 1 +

3 7 + or 40 + 1 + 0.3 + 0.07 10 100

289

18-12-2023 11:39:19

d. 400 + 80 + 6 + e. 800 + 70 + 9 + f. 900 + 70 + 6 + 6. a. 0.04 e. 70.03

7 2 + or 400 + 80 + 6 + 0.7 + 0.02 10 100

3 4 + or 800 + 70 + 9 + 0.3 + 0.04 10 100

7D 1. a. 25

b. 2 c. 5 d. 4 e. 125 f. 25 b. 0.32 c. 0.8 d. 0.34 e. 0.75 705 13 74 627 f. 0.475 3. a. b. c. d. 4 10 5 50

2. a. 0.9

7 or 900 + 70 + 6 + 0.07 100

b. 14.02 f. 867.17

Word Problem 1. 20 + 3 +

c. 43.48

d. 503.04

4 8 + or 20 + 3 + 0.4 + 0.08 10 100

7C 1. a. twenty-five and one hundred twenty-seven thousandths

or twenty-five point one two seven b. forty-two and eight hundred seventy-two thousandths or forty-two point eight seven two c. sixteen point zero one two or sixteen and twelve thousandths d. one hundred seventy-four and two hundred one thousandths or one hundred seventy-four and two zero one e. ninehundred sixty-five point zero four five or nine-hundred sixty-five and forty-five thousandths f. seven hundred sixty-five and four thousandths or seven hundred sixtyfive point zero zero four 2. a. 31.352 b. 52.402 c. 12.041 d. 302.005 3. a. 10 + 5 + 0.06 + 0.002 or 10 + 5 +

e. 400 + 80 + 6 + 0.1 + 0.04 + 0.005

51 ; 2.050 20

78 ; 3.120 25

Word Problems 1.

7E 1. None

3 8

803 25

4. a. 7 ; 0.35 20

5. a. >

4. a. 0.006 b. 14.025 c. 16.106 e. 506.307 f. 489.982 5.

1 ; 0.005 200

131 ; 0.524 250

c. =

d. <

e. >

f. >

0.5

0.6

3 1 + 100 1000

d. 106.008

;

6. Answers may vary. a . 45.60; 45.600 b. 187.20; 187.200 c. 87.020; 87.0200 d. 963.140; 963.1400 e. 12.7010; 12.70100 f. 189.2210; 189.22100 Word Problems 1. one and two hundred sixty-nine thousandths or one point two six nine 6 5 9 + + 10 100 1000

2. 0.625

1 1000

f. 100 + 50 + 4 + 0.03 + 0.001 or 100 + 50 + 4 +

b. <

21 ; 2.1 10

2. a.  b.  c.  d.  e.  f.  3. a. 1 b. 2 c. 1 d. 3 e. 1 f. 2 4. a. 13.15, 1.20 b. 3.48, 1.20 c. 48.00, 1.526 d. 1.400, 47.584 e. 53.230, 17.164 f. 1.002, 348.100 5. a. 0.6 and 0.2 b. 0.35 and 0.50 c. 0.625 and 0.750 d. 0.80 and 0.25 e. 0.500 and 0.125 Word Problem 1. 23.200, 26.987 and 24.540 7F 1. a.

6 4 2 + + 10 100 1000

1 4 5 + + 10 100 1000

2. 3 + 0.6 + 0.05 + 0.009 + or 3 +

c. f.

6 2 + 10 1000

d. 100 + 10 + 1 + 0.001 or 100 + 10 + 1 +

or 400 + 80 + 6 +

1151 500

6 2 + 100 1000

b. 50 + 4 + 0.6 + 0.04 + 0.002 or 50 + 4 + c. 70 + 8 + 0.6 + 0.002 or 70 + 8 +

0.9 0.89 b. 4.5 c. 15.67 d. 87.654 e. 186.6 2. a. 4.2 f. 294.98 3. a. 15.46 b. 89.14 c. 81.174 d. 853.6 e. 47.650 4. a. 14.01 < 14.1 < 14.101 < 14.14 b. 84.5 < 84.55 < 84.56 < 84.6 < 84.65 c. 184.1 < 184.112 < 184.2 < 184.23 d. 54 < 64.22 < 64.23 < 64.32 < 64.33 5. a. 7.54 > 7.5 > 7.44 > 7.4 b. 48.66 > 48.63 > 48.6 > 48.36 c. 921.41 > 921.4 > 921.35 > 921.26 d. 185.55 > 185.501 > 185.5 > 185.45 Word Problems 1. 0.9 litres 2. 33.7°C < 34.2°C < 34.5°C < 35.2°C 3. 24.68 > 24.54 > 24.36. 7G 1. a. True b. True c. False d. True 2. a. 3 b. 8 c. 48 d. 15 e. 71 f. 156 3. a. 32 b. 15 c. 123 d. 543 4. 22.9 5. a. < b. = c. = d. < e. < Word Problems 1. 16 2. 89

Chapter Checkup 1. a.

1.2 = 1 + 0.2 = 1 +

2 10

290

Maths Grade 5 Book_Chapter 13-17.indb 290

18-12-2023 11:39:20

Chapter 8 Let's Warm-up Do It Yourself

1. <

2. =

3. <

4. >

5. >

8A 1. a. 0.3 4 8 + 10 100

1.48 = 1 + 0.4 + 0.08 = 1 +

2. a. fifteen and two-tenths or fifteen point two b. seventyone and sixty-five tenths or Seventy-one point six five c. one hundred fifty-two and one tenths or one hundred fifty-two point one d. eight hundred fourteen and thirty-six hundredths or eight hundred fourteen point three six e. one hundred seventy-six and two hundred fiftyfour thousandths or one hundred seventy-six point two five four f. one hundred seventy-six and eight hundred one thousandths or one hundred seventy-six point eight zero one 3. a. 40.101 b. 63.4 c. 103.05 d. 420.011 e. 31.005 f. 95.223 1 or 10 + 1 + 0.1 10

4. a. 10 + 1 +

b. 100 + 80 + 9 + c. 10 + 6 +

5 4 + or 10 + 6 + 0.5 + 0.04 10 100

2 or 400 + 90 + 0.02 200

d. 400 + 90 + e. 10 + 7 +

5. a. 0.125

3 3 + or 800 + 40 + 3 + 0.03 + 0.003 100 1000

b. 0.8

63 5

b. 10;

e. 1000;

81 8

7. a. 12.3

c. 102.3

2. a. 3.2 b. 0.58 c. 19.101 d. 8.94 e. 41.837 f. 110.803 3. a. 19.7 b. 39.8 c. 550.584 d. 678.877 4. a. = b. > c. > d. > e. < f . > 5. a. 2.954 b. 77.617 Word Problems 1. 7.4L 2. 1.47 m 3. 3.45 km 4. ₹4939 8C 1. a. 0.8 b. 2.0 2. a. 456 b. 1547.8 c. 63,157 d. 10 e. 43.60 f . 179.34 g. 2.528 h. 438.987 3. a. < b. = c. < d. > e. > f. > 4. a. 10 b. 1000 c. 10 d. 1000 e. 9.87 f. 52.2533 5. 27.6 Word Problems 1. ₹206.15 2. ₹7516.7 3. ₹158.75 8D 1. ₹89,190 2. ₹633.05 3. ₹52,030 4. Suresh’s Father, ₹17,458 5. ₹18,970.02 Word Problem 1. ₹1344.89 8E 1. a. 0.2

6 8 4 + + or 10 + 7 + 0.6 + 0.08 + 0.004 10 100 1000

f. 800 + 40 + 3 +

6. a. 10;

3 or 100 + 80 + 9 + 0.3 10

b. 0.7 2. a. 0.4 b. 0.63 3. a. 35.9 b. 100.24 c. 79.19 d. 117.882 e. 826.818 f. 1052.518 4. a. 157.7 b. 53.9 c. 88.58 d. 51.003 5. a. < b. = c. > d. = Word Problems 1. ₹15.75 2. 31.54 grams 3. 5.862 km 4. a. 70.88 kg b. 83.735 kg b. 0.77 8B 1. a. 0.7

c. 0.6

d. 0.9

261 5

c. 10;

f. 1000;

5063 200

12.4

b. 14.5

> 102.25

e. 0.25

241 2

d. 222.22

e. 187.540 > 187.504

3. 100 + 50 + 6 + 0.2 + 0.05 or 100 + 5 + 6 +

Answers

Maths Grade 5 Book_Chapter 13-17.indb 291

222.02 <

b. 0.2

104 5

14.55 >

f. 3.003

8. a. 14.2 < 14.23 < 14.25 < 14.3 b. 87.50 < 87.60 < 87.64 < 87.7 c. 235.6 < 235.64 < 235.66 < 235.666 d. 888.08 < 888.8 < 888.88 < 888.888 9. a. 1.33 > 1.31 >1.3 > 1.2 b. 19.54 > 19.501 > 19.44 > 19.4 c. 555.555 > 555.55 > 555.5 > 555.05 d. 748.11 > 748.101 > 748.1 > 748.01 10. a. 1 b. 15 c. 100 d. 100 Word Problems 1. Utkarsh 2. 41

4. 12.9 < 12.99 < 13.01 < 13.1

d. 10;

f. 0.8

3.033

2. a. 1.56 b. 15.464 c. 0.5123 d. 5.1347 e. 0.012 f. 0.032561 g. 0.0231 h. 0.2 i. 0.011236 3. a. 20.1 b. 9.13 c. 45.37 d . 4.09 e. 5.415 f. 32.45 4. a. 42.05 b. 21.02 c. 0.534 d. 4.725 e. 16.512 f. 39.475 5. a. = b. > c. < e. > f. < 6. 100 Word Problems 1. 4.2 kg 2. 5.894 kg 3. 1000 pounds 4. $6.32

Chapter Checkup 1. a. 0.7

e. 107

f. 1000

2. a. 63.123 e. 518.755 3. a. 1.2

d. =

b. 0.78

b. 46.34 c. 33.48 f. 114.921

d. 51.792

2 5 + 10 100

291

18-12-2023 11:39:20

b. 3

9C 1. Answers may vary. Sample answers:

a. ∠XYZ, ∠ZYX b. ∠POR, ∠ROP c. ∠AOB, ∠BOA d . ∠LMN, ∠NML 2. a. 74° b. 115° c. 152° d. 120° 3. a. 53° b. 90° c. 123° d. 32° e. 57° f. 127° 4. a. b. c.

4. a. 10 b. 1000 c. 100 d. 10 e. 1000 f. 1000 5. a. 18.84 b. 58.518 c. 289.26 d. 25.632 e. 516.675 f. 1,449.612 6. a. 4.701 b. 100 c. 1000 d. 1000 e. 0.0142 f. 100 7. a. 33.07 b. 85.971 c. 55.33 d. 37.399 e. 50.605 f. 24.512 8. a. 18.4 b. 12.35 c. 30.02 d. 46.39 e. 23.65 f. 12.142 9. a. 25.23 b. 24.85 c. 24.68 d. 22.8852 10. a. 11.36 b. 11.245 c. 23.65 d. 52.25 e. 71.22 f. 89.245 11. a. ₹372 b. ₹13,378.5 c. ₹3,765.5 d. ₹2,262.255 e. ₹31,000 Word Problems 1. ₹4.2 2. 12.1 hours 3. 3.96 km 4. $1.22 5. ₹14,259.2 6. 5.15 days 7. 25.543 kg

Let's Warm-up 3. Standing Line

Do It Yourself

c. e.

160°

5. R

95°

N d.

   5 . a. A, B, C, D, E, F; AC , CB, AB b. Answers may  vary. Sample answers: FD c. 1 ray, BE . Word Problem Intersecting lines 9B 1. Answers may vary. Sample answers:

155°

180°

8. a. Quadrilateral; 4 sides, 4 angles b. Heptagon; 7 sides, 7 angles c. Quadrilateral; 4 sides, 4 angles d. Octagon; 8 sides, 8 angles 9. a, d 10. O

125°

2. a. ∠PQR, ∠RQP or ∠Q b. ∠LMN, ∠NML or ∠M c . ∠XYZ, ∠ZYX or ∠Y 3. ∠1 and ∠3 4. Acute, Obtuse, Straight, Obtuse 5. 8 right angles C

Chapter 10 Let's Warm-up 2.

10 cm

Word Problems 1. Parallel lines

145°

1. Slanting Line 2. Sleeping Line 4. Sleeping Line 5. Sleeping Line

b. Ray c. Intersecting lines 2. a. False b. False c. True 3. Line segment, Parallel lines, Point, Ray

120°

85°

Q 6 cm Word Problem 1. 90° 9D 1. PQ, QR and PR. 2. a. Quadrilateral b. Nonagon c. Hexagon d. Decagon 3. a. nine b. three c. seven d. five 4. c 5. a. False b. True c. True d. False 6. Polygons: a,c,d; Non-polygon: b Word Problem 1. Hexagon Chapter Checkup 1. a. Line b. Ray c. Line segment d. Line segment 2. a. True b. False c. False 3. Acute angle, Right angle, Straight angle, Obtuse angle 4. a. acute b. right c. straight d. obtuse e. acute 5. a. 150° b. 130° c. 170° 6. a. 120° b. 155° c. 129° d. 90° 7. a. b. c. d. 140° 115° 65° 75°

9A 1. a. Parallel lines

4. a.

100°

Chapter 9

d. Line d. True

50°

M 2. Pentagon; 5 sides and 5 angles

1. 3.

D 4.

E F Word Problem 1. 7:30 am: Acute angle 1:30 pm: Obtuse angle

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3. a.

Do It Yourself 10A 1. c and d 2.

A H T b.

4 . a. False b. False c. True 5. Answer may vary. Sample answer:

The pattern is rotating 90° in clockwise direction. 4 . a. 45° Clockwise b. 90° Clockwise c. Not a pattern. d. 180° Anti-Clockwise or 180° Clockwise. 5. a. b.

10B 1.

d. True

Word Problem 1.

Chapter Checkup 1. a.

Word Problem 1. d

d. Asymmetrical.

he number of sides of each figure in each T pentagon is in the pattern: 6, 5, 4, 3. 2. d, f, g, h. 3. a.

2. a.

b. c.

4. a. 5, 16, 27, 38, 49 b. 10, 21, 43, 87, 175 c. 1, 2, 6, 42, 1806 d. 15, 18, 23, 26, 31 5. 256, 241, 226; 15 is subtracted each time. 6. a. 45, 55, 65, 75, 85, 95 b. 729, 2187, 6561, 19,683, 59,049 c. 57, 64, 72, 81, 91 d. 75, 69, 62, 54, 45 7.

3. b 4. b.

5 . b

Word Problem 1. b

10C 1. a. Adding 4 to each term.

b. Adding 2 to each term. c. The pattern has squares of natural numbers. d. Adding 5 to each term e. Each term is the sum of the two preceding terms. 2. a. 53453 b. 89012 c. 14523 d. 67209 e. 537890 3. 5 red and 5 yellow flowers. 4 . 60, 59, 57, 54, 50, 45, 39, 32, 24, 15. 5. Answer may vary. Sample answer:

6 1 5

Word Problems 1. 84 2. 31 10D 1. a. 2 lines of symmetry b. Asymmetrical c. 1 horizontal line of symmetry d. Many lines of symmetry. 2. a. no line of symmetry

c. no line of symmetry

Answers

Maths Grade 5 Book_Chapter 13-17.indb 293

9. 2 1 4 16 8 10. Similarity: Same pattern rule. (×2) Difference: The first pattern are Multiples of 2 but the second pattern have multiples of 3. Word Problems 1. 125789 2. 17, 20, 23 3. 128, 256, 512 4. 300 steps 5. ₹2560

Chapter 11

1. Answers may vary. Sample answers: 20 cm 2. Answers may vary. Sample answers: 1m 3. 5 kg 4. 8000 mg 5. 2000

Let's Warm-up

293

18-12-2023 11:39:28

Do It Yourself 11A 1. Answers may vary. Sample answers:

a. 15 cm; 13 cm; 2 cm b. 3 cm; 5cm; 2 cm c. 4 cm; 4 cm; 0 cm d. 22 cm; 19 cm; 3 cm 2. a. 5.5 cm b. 2.7 cm c. 9 cm d. 4.5 cm 3. a. 4.5 cm b. 689.2 m c. 0.7698 km d. 1.056 km e. 0.547 m f. 203.4 m 4. a. 45.1 m b. 16,080 mm c. 280 m 5 dm 5. 0.0001 km Word Problems 1. 25 km 794 m 2. 3 cm 3 mm 3. 53 m 16 cm 4. 53 dm 11B 1. a. 30 g b. 85 g c. 70 kg d. 100 g 2. a. 0.079 kg b. 46,77,000 g c. 68.76 g d. 165.5 g e. 9,75,000 mg f. 8,40,000 mg 3. a. 6.01 kg; 6010g b. 16.08 g; 16,080 mg c. 0.5476 kg; 5,47,600 mg d. 3.08 g; 308 cg e. 87.06 kg; 8706 dag f. 12.42 hg; 1242 g 4. 1.688 kg 5. 25 eggs Word Problems 1. 6 kg 133 g 2. 6 kg 100 g 3. 2.34 kg 4. 4 kg 56 g 5. a. 2 kg 88 g b. 1 kg 478 g 6. 0.0051 kg Chapter Checkup 1. a. m; kg b. cm; g c. m; kg 2. a. 12.5 cm b. 8.5 cm c. 12 cm d. 13.5 cm e. 17.9 cm f. 22.2 cm 3. 14.2 cm 4. 4.5 cm 5. a. 0.008 km b. 4.35 hm c. 12,32,000 mm d. 89.7 dam 6. a. 5000 g b. 4.64 g c. 54,78,000 mg d. 43.07 kg 7. 380 g 8. 1,25,000 mg 9. 41 kg 10. Answers may vary. Sample answers: a. One 4 cm and one 8 cm b. One 4 cm and two 8 cm c. Three 8 cm Word Problems 1. a. 36 cm b. No 2. No, they cannot enter the lift together; Yes, We must follow the lift rules in respect of overloading to ensure the safety for all. 3. 20 kg flour, 7.5 kg butter and 10 kg sugar.

Chapter 12

Perimeter: 10 cm; 12 cm; 12 cm; 14 cm Area: 6 cm2; 6 cm2; 5 cm2; 6 cm2

Let's Warm-up

Do It Yourself 12A 1. a. 54 cm b. 280 m c. 76 m

2. a. 128 m b. 212 cm c. 108 m d. 312 cm 3. a. 80 cm b. 300 cm c. 60 cm d. 100 cm 4. a. 128; 144; 240; 150; 400 b. 84; 168; 220; 296; 384 5. a. 24 mm b. 37 mm Word Problems 1. 180 m 2. 8 cm 12B 1. a. sq. m b. sq. cm c. sq. m d. sq. m e. sq. km f. sq. m g. sq. m h. sq. m 2. a. 330 sq. cm b. 1692 sq. m c. 4060 sq. m d. 5456 sq. cm 3. a. 3136 sq. m b. 4489 sq. cm c. 576 sq. m d. 1600 sq. cm 4. a. 7.5 sq. units b. 17.5 sq. units c. 6 sq. units 5. a. 96 sq. cm b. 49 sq. cm c. 33 sq. cm 6. Answers may vary. Sample answers: a. b.

Perimeter = 22 cm `

Area = 80 sq. cm Word Problems 1. 144 sq. m 2. 400 sq. m

Chapter Checkup 1. a. 64 cm; 252 sq. cm b. 96 m; 567 sq. m

c. 164 m; 1617 sq. m d. 254 cm; 3990 sq. cm 2. a. 92 cm; 529 sq. cm b. 128 m; 1024 sq. m c. 180 m; 2025 sq. m d. 320 cm; 6400 sq. cm 3. a. 40 cm; 128 cm

b. 105 m; 11,025 sq. m c. 28 m; 952 sq. m 4. Largest area – B Smallest perimeter – A and C 5. a. 39 sq. m b. 78 sq. m c. 64 sq. cm 6. a. 12 sq. units b. 6 sq. units c. 4 sq. units d. 10 sq. units 7. d 8. c 9. 44 cm 10 No. Word Problems 1. 5000 sq. cm 2. 11 sq. m

Chapter 13

1. Jug

Let's Warm-up Do It Yourself 13A 1. a. 5 mL b. 3 L

4. 5000

2. 8 glass 5. 8000

3. 8 L

2. a. 0.658 L b. 84,37,000 mL c. 257.6 L d. 50.54 L e. 82.1 daL f. 1,44,000 mL 3. a. 8.060 L b. 3289 cL c. 4012 L 3 dL d. 9006 daL 4. Incorrect, 30.50 dL 5. Yes Word Problems 1. 0.75 L 2. 7 cups 13B 1.

2. a. 16 cu. units b. 8 cu. units c. 80 cu. units d. 36 cu. units 3. 900 cu. cm 4. 6660 cu. cm 5. Rinku’s lunch box is bigger in size. Word Problems 1. 6 cu. m 2. 4 bottles 3. 260 mL

Chapter Checkup 1. a. 50 mL b. 20 L c. 500 mL d. 250 mL

e. 5000 L 2. about 1 L 3. 800 mL, 200 mL, 300 mL, 700 mL, 500 mL 4. a. 0.980 L b. 68,69,000 mL c. 979.6 L d. 34.56 L e. 24.3 daL f. 9,12,600 mL 5. Students will colour the jars upto 50 mL, 40 mL, 10 mL and 30 mL mark in Set A. They will colour the jars upto 60 mL, 20 mL, 125 mL and 90 mL mark in Set B. 6. a. 4,4,4 b. 8,2,3 7. a. 12 cu. units b. 44 cu. units c. 15 cu. units d. 22 cu. units 8. All the solids are of equal volume. 9. a. 112 cu. unit b. 63 cu. unit c. 36 cu. unit 10. 37,500 cu. cm Word Problems 1. 765 mL 2. 2 L 100 mL 3. 26 L 4. 9 L 900 mL

Chapter 14

1. cube

Let's Warm-up Do It Yourself 14A 1. a.

4. sphere b.

2. cylindrical c.

3. cuboid d.

2. Cone: F = 2; E = 1; V = 1, Sphere: F = 1; E = 0; V = 0, Cuboid: F = 6; E = 12; V = 8 3. Square prism: 2 square faces and 4 rectangular faces. Cylinder: 2 circular identical faces and one curved face. 5. Answer may vary. Sample 4. answer: Pyramid has 1 base and prism has 2 bases. Both are solid figures with three dimensions. Word Problem 1. Similarity: 5 faces Difference: Rohan’s tent has a triangular base, and Simi’s tent has a rectangular base. 14B 1. a. 2. a. 3. a. Candle b. Pyramid c. Ice cream cone d. Ice cube

294

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4. a.

Word Problem 1. Megha 2.

14C 1. a.

3. a.

Top View

Front View

Side View

Front view

4. a.

Top view

Side view

c. 6. a. 32 km b. 64 km c. 120 km d. 200 km 7. Answers may vary. Sample answers:

e. 256 km

Word Problem 1. 15 building blocks

5. b

Front view

Side view

Top view

a. Yes

14d 1. a. south b. east c. west

2. 125 km 3. 24 km 4. Cupboard = 2 m, 8 m; table = 6 m, 4 m; desk = 2 m, 6 m 5. 12.5 km Word Problem 1. 35 km 2 . one

Window

Door

9. 3 times.

Window

Chapter 15 4. Hours

Door

Cube

Cylinder

Cuboid

2. a. Net b. Deep drawing c. Cube and cuboid d. Floor plan 3. a. Cylinder: Faces = 3; Edges = 2; Vertices = 0 b. Cube: Faces = 6; Edges = 12; Vertices = 8 c. Cone: Faces = 2; Edges = 1; Vertices = 1 d. Sphere: Faces = 1; Edges = 0; Vertices = 0 e. Cuboid: Faces = 6; Edges = 12, Vertices = 8 f. Square Pyramid: Faces = 5; Edges = 8; Vertices = 5

Maths Grade 5 Book_Chapter 13-17.indb 295

1. Minutes 5. Months

Let's Warm-up

5. Windows of the backside and either left or right side. Word Problem 1. 8 windows.

Answers

KITCHEN

Word Problem 1. 21 km

Door

Chapter Checkup 1. a.

10. Answer may vary. Sample

BEDROOM

4. kitchen

Window

LIVING ROOM

b. 3

Window

Door

8. 4 doors answer:

Window

14E 1.

Door

Window Window

2. Seconds

3. Days

Do It Yourself 15A 1. a. 180 minutes b. 240 minutes c. 140 minutes

d. 190 minutes e. 230 minutes f. 270 minutes 2. a. 300 seconds b. 480 seconds c. 550 seconds d. 620 seconds e. 760 seconds f. 950 seconds 3. a. 2 hours b. 3 hours c. 2 hours 30 minutes d. 3 hours 20 minutes 4. a. 4 minutes b. 8 minutes c. 5 minutes 50 seconds d. 7 minutes 20 seconds e. 8 minutes 40 seconds f. 9 minutes 10 seconds 5. 4800 seconds Word Problems 1. 1 hour 35 minutes 2. 510 minutes 3. 1 hour 45 minutes 15B 1 . a. 11 weeks 1 day b. 8 weeks 6 days c. 3 years 11 months d. 5 years 11 months 2. a. 7 weeks 1 day b. 6 weeks 3 days c. 9 years 3 months d. 2 years 6 months 3. 17 years 3 months 4. 15 months 21 days

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Number of Students

8 6 4 2 0

Mystery Adventure Fantasy Science Historical Fiction fiction Book Genre

5. a. 35 tickets b. 715 tickets Word Problem 1.

1. 370 p

Let's Warm-up

4. ₹3.05

2. ₹4.12

3. 610 p

c. 35 tickets

8 6 4 2 0

nda

y ay day sda rsd dne Thu We Day

c. 40 students 3. a. Activities

2. ₹25 3. ₹1026 4. ₹180 5. ₹36 Word Problems 1. Second Shopkeeper. 2. 8; 0 3. ₹41 16B 1. a. ₹1285 b. ₹145 c. ₹8400 d. ₹455 2. a. Loss has been incurred b. greater c. CP d. ₹500 3. ₹67 4. Profit, ₹565 5. Loss, ₹6950 Word Problems 1. ₹7; ₹16 2. ₹217.5 16C 1. ₹20 2. ₹7 3. Loss, ₹5 4. Loss, ₹14 5. Profit, ₹26 6. Neither profit nor loss 7. Neither profit nor loss 8. Loss, ₹20 9. ₹28,437 10. ₹20,250 Chapter Checkup 1. a. Unitary method b. ₹6 c. ₹260 d. Incurred a loss 2. a. False b. True c. True d. False 3. 10 Pencils for ₹50 4. 22 chairs at ₹14300. 5. a. Profit, ₹18 b. Loss, ₹7 c. Loss, ₹22 6. ₹20,860 7. ₹66.33 8. ₹166.67 9. ₹2200 10. ₹2750 Word Problems 1. ₹14.75 2. Loss; ₹5 3. Loss; ₹198 4. Profit; ₹7000

Studying

Watching TV

Reading

b. 80 students

Fraction

Angle

3 =1 12 4 4 =1 12 3 2 =1 12 6 3 =1 12 4

1. Kavya

Let's Warm-up 5. 25

Do It Yourself 17A 1. V-A, V-C

2. a. 55

2. Rehan

3. Kavya

b. buffalo, horse

4. 100

Playing 1 4

1 4

Watching TV

Studying

c. horse

Playing

360° × 1 = 90° 4 1 360° × = 120° 3 360° × 1 = 60° 6 360° × 1 = 90° 4

Hours Reading

Chapter 17

Frid

2. a. Walk

Hours

Playing

s Tue

17B 1. a. Food b. ₹25 c. ₹10

Do It Yourself 16A 1. a. 5 b. 60 c. 200 d. 40; 2200 e. ₹15

d. 35 tickets

16 14 12 10

Chapter 16

20 18 16 14 12 10

Number of Squirrels

5. 11 years 6 months old Word Problems 1. 2 months 15 days 2. 4 years 5 months 25 days 15C 1. a. 5 hours 55 min b. 10 hours 50 min c. 1 hour 35 min d. 9 hours 25 min 2. 1:40 p.m. 3. 3 hours 15 minutes 4. 3:15 p.m. 5. 1 hour 26 minutes 5 seconds Word Problems 1. 90 minutes 2. 3 hours 30 minutes 3. 2 hours 15 minutes 4. 5 hours 25 minutes 15D 1. a. ii b. i 2. 26°C, 35°C, 42°C, 12°C 3. a. Srinagar b. Chennai c. 29.7°C d. 3.8°C 4. 7.2°C 5. City B > City D > City C > City A Word Problem 1. 22.5°C. Chapter Checkup 1. a. 7200 seconds b. 5400 seconds c. 8400 seconds d. 320 seconds e. 400 seconds f. 415 seconds 2. a. 1.5 hours b. 1 hour 55 minutes c. 2 hours 40 minutes d. 4 minutes 40 seconds 3. 390 minutes 4. 375 minutes 5. 5 pm 6. 6 years 11 months 7. a. 12:20 p.m. b. 9:00 p.m. c. 3:05 p.m. 8. 16.3°C 9. 15 August 10.9 times Word Problems 1. Sunday, 10 July 2. 3 hour 43 minutes 3. 8 minutes 4. 9:30 a.m.

Studying

Watching TV Items Served

1 3 Reading

Curd Rice 1 4

Mixed juice 3 10

Day 5

Day

Day 4

Veg Soup 1 10

Day 3 Day 2

Curd Rice

Day 1 0

4 6 8 10 12 14 16 18 20 22 24 26 28 Number of Plastic Bottles Recycled

Salad

5. a. 50 students

Pulav 3 20 Pulav Veg Soup

1 5

c. Sports

Salad 1 5

Mixed juice

1 20

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Word Problem 1.

6. a. Friday b. 65 kg c. 5 kg 7. a. 5:00 p.m. customers 8. a. Jaipur b. Ooty c. 15°C d. Bangalore 9.

Favourite sports Soccer

Tennis

1 4

b. 25

December

November

Swimming Soccer

1 6

Basketball

17C 1. Wednesday and Thursday

Month

Basketball

Swimming

August July 0

10.

Favourite Instrument Others

Guitar

120

Number of Refrigerators

140

Number of absentees

September

Tennis

2. Most = Pune; Least = Goa 3. a. 8°C b. Week 1 c. 8°C 4. a. 2017 b. 800 c. 2018 5. a. Match 2 and Match 6 b. 10 runs c. 5 times d. 180 runs Word Problem 1. a. April b. August; 9000 items c. 2000 items Chapter Checkup 1. a. 15 b. 19 c. 2 more 2. a. Thursday b. Boys c. 14 pages d. 3 pages 3.

October

100 80 60 40 20 0

4. a. 200 5.

nda

b. 100

Others

y y day nesda da d urs e Th W

s Tue

c. Mystery

Day

a rd

u at

Piano

5 Guitar

Piano

Word Problem 1. a. Chocolate d. 22 children

Hours spent

Drums

1 4 Others

b. Blueberry

c. 14 children

8 School

Sleep 3 8

Play School

ida

Drums 1

Homework

Answers

Maths Grade 5 Book_Chapter 13-17.indb 297

24 Play

1 3

Homework 1 8 Sleep

Others

297

18-12-2023 11:39:32

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