Imagine_Maths_CB_Grade5_AY25 by Uolo

MATHEMATICS

Master Mathematical Thinking

Acknowledgements

Academic Authors: Animesh Mittal, Muskan Panjwani, Anjana AR, Anuj Gupta, Gitanjali Lal, Simran Singh

Creative Directors: Bhavna Tripathi, Mangal Singh Rana, Satish

Book Production: Sanjay Kumar Goel, Vishesh Agarwal

Project Lead: Neena Aul

VP, Learning: Abhishek Bhatnagar

All products and brand names used in this book are trademarks, registered trademarks or trade names of their respective owners.

First impression 2024

Second impression 2025

This book is sold subject to the condition that it shall not by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior written consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of both the copyright owner and the above-mentioned publisher of this book.

Book Title: Imagine Mathematics 5

ISBN: 978-81-979482-3-7

Published by Uolo EdTech Private Limited

Corporate Office Address: 85, Sector 44, Gurugram, Haryana 122003

CIN: U74999DL2017PTC322986

Illustrations and images: www.shutterstock.com, www.stock.adobe.com and www.freepik.com

All suggested use of the internet should be under adult supervision.

Preface

Mathematics is an essential tool for understanding the world around us. It is not just another subject, but an integral part of our lives. It shapes the very foundation of our understanding, personality and interaction with the world around us. Studies from across the globe have shown that proficiency in mathematics significantly influences career prospects and lifelong learning.

According to the NEP 2020, mathematics and mathematical thinking are crucial for empowering individuals in their everyday interactions and affairs. It focuses on competencies-based education, which essentially means actively and effectively applying mathematical concepts in real life. It also encourages innovative approaches for teaching maths, including regular use of puzzles, games and relatable real-world examples to make the subject engaging and enjoyable.

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 math exciting, relatable and meaningful for children.

Imagine Mathematics 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 NCF 2023 and other literature in learning educational pedagogies. Subsequent pages elaborate further on this approach and its actualisation in this book.

This book incorporates highly acclaimed, learner-friendly teaching strategies. Each chapter introduces concepts through real-life situations and storytelling, connecting to children’s experiences and transitioning smoothly from concrete to abstract. Clear explanations and simple steps are provided for problem-solving. Engaging facts, error alerts and fun activities are integrated throughout to keep learning interesting and holistic. Importantly, concepts are interconnected across topics and grades, providing a scaffolding that ensures comprehensive and meaningful learning.

This book supports learners at all levels, providing opportunities to build critical thinking skills through questions and activities aligned with Bloom’s Taxonomy. For those seeking a greater challenge, the book includes thought-provoking questions that push learners to apply, analyse and evaluate. Additionally, the problems are rooted in real-world contexts, making the learning experience both relatable and meaningful.

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 worksheets, among other things. Interactive exercises on the digital platform make learning experiential and help in concrete visualisation of abstract mathematical concepts. We invite educators, parents and students to embrace Imagine and join us in nurturing the next generation of thinkers, innovators and problem-solvers. Embark on this exciting journey with us and let Imagine be a valuable resource in your educational adventure.

Product Package at a Glance

Welcome to our comprehensive learning package designed to enhance educational experiences through three key components: print materials, digital resources and assessment tools. Our print materials provide in-depth and accessible information in a pedagogically suitable way, our digital resources offer interactive and engaging learning experiences, while our assessment tools ensure thorough understanding and progress tracking.

PRINT

Engaging Textbooks

Teacher Manuals

ASSESSMENT

Competency-based Model Assessments

Question Paper Generator

Student and Teacher Apps

Video Solutions

Interactive Tasks & Exercises

Byte Size Lesson Modules

Key NEP Recommendations

The National Education Policy (NEP) 2020, introduced by the Government of India, represents a transformative shift in the country’s education system. It aims to create a more holistic, dynamic and multidisciplinary approach to education. NEP 2020 focuses on fostering conceptual understanding, skills and values that align with the demands of the 21st century, while also preserving India’s rich cultural heritage. UOLO is fully committed to actualising the vision of NEP 2020 by meticulously adhering to its outlined recommendations.

1. Focus on conceptual understanding

2. 21st century skills, values and dispositions

3. Critical thinking and problem-solving

4. Application in real life

5. Holistic and integrated learning

6. Experiential learning

7. Enjoyable and engaging

8. Computational and mathematical thinking

9. Technology-based solutions

10. Factoids on India

Competency-based Education

NEP Pages 12, 17, and 22

Teaching and Learning Pedagogy

NEP Pages 3, 11, 12, 18, and 27

National Pride

NEP Pages 15, 16, and 43

11. Assessment of core concepts and application skills Assessments

NEP Pages 12, 18, and 22

Our Key Features: Aligning with the NEP

The GRR Approach

Pedagogical approach that empowers students to learn by the teacher progressively transitioning the responsibility to the students.

Competency-based Assessments

Test papers designed to evaluate understanding of core concepts and application of skills.

3 8 11

Contextual Learning

Introducing concepts through real-life situations and connecting them to students’ experiences.

Classroom Activity

A hands-on classroom activity to investigate and understand mathematical concepts in an engaging and concrete way.

Recall and Revisit

Introductory page with a quick recall of concepts done in previous grades.

Translating Words into Action!

Critical Thinking Questions

Intellectually stimulating questions designed to enhance problem-solving and analytical-thinking skills, promote deeper understanding and foster independent thinking.

Real-Life Math Problems

Scenario-based questions designed to help in applying theoretical knowledge to real-world scenarios, reasoning skills and improves reading comprehension & analytical abilities.

Common Misconceptions

Concise snippets of information designed to caution against potential errors and misconceptions

Think, Analyse and Answer

A quick, mathematical-thinking question

Fun Facts

Facts related to the concepts, or bite-sized information about the contribution of Indian scholars to mathematics

Gradual Release of Responsibility

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

Understanding Large Numbers

Dhruv was reading a newspaper. He came across news about different states in India that participated in the COVID vaccination drive and the number of vaccinations given until August 2023. Given below is the data of four states.

Real Life Connect

Theoretical Explanation

I do

Delhi Haryana Sikkim Goa 37409161 45546836 1360477 2874477

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.

All About 7-Digit and 8-Digit Numbers

Temperature Range (°C) Weather

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.

Temperature Range (°C)

Below 0 Very Cold 0–10 Cold 11–20 Mild 21–30 Warm 31–40 Hot

Examples

Did You Know?

Sikkim Goa Delhi Haryana 1360477 2874477 37409161 45546836

Did You Know?

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Above 40 Very Hot

7-digit Numbers 8-digit Numbers

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Normal body temperature = 37°C.

Place Value, Face Value and Expanded Form Reading and Writing 7-Digit and 8-Digit Numbers

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

Let us help Dhruv understand 7-digit and 8-digit numbers!

Normal body temperature = 37°C.

So, Aliya’s body temperature was 39.3°C.

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

We know that the greatest 6-digit number is 999999. Now, if we add 1 to this number, we get 1000000. 999999 + 1 = 1000000

Example 9: 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?

So, Aliya’s body temperature was 39.3°C.

1000000 is the smallest 7-digit number and is read as “Ten Lakhs”.

To find the difference, subtract the lowest temperature from the highest temperature.

37.5°C – 21.2°C = 16.3°C

Example 9: 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?

The temperature difference is 16.3°C.

We saw in the news article that the number of vaccinations administered in Sikkim was 1360477. Let us try to place this 7-digit number in the place value chart.

To find the difference, subtract the lowest temperature from the highest temperature.

Write the readings of the given thermometers.

37.5°C – 21.2°C = 16.3°C

The temperature difference is 16.3°C. Write the readings of the given thermometers.

Temperature = 45°C Temperature = Temperature = Temperature =

Converting Between Units of Temperature

Temperature = 45°C Temperature = Temperature = Temperature =

Celsius (°C) is like the universal language of temperature. Fahrenheit (°F) is another way to talk about temperature. We learnt that water freezes at 0°C and boils at 100°C. In Fahrenheit, water freezes at 32°F and boils at 212°F.

Converting Between Units of Temperature

GRR Steps Unit Component

Do It Yourself

You do

Chapter Checkup Challenge

Case Study

Do It Yourself 14B

1 Identify the shape for which the net is drawn. a b c d

Do as directed.

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

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

2 Look at the net and identify the object it belongs to.

3 Draw the net of the given shapes.

Critical Thinking & Cross Curricular

The Kumbh Mela is a major pilgrimage where Hindu pilgrims take the holy bath in the Ganges. The number of people who visited the Kumbh Mela in 1980 were 20,356,817 and those who visited in 1989 were 29,304,871. In which year did less than 25 crore people visit to the Kumbh Mela?

a 1980 b 1989 c Both years d None of the year

4 Rishi and Megha made the net of a square-based pyramid. Who made the net correctly? Explain your answer with reasons.

Comparing, Ordering and Rounding-Off Large Numbers

a Rishi’s drawing

Chapter Checkup

b Megha’s drawing

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.

Guess the best units of length (m or cm) and weight (kg or g) for the given objects. a b c

5 Draw the net of a hexagonal prism.

6 Show using nets how a rectangular prism is different from a rectangular pyramid.

Comparing and Ordering Numbers

Measure the objects.

Rahul: Which country did India supply the greatest number of vaccine doses to?

Challenge

Bran: We could compare the numbers to find the country to which India supplied the greatest number of vaccine doses.

Comparing Numbers

Critical Thinking

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

a The digit in the hundreds and ones place is 6.

What if Rahul wanted to compare the number of vaccines sent to Nepal and Australia? Let us find out.

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

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

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

d The face value of the digit in the thousands place is 5.

Remember!

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

Thus, Australia was donated more vaccine doses.

e The digit in the tens place is the biggest 1-digit number. What is the secret code?

What is the secret code?

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.

A number with more number of digits is always greater.

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

James has some cotton candy which is 3 cm longer than the cotton candy shown below. How long is James’ cotton candy?

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Chapter 1 • Numbers up to 8 Digits

Case Study

How much longer is the red straw than the blue straw?

Cross Curricular

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Convert the lengths.

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Country

Convert the weights.

into dam

a 5 kg into g b 4 g 64 cg into g c 5487 g into mg d 43 kg 7 dag into kg

United Kingdom 67,736,802

Poland 41,026,067

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?

1 Which country has the least population?

a Italy b Germany c Poland d United Kingdom

2 Which country has the greatest population?

a Poland b United Kingdom c Italy d Germany

Pearson, P. D., & Gallagher, G. (1983). Contemporary Educational Psychology. Fisher, D., & Frey, N. (2021). Better learning through structured teaching: A framework for the gradual release of responsibility. Fisher, D., & Frey, N. (2014). Checking for understanding: Formative assessment techniques for your classroom. 200

a Poland b United Kingdom c Italy d Germany

3 Which country has approximately double the population than that of Poland?

4 Arrange the countries in ascending order as per their population.

5 If all the digits in the population of each country is rearranged to form the greatest number, then which country will have the greatest population?

The NEP Tags

The National Education Policy (NEP), 2020, outlines essential skills, values, dispositions and learning approaches necessary for students to thrive in the 21st century. This textbook identifies and incorporates these elements throughout its content, activities and exercises. Referred to as “NEP Tags,” they are defined as follows:

Art Integration

Bringing creativity and fun into learning by combining art with maths. Students construct and demonstrate understanding through an art form.

Collaboration

Working effectively with others. Includes clear communication, teamwork, active listening and valuing diverse approaches.

Cross Curricular

Integrating mathematical concepts with other subjects to see the real-life applications of maths.

Experiential Learning

Gaining knowledge and skills through direct, hands-on experiences rather than a traditional classroom setting.

Communication

Explaining mathematical ideas, processes and solutions to others, whether through verbal explanations, written work or visual representations.

Value Development

Promoting ethics, and human & constitutional values, like empathy, respect for others, cleanliness, courtesy, democratic spirit, spirit of service, respect for public property, responsibility and equality.

Creativity

Using imagination and original thinking to solve problems and explore mathematical concepts in innovative ways. Also includes finding unique solutions, and designing original problem-solving strategies.

Education Standards as per the NCF 2023

The National Curriculum Framework for School Education (NCF), released in 2023, is based on the vision of the National Education Policy (NEP), 2020, and enables its implementation. The NCF provides guidelines for designing school syllabi and textbooks in India. It aims to improve the quality of education by making it more relevant, engaging, inclusive and learner-centric. To achieve this, the NCF has articulated precise Learning Standards through well-defined Curricular Goals and Competencies. These statements serve to harmonise the syllabus, content, pedagogical practices and assessment culture, ensuring a cohesive and comprehensive educational experience.

Curricular Goals: Statements that give direction to curriculum development and implementation in order to achieve the Curricular Aims. They are also specific to a School Stage (e.g., the Foundational Stage) and a Curricular Area (e.g., Mathematics).

Competencies: Learning achievements that are observable and can be assessed systematically. These Competencies are derived from the Curricular Goals and are expected to be attained by the end of a Stage.

Curricular Goals

CG-1

Understands numbers (counting numbers and fractions), represents whole numbers using the Indian place value system, understands and carries out the four basic operations with whole numbers, and discovers and recognises patterns in number sequences

CG-2

Analyses the characteristics and properties of two- and three-dimensional geometric shapes, specifies locations and describes spatial relationships and recognises and creates shapes that have symmetry

Competencies

C-1.1 Represents numbers using the place value structure of the Indian number system, compares whole numbers, and knows and can read the names of very large numbers

C-1.2 Represents and compares commonly used fractions in daily life (such as ½, ¼) as parts of unit wholes, as locations on number lines and as divisions of whole numbers

C-1.3 Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10×10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems

C-1.4 Recognises, describes, and extends simple number patterns such as odd numbers, even numbers, square numbers, cubes, powers of 2, powers of 10, and Virahanka–Fibonacci numbers.

C-2.1 Identifies, compares and analyses attributes of two- and three-dimensional shapes and develops vocabulary to describe their attributes/properties

C-2.2 Describes location and movement using both common language and mathematical vocabulary; understands the notion of map (najri naksha)

C-2.3 Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes

C-2.4 Discovers, recognises, describes and extends patterns in 2D and 3D shapes

The above is an abridged version of the curricular goals and competencies relatioship in Maths for the Foundational Stage (NCF 2023, pages 340–341). The next section shows the coverage of all these competencies across the chapters.

Mapping with NCF 2023

C-1.1 Represents numbers using the place value structure of the Indian number system, compares whole numbers, and knows and can read the names of very large numbers

C-1.2 Represents and compares commonly used fractions in daily life (such as ½, ¼) as parts of unit wholes, as locations on number lines and as divisions of whole numbers

C-1.3 Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10 × 10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems

C-1.4 Recognises, describes and extends simple number patterns such as odd numbers, even numbers, square numbers, cubes, powers of 2, powers of 10, and Virahanka–Fibonacci numbers.

CG-1

CG-2

C-2.1 Identifies, compares and analyses attributes of twoand three-dimensional shapes and develops vocabulary to describe their attributes/properties

C-2.2 Describes location and movement using both common language and mathematical vocabulary; understands the notion of map (najri naksha)

Analyses the characteristics and properties of twoand three-dimensional geometric shapes, specifies locations and describes spatial relationships, and recognises and creates shapes that have symmetry

C-2.3 Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes

C-2.4 Discovers, recognises, describes and extends patterns in 2D and 3D shapes

C-3.1 Measures in non-standard and standard units and evaluates the need for standard units

CG-3

Understands measurable attributes of objects and the units, systems and processes of such measurement, including those related to distance, length, weight, area, volume and time using nonstandard and standard units

C-3.2 Uses an appropriate unit and tool for the attribute (like length, perimeter, time, weight, volume) being measured

C-3.3 Carries out simple unit conversions, such as from centimetres to metres, within a system of measurement

C-3.4 Understands the definition and formula for the area of a square or rectangle as length times breadth

C-3.5 Devises strategies for estimating the distance, length, time, perimeter (for regular and irregular shapes), area (for regular and irregular shapes), weight, and volume and verifies the same using standard units

C-3.6 Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas

C-3.7 Evaluates the conservation of attributes like length and volume, and solves daily-life problems related to them

C-4.1 Solves puzzles and daily-life problems involving one or more operations on whole numbers (including word puzzles and puzzles from ‘recreational’ areas, such as the construction of magic squares)

C-4.2 Learns to systematically count and list all possible permutations or combination given a constraint, in simple situations (e.g., how to make a committee of two people from a group of five people)

CG-4 Develops problem-solving skills with procedural fluency to solve mathematical puzzles as well as daily-life problems, and as a step towards developing computational thinking.

C-4.3 Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation or paper-pencil calculation, in accordance with the context

C-5.1 Understands the development of zero in India and the Indian place value system for writing numerals, the history of its transmission to the world, and its modern impact on our lives and in all technology

CG-5 Knows and appreciates the development in India of the decimal place value system that is used around the world today

1 Numbers up to 8 Digits

Let’s Recall

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

Chapter at a Glance: Walkthrough of Key Elements

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.

1 Numbers up to 8 Digits

Number Lakhs Thousands Ones Lakhs (L) Ten Thousand (TTh) Thousands (Th) Hundreds (H) Tens (T) Ones (O) 2,01,301 2 0 1 3 0 1

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. periods place

Let’s Recall

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

Let’s Recall: Introductory page with a quick recall of concepts learnt in the previous grade

Dhruv was reading a newspaper. He came across news about different states in participated in the COVID vaccination drive and the number of vaccinations given August 2023. Given below is the data of four states.

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

Number name: Two lakh one thousand three hundred one

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.

Letʹs Warm-up

Fill in the blanks.

1 The place value of 8 in 8,60,765 is 2 The number 4,36,536 can be written in words as:

Let’s Warm-up: Quick 4–5 questions to test the pre-knowledge

3 The place value of the digit in 4,15,124 and 4,67,890 is the same.

4 8,76,504 has 6 in the place.

All About 7-Digit

and 8-Digit Numbers

I scored out of 4.

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

Dhruv was reading a newspaper. He came across news participated in the COVID vaccination drive and the August 2023. Given below is the data of four states.

Did You Know?

Temperature Range (°C) Weather

Below 0 Very Cold

0–10 Cold

I scored out of 4.

Place Value, Face Value and Expanded Form

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Reading and Writing 7-Digit and 8-Digit Numbers

Did You Know?

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

11–20 Mild

Normal body temperature = 37°C.

21–30 Warm

31–40 Hot

Let us help Dhruv understand 7-digit and 8-digit numbers! We know that the greatest 6-digit number is 999999. Now, if we add 1 to this number, we get 1000000.

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

All About 7-Digit and 8-Digit Numbers

Above 40 Very Hot

999999 + 1 = 1000000

So, Aliya’s body temperature was 39.3°C.

Example 8: Aliya’s body temperature was 2.3°C higher than normal. What was Aliya’s body temperature?

Normal body temperature = 37°C.

Examples: Solved problems showing the correct method and complete solution

While reading the news, Dhruv got confused and could data. The numbers of vaccinations given were either

Example 9: 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?

Aliya’s body temperature = 37°C + 2.3°C = 39.3°C

So, Aliya’s body temperature was 39.3°C.

1000000 is the smallest 7-digit number and is read as “Ten Lakhs”. We saw in the news article that the number of vaccinations administered in Sikkim 1360477. Let us try to place this 7-digit number in the place value chart.

To find the difference, subtract the lowest temperature from the highest temperature. 37.5°C – 21.2°C = 16.3°C

Example 9: 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.

The temperature difference is 16.3°C. Write the readings of the given thermometers.

37.5°C – 21.2°C = 16.3°C

The temperature difference is 16.3°C.

Do It Together: Guided practice for learners with partially solved questions

Write the readings of the given thermometers.

Place Value, Face Value and Expanded Form

Reading and Writing 7-Digit and 8-Digit Numbers

Let us help Dhruv understand 7-digit and 8-digit numbers!

We know that the greatest 6-digit number is 999999.

Temperature = 45°C Temperature = Temperature = Temperature =

Converting Between Units of Temperature

Now, if we add 1 to this number, we get 1000000. 999999 + 1 = 1000000 1000000 is the smallest 7-digit number and is read

We saw in the news article that the number of vaccinations 1360477. Let us try to place this 7-digit number in

What

Let

Example

Fill in the blanks to convert 5 m 230 mm to m. mm = 1 m

1 mm = 1 m

5 m 230 mm = 5 m + 230 × 1 m = 5 m + m = m

Word Problems on Length

Did You Know: Interesting facts related to the topic

Do It Yourself: Exercise at the end of each topic with practice questions

After making a pair of trousers with 2 m of cloth, Sana thought of buying 1 m 55 cm of cloth for making a shirt. Let us see what length of cloth she has together.

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

NEP Tags: To show alignment with NEP skills and values

Do

It Yourself

14B

Error Alert: Caution against misconceptions

1 Identify the shape for which the net is drawn.

2 Look at the net and identify the object it belongs to.

3 Draw the net of the given shapes. a b

4 Rishi and Megha made the net of a square-based pyramid. Who made the net correctly? Explain answer with reasons.

a Rishi’s drawing b Megha’s drawing

Chapter Checkup: Chapter-end practice exercises aligned to different levels of Blooms Taxonomy

5 Draw the net of a hexagonal prism.

6 Show using nets how a rectangular prism is different from a rectangular pyramid.

Picture-based Questions: Questions featuring visual stimuli to foster comprehension and interpretation

Challenge

Challenge: Critical thinking questions to enhance problem-solving and analytical-thinking skills

Case Study: Scenario-based questions designed to help apply theoretical knowledge to real-world situations

Real-life Questions: Questions that help make connections with real life or other subjects

Critical Thinking

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

a The digit in the hundreds and ones place is 6.

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

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

a The digit in the hundreds and ones place is 6.

d The face value of the digit in the thousands place is 5.

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

Thinking

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

e The digit in the tens place is the biggest 1-digit number. What is the secret code?

d The face value of the digit in the thousands place is 5.

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the

e The digit in the tens place is the biggest 1-digit number. What is the secret code?

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

a The digit in the hundreds and ones place is 6.

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

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Case Study

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

Case Study

What is the secret code?

d The face value of the digit in the thousands place is 5.

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

Cross Curricular

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Case Study

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

Country Population

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

1 Which country has the least population?

a Italy b Germany c Poland d United Kingdom

2 Which country has the greatest population?

1 Which country has the least population? a Italy b Germany c Poland d United Kingdom

a Poland b United Kingdom c Italy d Germany

2 Which country has the greatest population? a Poland b United Kingdom c Italy d Germany

3 Which country has approximately double the population than that of Poland?

1 Which country has the least population? Cross Curricular

4 Arrange the countries in ascending order as per their population.

3 Which country has approximately double the population than that of Poland?

Cross Curricular

1 Numbers up to 8 Digits

Let’s Recall

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

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 Ones

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. periods place

Lakhs (L) Ten Thousands (TTh) Thousands (Th) Hundreds (H) Tens (T) Ones (O) 2,01,301 2 0 1 3 0 1

Place Value 201301

Face Value

1 ones = 1 × 1 = 1

0 tens = 0 × 10 = 0

3 hundreds = 3 × 100 = 300

1 thousands = 1 × 1000 = 1000

0 ten thousands = 0 × 10000 = 0

2 lakhs = 2 × 100000 = 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 The place value of 8 in 8,60,765 is .

2 The number 4,36,536 can be written in words as: .

3 The place value of the digit in 4,15,124 and 4,67,890 is the same.

4 8,76,504 has 6 in the place.

Understanding Large Numbers

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, Face Value and Expanded Form

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.

The face value of a digit is the value of the digit itself.

Reading and Writing 7-Digit and 8-Digit Numbers

We know that the greatest 6-digit number is 999999. Now, if we add 1 to this number, we get 1000000.

999999 + 1 = 1000000

1000000 is the smallest 7-digit number and is read as “Ten Lakhs”.

Let us show the number of vaccinations administered in Sikkim, 1360477, in the place value chart.

9999999 is the greatest 7-digit number and if we add 1 to this number, we get 10000000.

10000000 is the smallest 8-digit number and is read as “One crore”.

Now let us show the number of vaccinations administered in Delhi, 37409161, in the place value chart.

Crores Lakhs Thousands Ones

Crores (C) Ten Lakhs (TL) Lakhs (L) Ten Thousands

Let us write the face value and place value of 37409161.

37409161 Face Value Place Value

1 ones = 1 × 1 = 1

6 tens = 6 × 10 = 60

1 hundreds = 1 × 100 = 100

9 thousands = 9 × 1000 = 9000

0 ten thousands = 0 × 10000 = 0

4 lakhs = 4 × 100000 = 400000

7 ten lakhs = 7 × 1000000 = 7000000

3 crores = 3 × 10000000 = 30000000

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.

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

The expanded form of 37409161 can be given as:

30000000 + 7000000 + 400000 + 9000 + 100 + 60 + 1 = 37409161

Form Standard Form periods place

Remember!

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

37409161

7000000 + 400000 seventy-four lakh Lakhs period 30000000 three crore Crores period 9000 nine thousand Thousands period 100 + 60 + 1 one hundred sixty-one Ones period

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

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

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

Example 1: Rewrite the numbers in a place value chart. Also, write them in words.

1 54879509 2 6509808

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 = 7809109

2 Nine crore five lakh ten thousand two hundred = 90510200

Example 3: Write the expanded form of the number 64870977.

Expanded Form: 60000000 + 4000000 + 800000 + 70000 + 0 + 900 + 70 + 7

Number name = Expanded form = Do It Together

Read the number. Write its number name and expanded form.

65790284

Indian and International Number System

The Indian number system includes the ones period, the thousands period, the lakhs period and the crores period.

There is another number system used globally, called the International number system.

In the Indian number system, we put commas after each period, starting with the ones. But the same number in the international system will be read differently.

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

So, from the above table, we can say that:

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

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

Indian Number System

Standard Form: 9,87,10,325

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

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

Standard Form: 98,710,325

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

Expanded form: 90,000,000 + 8,000,000 + 700,000 + 10,000 + 300 + 20 + 5 Indian Number System International Number System

Do It Together

Example 4: Write the numbers in the international number system using commas and number names. Write their expanded forms.

1 79027348 2 90710946

1 79,027,348 = Seventy-nine million twenty-seven thousand three hundred forty-eight = 70,000,000 + 9,000,000 + 20,000 + 7,000 + 300 + 40 + 8

2 90,710,946 = Ninety million seven hundred ten thousand nine hundred forty-six = 90,000,000 + 700,000 + 10,000 + 900 + 40 + 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:

2 65,780,245 = Expanded form:

Write the place value and face value of the underlined digit in the following numbers. a

Write the standard form of the numbers in the Indian and International number systems. a 21643332 b 1200621 c 46207219 d 95910158

Write the number names and expanded form of the given numbers.

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.

The Amazon Rainforest covers approximately 2722000 square miles. Write the number in the Indian and the international number system.

Do as directed.

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

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

India got its independence in 1947. Its population at that time was 353 million. Write the population as a numeral.

Challenge

Critical Thinking & Cross Curricular

a 1980 b 1989 c Both years d Neither year

Comparing, Ordering and Rounding-off Large Numbers

Comparing and Ordering Numbers

Rahul: Which country did India supply the greatest number of vaccine doses to?

Bran: We could compare the numbers to find the country to which India supplied the greatest number of vaccine doses.

Comparing Numbers

What if Rahul wanted to compare the number of vaccines sent to Nepal and Australia? Let us find out.

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

Thus, Australia was sent 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.

Remember!

A number with more digits is always greater.

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 until we find different digits. The number with the greater digit is greater.

Here, the digits are the same until the thousands place. Now, we compare the digits in the hundreds place. We see that 4 < 6.

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

Example 5: Compare the numbers 6,47,17,389 and 6,47,00,508.

6 = 6 4 = 4 7 = 7 1 > 0

So, 6,47,17,389 > 6,47,00,508

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

Did You Know?

Aryabhata (476–550 CE) was the first mathematician who was known for his significant contributions to the place-value system, where he used letters of the alphabet to denote numbers, expressing quantities.

Ordering Numbers

Do you remember the number of vaccine doses donated by India to different countries? Let us read them again.

Can you rearrange them in ascending order?

To arrange the number of vaccine doses in ascending order, follow the given steps.

1. Compare the two 7-digit numbers.

94,99,000 < 98,02,000

So, 94,99,000 is the smallest number.

2. Compare the two 8-digit numbers.

3,09,13,200 > 2,80,82,800

So, 3,09,13,200 is the greatest number.

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?

Example 6: Arrange 38,65,080; 10,00,000 and 4,50,50,809 in descending order.

Descending order

4,50,50,809

38,65,080

10,00,000

4,50,50,809 > 38,65,080 > 10,00,000 is the descending order.

Example 7: Arrange 8,15,64,205; 8,13,54,610; 65,99,090; 8,76,01,006 in descending order.

65,99,090 is the smallest number since it has the least number of digits. 8,76,01,006 is the largest number.

8,15,64,205 > 8,13,54,610.

Thus, the descending order of the numbers is 8,76,01,006 > 8,15,64,205 > 8,13,54,610 > 65,99,090.

Which of the following numbers is the greatest?

98,89,001; 1,02,35,849; 1,20,64,980; 99,47,099

Write the numbers in the place value chart.

The greatest number is .

Forming Numbers

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.

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?

Greatest number: Write the digits in descending order.

9 7 6 5 3 1 0

Smallest number: Write the digits in ascending order. 0 will appear in the second position; otherwise it forms a 6-digit number if 0 comes in the leftmost position.

1 0 3 5 6 7 9

Now, what if we want to form a number by repeating 1 digit?

Greatest number: Repeat the greatest digit. We get an 8-digit number.

9 9 7 6 5 3 1 0

Smallest number: Repeat the smallest digit. We get an 8-digit number.

Think and Tell

Why did we choose to repeat the greatest digit?

1 0 0 3 5 6 7 9

Example 8: 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.

Greatest number = 98654210; Smallest number = 10245689

Example 9: Find the greatest and smallest 8-digit number using the digits 5, 4, 7, 6, 0, 1 and 3 but repeating any one digit only once.

The given digits are 5, 4, 7, 6, 0, 1 and 3.

To form the greatest number, we will repeat the digit 7.

Greatest 8-digit number formed using the given digits = 77654310

To form the smallest number, we will repeat the digit 0.

Smallest 8-digit number formed using the given digits = 10034567

Find the second greatest number using the digits 8, 4, 6, 9, 0, 1, 3 and 7.

Do not repeat any digit.

The given digits are 8, 4, 6, 9, 0, 1, 3, 7.

The greatest number using the given digits = .

We will get the second greatest digit by interchanging . The second-greatest number using the given digits = .

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

d 6,24,58,110 6,24,59,211 e 82,60,154

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

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

Arrange the following numbers in ascending order and descending 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

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

a 5, 3, 4, 0, 8, 9, 1 b 5, 7, 6, 2, 1, 3, 8

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

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

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

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

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

Rearrange all the digits of the number 5,48,79,802 to form the largest and the smallest 8-digit number.

Write the greatest 8-digit number and the smallest 7-digit number using:

a two different digits

b four different digits

c five different digits

The total areas of 4 countries in sq. km are given below. Arrange the names in descending order of their area.

Russia - 17,098,242; India - 3,287,590; China - 9,706,961; Australia - 7,692,024.

Create a word problem on comparing two 8-digit numbers.

Consider the digits, 5, 0, 1, 3, 7, 9, 6, 2. Vijay formed the greatest and the smallest 8-digit numbers using these digits. He also formed the greatest and the smallest 7-digit numbers using any 7 digits. No digit was repeated. Arrange the 4 numbers formed by Vijay in descending order.

Use any of the digits 0 to 9, using each digit only once to form a 7-digit number that is closest to 40,00,000. 1 2

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 loud.

What if we just said that India donated about 3,00,00,000 vaccines to Bangladesh. It still gives a fair idea of how many vaccines were donated. This is called rounding off a number.

While rounding off numbers, terms like “about” and “approximately” are added to convey that the number is close to being exact.

Let us learn how to round-off numbers to different places.

Rounding-Off to the Nearest 10s

Did You Know?

The diameter of the Sun is approximately 14,00,000 km.

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

5 4, 7 0, 8 2 3

3 < 5

rounded off to

5 4, 7 0, 8 2 0

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 2 7

> 5

Rounding-Off to the Nearest 100s

rounded off to

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.

rounded off to

6 4, 3 0, 7 0 8

0 < 5

6 4, 3 0, 7 0 0

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.

rounded off to

5, 6 4, 3 0, 7 5 8 5, 6 4, 3 0, 8 0 0 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.

rounded off to

7 8, 5 1, 4 2 3 7 8, 5 1, 0 0 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.

rounded off to

2, 8 3, 4 9, 6 2 7 2, 8 3, 5 0, 0 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 10: Round off 3,76,87,519 to the nearest 10s, 100s and 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.

Round off 7,39,81,506 to the nearest 10s, 100s and 1000s.

To the nearest 10s: 7,39,81,506 is rounded off to 7,39,81,510.

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

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

Round off the following numbers to the nearest tens.

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

The distance of the Moon from the Earth is 238,855 miles. What is the approximate distance between the Moon and the Earth when rounded off to the nearest thousand?

Shekhar participated as a volunteer in the municipal corporation drive to plant trees and provide a habitat for various species of birds, insects and other wildlife. The municipal corporation spent ₹65,94,830 on the project. Rewrite the amount spent by rounding off the number to the nearest 1000s. Have you ever planted trees?

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 Repeat the greatest digit only once.

1 Challenge

b Repeat the smallest digit only once.

Critical Thinking

Rihan thinks of a number. He says that the number when rounded to the nearest thousands becomes 3,23,46,000. What is the smallest and the biggest possible number that Rihana could think of?

Points to Remember

• A 7-digit number has 7 digits, with ten lakhs as its highest place.

• 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 value of all the digits.

• A number with more digits is always greater.

• Numbers can be rounded off to give us an approximate value.

Math Lab

Collaboration & Experiential Learning

Mystery Number Puzzle

Objective: To solve puzzles involving large numbers by identifying missing digits based on the given clues.

Setting: In pairs

Method:

Materials Required: Puzzles or worksheets with missing digit problems, Pencils

Prepare puzzles where certain digits of large numbers are missing. Provide clues to identify the missing digits (e.g., “The digit in the millions place is double the digit in the ten thousands place”).

Students work in pairs to solve the puzzles and fill in the missing digits.

The team which solves the puzzle first wins.

Once the students have completed their puzzles, bring the class together to discuss the solutions.

Chapter Checkup

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

a 3507681

c 63565842

b 42087950

d 91500084

Write the numbers for the following number names in both number systems.

a Sixty million seven hundred fifteen thousand two hundred thirty-nine

b Eight crore nine lakh fifty thousand two

c One million one hundred thousand thirty-nine

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

a 6,45,87,123

Fill in the blanks using <, > or =.

a 6,56,52,567 6,48,90,650

c 34,57,879 34,57,879

Arrange the following numbers in ascending order.

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

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

Arrange the following numbers in descending order.

a 4,56,45,768; 5,36,45,787; 2,40,85,167; 43,56,787

b 80,88,428; 4,90,76,837; 9,09,87,897; 80,68,964

b 89,09,008

b 90,00,518 90,76,757

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

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

Under the Vaccine Maitri initiative, India supplied COVID-19 vaccines to various countries. India supplied around 3,151,324 vaccines to UK. Write the expanded form of the number in both numeral systems.

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 Write the number in figures and words using both the Indian and international number systems.

b Round off the number to the nearest tens, hundreds or thousands.

Challenge

1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.

a The digit in the hundreds and ones place is 6.

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

c The digit in the ten lakhs and ten thousands place is the smallest odd number.

d The face value of the digit in the thousands place is 5.

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

2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.

Case Study

Population of Countries

The population of different countries is shown using a table. Read the data carefully and answer the questions.

1 Which country has the least population?

a Italy

c Poland

b Germany

d United Kingdom

2 Which country has the greatest population?

a Poland

c Italy

b United Kingdom

d Germany

3 Which country has approximately double the population than that of Poland?

4 Arrange the countries in ascending order as per their population.

5 If all the digits in the population of each country are rearranged to form the greatest number, then which country will have the greatest population? Cross Curricular

2 Operations on Large Numbers

Let’s Recall

Tina buys beads for making ornaments. She has 1150 beads. We buys 450 more beads.

How many beads does she have now?

Th H T O

1 1 5 0 + 4 5 0

1 6 0 0

1150 + 450 = 1600 beads.

This is called addition.

Now, let us say that out of the 1600 beads, she uses 200 beads to make bracelets. How many beads are left with her now?

Th H T O

1 6 0 0

2 0 0

1 4 0 0

1600 − 200 = 1400 beads

This is called subtraction.

Tina makes 4 necklaces. She uses 150 beads in each necklace.

How many beads did she use?

Let us count.

150 + 150 + 150 + 150 = 600

Or, 4 × 150 = 600

This is called multiplication.

Division is equal sharing. 1

Letʼs Warm-up Match the

Now suppose Tina wants to pack 1000 equally in 4 boxes. How many beads will be there in each box?

Let us divide 1000 by 4.

1000 ÷ 4 = 250

This is called division.

Addition and Subtraction of Numbers up to 6 Digits

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

Father: We made ₹2,50,678 by selling cakes and ₹1,56,240 by selling cookies this year.

Sahil: Wow! How much did we make last year?

Father: Last year, we made a total of ₹3,15,500.

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.

Add: ₹2,50,678 + ₹1,56,240

So, Sahil’s father made ₹4,06,918 this year.

Example 1: Add 1,98,794, 52,250 and 21,000.

Using the steps, we can add the numbers as: 1,98,794 + 52,250 + 21,000 = 2,72,044.

Example 2: 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?

Boxes produced in 2019 = 4,56,360

Boxes produced in 2018 = 3,60,780

Boxes produced in 2017 = 90,995

Total number of boxes produced = 4,56,360 + 3,60,780 + 90,995 = 9,08,135

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

A publishing company published 1,26,716 poetry books, 2,43,120 non-fiction books and 80,133 fiction books. How many books has the company published?

Number of poetry books = 1,26,716

Number of non-fiction books = 2,43,120

Number of fiction books = 80,133

The total number of books published by the company are .

Subtracting Numbers up to 6 Digits

Remember!

Changing the order of addends does not change the sum.

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.

Let us subtract the sales made last year from the sales made this year.

₹4,06,918 − ₹3,15,500 = ₹91,418. Therefore, they earned ₹91,418 more this year than last year.

Example 3: Find the difference of 57,588 and 6765.

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

Therefore, 57,588 − 6765 = 50,823.

Example 4: 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.

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.

A store sold 2,85,586 shirts in a year. The sale of shirts for the first month was 19,678. How many shirts were sold in the remaining 11 months?

Shirts sold in 1 year (or 12 months) =

Shirts sold in the first month = 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.

Do It Yourself 2A

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 the numbers involved in subtraction can be changed.

Add the given numbers.

Subtract the given numbers.

Write the numbers in columns and add.

a 1,72,744 and 5,56,200 b 3,44,567, 78,456 and 39,894

c 5,89,569, 1,24,887 and 56,758 d 3,86,565, 2,34,567, 56,468 and 46,568

Find the difference of the given numbers.

a 87,687 and 5789

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

b 9,87,609 and 56,000

d 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.

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?

What is the difference between the diameters of Jupiter and Saturn if Jupiter’s diameter is about 142,984 kilometres and Saturn’s diameter is about 120,536 kilometres.

An NGO collected ₹2,89,230 for a charity fund in one year and ₹3,97,500 in another year. They used ₹3,05,700 out of the total amount collected in the two years. How much money are they left with now?

Do you volunteer for any charity activities?

Challenge

Critical Thinking & Value Development

Suhani adds two 5-digit numbers to get a 6-digit number. If the first number is 52,135, what range of values can the second number take?

Multiplication and Division of Numbers up to 6 Digits

Jay runs a toy store. He 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 by 2 to find the total number of marbles he will give. So, Jay will give 2450 marbles. Now let us say that a customer wants to give marbles from one jar to 5 of his friends. To find out how many marbles each friend will get, he will have to divide 1225 by 5.

Each friend will get 245 marbles.

Let us see some properties of multiplication and division.

1 Order Property: Two numbers can be multiplied in any order. The product will always be the same. For example: 12 × 4 = 48 and 4 × 12 = 48

2 Grouping Property: Two or more numbers can be grouped in any way. The product will be the same. For example: (3 × 5) × 6 = 15 × 6 = 90 and 3 × (5 × 6) = 3 × 30 = 90

2 Distributive Property: The product of a sum of two or more numbers is equal to the sum of the products. For Example: 4 × (2 + 3) = (4 × 2) + (4 × 3) = 8 + 12 = 20 and 4 × (2 + 3) = 4 × 5 = 20

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 by the number of jars he wants. He will have to multiply 1225 by 10.

Step 1

Multiply the non-zero digits.

1225 × 1 = 1225

So, he will get 12,250 marbles.

What if he buys 100 jars?

Step 1

Multiply the non-zero digits.

1225 × 1 = 1225

Step 2

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

1225 × 100 = 1,22,500

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

Step 2

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

1225 × 10 = 12,250

Remember!

The product of any number multiplied by 1 will always be the number itself. The product of any number multiplied by 0 will always be 0.

What if he buys 1000 jars?

Step 1

Multiply the non-zero digits. 1225 × 1 = 1225

Step 2

Put the remaining 0s at the end of the product. 1225 × 1000 = 12,25,000

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

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

Example 5: Multiply 7865 by 30.

Step 1

Multiply the non-zero digits.

7865 × 3

TTh Th H T O 2 1 1 7 8 6 5 × 3 2 3 5 9 5

Step 2

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

7865 × 30 = 2,35,950 Therefore, the product of 7865 × 30 is 2,35,950.

Example 6: A book contains about 1244 letters on one page. If there are the same number of letters on each page, then how many letters will there be on 200 pages?

Number of letters on 1 page = 1244

Number of letters on 200 pages = 1244 × 200 = 2,48,800

Therefore, the book has 2,48,800 letters in total.

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, from Jay’s store. Let us find the total number of marbles using multiplication.

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

Mr Sharma is a businessman. He visited Jay’s store and wants to buy 1121 jars of marbles.

To find the total number of marbles he is buying, we will have to multiply 1225 by 1121.

We will first expand 1121 = 1000 + 100 + 20 + 1 = 1 thousand + 1 hundred + 2 tens + 1 one.

Did You Know?

Zero was invented by the great Indian mathematician Aryabhata in the 5th century.

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

Remember!

Anything multiplied by zero is zero.

Do It Together

Example 7: Multiply 3479 by 452.

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. 3479 × 452 = 15,72,508

Example 8: 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. What is the total amount collected by the school?

We know that: Total amount collected = Total amount paid by 1500 students

The amount to be paid by 1 student is ₹555.

The amount to be paid by 1500 students = 1500 × ₹555 = ₹8,32,500

So, the total amount collected by the school is ₹8,32,500.

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 by 1313.

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.

Division by 10

Remainder Quotient

1 digit from the right is the remainder

Division by 100 2 digits from the right is the remainder

Division by 1000

3 digits from the right is the remainder

Remaining is the quotient

Number of marbles in each jar

40,000 ÷ 10

Remainder = 0

Quotient = 4000

40,000 ÷ 100

Remainder = 00

Quotient = 400

40,000 ÷ 1000

Remainder = 000

Quotient = 40

Jay will put 4000 marbles in each jar.

Jay will put 400 marbles in each jar.

Jay will 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, form the remainder. The remaining digits are the quotient.

Let us see some more properties of division.

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

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

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

Example 9: Divide 15,679 by 100.

15,679 ÷ 100 = 156, with a remainder 79.

Thus, quotient = 156 and remainder = 79.

Example 10: Divide 98,562 by 1000.

98,562 ÷ 1000 = 98, with a remainder 562.

Thus, quotient = 98 and remainder = 562

Divide 67,688 by 1000.

67,688 ÷ 1000

Quotient = Remainder

Thus, quotient = and remainder = .

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.

Dividing 5-Digit Numbers by 3-Digit Numbers

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

He can find out by division, using the given steps. Take the first 3 digit number to start dividing and bring down the next digit until you get a remainder

Thus, the quotient = 333, and the remainder = 40.

So, Jay can only put 333 marbles in each jar, and he will be left with 40 marbles.

Example 11: Divide 76,545 by 115.

Divisor = 115 and Dividend = 76,545

665

115 76545 –690 754 645 –690 –575 70

Since 70 is less than 115, the remainder is 70.

Thus, the quotient = 665, and the remainder = 70.

Example 12: A room is large enough for 256 people to sit. How many rooms will be required for 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.

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

Cost of 235 books =

Cost of one book = ÷

Therefore, the cost of one book is .

Do It Yourself 2B

Write True (T) or False (F).

a The divisor and remainder in a division sum can be the same.

b The result of multiplication is called the 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 47,852 c 82,587

Divide the given numbers by 10, 100 and 1000.

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

Divide the given numbers.

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

The book Romeo and Juliet by Shakespeare has about 24,645 words. Each page contains 155 words. How many pages are there in the book?

The owner of a shop makes a profit of ₹98,000. He decides to keep ₹30,000 for himself and distribute similar stationery kits among 100 kids in an orphanage. What is the cost of each stationery kit? Have you ever donated anything to an orphanage?

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

Choosing the Right Operator

On a school trip, there were 6 teachers who accompanied one group out of 2 equal groups in a class of 54 students. On the day of the trip, 2 groups of 5 students each were absent. Ram, a student on the trip, wonders how many people including the teachers, are on the trip. Let us help him find out.

Solving Expressions Using DMAS

Ram tries putting the number and operators 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. 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 DMAS.

Now, let us help Ram figure out how many people are there on the trip.

Using DMAS

Let us 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

Simplify division (÷)

= 6 + 54 ÷ 2 − 2 × 5

= 6 + 27 − 2 × 5

Step 2

Simplify multiplication (×)

= 6 + 27 − 2 × 5

= 6 + 27 − 10

Addition + S Subtraction

Step 3

Simplify addition (+)

= 6 + 27 − 10

= 33 − 10

Therefore, there are 23 students and teachers on the trip.

Example 13: Simplify: 42 ÷ 7 + 3 × 9 − 1

42 ÷ 7 + 3 × 9 − 1

= 6 + 3 × 9 − 1

= 6 + 27 − 1

= 33 − 1 = 32

Simplify: 100 72 ÷ 8 + 4 × 3

100 − 72 ÷ 8 + 4 × 3 = 100 − + 4 × 3

Multi-step Word Problems

Step 4

Simplify subtraction (−)

= 33 − 10 = 23

A teacher has 72 apples. He distributes them equally among 4 bags and keeps only 1 bag for himself. Then, 4 students give the teacher 10 apples each. Finally, all of them together eat 20 apples from the bag.

Calculate the number of apples left with the teacher.

Do It Together

The teacher puts the apples equally in 4 bags. That is, 72 ÷ 4. 4 students give 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 left in the bag are:

72 ÷ 4 + 4 × 10 − 20

= 18 + 4 × 10 − 20

= 18 + 40 − 20

= 58 − 20 = 38

Example 14: Tarun has 2 sets of 10 toy cars. His father gives him 5 more toy cars. Tarun 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

Tarun gives 4 cars to his friend = 2 × 10 + 5 − 4

Number of cars left = 20 + 5 − 4 = 25 − 4 = 21

Therefore, Tarun will be left with 21 toy cars.

Think and Tell Are multiplication and division related?

Lisa has 2 bunches of 50 flowers. Each bunch is of a different colour. 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 =

50 × 2

Think and Tell

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

Therefore, Lisa is left with flowers.

Do It Yourself 2C

Tick () the correct answer.

a The order of DMAS is:

c 100 ÷ 10 + 10 × 10

Write True (T) or False (F).

a In DMAS, we first perform addition/subtraction and then division/multiplication.

b In DMAS, the last step is subtraction.

c 5 × 4 + 12 ÷ 3 = 24

d 36 ÷ 6 – 3 × 2 = 2

Fill in the blanks.

a 82 × 3 + 20 =

55 ÷ 11 + 7 = d 20 + 50 ÷ 2 – 5 =

Simplify. a 120 – 12 + 36 ÷ 3 × 5 b 279 + 321 ÷ 3 –

676 + 835 × 45 ÷ 15 – 10

Alex has saved ₹120. He wants to buy a toy for ₹28 and a book. The book is half the price of the toy. How much money will he have left after buying the toy and the book?

Anish had 50 stickers. He gave 3 stickers to each of his 4 friends. He then bought 10 more stickers. How many stickers does Anish have now?

Sahil has 15 pencils. He wants to distribute them among his 3 friends. All three friends already have 2 pencils. If he gives each friend the same number of pencils, find the total number of pencils each friend has. Do you also share things with your friends?

Solar Panels are devices that convert sunlight into electricity using photovoltaic (PV) cells made of semiconductor materials like silicon. In a school with 350 students, the administration wants to install solar panels to help the environment and teach students about renewable energy. On Monday, 25 students were absent. The rest of the students brought ₹230 each for the project. How much money did the students bring?

Critical Thinking

The difference of 81 and 3 is added to the sum of their product and quotient. What is the resultant number?

Points to Remember

• When we multiply a number by 10, 100, 1000 and so on, we add 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 forms the remainder. The remaining digits are the quotient.

• When more than two operations are given together in a problem, we use DMAS.

Setting: In pairs

Roll, Multiply, Win: The Dice Challenge

Materials Required: 3 dice, a pencil and paper.

Method:

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

Note down the number obtained on a sheet of paper.

Keep repeating the above process, and each time add the results obtained after multiplication to find the sum.

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

Chapter Checkup

Find the sum of the given numbers.

a 56,789 and 23,456 b 9,87,654 and 45,774

c 6,73,778 and 5,67,433 d 2,53,621, 12,365 and 36,586

Subtract the given numbers.

a 54,676 and 34,575 b 8,64,826 and 96,537

c 8,68,636 and 65,365 d 9,54,863 and 8,45,622

Multiply the given numbers by 10, 100 and 1000.

a 896 b 4546 c 6457 d 9876

Divide the given numbers by 10, 100 and 1000.

a 2100 b 4086 c 51,200 d 74,562

Find the product of the given numbers.

a 6789 and 432 b 8646 and 200

c 4321 and 2981 d 1256 and 3251

Divide the given numbers using long division.

a 54,767 and 231 b 56,785 and 314

c 88,757 and 768 d 99,866 and 982 1 2

a 18 ÷ 2 × 3 b 12 × 6 − 2 + 18 ÷ 3

c 34 + 2 × 5 − 9 ÷ 9 d 63 − 4 + 8 ÷ 2 × 30

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.

Raj and his team are census workers, whose job is to collect the population. In Ghaziabad city, there are 3,81,53,154 men, 3,15,31,873 women and 6,81,231 children. How many people are there in Ghaziabad?

Isha 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?

Ooty organises a botanical flower show every year. A garden is divided into 60 equal sections. Each section has 1280 flowers. 30 of 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?

An auditorium has a capacity of 64,070 people. If one row can seat 430 people, how many rows are there in the auditorium?

Maya has 80 marbles. She gives 5 marbles each to 6 of her friends. Then, she finds 15 more marbles. How many marbles does Maya have now?

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

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

There are 5565 mangoes. The number of oranges is twice the number of mangoes. How many oranges will each box contain if the shopkeeper keeps an equal number of oranges in 5 cartons?

Shreya had ₹45,000 in her bank account. She got her salary and the amount in her bank account became three times what it was before. She donated ₹14,500 to an emergency relief fund for the soldiers on the Indian border. How much money was left in her bank account? Have you ever collected funds for a charity event?

Write a word problem with one 6-digit number and two 5-digit numbers. The word problem should have addition, division and subtraction.

1 The population of Hassan is 1,88,000. The population of Mysore city is 12,61,000.

Read the statements and answer the question given below.

Statement 1: The population of Belgaum is 7,83,000, which is 5,95,000 more than the population of Hassan.

Statement 2: The population of Hassan and Belgaum together is less than the population of Mysore.

Is statement 1, 2 or both true? Show your working.

2 Ravi has two bank accounts A and B. The combined balance of the two accounts is ₹8,75,632. What is the balance in account B?

Read the statements and choose the correct option.

Statement 1: The balance in Account A is ₹3,56,289.

Statement 2: Account B has more balance than account A.

a Statement 1 alone is sufficient to answer.

b Statement 2 alone is sufficient to answer.

c Both statements together are sufficient to answer.

d Both statements together are not sufficient to answer.

Case Study

Cross Curricular

Rice Production

Our country is the largest rice-producing country. Most of the states in our country produce rice, and most of the rice varieties are exported to different parts of the world. The table shows the amount of rice produced by the state in 2 months.

State

Karnataka

Tamil Nadu

Bihar

Amount of Rice Produced

5,04,975 kg

8,20,965 kg

7,05,136 kg

Madhya Pradesh 4,94,814 kg

1 The total weight of the rice produced by Karnataka and Bihar is .

2 Write True (T) or False (F).

a Tamil Nadu produces 4,67,870 kg more rice than Madhya Pradesh.

b Karnataka produces 3,15,990 kg less rice than Tamil Nadu.

3 Tamil Nadu wants to deliver 78,432 kg of rice in 456 bags to a relief camp. What will be the weight of rice in each bag?

4 24,430 kg of rice from Bihar and 20,510 kg of rice from Karnataka are to be mixed and put equally in 214 containers. What will be the weight of the rice in each container?

33 Factors and Highest Common Factor

Let’s Recall

We know about multiplication facts of numbers. Let us look at the multiplication facts of the number 16.

Imagine we have 16 tiles that need to be arranged in rows and columns. We can do it in the following manner:

× 8 = 16

× 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, as 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 12

2 27

3 30

4 46

5 55

5, 11

3, 5

2, 4

3, 9 4 × 4 = 16

I scored out of 5.

Understanding Factors

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 fit on the stage.

So, she decided to try different ways in which the dancers could fit onto the stage.

2 rows of 6 dancers 4 rows of 3 dancers 3 rows of 4 dancers

She noticed that all the dancers could fit when she made 3 rows of 4 dancers.

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

1 × 12 2 × 6 4 × 3 3 × 4

Factors of a Number

Think and Tell

Can there be more ways to arrange the dancers in lines having equal rows and columns?

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.

2 × 6 Product 12 =

Factors

Let us look at the factors of 12 once more.

• 1 is its smallest factor. 1 is the smallest factor of every number.

• 12 is the greatest factor of itself. Every number is the greatest factor of itself.

• Every factor of 12 is less than or equal to 12. The factors of a number are always equal to or less than the 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.

1 × 18 = 18; 2 × 9 = 18; 3 × 6 = 18

The multiplication table of 4 does not give 18 as a product.

The multiplication table of 5 does not give 18 as a product.

Remember!

When finding factors by multiplication, always:

• Start with the multiplication table of 1. • Stop when any factor starts repeating.

6 × 3 = 18. This 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.

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

36 ÷ 1 = 36 0 1 and 36 are factors of 36.

36 ÷ 2 = 18 0 2 and 18 are factors of 36.

36 ÷ 3 = 12 0 3 and 12 are factors of 36.

36 ÷ 4 = 9 0 4 and 9 are factors of 36.

36 ÷ 5 = 7

36 ÷ 6 = 6

36 ÷ 7 = 5

5 and 7 are not factors of 36.

6 is a factor of 36.

7 and 5 are not factors of 36.

36 ÷ 8 = 4 4 8 and 4 are not factors of 36.

36 ÷ 9 = 4 0 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.

1 × 20 = 20

2 × 10 = 20

The multiplication table of 3 does not give 20 as a product.

4 × 5 = 20.

5 × 4 = 20. This 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.

30 ÷ 1 = 30 0 1 and 30 are factors of 30.

30 ÷ 2 = 15 0 2 and 15 are factors of 30.

30 ÷ 3 = 10 0 3 and 10 are factors of 30.

30 ÷ 4 = 7 2 4 and 7 are not factors of 30.

30 ÷ 5 = 6 0 5 and 6 are factors of 30.

30 ÷ 6 = 5 0 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.

Find the factors of 24 using the division method.

24 ÷ 1 = 24 0 1 and 24 are factors of 24. 24 ÷ 2 = 0 and are factors of 24.

24 ÷ 3 = 8 and are factors of 24.

24 ÷ 4 = 0 and are factors of 24.

24 ÷ 5 = 4 4 5 and 4 are not factors of 24.

24 ÷ 6 = 0 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 with more than two factors are called composite numbers.

Example 3: Which is a prime number, 16 or 17?

Factors of 16 = 1, 2, 4, 8 and 16.

Factors of 17 = 1 and 17

17 has 2 factors, so it is a prime number.

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

Remember!

• 0 and 1 are neither prime nor composite.

• 2 is the lowest and the only even prime number.

Do It Yourself 3A

Find the factors of the numbers using the multiplication method.

Find the factors of the numbers using the division method.

Sort the given numbers as prime or composite numbers.

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

Mycobacterium tuberculosis is a bacteria which is responsible for tuberclosis. It has a slower replication time and typically divides every 17 minutes (approx.). How many factor are there for the number 17?

An event manager is getting chairs arranged for a stage show. He wants to put the same number of chairs in rows as the total number of rows. How many chairs will be there in each row, if he has 169 chairs? Create a

Write if the given statement is true or false. Give a reason.

If 10 is a factor of a number, 2 is also a factor of that number.

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 looking at 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

A number is divisible by 2 and 5 if it is divisible by 10.

Example 4: Is 64 divisible by 2?

64 ends with the digit 4.

So, 64 is divisible by 2.

Example 5: Is 45 divisible by 5?

45 ends with the digit 5.

So, 45 is divisible by 5.

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

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, 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 also divisible by 3 if it is divisible by 9.

Example 6: Is 54 divisible by 3?

5 + 4 = 9

9 is divisible by 3.

So, 54 is divisible by 3.

Example 7: Is 452 divisible by 9?

4 + 5 + 2 = 11

11 is not divisible by 9.

So, 452 is not divisible by 9.

Do It Yourself 3B

Circle the numbers that are divisible by 2.

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

Circle the numbers that are divisible by 3.

Circle the numbers that are divisible by 9.

A secret passage has a number lock. Below are the clues to the correct number needed to open the lock. Which of the given numbers shows the secret code?

• The number is divisible by 3 and 5.

• The number has 0 as the last digit.

Anu is distributing gifts to the poor kids in her neighborhood. There are 10 kids in all. She has 40 pencils, 20 erasers, 55 candies and 20 notebooks. Which are the items that she can divide equally among the kids?

Challenge

Critical Thinking

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

Step 2: Fill in the branches with a factor pair of the number above. We chose 12 and 10 as the factor pair.

Step 3: Continue until each branch ends in a prime number.

Prime factorisation of the number = Product of prime numbers found by factor tree

Prime factorisation of 120 = 2 × 2 × 2 × 3 × 5

Remember!

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

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 it by prime numbers until we reach 1.

Let us factorise 81 by the division method.

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.

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

Example 8: Draw a factor tree for the number 36 and then express the number as a product of its prime factors.

Example 9: Use repeated division to find the prime factors of 84. Express the number as a product of its prime factors.

So, 36 = 2 × 2 × 3 × 3.

So, 84 = 2 × 2 × 3 × 7.

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

So, 64 = × × × × × .

Which of the following represents the prime factorisation of 80?

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

Draw factor trees to find the prime factorisation of the given numbers.

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

The Earth completes its revolution around the Sun in 365 days. What are the prime factors of 365?

Samantha has 132 flowers that she wants to arrange in vases. She wants to place an equal number of flowers in each vase. How many vases did she use if the number of vases that she used was the largest prime factor of 132?

Highest Common Factor

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 that requires groups of equal size.

The teachers aim to maximise the number of students in each group.

Finding the HCF

The teachers want to divide their students into groups. Thus, 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

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

Possible group size = 18, 9, 6, 3, 2 and 1.

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:

Thus, 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.

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

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 18 Factors of 27

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 10: 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.

The common factors of 15 and 20 are 1 and 5.

Example 11: 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.

Find the common factors of 10 and 14.

Factors of 10: 1, 2, , 10.

Factors of 14: .

Remember!

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

Remember!

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

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 numbers 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:

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 highest common factor (HCF) of 12 and 18 is 6.

The largest of the common factors is the HCF of the two numbers.

Example 12: 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.

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

Remember!

The HCF of the given numbers cannot be bigger than any one of the given numbers. Do It Together

Example 13: Mike has 16 blue marbles and 8 white marbles. If he wants to place them in identical groups without any marbles being left over, what is the largest number of groups Mike can make?

Number of blue marbles: 16

Number of white marbles: 8

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 = 1, 2, 4, 8

Highest common factor = 8

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

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

Let us now use the prime factorisation method to find the HCF of 24 and 32.

1. Find the prime factorisation of both numbers.

24 = 2 × 2 × 2 × 3

32 = 2 × 2 × 2 × 2 × 2

2. Find the common factors of both numbers.

24 = 2 × 2 × 2 × 3 32 = 2 × 2 × 2 × 2 × 2

3. Multiply the combination of common factors to get the HCF of the numbers.

HCF = 2 × 2 × 2 = 8.

Example 14: Find the HCF of 12, 18 and 24.

12 = 2 × 2 × 3

18 = 2 × 3 × 3

24 = 2 × 2 × 2 × 3

Remember!

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

The common factors of 12, 18, and 24 are 2 and 3.

Hence, HCF = 2 × 3 = 6.

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

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 the remainder is 0. 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.

Remainder 12 becomes the new divisor.

Remainder is 0. So, the last divisor 6 is the HCF.

Divisor 18 becomes the new dividend.

Example 15: Find the HCF of the given numbers using the long division method.

20 and 24

Example 16: Three pieces of fabric, measuring 42 m, 49 m and 63 m long have to be divided into curtains of the same length. What is the greatest possible length of each curtain?

Length of pieces of fabric = 42 m, 49 m and 63 m.

Greatest possible length of each curtain = HCF of 42 m, 49 m and 63 m. 42 6 3 1 – 4 2

4 9 2

Hence, the greatest possible length of each curtain is 7 m.

Find the HCF of 52 and 68.

52 6 8 1

Since the last divisor is , the HCF of 52 and 68 is . Do It Together

Find the common factors for the 2 numbers.

a 13, 36

b 21, 16 c 28, 35 d 24, 27

e 30, 45 f 30, 54 g 42, 22 h 50, 45

Find the HCF by finding all the factors.

a 16, 24 b 15, 90 c 25, 45 d 27, 47

e 27, 81 f 28, 42 g 40, 50 h 33, 55

Draw factor trees to find the HCF of the given numbers.

a 12, 15 b 26, 65 c 36, 54 d 24, 40, 56

Find the HCF of the following numbers by using the long division method.

a 18, 24

b 14, 84 c 20, 120 d 81, 108

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.

Rice takes 120 days to mature, while maize takes 150 days to mature. A farmer wants to plant and harvest both crops in such a way that they can start planting both crops again on the same day after the crops have matured and been harvested. How many days will it take for this cycle to repeat?

Critical Thinking & Cross Curricular

The Central Food Technological Research Institute primarily focuses on helping people preserve and create new organic food and liquids. Siya, a research student, wants to create a juice made from 36 litres of lime juice, 45 litres of lemon juice and 72 litres of mint juice. Find the largest container that can measure the contents of the juice 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.

• 0 and 1 are neither prime nor composite numbers.

• 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

Experiential Learning & Art Integration

Prime and Composite Numbers from 1 to 100

Setting: In groups of 4

Materials Required: Number chart for numbers 1 to 100, coloured pencils

Method:

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

List the numbers that have dots of all the colours.

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

Write the number of prime numbers between 1 and 100.

Write the number of composite numbers between 1 and 100.

Chapter Checkup

Sort the numbers as prime or composite numbers.

a 49 b 61 c 73 d 99

Find the factors of the numbers using the multiplication method.

a 16

b 20 c 24 d 42

Find the factors of the numbers using the division method.

a 28

b 36 c 56 d 80

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

a 35

b 93 c 450 d 700

Draw factor trees to find the prime factorisation of the given numbers.

a 88 b 102 c 112 d 140

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

a 75 b 21 c 128 d 164

Find the common factors of each pair of numbers.

a 25, 45 b 75, 125 c 33, 55 d 120, 156

Find the HCF by finding all the factors.

a 16, 60 b 25, 65 c 48, 120 d 150, 225

Find the HCF using the prime factorisation method.

a 34, 38 b 34, 51 c 60, 225 d 105, 180

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

Write a 3-digit number such that both 420 and the number have 2 as a common factor.

Rainwater harvesting is a simple technique of collecting, filtering and subsequently storing rainwater into reservoirs or tanks. Two tanks contain 250 litres and 425 litres of rainwater, respectively. What will be the maximum capacity of a bucket that can measure the water in both tanks an exact number of times?

Kirti is making identical balloon bunches for a party. She has 24 white balloons and 16 orange balloons. She wants each bunch to have the same number of balloons of each colour. What is the greatest number of bunches she can make if she uses every balloon?

Jose is making a game board that is 66 inches by 24 inches using square tiles only. What is the largest square tile he can use, and how many tiles will he need?

Neeta wants to distribute refreshments at a party. She has 240 cupcakes and 160 sandwiches. She wants to equally distribute the food items among her classmates in packets. What is the maximum number of packets she can make, and what will be the contents of each?

Anita baked 30 oatmeal cookies and 48 chocolate chip cookies to package in plastic containers to giveaway at an old age home. She wants to divide the cookies into identical containers with each container having the same number of each type of cookie. If she wants each container to have the greatest number of cookies possible, how many plastic containers does she need? Have you ever visited an old age home?

Challenge

Critical Thinking

1 Find the greatest number that divides 178 and 128, leaving a remainder of 8 in each case.

2 ‘I am twice the number that divides two numbers without leaving any remainder. One number is 2 more than half a century, and the other number is 16 more than a century.’

Who am I?

Case Study

Cross Curricular

Colourful Gems

Jewellery making is an art. The stones used in making a jewel goes through a lot of chemical processes. Rubies are red in colour and are made from the corundum mineral. Emerald is made from beryl mineral and it is green in colour. Pearls are obtained below the sea and are white in colour. Jade comes in vibrant colours and is obtained from jadeite.

A jeweller has a few pearls, emerald, ruby and jade. He customises jewellery according to the needs of the customer.

1 A customer asks him to make a pearl bracelet with an odd number of strings. The total number of pearls in the bracelet is 72. If the number of strings is also a prime number, then which of these can be the number of strings? a 8 b 9 c 7 d 3

2 A girl asks the jeweller to use emeralds and rubies to make bangles. The jeweller has 18 rubies and 24 emeralds. What is the maximum number of stones he can use in each, such that the number of rubies and emeralds are equal in the bangles?

a 2 b 6 c 3 d 9

3 A girl orders 32 pairs of earrings. She also asks the jeweller to pack them per his choice. If the jeweller decides to pack them in prime factors, draw the factor tree for the same.

4 The jeweller decides to use 24 jades and 36 emeralds to make identical necklaces without any stones left. What is the greatest number of necklaces that the jeweller can make?

4 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.

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

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

Skip counting by 7 gives the following result.

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

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

1 Multiplication table of 8 60, 96, 108

2 Multiplication table of 9 40, 70, 80

3 Multiplication table of 10 24, 48, 64

4 Multiplication table of 12 36, 54, 63

I scored out of 4.

Understanding Multiples

Ajay is very fond of music. He attends music classes on every fifth day of the month. The rest of the month, he practises 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.

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.

For example, 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, …

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?

Look at the multiplication sentence given below. 7 and 2 are the factors of 14. 14 is the multiple of 7 and 2.

Factors of 14 7 × 2 = 14

Multiple of 7 Multiple of 2

Remember!

Multiple of a number = Number × Any number

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.

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.

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

25 × 1 = 25

25 × 2 = 50

25 × 3 = 75

25 × 4 = 100

25 × 5 = 125

Using Repeated Addition

25 + 0 = 25

25 + 25 = 50

25 + 25 + 25 = 75

25 + 25 + 25 + 25 = 100

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.

25 is a multiple of 1.

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

There are countless multiples of 25.

A number is a multiple of itself.

Every number is a multiple of 1.

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

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

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).

For example:

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

Do It Together

Example 1: Show the first eight multiples of 8, on a number line.

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

Example 2: What is the 9th multiple of 7?

The 9th multiple of 7 is 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.

Find the first five multiples of 3 using the multiplication tables. Show them on a number line.

3 × 1 = 3 3 × 2 = 3 × 3 = 3 × 4 = 3 × 5 =

So, 3, , , , and are the first five multiples of 3.

Do

It Yourself 4A

Write the first 5 multiples of the given numbers.

2 Is 7209 a multiple of 9? Justify.

3 The kumbh mela is a major pilgrimage and festival in Hinduism. The ‘Purna Kumbh mela’ is organised every 12 years. The last Purna Kumbh mela was organised in 2013. When will the next three Purna Kumbh melas be organised?

4 Select any number with 0 in its ones place and write its first ten multiples. State whether the multiples are odd or even.

5 Cicadas are large and winged insects known for their distinct, loud sound. Two species of cicadas emerge from the ground every 13 years and 17 years respectively. If both species emerged in 2021, will they emerge together over the next century?

1 A and B are two numbers. A is the factor of B. Can A also be the multiple of B? Give an example to explain your answer.

Least Common Multiple

Apart from music classes, Ajay also attends art classes every fourth day in the month of August.

Ajay marks every fourth day on the calendar and completes the list of the days on which he will attend his music and his art classes in the month of August.

Ajay looks at the lists to see if there are any days on which he has to attend both—his music and his art classes. He finds that 20th August is the date when he would have to attend both the classes.

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.

30 is a common multiple of 5 and 6. 60 is a common multiple of 5 and 6.

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.

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.

Example 4: Find the LCM of 3 and 5.

15 is a common multiple of 3 and 5.

Did You Know?

Aryabhata was an Indian mathematician who helped us understand how to find the least common multiple (LCM) of numbers, which is useful for solving problems where we need to find a common time or number for different things to happen together.

30 is a common multiple of 3 and 5.

of 3:

15 is the lowest of all common multiples. So, 15 is the LCM of 3 and 5.

15 is the lowest among the common multiples, it is the LCM of 3 and 5.

Example 5: Find the LCM of 6, 4 and 8.

Common multiples of 6, 4 and 8 = 24, 48, …

Since 24 is the lowest among all the common multiples, so it is the LCM of 6, 4 and 8.

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

Write the multiples of 12 and 16. Colour the common multiples. Find the LCM.

Multiples of 12:

Multiples of 16: …

Which of the following is a common multiple of 25, 75 and 50? a 75 b 100 c 125 d 150

2 Find the common multiples of the given numbers. Write their LCM. a 12 and 15 b 9 and 63 c 5 and 55 d 8 and 10 e 6, 12 and 18 f 6, 10 and 15 g 12, 15 and 20 h 15, 25 and 30

3 Which of these will have the same lowest common multiple as 4 and 15? a 4 and 20 b 6 and 25 c 5 and 12 d 8 and 30

4 The U.S. presidential elections are held every 4 years, while Senate elections are held every 6 years. If both elections are held in 2024, in which year will they next occur together again?

5 A machine requires maintenance every 12 days, and another machine requires maintenance every 18 days. If both machines are maintained today, in how many days will they both need maintenance on the same day again?

6 Two sisters, one in a primary school and the other in a middle school, have lunch together when their school bells ring at the same time. The bell of the primary school rings every 30 minutes, while the bell of the middle school rings every 40 minutes. If both bells ring together at 8:30 a.m., when will they ring together again?

7 Create a question based on LCM.

Challenge

Critical Thinking

1 Ravi thinks of a number. He says that the third multiple of that number is equal to half the third multiple of 24. What number was Ravi thinking of?

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

Step 2: Multiply each factor the greatest number of times it occurs in the prime factorisation.

We take 2 two times, 3 two times, and 5 one time and multiply

60 = 2 × 2 × 3 × 5

45 = 3 × 3 × 5

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

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.

36 = 2 × 2 × 3 × 3 48 = 2 × 2 × 2 × 2 × 3

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

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 =

= 2 × 2 × 2 × 7

Do It Yourself 4C

Find the prime factorisation of the numbers.

a 35 and 75 b 44 and 88 c 48 and 12 d 25 and 115

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

8 = 2 × 2 × 2 25 = 5 × 5

= 2 × 2 × 2 × 2

a 8, 10 b 8, 16 c 16, 10 d 25, 10

3 Find the LCM of the given numbers using the prime factorisation method.

a 16, 24 b 25, 35 c 36, 45 d 63, 105 e 18, 40 and 45 f 72, 96 and 108 g 48, 56 and 70 h 30, 60 and 90

4 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?

5 The T20 cricket world cup is held every 2 years while the FIFA world cup is held every 4 years. Both the world cups were last held together in 2022. How many times will both world cups be held together till 2040 after 2022?

6 Create a word problem on finding the LCM of 2 numbers.

1 This year, my age is a multiple of 7. Next year, it will be a multiple of 8. I am older than 20 but younger than 80. How old am I?

LCM by Short Division Method

Let us see how we can find the LCM of 25 and 45 using the short division method.

Step 1: Arrange the numbers in a line.

Step 2: Start dividing the numbers by common prime number, which, in this case, is 5.

25, 45 5 5, 9 3 1, 9 3 1, 3 1, 1

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

Step 4: Multiply all the prime factors to get the LCM.

LCM = 5 × 5 × 3 × 3 = 225

Example 8: Find the LCM using the short division method.

1 6, 12 and 15

3 6, 12, 15

2 2, 4, 5

2 1, 2, 5

5 1, 1, 5 1, 1, 1

LCM = 3 × 2 × 2 × 5 = 60

2 12, 16 and 20

2 12, 16, 20

2 6, 8, 10

2 3, 4, 5

2 3, 2, 5

3 3, 1, 5

5 1, 1, 5 1, 1, 1

LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240

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

3 24, 15

2 8, 5

LCM =

Do It Yourself 4D

Find the LCM using the short division method.

a 21, 24 b 25, 30 c 32, 48 d 60, 75

e 28, 42, 56 f 75, 100, 150 g 21, 63, 105 h 90, 135, 180

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

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

3 Match the following.

LCM of 42, 70

of 63, 105

of 30, 45

LCM of 9, 15

LCM of 12, 48

4 Jupiter takes around 12 years to revolve around the Sun while Saturn takes around 30 years to revolve around the Sun. If Jupiter, Saturn and the Sun are in a straight line at the beginning of their revolution cycle, after how many years will they be in their original positions again?

Challenge

1 Read the statements and choose the correct option.

Critical Thinking

Assertion (A): Anna takes 8 minutes to complete one round, and Sylvia takes 12 minutes. If both start cycling at the same time and move in the same direction, they will meet at the starting point after 24 minutes.

Reason (R): The least common multiple (LCM) of 8 and 12 is 24.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

Word Problems

When do we use LCM?

When do we use HCF?

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

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

Let us look at some examples to understand this better.

An 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.

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.

An 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.

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.

The HCF of 32 and 40 is 8.

So, the greatest number of teams that can be formed is 8.

Example 9: In a math quiz, Ajay is tasked with finding the LCM of some numbers. Ajay finds the LCM of 16, 24, 36 and 54 using the short division method. He says that the LCM of 16, 24, 36 and 54 is 72. Is he right? Why or why not?

To find the prime factors of 16, 24, 36 and 54, we find the LCM.

LCM of 16, 24, 36 and 54 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432

So, 72 is not the LCM of the given numbers.

Thus, Ajay is not correct.

2 16, 24, 36, 54

2 8, 12, 18, 27

2 4, 6, 9, 27

2 2, 3, 9, 27

3 1, 3, 9, 27

3 1, 1, 3, 9

3 1, 1, 1, 3 1, 1, 1, 1

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

Let us find the LCM of 4 and 10.

4 = 2 × 2

10 = 2 × 5

So, LCM of 4 and 10 = 2 × 2 × 5 = 20

Thus, the 20th student will get both the chocolate and the candy.

Example 11: For a morning walk, three people step out together. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance that each should walk before they are together again?

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.

2 80, 85, 90

5 40, 85, 45

2 8, 17, 9

2 4, 17, 9

2 2, 17, 9

3 1, 17, 9

3 1, 17, 3

17 1, 17, 1 1, 1, 1

Example 12: 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

HCF = 2 × 7 = 14.

98 = 2 × 7 × 7

Therefore, the required length of each table mat is 14 cm.

126 = 2 × 3 × 3 × 7

What is the smallest possible length that can be measured exactly by the scales of lengths 3 cm, 5 cm and 10 cm?

5 3, 5, 10

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.

LCM of 3, 5 and 10 =

So, the smallest possible length that can be measured by each of the given scales is cm.

Do It Yourself 4E

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 that he will mark for the classes scheduled in the month of August.

2 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?

3 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?

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

5 A school is organizing a sports event. There are 72 soccer balls and 96 basketballs. The organizers want to divide the balls into the largest possible equal groups, such that each group has the same number of soccer balls and basketballs. How many soccer balls and basketballs will be in each group?

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

7 Three different species of flowers have different blooming cycles, depending on their species and types. The sunflowers bloom every 10 weeks, the roses bloom every 5 weeks and the tulips bloom every 3 weeks. If all three species bloom today, after how many weeks will they all bloom together again?

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

9 Create a word problem on finding the HCF.

1 On every fifth visit to a restaurant, you receive a free beverage. On every tenth visit you receive a free appetizer.

a If you visit the restaurant 100 times, on how many 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?

Points to Remember

• The Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple 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.

• 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.

Math Lab

Setting: In groups of 3

Locate the LCM

Experiential Learning & Collaboration

Materials Required: Crayons of three colours—blue, orange, and green, and a notebook.

Method:

The students will make a grid of 10 rows and 10 columns in their notebooks. The students will write the numbers from 1 to 100 in them.

The teacher will ask the students to find the LCM of three numbers, say, 3, 6 and 8. 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.

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.

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

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

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

a 7 b 9 c 11 d 13

4 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, …

a Only 180 b 90, 180 c Only 90 d 30, 45

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

a 2 × 3 × 3 2 × 5 × 7

b 2 × 3 × 5 3 × 3 × 5

c 2 × 3 × 3 × 7 2 × 3 × 11

d 3 × 3 × 5 × 7 2 × 3 × 5 × 11

e 2 × 3 × 3 × 5 2 × 2 × 3 × 5

6 Find the LCM of the given numbers using prime factorisation.

a 18, 27 b 35, 14 c 5, 9, 15 d 25, 40, 60

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

a 360 b 90 c 120 d 180

8 Find the LCM by the short division method.

a 12, 20 and 32 b 93, 62 and 120 c 15, 36 and 40 d 45, 18 and 63

9 Aakash noticed that the number of questions given for homework is divisible by both 3 and 13. What is the smallest possible number of questions that could have been given?

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

11 El Niño and La Niña are opposite phases of a natural climate pattern across the tropical Pacific Ocean. If El Niño occurs every 3 years and La Niña every 7 years, starting from 2020 when both were observed, in what future year will both phenomena occur together again?

Challenge

1 A teacher asks the students to name any number that has 75 as a multiple. Rahul says 15, Sahil says 25, and Ajay says 12.

Which of the statements given below is correct regarding this scenario?

a Rahul and Ajay are correct. b Ajay and Sahil are correct.

c Rahul and Sahil are correct. d Only Rahul is correct. Critical Thinking

2 Read the given statements. Choose the correct option.

Assertion (A): The LCM of 3 and 9 is 9.

Reason (R): The LCM of any two numbers is always greater than their HCF.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

Case Study

d A is false, but R is true. Value Development

Coordinating for a Cause

Riya and Samir are school friends. They live at equal distance from their school but in opposite directions. They use the local bus to go to their school. Riya’s bus arrives every 15 minutes, while Samir’s bus arrives every 20 minutes.

For Earth Day, they decided to plant trees together, outside the school, to increase awareness about saving the environment. For this, they wanted to reach the school at the same time. The buses arrived at their respective bus stands at 8:00 a.m. but both of them missed their buses.

Now, answer the following questions.

a Every 30 minutes

c Every 75 minutes

1 How often do the buses arrive at the same time?

b Every 60 minutes

d Every 90 minutes

2 When should Riya and Samir take the next bus so as to reach school at the same time?

a 8:15 a.m.

c 8:45 a.m.

b 8:30 a.m.

d 9:00 a.m.

3 When they finished planting trees, Riya said, “The number of trees I planted is a multiple of 2.” Samir said, “The number of trees I planted is a multiple of 5.” They both planted the same number of trees. What is the lowest multiple of trees planted that is common to both?

4 On the way back home, Riya’s bus arrives every 10 minutes and Samir’s bus arrives every 15 minutes at the school’s bus stand. If the last time that both the buses arrived together was at 3:30 p.m., when will they arrive at the school bus stop together again?

5 Have you ever taken part in such plantation drives? Give a reason for your answer.

5 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.

Each equal part represents a half 1 2

An apple has been cut into 3 equal parts. Each slice represents a third 1 3

of the apple.

A pear has been cut into 4 equal parts.

Each slice represents a quarter 1 4

of the pear.

In each fraction, the number at the top is called the numerator and the number at the bottom is called the denominator.

Let’s Warm-up

Understanding Fractions

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.

Reviewing Fractions

Look at the pattern on the origami sheet carefully. What do you see?

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 divided into 8 equal parts, one part is 1 out of 8 or 1 8 of the square sheet.

What about Amit’s sheet? Here, 4 equal triangles divide the square.

So, each part is 1 out of 4 or 1 4 of the square sheet.

Richa’s Sheet

Amit’s Sheet

Do It Together

Like Fractions: Same denominators

Examples: 23456 ,,,, 77777

Unlike Fractions: Different denominators

Example 1: Which of the following are like fractions?

1 41 and 55 2 3 6 1 3 and

Examples: 46263 ,,,, 5791113

1 As the denominators of 41 and 55 are the same, they are like fractions.

2 As the denominators of 31 and 63 are different, they are unlike fractions.

Example 2: Classify the fractions 25179 , , 3, , 537105 as proper fractions, improper fractions or mixed numbers.

Proper Fractions 2 5 and 7 10 Improper Fractions 59 and 35

Write the fractions 15 20 ; 15 1 2 ; 15 10 ; 2 10 ; 1 3 in the table.

Numbers 3 1 7

Converting Between Improper Fractions and Mixed Numbers

Let΄s see how we can convert fractions with the help of examples.

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. Here we see each square divided into 4 equal parts. Mixed Number

Improper Fraction

Parts each shape is divided into = 4

Total parts shaded = 11 11 4

Wholes shaded = 2

Parts shaded = 3 out of 4 2 3 4

Convert 4 4 5 into an improper fraction.

1 Split the mixed number into a whole number and proper fraction.

Quick way:

2 Write the whole number part as a proper fraction.

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

Step 2: Add the result obtained in step 1 to the numerator.

3 Multiply the numerator and denominator by the denominator of the fractional part.

4 Add the numerators to get the improper fraction.

Conversion of Improper Fractions to Mixed Numbers

Write 24 5 as a mixed number.

Divide the numerator by the denominator.

The divisor is the denominator of the fraction.

The quotient becomes the whole number part. Example 3:

Step 3: Write the result obtained in step 2 over the denominator.

Remember!

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

Are the given pairs of fractions like or unlike?

a 4 7 and 1 4 b 5 8 and 4 8 c 1 4 and 1 5 d 3 7 and 5 7 e 6 18 and 7 18

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

a 461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797

b461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797

c 461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797

f 461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797 g 461585171297122684 , , 4, , 5,

d 461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797 e 461585171297122684 , , 4, , 5, , , , 6, , , 7,1 52758111539291797

461585171297122684

7,15 52758111539291797

, 4, , 5, , , , 6, , , 7,15 52758111539291797 j 461585171297122684 , , 4, , 5, , , , 6, , , 7,15 52758111539291797

Match the mixed numbers and the improper fractions of the same value.

Convert between improper and mixed fractions.

87 rice packets are distributed among 5 families during floods. How many packets of rice does each family get? Express your answer as a mixed number. Have you ever helped flood victims? Use fraction strips to represent 15 6 as an improper fraction and a mixed number.

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

a The fraction has a prime denominator and is an improper fraction.

b The mixed number 7 2 6 shows the same fraction.

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.

Equivalent Fractions Using Multiplication

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

Amit has cut 1 out of 4 parts. So, he now has 1 4 of the entire sheet. Since both Amit and Richa have the same size of paper, we can say 1 4 = 2 8 .

These are called equivalent fractions.

We can multiply the numerator and the denominator by the same number to find any equivalent fraction.

2 8 and 1 4 are equivalent fractions.

Example 4: Write three equivalent fractions for 3 7 . 3

Equivalent Fractions Using Division

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

Richa Amit

Fraction in its Simplest Form

To convert a fraction to its simplest form, divide the numerator and the denominator by the highest common factor.

Convert 4 8 to its simplest form.

HCF of 4 and 8 = 4. ÷ == ÷ 4 4 41 88 42

Checking for Equivalent Fractions

Check whether 4 5 and 12 15 are equivalent or not.

1 Cross multiply the denominator of one fraction with the numerator of the other fraction and vice versa. × 4 5 12 15

Another way:

2 Check if the products of both the multiplications are the same. If the products are the same, the fractions are equivalent.

4 × 15 = 60 5 × 12 = 60 Hence, 4 5 and 12 15 are equivalent.

Check whether 4 5 and 12 15 are equivalent or not.

Reduce both the fractions to their lowest form and check if they are equal.

4 5 is already in its lowest form.

12 15 = ÷ ÷ 123 153 = 4 5

Both the fractions are equal, so 4 5 and 12 15 are equivalent.

Example 5: Write any three equivalent fractions for 90 120 using division.

Example 6: Express 42 91 in the simplest form.

Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42; Factors of 91 = 1, 7, 13, 91.

HCF of 42 and 91 = 7.

42 91 = 427 917 ÷ ÷ = 6 13

Thus, 6 13 is the simplest form for 42 91 .

Example 7: Check whether the 602 and 903 are equivalent or not.

÷ = ÷ 603022 and 903033

As = 22 , 33 hence 602 and 903 are equivalent.

Example 8: Fill in the box to make the given fractions equivalent. = 5 1155

Since the denominator 11 × 5 = 55; hence, the numerator 5 × 5 = 25.

Thus, = 525 1155 .

Express 48 96 in the simplest form.

Thus, the simplest form for 48 96 is . Do It Together

Factors of 48 = 1, 2, 3, , , , 12, , 24, .

Factors of 96 = .

HCF of 48 and 96 = .

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

Express the given fractions in their lowest form.

Check whether the given fractions are equivalent or not.

Look at the coloured grid. Write the fraction shown by each colour. Which set of fractions are equivalent?

Comparing and Ordering Fractions

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

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!

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 4 = 2 8

To compare like fractions, compare the numerators.

Greater the numerator, greater the fraction.

2 > 1. Thus, 2 8 > 1 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 7 and 2 8 . 7 < 8. Thus, 22 78 > .

Richa Amit

Richa’s sheet Amit’s sheet

But, what if both the numerators and the denominators are different?

Method 1: Make the fractions like, using the LCM Method.

Compare 1 4 and 2 6 .

Step 1: Find the LCM of the denominators.

The LCM of 4 and 6 is 12.

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

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

< 4 So, 34 1212 < . Thus, 12 46 < .

Method 2: Cross-multiplication Method

Compare 1 5 and 2 7 .

Cross multiply the denominators with the numerators.

1 × 7 = 7 and 5 × 2 = 10

7 < 10. Thus, 1 5 < 2 7 .

Example 9: Which fraction is smaller? 6 8 or 10 15

LCM of 8 and 15 = 120 6 8 = 615

120 > 80 120 .

Hence, 10 15 is the smaller fraction.

Example 10: Which fraction is bigger? 4 7 or 5 8

Apply the cross-multiplication method.

4 × 8 = 32

5 × 7 = 35

32 < 35

Thus, 5 8 is bigger than 4 7 .

Example 11: Lila has two pieces of ribbon. One piece is 3 4 of a metre long, and the other piece is 2 3 of a metre long. Which piece of ribbon is longer?

To compare 3 4 and 2 3 , we find a common denominator, which is 12.

× == × 3339 44312 ; × == × 2248 33412

Since 9 12 is greater than 8 , 12 piece of the ribbon that is 3 4 of a metre long is longer.

Compare using >, <, =. 3 11 2 6

Apply the cross-multiplication method.

3 × 6 = 18 2 × 11 = . 18 . Thus, 3 11 2 6 .

Ordering Fractions

Arrange 1324 ,,, 5757 in ascending order.

Step 1

Convert all the fractions into like fractions.

LCM of 5 and 7 is 35.

1 5 = 17 57 ×× 17 57 × = 7 35 3 7 = 3 7 × 5 5 = 15 35

2 5 = × 27 57 ×× 27 57 = 14 35 4 7 = 45 75 ×× 45 75 × = 20 35

Example 12: Arrange the fractions 3 4 , 9 13 , 4 26 , 1 2 in descending order.

LCM of 2, 4, 13 and 26 = 52. 331339 441352 =×= 99436

=×= 4428 2626252 =×= 112626 222652 =×= 39 > 36 > 26 > 8

So, 3936268 52525252 >>> . Thus, 3914 413226 >>> .

Step 2

Since the denominators are the same, compare the numerators and order the fractions.

7 < 14 < 15 < 20 So, 7141520 35353535 <<< . Thus, 1234 5577 <<<

Arrange the fractions 5323 ,,, 6432 in ascending order.

LCM of 2, 3, 4, 6 = . =×=

Thus, the ascending order is

Compare the following fractions using >, <, =.

Put a tick () for the statements which are correct and a cross () for the statements which are wrong. a 4 5 > 2 3

Arrange the following in ascending order.

Arrange the given fractions in descending order.

3131 ,,, 8456

Represent visually 2 fractions greater than the fraction 1 4 .

Shashank studied for 6 13 hours while Reshma studied for 11 23 hours. Who studied for a longer duration?

Create a word problem on your own on comparing two fractions.

Challenge

Critical Thinking

Rakesh, Roshan, Swati, and Prerna were assigned a task each. They completed the task in 4 6 hours, 7 8 hours, 1 hour and 2 3 hours, respectively.

Which of the following statements is false regarding the given scenario?

Statement I: Rakesh and Prerna took the same amount of time.

Statement II: The time taken by Roshan is less than that of Prerna but more than that of Swati. Statement III: Swati took the most amount of time.

Statement IV: The time taken by Roshan is less than that of Swati but more than that of Rakesh or Preeti.

Points to Remember

• A fraction indicates one or more parts of a whole.

• Fractions with the same denominator are like fractions.

• Fractions with different denominators are unlike fractions.

• Proper fractions have a smaller numerator than denominator.

• Improper fractions have a numerator equal to or greater than the denominator.

• Mixed fractions combine a whole number and a fractional part.

• A fraction is simplest when the HCF of its numerator and denominator is 1.

• When comparing fractions with the same denominator, the one with the greater numerator is greater.

Experiential Learning & Collaboration

Equivalent Fractions

Aim: To understand how to represent equivalent fractions.

Materials Required: Paper strips.

Setting: In groups of 4

Method:

Each group should choose a fraction like 1 2 1 3 1 4 1 5 ,, , etc.

Fold one strip to show the chosen fraction (e.g., fold in half for 1 2 )

Mark the folds and label each part.

Fold and mark the other three strips to show three equivalent fractions.

Compare the strips and discuss how they show equivalent fractions.

Chapter Checkup

Convert the given mixed numbers into improper fractions. a 1 7

Convert the given improper fractions into mixed numbers.

Find any three equivalent fractions using multiplication.

a 2 6 b 1 9 c 5 8

Find any three equivalent fractions using division.

Write the following fractions in the simplest form.

Compare the fractions and put the correct symbol >, <, =.

a 1 2 2 and 14 5 b 4 6 5 and 34 4 c 24 3 and 8

Arrange the following fractions in ascending order.

9 and 1 6 3

a 5141 ,,, 12256 b 1331 ,,, 5582 c 1262 ,,, 2375 d 11134 ,,, 12257

Reading books is a relaxing activity, that reduces stress levels and enhances brain connectivity. Three friends, Alex, Bailey, and Casey borrowed some books from the library. Alex took 1 4 , Bailey took 3 8 , and Casey took 3 12 of the total borrowed books. Who took the least fraction of the borrowed books and

Asia: 3 5 , Australia: 1 80 , Europe: 7 80 , Africa: 7 40 , South America: 1 20 , North America: 3 40 9 Cross

The approximate population of the 6 continents of the world as fractions are given below. Arrange them in ascending order.

Challenge

1 If 4 fractions 1 2 , 3 8 , 5 4 and 5 6 are placed on the number line, which fraction will be to the extreme left?

2 Read the statements and choose the correct option.

Assertion (A): 1 2 > 1 3 > 1 4 .

Reason (R): In unit fractions, the denominators are compared. The smaller the denominator, the smaller the fraction.

a Both A and R are true and R is the correct explanation of A.

b Both A and R are true but R is not the correct explanation of A.

c A is true but R is false.

d A is false but R is true.

Case Study

Value Development & Cross Curricular

Sorting Recyclables

Mrs Gupta’s class is learning about recycling and the importance of sorting waste correctly. For their science project, they decide to organise a recycling drive. The students need to sort paper, plastic, and glass into different bins.

The bins are labelled with different fractions representing how full they can be before being emptied:

Paper Bin: 3 6 full; Plastic Bin: 3 4 full; Glass Bin: 4 5 full

Tanvi, one of the students, notices that the fractions on the bins look different but wonders if they might actually be the same.

Answer these Questions:

1 Which of the following fractions is equivalent to the paper bin?

2 Which option shows the fractions from the bins arranged in ascending order?

3 The fraction of the glass bin is equivalent to 10

4 The fraction of the plastic bin is an improper fraction. (True/False)

5 Tanvi realises that sorting recyclables correctly helps keep the environment clean and conserves resources. Why is it important to recycle and sort waste properly?

6 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 .

All the parts, each representing 1 5 th of the chocolate, can be added to make one whole. That is, if we add 1 5 five times, we get 1.

All equal parts add together to make a complete

Let’s Warm-up

Fill in the blanks.

Addition and Subtraction of Fractions

Shagun celebrated her birthday. Her mother made apple pies for the party.

Shagun: I will eat 2 slices of the apple pie.

Arvind: I will eat 3 slices of the apple pie.

All her other friends also had the apple pie.

Adding and Subtracting Fractions

There are 8 slices in one apple pie.

Shagun ate 2 8 of the apple pie.

Arvind ate 3 8 of the apple pie.

What part of the apple pie did they both eat together?

Adding Unlike Fractions

What if the denominators of the two fractions 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.

Let us add the unlike fractions 2 5 and 1 15 .

Sum of like fractions = Sum of numerators Common denominator Remember!

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 the equivalent fractions of the two fractions so that the denominators of both become the LCM.

Step 3

Add the numerators after the denominators become the same.

6 15 + 1 15 = 6 + 1 15 = 7 15

Step 4

Reduce the fraction to its simplest form, if required. 7 15 is already in its simplest form.

Thus, 2 5 + 1 15 = 7 15

Example 1: Add 6 9 and 5 12 .

LCM of 9 and 12 = 36

6 9 = 6 9 × 4 4 = 24 36

24 36 + 15 36 = 24 + 15 36 = 39 36

Example 2: Add 2 7 and 5 9 . LCM of 7 and 9 = 63

Example 3: Suhani spent 5 7 hours on a project on Saturday and 11 14 hours on Sunday. How many hours did she spend on the project together?

Fraction of hours spent on Saturday = 5 7 hours

Fraction of hours spent on Sunday = 11 14 hours

Total hours spent = 5 7 + 11 14 = 10 + 11 14 = 21 14 = 3 2 hours = 11 2 hours [LCM of 7 and 14 = 14]

Add 2 7 and 9 14. LCM of 7 and 14 = . 2 7 = 2 7 × =

Adding Mixed Numbers

Remember Shagun, who had a birthday party?

Her friends ate 2 whole apple pies and 5 6 of an apple pie. 3 3 6 of the apple pies was left.

How much apple pie was there in total? Here the two mixed numbers are like.

Method 1: When the mixed numbers to be added are like, we can add the whole number to the whole number and the fraction to the fraction.

Add 5 2 6 and 3 3 6 .

Add the whole number to the whole number.

Add the fractional part to the fractional part.

Convert the fraction to a mixed number.

Add the results of step 1 and step 3.

What if the total apple pie eaten was 1 2 3 and the apple pie left was 1 3 2 , then how much apple pie was there in total?

Method 2: When the mixed numbers to be added are unlike, we convert the mixed numbers to improper fractions and then add.

Add 1 2 3 and 1 3 2 .

Convert the mixed numbers to improper fractions.

Example 4: Add 1 3 3 and 1 5 4 .

Example 5: Rakesh has two ropes of length 1 3 3 m and 5 1 6 m. What is the total length of the ropes?

Length of both the ropes = 1 3 3 m and 15 6 m

Total length of the ropes = 3 + 1 +

Hence, the total length of rope is 1 5 6 m. Add 3 2 6 and 3 1 4 . On converting to improper fractions, we get:

Subtracting Unlike Fractions

Remember, Shagun and Arvind had eaten 5 8 of an apple pie. Let us see how much apple pie was left with them. A whole is represented as 1. Apple pie

Thus, 3 8 of the apple pie was left.

What if 3 4 of the apple pie was left and Shagun ate 1 3 of the apple pie from the leftover?

Let us find out.

Make the fractions like using the LCM method.

To find the leftover pie, we will subtract 1 3 from 3 4 . LCM of 3 and 4 = 12

Subtract the like fractions.

Find 7 8 1 4 .

Convert the fractions into like fractions.

of 8 and 4 = 8

7 8 1 4 = 5 8 . Find the difference: 7 16 1 14

and

Subtracting Mixed Numbers

Subtracting Like Mixed Numbers

If the total apple pie available was 1 5 3 and the portion eaten was 2 2 3 , then how much apple pie would be left?

To find the leftover pie, we will subtract 2 2 3 from 5 1 3 .

Subtracting Unlike Mixed Numbers

If the total apple pie available was 2 4 5 and the portion eaten was 1 2 3 , then how much apple pie would be left?

In this case, we will subtract 1 2 3 from 2 4 5 .

Did You Know?

Mahavira, in his work "Ganita Sara Samgraha," provided systematic rules for dealing with fractions. He discussed how to reduce fractions to their lowest terms and how to handle operations with mixed numbers.

Example 8: Niharika bought 3 4 4 litres of milk from the milk parlour. She was left with 5 2 8 litres of milk after preparing a sweet dish. How much milk did she use in the sweet dish?

Amount of milk bought = 3 4 4 litres

Amount of milk left = 52 8 litres

Amount of milk used in the dish = 42351921 4848 −=−

Circle the pair of fractions in each problem which add together to make 1 .

Add the fractions and write the answer in the simplest form.

Answer the questions.

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

b Subtract the difference of 1 2 2 and 1 1 2 from the difference of 1 5 3 and 1 3 3 .

A basket has 2 1 5 kg of mangoes. Rajiv puts 2 2 5 kg more mangoes in it. What is the total weight of the mangoes in the basket?

Earth’s atmosphere is composed of many gases. The atmosphere is composed of about 39 50 nitrogen and about 21 100 oxygen. What is the fraction of all the other gases in the atmosphere?

Sanjay sent 50 1 2 kg of wheat to help the flood victims. 10 3 4 kg of wheat was spoilt. What quantity of wheat reached the victims? What else can be done to help flood victims?

Create a multi-step word problem on addition and subtraction of fractions.

Challenge

Critical Thinking

Raghuveer purchased 4 types of fruit weighing 3 7 4 kg. He purchased 1 1 2 kg of apples, 1 2 4 kg of pears, 2 1 3 kg of oranges and some litchis.

Statement 1: The total weight of apples and pears is 3 1 2 kg.

Statement 2: The weight of litchis bought is 2 1 3 kg.

a Only statement 1 is true.

b Only statement 2 is true.

c Both the statements 1 and 2 are true.

d Both the statements 1 and 2 are false.

Multiplication and Division of Fractions

Mithun, a shopkeeper, sells dairy products like milk, curd and paneer.

He packs milk in 1 4 litre packets.

Multiplying and Dividing Fractions

Mithun packed 8 packets of 1 4 litre of milk each. Let us see what quantity of milk he packed.

Multiplying Fractions and Whole Numbers

To show this using fraction strips, we can say that there are 8 groups of 1 4 . What will we get when 8 one-fourths are put together?

8 groups of 1 4 is the same as 8 × 1 4 .

The steps to multiply the whole number 8 by a fraction 1 4 are given below.

Step 1

To multiply a whole number with a fraction, multiply the whole number by the numerator of the fraction. The denominator remains the same.

8 × 1 4 = 81 4 × = 8 4

Step 2

Convert the improper fraction into a mixed number.

8 2 4 =

Thus, the total quantity of milk packed is 2 litres.

Example 9: Multiply 5 and 6 7 . 5 6 6 5 77 × ×= 30 7 = ×+× ==+= (47)247 30 22 4 77777

Example 10: Mahesh reads 2 5 pages of a book. The total number of pages in the book is 100. How many pages has he read?

Total number of pages = 100

Fraction of pages read = 2 5

Number of pages read = 2 5 of 100 = 2 10040 5 ×= pages.

Hence, Mahesh read 40 pages.

Multiply 5 9 by 21.

5 9 × 21 = 5 9 × 21 = =

Multiplying Two Fractions

Multiply 2 5 and 3 7

Let us multiply the above 2 fractions using pictures.

1 To show 2 5, divide the rectangle vertically into 5 equal parts. Shade 2 out of 5 parts.

2 Then, to show 3 7, divide the rectangle horizontally into 7 equal parts.

3 Draw crosses in 3 out of 7 parts, horizontally.

4 Count the number of equal parts which have both the shading and the cross (×). 6 out of 35 parts, both have the green colour and the cross (×).

Fractions can be multiplied using two methods.

Method 1: Multiplying the numerators and multiplying the denominators.

Method 2: Cancelling out the common factors in the numerators and denominators.

Fraction of a fraction:‘of’ means multiplication when multiplying fractions.

Example 11: Multiply 2 3 by 7 8 . 27 2714 383824 × ×== × ÷ == ÷ 141427 2424212

Thus, 277 3812 ×=

Example 12: Find 4 9 of 7 2 . 4 9 of 7 2 = 47 92 × ×=×=×=

Example 13: In a grid 4 5 of the squares are shaded. If 3 7 of the shaded squares are blue, what fraction of the entire grid is made up of blue squares?

Fraction of square shaded = 4 5

Fraction of shaded squares that are blue = 3 7

Fraction of grid made up of blue squares = 4312 5735 ×=

Hence, 12 35 of the grid is blue.

Multiply 5 8 by 4 7 using the fraction strips and cancel out the common factors.

5 8 × 4 7 =

Dividing a Whole Number by a Fraction

Lalit has to pack 2 cakes in boxes. Each box can hold 1 2 of the cake.

How many cake boxes are needed?

We need to find 2 ÷ 1 2 .

To find the number of boxes packed by Lalit, let us first 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.

What is the reciprocal of 5? 5 5 1 =

Reciprocal of 5 = 1 5

What is the reciprocal of 1 5 ?

Reciprocal of 1 5 = 5 1 or 5.

Two numbers are said to be reciprocals of each other when their product is 1. Thus, reciprocals are the multiplicative inverse of each other.

Now, let us find the number of boxes packed by Lalit.

There are 2 boxes. Each box can hold 1 2 of the cake.

So, we need to find 2 ÷ 1 2 or how many halves will fit into 2 wholes.

Let us see, using the figures.

2 ÷ 1 2 means how many halves will fit into 2 wholes.

Think and Tell

What is the reciprocal of 1?

Thus, 4 boxes of cake were packed by Lalit.

Example 14: Divide 5 by 1 5

Example 15: Sarah follows an exercise routine. She runs 3 4 miles in a day. How many days will it take her to achieve her running goal of 12 miles?

Running goal of Sarah = 12 miles

Number of miles she runs in a day = 3 4 miles

Number of days required to reach the goal = ÷=×=

Hence, Sarah will take 16 days to achieve her goal.

by 14

using the keep, change and flip method.

Dividing a Fraction by a Whole Number Divide 1 2 by 4 .

Let us understand this in another way.

1 Write the whole number as a fraction.

2 Reverse the ‘÷’ symbol to ‘×’.

Example 16: Divide 1 3 by 6.

3 Write the reciprocal of the fraction.

4 Multiply the fractions to get the answer.

Example 17: Divide 4 8 by 2.

Thus, ÷= 11 6

Example 18: Megha has 5 6 of a watermelon. She wants to share it equally with her

5 friends. What fraction of watermelon will each of her friends get?

Fraction of watermelon with Megha = 5 6

Number of friends she wants to divide the watermelon among = 5

Fraction of watermelon each friend will get = ÷=×==

Hence, each friend will get 1 6 of the watermelon.

Divide 3 4 by 9 using the fraction strips and the reciprocals. 3 9 4

Dividing a Fraction by a Fraction

Divide 1 2 by 1 6 .

Method 1

How many one-sixths will fit into one-half?

Example 19: Divide 1 3 by 1 6 . How many one-sixths will fit into one-third? 2

Example 21: A water tank is filled to 2 3 of its capacity. Rahul wants to empty the tank using buckets that can hold 1 6 of the tank’s capacity. How many buckets are needed to empty the tank?

Volume of water in the tank = 2 3 of its capacity

Volume of water each bucket can hold = 1 6 of the tank’s capacity

Number of buckets needed to empty the tank = 212 64 363 ÷=×= buckets

Hence, 4 buckets are needed to empty the tank. Divide 3 4 by 1 8 using the fraction strips and the reciprocals.

Do It Yourself 6B

Find the reciprocal of the given fractions.

Multiply the fractions using fraction strips. a 52 65 ×

Multiply.

1 2 4 ×

Divide.

13 43 ×

75 86 ×

Answer the questions.

a What is 3 4 of 2 hours?

b How many days are 2 7 of 1 week?

Sushen and his friends ate 3 4 of a cake at his party. The total weight of the cake was 2 kg. What weight of cake was eaten?

Ravi has 21 kg of rice. He packs the rice in packets of 3 4 kg each. How many such packets does he pack?

A car has 50 litres of petrol in its fuel tank. 3 5 of the fuel is used up to travel from one place to another. What quantity of fuel is left in the fuel tank? What should we do to utilise fuel efficiently?

How many bottles of capacity 3 4 L can be filled from a container of capacity 57 4 L? Create a word problem to multiply a whole number by a fraction.

The capacity of a cylindrical tank in a kitchen is 40 litres. It is 4 5 full of water. Nisha used a quarter of this volume of water while cooking food. If Nisha needs 28 litres of water to wash all the utensils, will she be able to do so?

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.

• 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.

Setting: In pairs

Collaboration & Communication

Method:

Materials Required: Fraction bingo cards, markers or coloured pencils

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

Each pair gets a fraction bingo card and a coloured pencil or marker. Call out the fractions along with the operation that needs to be done.

The pairs will perform the operation and check if the answer is on their bingo card. If found they will circle the answer.

The pairs that gets all the answers in a row, column or diagonal says Bingo and wins!

Chapter Checkup

Find the multiplicative inverse of the given fractions.

a 4 5 b 18 17 c 78 14 d 1 5

Write True or False.

a The reciprocal of every fraction is a proper fraction.

b 0 ÷ any fraction = 0.

c The reciprocal of 0 is 0.

d The multiplicative inverse of any fraction is always greater than 1.

Add the given fractions.

Subtract the fractions and express the answer in its simplest form.

Multiply to find the product. Also, write the answer in its simplest form.

Divide and find the quotient.

2 5 5 ÷

Answer the following questions.

a Which is more: the difference of 1 2 8 and 1 8 4 or the sum of 1 2 4 and 1 4 3 ?

b Which is less: the product of 5 and 4 5 or the sum of 1 1 4 and 4 2 5 ?

c What is the difference between the product of 4 and 1 5 , and the product of 2 5 and 3?

Delhi is the capital city of India. The age distribution in Delhi is given as; Children29 100 , Adults13 20 and Elderly people3 50 . What fraction of the population of Delhi are not children?

A boy was travelling to his grandmother’s home. He travelled 41 7 km by bicycle, 23 4 km on foot and 101 2 km by car. What is the total distance covered by him?

The Delhi metro is a rapid rail system that connects Delhi to its adjoining cities. About 65 km of the total length was constructed in phase 1, about 125 km in phase 2 and about 167 km in phase 3. About 11 2 times of the length of phase 1 was constructed in phase 4. What is the length constructed in Phase 4?

3 4 of a parking lot is full when there are 66 cars in it. How many cars can be parked inside the parking lot?

14 cans can hold 1441 2 litres of water. What is the capacity of each can?

Radhika purchased 20 m of cloth. She used 112 5 m of cloth for the curtains and 35 6 m of cloth for a bed sheet. How much cloth does she have left?

Create a word problem to divide a fraction by a whole number.

Challenge

1 A recipe requires 5 6 cup of grated cheese. With the help of 6 cheese cubes, you get 3 5 of the amount you need. How much more grated cheese (in cups) do you need? How many more cheese cubes should you grate?

2 Read the given statements. Choose the correct option.

Assertion (A): Shefali has 2000 marbles, out of which 1 4 are red and the rest are green. 2 5 of the red marbles are defective. The total number of non-defective red marbles is 300.

Reason (R): To add two unlike fractions, first the denominator is made the same by finding the equivalent fractions and then the numerators are added.

a Both A and R are true and R is the correct explanation of A.

b Both A and R are true but R is not the correct explanation of A.

c A is true but R is false.

d A is false but R is true.

Case Study

Value Development & Creativity

Sonia's Summer Diary

It is summer holidays for Sonia. Her mother signed her up for different social service classes. Sonia goes to a different class each day of the week. The table below shows how much time Sonia spends in each class. Read the table and answer the questions.

1 How much time does Sonia spend in Community Clean-up Programs in the entire week?

2 If Sonia wants to add another 1 1 2 hours for a gardening and farming class on Sunday, how many fewer hours will she spend in the class on Sunday than on Wednesday? a 1 2 4 hours b 3 4 hours c 1 4 hours d 3 3 4 hours

3 Sonia’s friend Misha goes to animal care and welfare classes but spends 45 minutes less time there than Sonia. How much time does Misha spend in the class?

4 Sonia learnt how to create art from recycled materials and made small pots for gardening. Look for some recyclable materials around you and use them to create something new.

7 Introduction to Decimals

Let’s Recall

Fractions are parts of a whole.

Rajesh has an apple pie which is divided into 10 equal parts.

He eats 1 part of it.

The fraction of pie eaten by him = 1 10

All the equal parts join together to form a whole.

1 10 + 1 10 + 1 10 + 1 10 + 1 10 + 1 10 + 1 10 + 1 10 + 1 10 + 1 10 = 1

If a whole is divided into 10 equal parts, then each part is equal to 1 10. 1 whole

Letʼs Warm-up

Fill in the blanks.

1 1 10 + 1 10 + 1 10 = .

If a whole is divided into 100 equal parts, then each part is equal to 1 100

2 Each equal part in a hundred grid is equal to .

3 When a whole is divided into 10 equal parts, each part is equal to .

4 1 5 + 1 5 + 1 5 + 1 5 + 1 5 = .

5 The fraction for 3 out of 10 = .

I scored out of 5.

Understanding Decimals

Alfred, the carpenter, goes to the market to purchase nails and screws.

The weight written on the packet of nails is 2.5 kg.

Alfred gets confused when he sees a weight which is not a whole number. Let us help him out by reading the number!

Reading and Writing Decimals

Tenths

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.

Reading and Writing Tenths

Decimal point

When 1 whole is divided into 10 equal parts, then each part is called one-tenth.

Fractional representation = 1 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

Let us see how to read 0.6. It is written as zero and six tenths.

Combining Whole Numbers and Tenths

Write the decimal for the shaded part.

Whole number part

Decimal point

Decimal part

Expanded Form

Decimals up to Tenths

Remember!

The decimal part after the decimal point is always less than 1 whole.

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.

Ones (value of 4 is 4) Tens (value of 1 is 10)

Expanded form: 10 + 4 + 6 10 Or 14.6 = 10 + 4 + 0.6

Expanded form of 45.7 = 40 + 5 + 0.7

Example 1: Write the decimal number of the coloured portion. Also, write the decimal number in words.

Thus, the coloured part represents 2.2 in decimal form. In words: Two point two or two and two tenths.

Example 2: Fill in the blanks.

Figures

6.2

11.6

Words

Six and two tenths

Eleven and six tenths 51.9

Fifty-one point nine

Seven hundred sixty point five

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 is written as fifteen and seven tenths.

15.7 = 10 + 5 + 7 10 Or 15.7 = 10 + 5 + 0.7

23.9 is written as twenty-three and nine tenths.

23.9 = 20 + 3 + 9 10 Or 23.9 = 20 + 3 + 0.9

Shade the figure to show the given decimal. Write it in expanded form.

3.4 = 3 + __________ or __________ +

Hundredths

When 1 is divided into 100 equal parts, then each part is equal to one-hundredth.

Fractional representation = 1 100

Decimal representation = 0.01

One-hundredth is the same as one-tenth further divided into 10 equal parts.

Let us see a few more hundredths.

Shaded part = 8 100

Shaded part = 35 100

Shaded part = 79 100

Reading and Writing Hundredths

Let us learn how to read decimal numbers up to hundredths. 15.54 fifteen fifty-four hundredths and fifteen point five four

Expanded Form of Decimals up to Hundredths

Let us note down the above number in the place value chart and write its expanded form.

1 5 . 5 4 Expanded form: 10 + 5 + 5 10 + 4 100 Or 15.54 = 10 + 5 + 0.5 + 0.04

Example 4: Fill in the blanks.

Figures

Words 3.14

Three and fourteen hundredths 14.36

Fourteen and thirty-six hundredths

Twelve point nine one 104.05 One hundred four point zero five

Example 5: Write 34.18 in expanded form.

Expanded form: 30 + 4 + 1 10 + 8 100 Or 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.

Thousandths

When 1 is divided into 1000 equal parts, then each part is equal to one-thousandth.

Fractional representation = 1 1000

Decimal representation = 0.001

One-thousandth is the same as one-hundredth further divided into 10 equal parts.

Reading and Writing Thousandths

Let us read and write decimal numbers in thousandths.

Write 14.179 in the place value chart.

Expanded Form of Decimals up to Thousandths

The expanded form of the above number using the place value chart can be given as:

Equivalent Decimals

Equivalent decimals are decimal numbers which have the same value. They are also called equal decimals. For example:

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.

For example: 0.5 and 0.05 are not equivalent decimals.

Remember!

When we place zeroes on the right-hand side of a decimal number, its value remains the same.

Example 6: Fill in the blanks.

Figures

14.814

16.005

514.047

5.407

Words

Fourteen and eight hundred fourteen thousandths

Sixteen and five thousandths

Five hundred fourteen point zero four seven

Five point four zero seven

Example 7: Write 76.103 in the expanded form.

76.103 = 70 + 6 + 1 10 + 3 1000 Or 76.103 = 70 + 6 + 0.1 + 0.003

Example 8: Write any three equivalent decimals for 14.3.

14.3 = 14.30 = 14.300 = 14.3000

Write the decimal for the expanded form.

Aryabhata, an astronomer and mathematician from ancient India, represented fractional values in a way that contributed to the development of the modern decimal system. Do It Together

1 7 tens + 5 ones + 9 hundredths + 2 thousandths =

2 600 + 20 + 2 + 7 1000 =

3 5 tens + 2 ones + 5 hundredths + 1 thousandths = 50 + 2 + 5 100 + 1 1000 =

Conversion Between Fractions and Decimals

Did You Know?

Now, let us learn how to convert fractions to decimals and vice versa.

Converting Fractions to Decimals

Let us convert 1 5 into a decimal number.

Step 1: Multiply the denominator and the numerator by the same number so that we get a power of 10 in the denominator.

Step 3: Insert a decimal point before the number of places equal to the number of zeroes in the denominator from the rightmost side. 1 5 = 1 × 2 5 ×

Converting Decimals to Fractions

Let us now convert 0.25 to a fraction.

Step 1: Write the decimal with a denominator of 1.

Step 2: Write a fraction with the denominator as 10, 100 or 1000.

So, 1 5 is the same as 0.2.

Step 2: Convert the denominator into multiples of 10, 100 or 1000 to eliminate the decimal point.

0.25 = 0.25 1 = 0.25 × 100 1 × 100 = 25 100 = 25 ÷ 25 100 ÷ 25 = 1 4

Step 3: Express the fraction in its simplest form.

Example 9: Convert the given fractions into decimals.

Think and Tell

Do we convert all the fractions into fractions with denominator of 10, 100, 1000 to convert them into decimals?

Example 10: Convert the decimals into fractions.

1 Convert the given fractions into decimals.

2 Convert the decimals into fractions. a 14.50 b 1.75 14.50 = 14.5 = 145 10 = 1.75 = =

Percentages

Let us look at the given grid. How will we write it as a fraction or a decimal?

40 out of 100 squares are shaded.

Fraction of the shaded part = 40 100

Decimal of the shaded part = 0.40

40 100 can also be written as 40 percent since percent is per hundred.

Percentage is represented by ‘%’. So, 40 percent is also written as 40%.

Converting Fractions to Percentages

Fraction Multiply by 100% Percentage

For example:

100 = 53 100 × 100% = 53%

4 5 = 4 5 × 100% = 80%

Converting Decimals to Percentages

Multiply by 100 and shift the decimal point two places to the right.

For example:

0.45 = 0.45 × 100% = 45%

3.25 = 3.25 × 100% = 325%

Converting Percentages to Fractions

Percentage Divide by 100 Fraction

For example:

47% = 47 100

48% = 48 100 = 12 25

Converting Percentages to Decimals

Divide by 100 and shift the decimal point two places to the left.

For example:

69.4% = 69.4 ÷ 100 = 0.694 84% = 84 ÷ 100 = 0.84

Example 11: Convert the given percentage to a fraction and a decimal.

1 18%

As a fraction: 18% = 18 100 = 18 ÷ 2 100 ÷ 2 = 9 50

As a decimal: 18% = 18 ÷ 100 = 0.18

Example 12: Do as directed.

1 Convert 5 into a percentage.

5 = 5 × 100% = 500%

2 Convert 12.306 into a percentage.

12.306 = 12.306 × 100% = 1230.6%

As a decimal: 12.3% = Do It Together

Express 12.3% as a fraction and a decimal.

As a fraction: 12.3% = 12.3 100 =

2 25%

As a fraction: 25% = 100 100 = 1 4

As a decimal: 25% = 25 100 = 0.25

Did You Know?

Percent is derived from a Latin word ‘per centum’ which means ‘per hundred’.

Write the decimals in words. a

Write the decimal numbers.

a Four hundred thirteen and two tenths b Seventy-two point three three

c Three hundred sixty-one point zero four d Fifty-two and four hundred two thousandths

Shade to show the decimals and then write them in words.

Write the expanded form of the given decimal numbers.

Write in decimal form.

Convert the fractions into decimal numbers and percentages.

Rewrite the decimals as fractions and percentages.

Write any two equivalent decimals for the given decimal numbers.

Write as fractions, decimals and percentages.

a Two and one tenth b Thirty-five hundredths

c Five hundred twenty-four thousandths d Three and one hundred twenty thousandths

Insert >, < or = in the blanks.

In 2021, Shivraj Dhananjay Thorat registered his name in the Asia Book of Records by covering a distance of 455.89 km in a day. Express the distance covered in expanded form.

Rakesh a contractor got a contract to build a ramp of a house. He completed 5 8 of the work in 7 days. What is the decimal form for the work completed?

The tiger is the national animal of India. There are approximately 3600 tigers in India whereas there are about 6000 tigers in the world. What percentage of the world’s tiger population is there in India?

Use the digits 9, 0, 8, 9, 7 and write two numbers between 0.9975 and 0.9998.

Types of Decimals

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 and Unlike Decimals

Let us learn about the types of decimals. Like Decimals

1.32 and 1.26 have 2 digits after the decimal point. 1.2 and 1.4 have 1 digit after the decimal point.

Same number of digits after the decimal point.

Unlike Decimals

Decimal numbers which have the same number of digits after the decimal point are called like decimals.

Thus, 1.32 and 1.26 are like 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.

Do It Together

Example 13: 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

Example 14: Which option shows unlike decimals?

a 14.2, 54.8, 23.6 b 71.25, 89.35, 41.51 c 13.147, 879.78

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.

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.

Converting Unlike Decimals to Like Decimals

Let us learn how to convert unlike to like decimals.

Consider 1.3 and 1.46.

Same number of digits after the decimal point Add 1 zero

Unlike decimals can be converted into like decimals by finding the equivalent decimal. Put zeroes at the end of the decimal number such that the number of digits after the decimal point is the same.

Example 15: Convert 14.5, 74.543 and 15.64 into a group of like decimals.

Convert each of the decimal numbers into equivalent decimals with 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 group of like decimals.

Convert 189.474 and 1.2 into like decimals.

189.474 = 189.474 1.2 =

Thus, and are a group of like decimals.

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 () 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

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

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

Convert the fractions into like decimals.

a 12 20 , 1 5 b 14 40 , 5 10 c 5 8 , 3 4 d 4 5 , 5 20

The value of different currencies in terms of Indian rupees can be given as:

Japanese Yen = ₹0.54; Canadian Dollar = ₹60.9; US Dollar = ₹83.394. Convert all decimals into like decimals.

Create a word problem to convert two unlike decimals to like decimals.

Challenge

d All the statements are correct. 1

Critical Thinking

Consider the fractions 1 2 , 8 25 , 7 8 , 29 40 . Which of the following options is correct for the given fractions?

Statement 1: 8 25 = 0.320 and 29 40 = 0.725

Statement 2: 1 2 and 7 8 when converted to decimal numbers are like decimals.

Statement 3: The descending order for 1 2 , 8 25 , 7 8 , 29 40 arranged as like decimals is 0.875 > 0.725 > 0.500 > 0.320

a Only statement 1 is correct.

b Both statements 1 and 2 are correct.

c Both statements 1 and 3 are correct.

Comparing, Ordering and Rounding-off Decimals

Bhawna volunteered at her school’s sport event, where a long jump competition took place.

She diligently recorded the jumps of the participants.

Suhani jumped 3.2 m, Sobhita jumped 3.15 m, Kiran jumped 3.18 m and Mridula jumped 3.7 m.

Bhawna wanted to compare the lengths of the jumps of the participants. Let us see how we could do this!.

Comparing Like Decimals

Let us compare the jumps of Suhani and Mridula.

The coloured part in 3.2 is less than the coloured part in 3.7

Since 3.2 < 3.7, Mridula jumped longer than Suhani.

Let us learn a quicker way to compare decimal numbers.

Compare 3.15 and 3.18.

Compare the whole numbers. The number with a larger whole number will be bigger. If the whole numbers are the same, compare the tenths. If the tenths are the same, compare the hundredths.

Follow the same process until you find unequal digits.

Example 16: Which of the given numbers is smaller—16.47 or 16.42?

Look at the digit in the hundredths place: 7 > 2.

So, 16.42 < 16.47. 16.42 is smaller.

Which is smaller—174.54 or 174.05?

Comparing Unlike Decimals

Remember Bhawna? She now wants to compare the jumps of all the participants and announce the winner.

Let us see how she finds the winner!

Let us compare 3.2 m, 3.15 m, 3.18 m and 3.7 m.

Error Alert!

If the number of digits in a number is more, then the number is not necessarily bigger.

< 23.65

To compare unlike decimals, make all the decimals like decimals. Then, compare the like decimals.

3.70 is the biggest number. Thus, Mridula jumped the farthest and is the winner.

Example 17: Which of the following decimals is bigger—48.25 or 48.251?

Look at the thousandths digit: 0 < 1.

So, 48.251 is bigger.

Which is greater—145.14 or 145.4?

is greater.

Ordering Decimals

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 decimals into like decimals.

3.2

Compare the decimals and arrange them.

Thus, 3.70 > 3.20 > 3.18 > 3.15

Example 18: Arrange 12.14, 12.4, 12.04, 12.41 in ascending order.

Make the decimals like:

So, 12.04 < 12.14 < 12.40 < 12.41.

Arrange 15.47, 15.4, 15.744 and 15.3 in descending order. Make the decimals like:

Descending order: .

Rounding off Decimals to the Nearest Whole Number

In a 100 m race, Rupali completed the race in 20.3 seconds and was the winner. Rupali’s coach wanted to know approximately how many seconds it took her to complete the race.

Mark the decimal number on the number line.

Check which whole number is closer to the decimal number.

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 now understand the other way for rounding off decimal numbers to the nearest whole number.

Check the digit in the tenths place. The digit in the tenths place = 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 is rounded off to 20.

Example 19: 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 to 15.

Write the decimal numbers on the number line. 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. Therefore, 263.5 rounds to .

Write if True(T) or False(F).

a If the digit in the tenths place is 5, then the number rounds up to the nearest whole number.

b 12.2 rounds up to 13 when rounded to the nearest whole number.

c 99.9 rounds up to 100 when rounded to the nearest whole number.

Find the greater decimal number. a 4.2 or 4.15 b 15.64 or 15.67 c 87.654

Round off the numbers to the nearest whole number.

a 2.6 b 7.9 c 48.2 d 15.1

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

b 84.56, 84.5, 84.6, 84.55, 84.65

d 64.23, 54, 64.32, 64.22, 64.33

Round off the numbers to the nearest whole number first and put the correct symbol <, >, =.

In May 2023, Bangalore Urban recorded 205.8 mm of rainfall. What is 205.8 rounded off to the nearest whole number?

In a racing competition, three athletes finish the race in 24.54 minutes, 24.68 minutes and 24.36 minutes, respectively. Arrange the numbers in descending order.

The population (in crores) of different states in India (in 2011) is given. Arrange the states in ascending order according to their population.

Challenge

Decide whether the given statement always answers the given question. Statement: A decimal number a is greater than 14.5. Is a equal to 15 when rounded to the nearest whole number?

Points to Remember

Critical Thinking

• Decimals are numbers between whole numbers.

• The decimal part of a number is always less than the 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; otherwise, they are called unlike decimals.

Collaboration & Experiential Learning

Cross Out Decimals

Aim: To understand and practise recognising with decimal numbers.

Setting: Whole class

Materials Required: Small sheets of paper with decimal numbers written on them, a pencil,

Method:

Prepare a sheet with a grid of decimal numbers from 1.1 to 6.6, spaced out evenly. Each decimal number should appear only once on the sheet.

Each student takes a turn to roll the dice twice.

The first roll gives the whole number part of the decimal (e.g., rolling a 3 means the whole number is 3).

The second roll gives the decimal part (e.g., rolling a 5 means the decimal part is 0.5).

Combine these rolls to form the decimal number (e.g., 3 and 7 become 3.7).

Students then cross out this number on their sheet.

If a student rolls a number that has already been crossed out, they roll again. The game continues until all the decimal numbers on the sheet have been crossed out by any one student.

Chapter Checkup

Shade the grid and write the expanded form of the decimals.

Write the decimal number in words.

Write as decimal numbers.

a Forty point one zero one b One hundred three and five hundredths

c Four hundred twenty and eleven thousandths d

and five thousandths

Place the numbers in the place value chart and write the expanded form of the decimal numbers.

Convert the fractions into decimal numbers and percentages.

a 1 8 b 4 5 c 12 20 d 36 40

Identify the number by which the decimal has to be multiplied and then convert it into a fraction and percentage.

a 12.6 b 52.2 c 20.8 d 25.315

Circle the smaller decimal number and put a square on the greater decimal number.

a 12.3 and 12.4 b 14.5 and 14.55

c 222.22 and 222.02 d 3.003 and 3.033

Round off the decimal numbers to the nearest whole number.

a 1.3 b 99.6 c 100.2 d 999.9

Arrange the numbers in descending order.

a 1.3, 1.31, 1.2, 1.33 b 19.4, 19.44, 19.54, 19.501

c 555.5, 555.55, 555.05, 555.555 d 748.01, 748.101, 748.11, 748.1

Show two like decimals using square grids.

Regular cycling strengthens leg muscles, improves joint mobility, and enhances overall body flexibility. Mr Raj and his family took part in a bicycle race. The time taken by each of them is given below. Who won the race?

Raj: 14.3 minutes Rekha: 15.2 minutes Utkarsh: 13.92 minutes Ali: 13.99 minutes

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.

Challenge

Critical Thinking

1 Look at the numbers shown below. Which number does not follow the pattern?

0.9, 0.91, 0.93, 0.97, 1 and 1.05

2 Read the statements and identify which of the following options is correct.

Assertion (A): The decimal number 0.78 is less than the decimal number 0.7801.

Reason (R): When comparing two decimal numbers, the number with more digits to the right of the decimal point is always larger.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, and R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

e Both A and R are false.

Case Study

Vertical Marvels

John got a school project in which he had to find the heights of multiple skyscrapers of the world. John made a list of 6 skyscrapers. Based on the list of the height of the skyscrapers, answer the questions given below.

1 Which of the following skyscrapers is the shortest?

a Merdeka 118

c Jin Mao Tower

b CITIC Tower

d Wuhan Greenland Centre

2 Which of the following skyscrapers is the tallest?

a Shanghai Tower

c Lotte World Tower

b CITIC Tower

d Merdeka 118

3 What is the ascending order of the heights of the given skyscrapers?

4 What are the heights of the towers when rounded to the nearest whole number?

5 Write the expanded form of the heights of the skyscrapers.

8 Operations with Decimals

Let’s Recall

A decimal number is a number which consists of a whole number part and a decimal part. The decimal part is always less than 1.

Decimal point

Whole number part Decimal part

Tarun and his friends measured their heights and wrote them in decimal form.

Tarun is 1.47 m tall, Ajay is 1.06 m tall, Rahul is 1.34 m tall and Raj is 1.2 m tall.

The increasing order of heights of the 4 friends is 1.06 m < 1.2 m < 1.34 m < 1.47 m.

Thus, Tarun is the tallest, followed by Rahul, Raj and Ajay.

Tarun is 1.47 m tall, Ajay is 1.06 m tall, Rahul is 1.34 m tall and Raj is 1.2 m tall.

The increasing order of heights of the 4 friends is 1.06 m < 1.2 m < 1.34 m < 1.47 m.

Letʼs Warm-up

Fill in the blanks using >, < or =.

I scored out of 5.

Ajay Rahul Tarun Raj

Addition and Subtraction of Decimals

Ana and her father went to order sweets for a wedding.

Ana: Father, how many sweets do we want to give to each guest?

Father: Ana, we will give 1.2 kg of laddoos and 1.5 kg of barfi to each of our guests.

Let us find the total weight of sweets that they are giving to each guest.

Addition of Decimals

To find the total weight of sweets, add 1.2 and 1.5.

Step 1

Represent 1.2 and 1.5 visually.

Step 2

Add the wholes with the wholes and tenths with the tenths. 1 whole + 1 whole 2 tenths + 5 tenths = 7 tenths

Thus, the total weight of sweets that each guest gets 2.7 kg.

Let us learn another way to add decimals.

Write the digits in columns and align the decimal points.

Add the numbers. Use regrouping, if required.

Put the decimal point in the answer at the same place as it is in the numbers above it.

Let us add some more decimal numbers.

Arun Sweet Shop

Example 1: Find the sum of the given numbers using the column method.

1 0.5 + 0.3 2 0.58 + 0.42

0.5 + 0.3 = 0.8

+ 0.42 = 1.00

Example 2: 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.

Remember!

Decimal numbers with the same number of decimal places are called like decimals.

Add the numbers visually and also show the column method.

1 0.8 + 0.6 0.8 + 0.6 = 2 0.14 + 0.93

+ 0.93 =

Subtraction of

Decimals

Let us see how can we subtract two decimal numbers using the column method.

Subtract: 17.2 – 16.84

Write the digits in columns 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 it has been put in the numbers above it.

Let us see how to subtract decimals using a grid.

Example 3: Find the difference of the numbers visually.

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

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.

Subtract the numbers visually and also show the column method, in your notebooks.

1 1.4 – 0.3 1.4 – 0.3 =

Add the tenths and hundredths visually and find the sum.

+ 0.5

Subtract visually and find the difference.

a 1 – 0.3 b 1.23 – 0.46

Add the numbers using the column method.

Subtract using the column method.

Compare using >, < or =.

Ramesh is a milkman. He has two cows. One is a Gir cow and the other is a Sahiwal cow. The Gir cow yields 13.4 litres of milk in a day while the Sahiwal cow yields 8.55 litres of milk in a day. How much more milk does the Gir cow yield than the Sahiwal cow?

Raman is 1.65 m tall and his brother is 0.18 m shorter than him. What is his brother’s height?

The weight of a frog is around 3.3 kg while the weight of a tadpole is 0.005 kg. What is the total weight of the frog and the tadpole?

In a relay race, Rashi covers 1.548 km, Prerna covers 2.328 km and Navya covers 1.986 km. What is the total distance (in km) covered by the three girls?

Read the table showing the weight of 4 students. Answer the questions below.

a What is the total weight of Rakesh and Ismail?

b What is the total weight of John and Shahid?

Priya travelled from point A to point B via bus and from point B to point C via auto. How much distance does she need to cover if she walks from point C to point D?

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 buy a packet of biscuits to feed a hungry dog. What was the total amount of money she had left?

Rahul was adding two numbers whose sum was 10. The first number was greater than 4 while the second number was greater than 5. Both the numbers have only one decimal place. How many combinations such as these could Rahul have added?

Multiplication and Division of Decimals

Ana’s father wanted to give sweets to 10 of his staff members, with each box weighing 0.5 kg.

Multiplying Decimals

Ana wanted to know the total weight of sweets given to the staff members.

How will they find the total weight? Let us learn!

Multiplying Decimals by 10, 100, 1000…

Weight of each box = 0.5 kg

Total number of boxes = 10

Total weight of 10 boxes = 10 × 0.5 = 5 kg

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 0 . 6 1 4

Multiplication by 100

When we multiply any decimal number by 100, we move the decimal point 2 places to the right.

100 × 0.614 = 61.4 0 . 6 1 4

Multiplication by 1000

When we multiply any decimal number by 1000, we move the decimal point 3 places to the right.

1000 × 0.614 = 614 0 . 6 1 4

Thus, we move as many decimal places to the right as there are 0s in the multiplier (10, 100 or 1000).

Remember!

614 is never written as 614.0 as we do not put a decimal point at the end of the number that has no decimal value.

Example 5: Solve.

Error Alert!

Adding 0s at the end of the decimal number does not change the value. Move the decimal point to the right while multiplying by 10, 100 or 1000. 12.54 × 100 = 12.5400

× 100 = 1254

1 4.52 × 10 2 23.48 × 100 3 1.413 × 1000 4 32.4 × 1000

Move the decimal point 1 place to the right.

4.52 × 10 = 45.2

Move the decimal point 2 places to the right. 23.48 × 100 = 2348

Move the decimal point 3 places to the right. 1.413 × 1000 = 1413

Move the decimal point 3 places to the right. 32.4 × 1000 = 32400

Fill in the blanks. 1 1.415 × 10 = 2 2.547 × = 254.7 3 1.01 × 1000 = 4 256.47 × = 25647

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 = 1.6

Steps to multiply 4 by 0.4 using the column method.

1. Ignore the decimal point in the decimal number and multiply the whole numbers.

4 × 4 = 16

2. Count the total number of digits after the decimal point in both the numbers. There is 1 digit after the decimal point for this question.

3. Place the decimal point in the product so as to obtain as many decimal places as there are in the decimal number. So, 4 x 0.4 = 1.6

Example 6: Multiply 0.6 and 3 visually.

Thus, 3 × 0.6 = 1 + 0.8 = 1.8

Example 7: Multiply using the column method. 506 × 1.06

Did You Know?

Bhaskara II’s work "Lilavati" (1150 CE) includes operations involving decimal fractions. He provided methods for calculations that made use of the decimal point to represent fractions more accurately.

Put the decimal point in the product so that it has 2 decimal places. 506 × 1.06 = 536.36

Multiply 0.7 and 4 visually and then show column multiplication.

0.7 × 4 =

Multiplying Two Decimal Numbers

We can multiply a decimal number with another decimal number as well. Let us multiply 1.5 by 2.6.

Steps to multiply 1.5 by 2.6 using the column method.

1. Ignore the decimal point in the decimal numbers and multiply them as whole numbers.

15 × 26 = 390

2. Count the total number of digits after the decimal point in both the numbers. Sum of the decimal places in the given decimal numbers = 1 + 1 = 2

3. Put the decimal point in the product so that it has 2 decimal places.

So, 1.5 × 2.6 = 3.90

Example 8: Multiply.

1 15.46 × 36.1

1 . 5 1 decimal place

1 decimal place 3 . 9 0 (1 + 1) = 2 decimal places

Think and Tell

15.46 × 36.1 = 558.106

Example 9: 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.

What is the product of the smallest 4-digit number and 45.623? 2 + 1 = 3 decimal places. 1 + 1 = 2 decimal places.

What is the product of 23.5 and 45.12?

Thus, 23.5 × 45.12 =

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.

Consider the chart which shows how many Indian rupees we can get when we exchange the money of different countries.

*rates tabulated are not specific and subject to change

How many Indian Rupees are there in 5 Nepalese Rupees? (1 NPR = ₹0.62)

1 NPR = ₹0.62

5 NPR = 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

Thus, $6 = ₹496.02

Example 11: Rihan’s father came back from South Korea. He wanted to exchange the South Korean currency (Won) at the airport. He had 15,51,000 Won with him. How much money will he get in Indian currency? (1 Won = ₹0.062)

Amount of money with Rihan’s father in Won = ₩15,51,000

We know that, 1₩ = ₹0.062

So, ₩15,51,000 = 15,51,000 × ₹0.062 = ₹96,162

Thus, Rihan’s father will get ₹96,162 in exchange for ₩15,51,000.

Read the exchange rate table given above. Complete the table.

Rupees (₹)

Do It Yourself 8B

Multiply the decimal number by the whole number visually.

0.2 × 4 b 0.4 × 5

Multiply.

Compare using >, < or =.

Fill in the blanks.

How many Indian rupees are the same as 1000 Euros? (1 Euro = ₹89.19)

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?

There are 24 workers at a factory. 11 workers are given ₹252.54 each and the rest of them are given ₹364.52 each. How much money is given to the workers in total?

Suresh’s father is working in U.A.E. He gets 800 dirham as a salary. Saurav’s father who is working in Sri Lanka gets 2200 Sri Lankan rupees. Who gets more Indian rupees as salary and by how much? (1 dirham = ₹22.51 and 1 LKR = ₹0.25)

New dustbins were to be purchased for Sanchita's school. The school ordered 96 dustbins and each dustbin costs ₹123.65. What was the total cost of the dustbins? Why do we need dustbins at home, in schools, and in public places?

Anjali is fond of collecting different currency notes. She has 2 Pounds, 5 Rand, 6 US Dollars, 5 Euros 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)

Challenge

Rashmi and Sneha are solving a maths problem. Each of them thinks of a number between 1 to 5 and writes it on a piece of paper. The division of their number was 1.25. They forgot to mention who thought of which number. Rashmi said she thought of a whole number while Sneha mentioned she thought of a number which had two decimal places. Which numbers did they both think of? 1 Critical Thinking

Dividing Decimals

If Ana’s father bought 92.5 kg of fruit. He wants to make 10 baskets of fruit. How much fruit will be there in each basket?

Let us learn how to find the answer!

Dividing Decimals by 10, 100, 1000, …

Total fruit bought = 92.5 kg, Total baskets = 10

Fruit in each basket = 92.5 ÷ 10 = 9.25 kg

Let us now divide 987.1 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.

987.1 ÷ 10 = 98.71

9 8 7 . 1

Division by 100

When we divide any decimal number by 100, move the decimal point 2 places to the left.

987.1 ÷ 100 = 9.871

9 8 7 . 1

Division by 1000

When we divide any decimal number by 1000, move the decimal point 3 places to the left.

987.1 ÷ 1000 = 0.9871

9 8 7 . 1

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 12: Divide.

1 453.2 ÷ 10 = 45.32

2 471.12 ÷ 1000 = 0.47112

3 53 ÷ 100 = 0.53

4 65.001 ÷ 1000 = 0.065001

Find the quotient.

1 15 ÷ 10 2 346.4 ÷

15 ÷ 10 = 1.5

Dividing Decimals by Whole Numbers

Divide 0.6 by 3.

Adding a 0 before any number does not change its value.

12.3 ÷

Step 1: Place the decimal point directly above the decimal point in the dividend.

Step 2: Divide as we divide whole numbers. Divide the ones.

6 tenths are put in 3 equal parts which is equal to 2 tenths in each part.

0.6 � 3 = 0.2

Steps

Step 3: Divide the tenths.

Step 4: Divide the hundredths. When

When there is a remainder while dividing (4.5 ÷ 2)

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.

Example 13: Find the quotient.

Thus, 5.62 ÷ 4 = 1.405

Thus, 5.25 ÷ 5 =

Yourself

Divide the given decimal numbers visually.

the blanks.

Find the quotient.

Compare using >, < or =.

How many pounds should Hari take with him to England, if the total expenses for the trip add up to ₹1,04,200? (Given: 1£ = ₹104.2 approximately)

Rajat works at a grocery store. He received nine cartons of items at his store. The total weight of 9 cartons is 37.8 kg. Find the weight of one carton.

Sakshi helps her mother in saving money by managing the grocery expenses of her house. She sees an offer on the internet where six 1-litre bottles of oil cost $37.92. Should she buy it if the rate of the same oil bottle is $6.5 per bottle in the market?

Create a word problem for division of a decimal number by a whole number.

Challenge

1

Critical Thinking

Mr Agarwal went from England to Germany. He had 1000 Pounds with him. How many Euros will he get in exchange for 1000 pounds? (Refer to the table for conversion of currencies)

Points to Remember

• To add or subtract decimal numbers, always remember to convert them into like decimals first and then 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.

Operations on Decimals

Objective: To reinforce the understanding and application of decimal operations

Setting: In groups of 4.

Materials Required: Printed maze worksheets with each intersection or decision point in the maze having a decimal operation problem (addition, subtraction, multiplication, or division), pencils

Method:

1 Distribute the maze worksheet to each group.

2 Ask the students to start at the “Entrance” of the maze.

3 One student from each group solves one decimal problem in their notebooks, which leads to next problem.

4 The group which performs the correct decimal operations first, and reaches the exit of the maze, wins.

Chapter Checkup

Add or subtract the numbers visually. a 0.23 + 0.47

Find the product of the numbers visually.

a 6 × 0.2

Add or subtract the numbers using the column method.

0.6

Fill in the blanks.

Find the product of the numbers.

a 12 × 1.54 b 18 × 3.251 c 9 × 32.14 d 1.2 × 21.36 e 41.5 × 12.45

120.2 × 12.06

Divide to find the quotient.

a 110.4 ÷ 6

Simplify.

a 12.145 + 18.415 + 2.51 b 41.54 + 56.81 – 12.379

15.47 + 81.415 – 41.555 d 31.23 + 17.28 – 11.111

Match the following.

a 1.51 + 23.72

c 4 × 6.17

d 3.26 × 7.02

Divide to find the quotient.

Read the exchange rate chart of different countries given in the chapter. Answer the questions.

a What is 6000 Won in Indian Rupees? b What is €150 in ₹?

c Convert 850 Rand into Indian Rupees. d What is 100.5 Dirham in ₹?

Jason bought an item for ₹45.8. He gave the shopkeeper ₹50. How much money will he get back from the shopkeeper?

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?

George’s friend returned from the USA. He exchanged $168 for Indian currency. If 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)

Rohan owns a carpool agency that takes people from Meerut to Gurgaon everyday. If the cars run for 4 hours each day and consume 6.5 L of petrol each hour, for how many days will the cars run with 130 litres of petrol?

decimal numbers.

1 John’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?

2 Read the statements and choose the correct option.

Assertion (A): When multiplying two decimal numbers, the product is always less than both of the original numbers.

Reason (R): Multiplying any two numbers between 0 and 1 results in a number that is smaller than each of the original numbers.

Options:

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

e Both A and R are false.

Case Study

Cross Curricular & Value Development

Patterns and Impacts: Analysing Rainfall in Indian Cities – 2023

In 2023, India experienced diverse weather patterns across its various cities. The government meteorological department recorded the rainfall in several major cities to analyse the effects of these patterns. The data is essential for urban planning, agricultural forecasts, and water resource management.

Read the questions and answer.

1 What is the total rainfall in Mumbai, Delhi and Bangalore? a 3981.4 mm b 3871.4 mm c 3791.4 mm d 3971.4 mm

2 Find the difference in rainfall between Chennai and Kolkata. a 256.6 mm

3 If the rainfall in Pune doubles next year, how much rainfall is expected in the next year?

4 Find the difference between the rainfall in Mumbai and the total rainfall in Delhi, Bangalore and Chennai combined.

5 Find the average rainfall of the 6 cities? Hint: Average = Total Rainfall Total number of cities

6 Why is it important for us to save rain water?

9 Lines and Angles

Letʼs Recall

Lines can be seen everywhere! Lines can be sleeping, standing or slanting.

Sleeping Line

For example,

Standing Line Slanting Line

We see standing and sleeping lines in this book.

We see a slanting line in this pen.

Letʼs Warm-up

Look at the pictures. Write the type of lines (sleeping, standing, slanting) these things show.

Lines and Line Segments

Vandana is going to New Delhi with her father. They are travelling by train. She is sitting next to the window.

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 places.

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 and the different elements in geometry.

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.

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.

A line segment is named by its end points. The line segment shown on the ruler can be named as AB or BA. The length of the line segment is 6 cm.

Line

A line is a collection of points along a straight path that may extend endlessly in both directions. For example, railway tracks.

A line has no beginning and no end point.

A line can be represented by a two-headed arrow showing that it can be extended in both directions.

P

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 and are equidistant from each other.

Parallel Lines

AB and CD are parallel. ABCD and AC = BD.

Arrow heads at both ends.

A vertical line standing exactly above a horizontal line.

Intersecting Lines

AB and CD are a set of intersecting lines.

Point X is the point of intersection.

Perpendicular Lines

AB and CD are perpendicular lines.

The symbol of a perpendicular line is ⊥.

Two lines that cross each other.

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.

AB is a ray with A as the starting point and B as a point on the ray.

Example 1: Name each given figure as a ray or line segment.

A pair of scissors A ladder Headlight of a car

The beam of light coming from the headlight of the car represent a ray. The pair of scissors and the ladder represent line segments.

Example 2: Look at the figure.

a Join dots to form a line segment and name it.

Join dots M and N to make a line segment MN.

b Draw a dot and show parallel lines.

Draw a dot at Q and join dots L and Q. LQ is parallel to MN.

Look at the figure and answer the questions.

1 Number of points outside the square =

2 Name 2 line segments = WX, XY

3 Length of line segment ZY is

4 The line parallel to WZ is

5 Draw a line passing through point E.

6 Draw a ray from point B.

Match the given figures with their names.

a b c d

Line Intersecting lines Parallel lines Ray

Write T for true and F for 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 Curve Ray Parallel lines Point A bridge Sides of a railway track Tip of a nail Sunlight

Read and draw.

A point

e Two intersecting lines UV and WX with O as the point of intersection

Look at the figure and answer the given questions.

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.

Disha visited the National Museum in Delhi where a collection of Indian weapons are on display. She saw a pair of swords hanging on the wall, like the ones shown in this picture. What kind of lines does the pair show?

True or False: Only one line segment can be drawn between two given points. Justify the statement using a figure.

Rishi and Rani are playing a game of hitting the targets. They are standing next to each other with some distance between them.

a Rishi and Rani hit the same target at the same time. Draw the type of line that will be formed.

b Rishi and Rani are standing next to each other, facing the targets. They have 2 different targets in front of them. They hit the targets at the same time. Draw the type of lines formed. 1 2

Understanding Angles

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 saw the hands of the clock make.

Types of Angles

How can the same hands make different shapes? This happens because of 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 its parts making 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.

An angle is named using the symbol ∠. The angle in the picture can be named as ∠AOB or ∠BOA or ∠O.

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 are called right angles. Let us look at some right angles.

Straight Angle

The angle formed on a straight line is called a straight angle.

A O B

Acute Angle

Angles that measure less than a right angle are called acute angles. Given below are some acute angles.

Obtuse Angle

Angles that are more than a right angle are called obtuse angles.

Example 3: Look at the pictures. Write A for acute angle, O for obtuse angle, S for straight angle, and R for right angle. 1 2 3

1 Draw the hands of the clock to show the angles.

Label the arms and vertex of the given angle. Name the angles of the given figures.

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? Do you reach the school on time? The snowflakes formed by water molecules are in the shape of a hexagon. Look at the figure and identify the types of angles in the figure.

Look at the given figure and answer the questions.

a What type of angle do you get if you remove sticks OC, OD, OF, OG and OA?

b If the stick OA is removed, what type of angle is formed by sticks OB and OG?

c Count the number of angles formed between OG and OE. Name them.

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; 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

Each line on the protractor measures 1°.

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.

2 Adjust the protractor (without shifting the centre from the vertex) so that one arm of the angle is along the baseline.

3 Look at the scale where the baseline arm points to 0° (inner scale in this case).

4 Read the measurement of this angle where the other arm crosses the scale.

The measure of ∠ AOB = 50°

Example 4: Measure ∠ LMN using a protractor.

The measure of ∠ LMN = 60°

Read the protractor and write the correct measure for the given angles.

Remember!

The length of the arms of an angle does not affect the measure of the angle.

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.

Using the inner scale of the protractor Using the outer scale of the protractor

Step 1

Draw a ray of any name such that it points to 0° of the inner scale. Here we are drawing a ray with the name OC.

Step 1

Draw a ray of any name such that it points to 0° of the outer scale. Here, we are drawing a ray with the name OC.

Step 2

Place the centre of the protractor on the vertex O and 0° on the arm OC. Mark a point A at 80°.

Step 2

Place the centre of the protractor on the vertex O and 0° on the arm OC. Mark a point A at 80°.

Step 3

Remove the protractor and draw a ray from O to A using a ruler. Thus, ∠AOC = 80°.

Step 3

Remove the protractor and draw a ray from O to A using a ruler. Thus, ∠AOC = 80°.

Example 5: Draw a 105° angle using the inner scale of a protractor.

Draw the given angles using the protractor. 1 30° using the outer scale 2 150° using the inner scale

Do It Yourself 9C

Name the angles as right, acute, obtuse or straight.

Measure each angle using a protractor.

Use a protractor to measure the angles.

Draw the given angles using the inner scale of a protractor.

According to the playground safety rules, an angle of 30 degrees is ideal for a playground slide. Draw a playground slide with the given angle.

Look at the figure. Identify and estimate the angle formed when the striker hits the coin through the centre.

Coin

Striker

2-D Shapes

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.

Kavya’s shape Rishi’s shape

All of them wondered how the same hints led to the same shape with different measures.

Triangles, Quadrilaterals and Polygons

Kavya and Rishi made the shapes of different measures 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 symbol ∆ is used to show the triangle: ∆ABC.

A quadrilateral has 4 sides and 4 angles.

A quadrilateral is named using the names of its vertices.

Quadrilaterals

The sides can be named as AB, BC and CA.

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.

Let us see some quadrilaterals. All the quadrilaterals have 4 sides, 4 angles and 4 vertices.

Polygons: A polygon is a closed figure that has 3 or more straight sides.

Polygons can be categorised on the basis of the number of sides and angles.

Think and Tell

Are triangles and quadrilaterals, polygons too?

Given below are some polygons along with their names, number of sides and number of angles.

Example 6: Which of the following is not a polygon?

We know that polygons are closed figures with three or more straight sides, hence figure 3 is not a polygon.

Write the number of sides and angles for each of the shapes.

Number of sides - 8 Number of angles -

of sidesNumber of angles - 4

of sidesNumber of anglesNumber of sides - 4

Write the names of each of the polygons.

Match the shapes with the number of angles they have.

three five nine seven

Write T for True and F for False.

a A quadrilateral has 4 sides.

c A pentagon has five sides.

b An octagon has eight angles.

d A rhombus has five sides.

The Tower of Winds in Athens was constructed in 50 BCE. The shape of the building has 8 sides and 8 angles. Name and draw the shape.

Answer the riddle: I am a closed shape. I have twice the number of sides and angles than a triangle has. Who am I?

Draw and name the sides in a triangles with vertices labelled as P, Q and R.

Challenge

Count the number of triangles and quadrilaterals in the given figure. 1

Points to Remember

Critical Thinking

• 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 extends 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.

Setting: In pairs

Let’s Make A Humanoid!

Materials Required: Chart paper, coloured paper, scissors, pencil, eraser.

Method:

Draw a humanoid using different shapes as shown in the figure.

Take the coloured paper, cut the shapes required for the body, hands, legs, face and the hat using 5 different 2-D shapes.

Stick the cut shapes on the figure drawn.

Measure the angle formed between the hands and the body.

Measure the angle formed by the shirt and skirt.

Discuss the type of angles that you observe in the figure.

Chapter Checkup

Which word among line, line segment and ray best describes the figure?

Look at the figure and write if true or false.

a AC is a line segment.

b PQ and RS are sets of parallel lines.

c CS is a line.

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.

Write the measure.

Measure the given angles using a protractor.

Draw the given angles using the outer scale of a protractor. a 65° b 115° c 140° d 155°

Write the names of the polygons. Also, write the number of sides and angles they have.

Tick () the polygons that have one or more right angles.

Draw a line segment MN equal to 10 cm. At N, draw ∠MNO = 125° using the inner scale.

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? Do you follow traffic rules?

The Museum of Deva has displayed an over-2000-year-old bronze matrix of Sarmizeugetusa, which was found in 2013. Name and draw the shape of the matrix shown.

1 Read the statements and choose the correct option.

Assertion: If two lines are perpendicular, then they intersect at a right angle.

Reason: Perpendicular lines are lines that intersect at a 90° angle, forming four right angles at their intersection point.

a Both assertion and reason are true.

b Both assertion and reason are false.

c Assertion is true but reason is false.

d Assertion is false and reason is true.

2 Look at the figure. Rearrange the sticks to form 5 squares. You can move only 2 sticks.

Case Study

Art Integration

A tangram is a puzzle made of 7 polygons. It is widely used among children to create different shapes and designs. Look at the design given below and answer the questions.

1 How many triangles are there in the figure?

a 3 b 2 c 4 d 6

2 What shape is formed when 2 right angle triangles and a parallelogram are placed together?

a Square b Rectangle c Pentagon d Hexagon

3 Can a circle be formed using the shapes in a tangram?

a Yes b Can’t say c No d None of these

4 Draw a fish using a square and 4 triangles.

10 Patterns and Symmetry

Let’s Recall

We can see patterns all around us.

In Nature In Fabric In Shapes In Numbers

Let us now see some patterns and try to extend them!

We can see that the pattern is in the order of circle, triangle, circle and triangle. So, our unit will consist of a circle, then a triangle, and so on.

Now, look at this pattern.

Patterns Around Us

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.

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 and sizes.

We see patterns in historical monuments, buildings and art forms of different states around us.

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 identify the rule in a given pattern and then use it to extend the pattern.

Colour Pattern

A A A Unit of Repeat

A B B B B

Rule = AABBAABB

The next 2 terms will be 2 red stars.

Shape Pattern

A A B Unit of Repeat

B B B C C

Rule = ABBCABBC

The next 2 terms will be a triangle and a circle.

Example 1: Colour to extend the repeating pattern.

Unit of Repeat

Example 2: Create a pattern using the given figures with the rule—ABCCBABCCBABCCB.

The pattern can be:

Extend the given patterns by drawing the next 3 shapes or figures. Colour them.

Rotating Patterns

Sometimes the shapes that are repeating are turned around at every step.

Observe the dot on each triangle.

A B C

90° When the triangle makes a quarter turn or 1 4 turn, it rotates by 90° which is a right angle.

180° When the triangle makes a half turn or 1 2 turn, it rotates by 180°, equal to two right angles.

270° When the triangle makes three quarters or 3 4 turns, it rotates by 270° which is equal to three right angles.

360° When the triangle makes a complete turn, it rotates by 360°, equal to four right angles.

A unit in a rotating pattern can turn either clockwise or anti-clockwise.

Clockwise Direction Anti-clockwise Direction

Example 3: What will be next?

Example 4: Create a pattern by turning the given figure by 60° at every step in clockwise direction.

The given shapes are rotating clockwise. Fill in the missing shapes.

Which of these do not show a pattern?

Observe and extend the pattern.

Identify the rule for the given rotating pattern. Draw the next three units and colour them.

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.

Rakesh has made a rotating pattern where each shape rotates at half turns. Which of the given patterns has been made by Rakesh?

What is the next term in the given pattern? Draw and colour to show.

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.

We see that both the purple and the black triangles keep increasing by one unit.

The increasing unit is:

This is a growing pattern.

On the other hand, in this painting, the triangles are forming a tree-like pattern.

A triangle is reduced at every next step until there is only one triangle left.

The decreasing unit is:

This is a reducing pattern.

Example 5: Extend the pattern by choosing the correct toy train that will come next.

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: . So, the correct option is 3.

Identify and write whether the following are growing or reducing patterns.

Tiling Patterns

Now, Rani is well aware of what patterns are. As she walks 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. It forms when a single unit or shape repeats over and over again without leaving any gaps.

When this pattern is repeated, it fills the area seamlessly with no gaps and no overlaps.

This is also called tessellation.

Types of Tessellation

Did You Know?

The structure of a beehive is also an example of a tiling pattern.

Rectangular Tessellation Triangular Tessellation Square Tessellation

The arrangement of bricks in a wall.

arranged in a tiling pattern.

Look at the given images. Are these tiling patterns?

and white squares arranged in tiling.

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.

Draw and colour to complete the pattern. A small part of it is done for you.

Triangles

Pink

Do It Yourself 10B

Which of these is not a tiling pattern? Give your answer with a reason.

Colour to complete the tile.

Draw the picture that comes next in each growing or reducing pattern.

Which tile does not belong in the tiling pattern? Create a tiling pattern that contains all four tiles.

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.

Now look at some patterns.

Pattern 1: The numbers are the same, although the order of the numbers is reversed.

25 + 80 +10 10 + 80 + 25

Pattern 2: One more number is getting added each time.

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 28

Now interchange the digits 82

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.

Example 6: Describe the rule for the given pattern. Extend the pattern for the next two terms. 4 10 16 22

Rule—The given number pattern is increasing by 6 each time.

Example 7: Generate number patterns with 5 terms following the given rule.

a Start with 5 and add 11.

5, 16, 27, 38, 49 +11 +11 +11 +11

Pattern: 5, 16, 27, 38, 49

b Start with 10, multiply with 2 and add 1. 10, 21, 43, 87, 175 ×2 +1 ×2 +1 ×2 +1 ×2 +1

Pattern: 10, 21, 43, 87, 175

Example 8: A swimming pool has small chlorine tablets added every day to keep it clean. On Monday, 1 tablet was added. On Tuesday, 3 tablets were added. On Wednesday, 7 tablets were added. On Thursday, 15 tablets were added. How many tablets are added on Friday?

Number of tablets added on Friday = 15 + 16 = 31.

Therefore, 31 tablets were added on Friday.

Coding and Decoding Patterns

Different methods are used for the coding and decoding of passwords. Look at the letters and their corresponding numbers, in the table below.

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?

Example 9: Look at the letters and their corresponding numbers table given above. Decode the given code. 9 12 9 11 5 13 1 20 8 19

From the table, 9 I, 12 L, 11 K, 5 E, 13 M, 1 A, 20 T, 8 H, 19 S

Thus, the code is I LIKE MATHS.

In a certain language, the letters are coded as follows:

Decode the word: CNGGREA.

We can see that:

Therefore, the given word is .

Patterns in Numbers

Look at the sequence: 1, 1, 2, 4, 7, 11, 16, 22 and 29.

If we observe the numbers, the rule being followed is +0, +1, +2, +3, ... and so on, hence making a pattern.

We can represent some numbers in the form of an equilateral triangle arranged in a sequence. These numbers are called triangular numbers. Numbers like 1, 3, 6, 10 and 15 are some examples.

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…

Triangular Numbers

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.

The numbers from 1 to 6 are written such that the sum of each side is 9.

The sum of the numbers in each row, column and diagonal is always the same.

It begins with 1 at the top and 1s on the sides. For each point inside, the two numbers above it are added to get a new number.

Magic Triangle Magic Squares Pascal’s Triangle

Patterns with Odd Numbers

When we add a certain number of odd numbers, their sum is equal to the product of that certain number with itself.

Let us see some more patterns. Start with any number. Multiply it by increasing numbers (1, 2, 3,...) and add 2 each time.

10 ⨯ 1 + 2 = 12

10 ⨯ 2 + 2 = 22

10 ⨯ 3 + 2 = 32

10 ⨯ 4 + 2 = 42

Can we see a pattern?

Yes. The numbers 12, 22, 32, 42, are in a pattern. The terms increase by 10 at every step.

Example 10: Observe the pattern and write the next two lines of the number patterns.

25 + 11 = 36

36 + 13 = 49

49 + 15 = 64

So, the next two terms will be:

8 × 8 + 17 = 81

9 × 9 + 19 = 100

Example 11: 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.

= 64

The stars are forming squares— 4 stars, 9 stars, 16 stars and so on. Draw stars and complete the pattern. 1 + 3 = 6 + 10 = + = 3 + 6 =

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 c 1, 4, 9, 16, 25, 36, 49

Use the given code to find the numbers.

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? Do you water the plants in your house?

A bakery is receiving a lot of orders. They got 12 orders in January, 24 in February and 36 in March. If the same pattern continues, how many orders will they get in July?

Create a magic triangle with a sum of each side that is equal to 12.

Challenge

What will come in place of the question mark?

Symmetry and Reflection

Kush was at a pottery shop.

He noticed that there were a lot of different pots.

He looked at a pot and realised that no matter from which angle he looked at the pot, it looked exactly the same.

How is that possible? Let us find out!

Symmetry in Plane Shapes and Solids

We can divide or fold a shape, letter or a number into two identical halves along a straight line. This is called symmetry.

Hoysala Temple, Belavadi, Karnataka

Lotus Temple, Delhi

The 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.

These two halves are said to be symmetrical. Let us look at some types of symmetry. Vertical Symmetry

Symmetry

Drawing Lines of Symmetry

Let us see symmetry and lines of symmetry in some shapes.

Symmetry

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 M C D

H O X I

Every figure does not always need to be symmetrical when divided into 2 parts. Error Alert!

Here O has infinite lines of symmetry and letters like F, G, J and L have no line of symmetry. Now, observe these pictures.

Do the dotted lines drawn represent 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?

Figures with no line of symmetry are called asymmetrical figures.

Example 12: Draw all possible lines of symmetry.

In a square, we have 4 lines of symmetry.

Reflections

Reflection means flipping a figure over a line so that it looks like a mirror image. Imagine folding a piece of paper and seeing the same shape on both sides.

Look at some figures and their reflection.

Example 13: Draw the reflection of the letter J.

JJSymmetry in Solids

Imagine you cut a solid in half, both sides will look the same. A line of symmetry can also be drawn in 3-D figures and the figures can be completed along the line of symmetry.

Line of symmetry

Example 14: Imagine cutting a cone (like an ice cream cone) in half. Would the two halves be symmetrical?

Yes, cutting a cone in half vertically, from the pointy tip straight down through the centre to the base, would be symmetrical.

Draw the other identical halves of the given figures.

Match the figures with their lines of symmetry.

1 horizontal line of symmetry

Many lines of symmetry d

2 lines of symmetry

Draw the lines of symmetry for the given figures.

Write True or False.

Draw the lines of symmetry (if any) for the given letters.

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.

Complete the missing half of each of the shapes across the mirror line.

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

Critical Thinking

• 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.

• The pattern formed by the repetition of a single unit or shape over and over again without any gaps is called tiling.

• 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.

Art Time!

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 Have you got a mirror image? Why/why not?

Tick () the figures that are symmetrical.

Draw all the possible lines of symmetry for the given figures. a b c d

Draw the reflection of the figures across the mirror line.

In a certain language, the letters are coded as follows.

What will come next in the given patterns? Draw two figures for each.

a b c

Complete the tiling patterns by drawing 4 more shapes.

a b c

Examine the number pattern below.

331, 316, 301, 286, 271,……

Write the next three numbers in the pattern. Also, write the rule followed.

Generate number patterns with 5 terms following the given rules.

a Start with 1 and add the double of itself.

b Start with 15 and add 3 and 5 alternately.

Fill in the missing numbers.

a 37 39 42 46 51

b 90 89 87 84 80

Draw the reflection of this toy house. If you cut this house vertically in half, would both halves look the same?

Which of these solids has a line of symmetry? Write “Yes” below the correct one.

Honeybees use beeswax, a substance secreted from glands in their abdomen, to create honeycomb. A bee is building a honeycomb. It starts with 2 cells, then 4, 8, 16, 32, 64 and so on. What will be the next three numbers in this pattern? Also, write the pattern rule.

Mihir gets pocket money every month. He saved ₹20 in January, ₹40 in February, ₹80 in March, ₹160 in April and so on. How much money did he save in August?

Emily is hiking in the mountains. On the first day, she takes 100 steps. On the 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?

Challenge

1 Look at the pattern below. Which number will replace the question mark?

2 In the question given, read the two statements marked as Assertion (A) and Reason (R). Choose the correct option.

Assertion: Tara is following a fitness regime. On the first day, she takes 20 steps. On the second day, she takes 25 steps, then 32 on the third day, 41 on the fourth day and so on. She will take 65 steps on the sixth day.

Reason: Tara is increasing the number of steps by 5 each day.

a Both A and R are true, and R is the correct explanation of A.

b Both A and R are true, but R is not the correct explanation of A.

c A is true, but R is false.

d A is false, but R is true.

Case Study

Cross Curricular & Value Development

Imagine you are visiting the magnificent Taj Mahal. This 17th-century white marble mausoleum located on the banks of the Yamuna River in Agra, India, is a UNESCO World Heritage site famous for its beauty and intricate details. Explore how symmetry plays a role in its design by answering these questions.

1 Which of the following shapes is most prominent in the Taj Mahal's design? a Triangle b Rectangle c Pentagon

2 The Taj Mahal has four identical minarets surrounding the main dome, reflecting perfect rotational symmetry. True or False?

3 If you draw an imaginary line down the centre of the Taj Mahal, both sides will mirror each other. This is called

4 The Taj Mahal complex has 3 lines of symmetry. (Yes/No)

5 What can you do to help maintain cleanliness during your visit to Taj Mahal?

11 Length and Weight

Let’s Recall

Look at the given picture. How do we measure the length of the desk or the weight of the pot?

Length

We use “length” to describe how long or short an object is. For example, take this laptop and table. The laptop is shorter in length than the table. We measure length in different units. Centimetres (cm) and millilitres (mm) are used to measure shorter lengths. Metre (m) is the standard unit of length. It is used to measure larger lengths. Kilometres (km) are used to measure distances.

Weight

We use weight to describe how light or heavy an object is. For example, is the laptop heavier or the table? The table is heavier, so the weight of the table is more. The units for measuring weight are grams (g) and kilograms (kg). Gram (g) is used to measure the weights of smaller and lighter objects. Kilogram (kg) is the standard unit of weight. It is used to measure the weights of bigger and heavier objects.

We can estimate the unit of length or weight that can be used to measure an object by looking at the object.

Letʼs Warm-up

Fill in the blanks with the units of measurements.

1 The length of the laptop

2 The length of the table shown above

3 The weight of the coffee cup

4 The weight of the chair

5 The height of the plant in a small pot

I scored out of 5.

Understanding Lengths

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 meets her requirement. So, she chooses to buy 2 m of cloth.

This is exactly what estimating length is! Estimation helps us to find the approximate measure of things. It gives us a fair idea of how long or tall (length) an object is just by looking at it. It also gives us an approximate measure of how far away (distance) a place is. For example, look at the given objects.

The width of a grain of rice is about 6 mm.

The thickness of a notepad is about 1 cm.

The distance from the floor to the door handle is about 1 m.

Did You Know?

The height of Mount Everest is about 8848 m.

The distance between Sana's home and the school is about 10 km.

Example 1: The rope in the figure is 8 cm long. What is the estimated length of the battery?

Estimated length of the battery = Half the length of the rope.

So, the length of the battery is about 4 cm.

8 cm

Circle the best estimate.

4 Length of your shoe

Measuring Lengths

Now, Sana takes the cloth to the tailor for stitching. She wants front pockets of width 4 cm on her trousers. Let us see how we can measure the width 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 are the measurements in cm.

1 millimetre 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?

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 long = 4.5 cm long.

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.

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.

Converting Between Units of Length

Do you remember 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.

Look at the picture of the metre ruler given below. On this ruler, there are short lines and long lines. The distance between two short lines is 1 centimetre (cm). So, from the ruler given below, we see that 1 m = 100 cm.

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) less than a metre

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.

Did You Know?

P.N.Seth, founder and secretary of the Indian Decimal society, aimed for the introduction of the metric system in India, which was accepted by the parliament in 1955.

Divide by 10 for every step as you move left

Multiply by 10 for every step as you move right

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

Divide by 10 for every step as you move left

Let us now convert 40 cm to m.

100 cm = 1 m. So, 1 cm = 1 100 m

40 cm = 1 100 × 40 m = 0.4 m

What if Sana had bought 1 m 85 cm of cloth?

Let us see how you would write it in metres.

1 m = 100 cm

1 m 85 cm = 1 m + 85 100 cm = 1.85 m.

Example 3: Convert 2356 dm to m.

10 dm = 1 m

1 dm = 1 10 m

2356 dm = 1 10 × 2356 m = 2356 10 = 235.6 m

Fill in the blanks to convert 5 m 230 mm to m. mm = 1 m

1 mm = 1 m

5 m 230 mm = 5 m + 230 × 1 m

= 5 m + m = m

Word Problems on Length

Did You Know?

There are other units of length, like inch and foot, which are used to measure the length of things like wire and cloth; and miles and yards are used to measure longer lengths like distances.

Example 4: Convert 6.547 hm to m. 1 hm = 100 m

6.547 hm = 6.547 × 100 m = 654.7 m

After making a pair of trousers with 2 m of cloth, Sana thought of buying 1 m 55 cm of cloth for making a shirt. Let us see what length of cloth she has together.

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

So, the total cloth required by Sana is 3.55 m.

Think and Tell

How do you think 1 m 55 cm is converted into 1.55 m?

Example 5: Samantha buys two rolls of tape. One is 9.56 m long and the other is 6.80 m long.

1 What is the total length of the two rolls?

2 What is the difference between 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 Total length = 9.56 m + 6.80 m

2 Difference in the

So, Samantha buys 16.36 m in total.

So, the difference between the two rolls of tape is 2.76 m.

Example 6: A 34 m long rope is divided into 10 equal pieces. What is the length of each small piece?

Length of a rope = 34 m

Number of equal pieces of rope needed = 10

Length of each piece of rope = 34 m ÷ 10

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 ×

m of cloth is required for making 5 sarees.

Do It Yourself 11A

1 Esimate the length of these items. Measure the actual lengths. Find the difference. a Pencil b Eraser c Sharpener

2 Write the lengths of the given objects.

3 Convert the given measurements.

4 Complete the table.

S. No.

5 Find the measurement of the pencil. Express your answer in km.

6 Rajni went to buy skirts. If 3 m 586 cm of cloth is required for one skirt, then how much cloth is needed for 6 such skirts?

7 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? How does riding a bicycle help the environment?

8 If the length of a door is 2 m 1 dm and that of a wall is 3 m 2 dm, what is the total length in dm?

9 Create a word problem on subtracting the length of two things.

Challenge

1 A rat was chewing up 2 cm of a cardboard each day. The owner removed the cardboard after one week. How much of the cardboard sheet was eaten in 1 week?

Understanding Weights

Vaishali went to the market to buy sugar to make dessert for a family get-together.

Vaishali: I want 1500 g of sugar, please.

Shopkeeper: I have only packets of 1 kg of sugar.

Vaishali was confused about how many packets of sugar she should buy.

Estimating Weights

Vaishali 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.

About a milligram

About a gram

grain of sugar

small paper clip

Example 7: Which of these objects measures about a gram?

About a kilogram

The weight of the leaf is about a gram.

Circle the best estimate.

3 A photo album

4 A tube of toothpaste

Converting Between Units of Weight

Look at the picture of the weighing scale. Starting from 100 g, read the weights on the scale, clockwise. What do you notice?

Here, 1 kg = 1000 g

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.

Remember!

The units of measuring weight, such as milligrams (mg), grams (g) and kilograms (kg) are used to measure light as well as heavy objects.

Book

Ball Leaf

Shoes

Divide by 10 for every step as you move left

Base Unit Bigger Units

Value increases 10 times Value decreases 10 times

Multiply by 10 for every step as you move right

Convert the quantity of sugar (2 kg) that Vaishali bought into g.

1 kg = 1000 g. So, 2 kg = 2 × 1000 = 2000 g

Divide by 10 for every step as you move left

Convert 400 mg to g.

1000 mg = 1 g. So, 1 mg = 1 1000 g

400 mg = 1 1000 × 400 g = 400 1000 = 0.4 g

What if she bought 1500 g of sugar? Let us see how we would write it in kilograms.

Quantity of sugar purchased by Vaishali = 1500 g

Since 1000 g = 1 kg, 1500 g = 1500 1000 = 1.500 kg

Example 8: Convert 7695 dg to g. 10 dg = 1 g

1 dg = 1 10 g

7695 dg = 1 10 × 7695 g = 7695 10 = 769.5 g

Fill in the blanks to convert 5 g 230 cg to cg.

1 g = cg

5 g 230 cg = 5 × cg + 230 cg = cg

Example 9: Convert 3.476 hg to g. 1 hg = 100 g 3.476 hg = 3.476 × 100 g = 347.6 g

Think and Tell

How do you think 2 kg 106 g is converted into 2.106 kg?

Error Alert!

Word Problems on Weights

Do you remember that the sugar bought by Vaishali 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

Weight of rice = 2 kg 106 g = 2.106 kg

Here, 2 < 2.106.

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: 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

Total weight of vegetables = 8 kg 250 g = 8.250 kg

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.

Example 11: 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.

Example 12: An egg has a weight of about 65 g. What is the weight of 3 dozen eggs?

The weight of an egg = 65 g

1 dozen = 12 eggs

3 dozen = 12 × 3 = 36 eggs

The weight of 3 dozen eggs = 36 × 65 = 2340 g = 2.34 kg

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?

Total weight of books removed from the suitcase = Weight of the suitcase after removing the books =

9 . 2 0 0

o It Yourself 11B

Circle the best estimate of the weight of the objects.

a Frog

Convert the measurements.

a 79 g into kg b 975 g into mg c 4677 kg into g

d 1655 dg into g e 6876 cg into g f 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 kg 6 g

d 3 g 8 cg e 87 kg 6 dag f 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?

A truck can carry 8 cartons of packed food. If they weigh 32 kg 448 g in total, then what is the weight of each carton?

Siya bought 2 kg 450 g of apples, 1 kg 547 g of guavas and 2 kg 136 g of pears. What is the total weight of fruits Siya bought?

The weight of 2 chairs is 16 kg 400 g. If the weight of one chair is 10 kg 300 g, then what is the weight of the other chair?

A man carried 4 conical flasks weighing 260 g each and 3 beakers weighing 150 g each.

a What is the total weight he carried?

b If 1 conical flask and 1 beaker had broken, what is the weight of the remaining conical flasks and beakers?

Three dg of baking soda is used in 1 kg of cake. If a baker prepares 17 cakes each weighing 1 kg in a day, then what quantity of baking soda is used in total?

1 Quintal and tonnes are also units of measuring weight. 1 quintal is 1,00,000 g and 1 tonne is 1000 kg. How many quintals is 1 tonne? Write the answer in kg.

Points to Remember

• Estimation is a fair idea of how far away (distance) something is, and how long or tall (length) an object is just by looking at the object.

• To measure the lengths of small objects, we use a centimetre (cm) ruler. The distance between two short 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!

Creativity & Collaboration

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.

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. using clay and building blocks, and measure the lengths of these objects.

Estimate the weights of these objects by holding them.

According to the lengths and weights of the dummy objects, assemble and build a playland 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.

Guess the best units of length (m or cm) and weight (kg or g) for the given objects.

a b c

Measure the objects.

a b c James has some cotton candy which is 3 cm longer than the cotton candy shown below. How long is James’ cotton candy?

How much longer is the red straw than the blue straw?

Convert the lengths.

Convert the weights.

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?

A lift in a building is allowed to carry up to 260 kg in weight. The people listed below want to enter the lift. Their weights are mentioned.

Amit: 85 kg Priya: 70 kg

Can they enter the lift together? Why?

Priyanshi: 58 kg Raju: 80 kg

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 the lines of the following lengths. Each strip can be used any number of times.

Rimi has 12 stamps that are each 3 cm long.

a What is the total length of all the stamps?

24 cm

b Will all 12 stamps fit into a single row across a stamp album that is 24 cm across?

To bake one cake, a baker needs the following items: 200 g of flour, 3 eggs, 75 g of butter, 100 g of sugar, and some milk. How many kilograms of flour, butter and sugar would be needed to bake 100 such cakes?

Create a word problem to add weights of two objects.

Critical Thinking

1 What is Meenakshi’s weight in grams? Read the clues and answer.

a Meenakshi’s weight is half of her grandmother’s weight.

b If you add the smallest 2-digit number to 28 less than the smallest 3-digit number, you will get the grandmother’s weight in kilograms.

2 What is Riya’s father’s weight? Choose the correct option that answers the question.

Statement 1: The father’s weight is three times the difference between Nita’s weight and Riya’s weight.

Statement 2: Nita’s weight is 36 kg and Riya’ weight is one-third of Nita’s weight.

a Statement 1 alone is sufficient to answer.

b Statement 2 alone is sufficient to answer.

c Statements 1 and 2 together are sufficient to answer.

d Statements 1 and 2 together are not sufficient to answer.

Case Study

Cross Curricular & Value Development

Bannerghatta National Park is a famous national park in India. A small portion of the national park has become the Bannerghatta Biological Park. The fauna in the park include lions, tigers, bears, elephants and many more.

1 Write the unit used to measure the lengths of the given animals.

a The unit used to measure the length of a tiger.

b The unit used to measure the length of a butterfly.

2 The weight of a male leopard is 72 kg and the weight of the hyena is 55,000 g. Which of these is true?

a The difference in weight is 1700 g.

b The difference in weight is 1.7 kg.

c The difference in weight is 170 g.

d The difference in weight is 17,000 g.

3 The park has two enclosures that were merged. The length of enclosure 1 was 8894 m and the length of the second enclosure was twice the length of enclosure 1. The new enclosure is divided into 2 pieces where one piece has a length of 20,000 m and the other has the play area. What is the length of the play area?

4 The tigers and lions have to be shifted to sanitise the place. The van can accommodate animals up to 300 kg. The weight of a tiger is 72 kg and that of a lion is 176 kg. How many animals can be shifted in one trip?

5 How can you help to save flora and fauna near you?

12 Perimeter and Area

Letʼs Recall

Perimeter is the total distance around the boundary of a closed shape. We can calculate the perimeter by adding the lengths of all the sides of a closed shape.

Let us find the perimeter of the given shape. This shape has 5 sides, each measuring 2 cm, 3 cm, 5 cm, 5 cm and 4 cm, respectively. So, its perimeter would be,

2 cm + 3 cm + 5 cm + 5 cm + 4 cm = 19 cm.

Area is the total surface occupied by a closed shape. We can determine the area of a shape by drawing it on a square grid, and counting the number of squares it covers. Let us find the area of the given shape. We can see that the rectangle occupies 12 squares. Therefore, the area of the rectangle is 12 square units.

Letʼs Warm-up

Look at the figures drawn on a 1 cm square grid. Fill in the blanks.

Perimeter

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

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.

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

Remember!

Perimeter is the total distance around the boundary of a closed shape.

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

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

So, the perimeter of a rectangle = 2l + 2b = 2(l + b)

The teacher and students together measured the length and breadth of the blackboard. They found the length of the blackboard as 100 cm and breadth as 80 cm.

So, the perimeter of the blackboard = 2 × 100 + 2 × 80 = 200 + 160 = 360 cm

Hence, 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 = 2(l + b)

= 2 × (38 + 28) = 2 × 66 = 132 cm

So, the perimeter of the window is 132 cm.

Example 2: Find the length of a rectangle whose breadth is 2 cm and perimeter is 16 cm.

We know that: Perimeter of a rectangle = 2l + 2b

So, 16 = 2 × l + 2 × 2

16 = 2 × l + 4

16 − 4 = 2 × l

12 = 2 × l = 12 ÷ 2 = 6

Therefore, the length of the rectangle is 6 cm.

Find the perimeter of the given pillow.

Here, length (l ) = 60 cm and breadth (b) = cm

Perimeter of a rectangle = 2(l + b)

The perimeter of the pillow = 2 × ( + ) = 2 × = cm

Perimeter of a Square

Did You Know?

Carl Friedrich Richard Foerster came up with the idea of measuring perimeters, and way back in ancient Greece, Hippocrates of Chios figured out how to measure areas!

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.

Let us consider a square with side length s.

Perimeter is the sum of all the sides of a figure.

So, the perimeter of the square = s + s + s + s = 4s

Perimeter of a square = 4 × side

Now, the teacher and the students together measured the sides of the bulletin board as 40 cm each.

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 a ribbon of length 160 cm to decorate the bulletin board.

Example 3: Find the perimeter of a square of side 19 m.

Here, s = 19 m

Since the perimeter of a square = 4 × s, perimeter of the square = 4 × 19 m.

Therefore, the perimeter of the square is 76 m.

Example 4: Rohan runs around a square park and covers a distance of 800 m. What is the length of each side of the park?

We know that: Perimeter of a square = 4 × s

800 = 4 × s

s = 800 4

s = 200

So, the length of each side of the square park is 200 m.

Find the perimeter of the chessboard.

Here, side (s) = 45 cm.

Perimeter of a square =

So, the perimeter of the chessboard is cm.

Do It Yourself 12A

Find the perimeter of the given objects.

Find the perimeter of the squares of the following measurements.

Find the perimeter of the rectangles of the following measurements.

Find the missing sides.

a Rectangle with length = 32 mm, breadth = ?, Perimeter = 112 mm

b Square with Perimeter = 148 mm, side = ?

A farmer needs to fence his square-shaped garden. The distance from one corner to the next corner is 45 m. How much wire would he require to fence the entire garden?

Sana gets an order to put lace around rectangular table mats measuring 50 cm × 20 cm. How much would she charge for each table mat if the rate is ₹10 per cm?

The Taj Mahal is a marble mausoleum on the bank of the Yamuna, in Agra. It was commissioned by the fifth Mughal ruler, Shah Jahan, in memory of his beloved wife, Mumtaz Mahal. The mosque measures 60 m in length and 30 m in width. What is the area covered by the mosque?

Seeta used a length of string to first form a square of side length 10 cm. She then used the same string to form a rectangle. The length of the rectangle is 12 cm. What would the breadth of the rectangle be?

Area

The length of a rectangle is three times its breadth. If the perimeter of the rectangle is 32 metres, find its length and breadth.

When you look around your house, you will notice that some walls are larger than others. When it comes to painting those walls, the one with more surface area requires more paint.

We can use the term �area� to describe how big a wall is. It is a way of measuring the amount of surface available for painting.

Area of Squares, Rectangles and Triangles

Now let us look at different shapes of walls and their areas.

Area of a Square

Let us find the area of the square of side 3 cm given on the square grid.

The area of each small square in the grid is equal to 1 square unit.

We can count all the small squares inside the big square to find the area.

Area of the square = Area of 9 small squares = 9 square units.

We can say that the area of a square = side × side or, the area of a Square = s × s

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?

Units of Area

Area is always measured in square units (sq. units). When the length of the side is in m, the area will be in sq. m. When the length of the side is in cm, the area will be in sq. cm.

A bed occupies a large part of the room and its sides are measured in metres. Therefore, the area it occupies will also be measured in sq. m.

Compare this with an eraser which is only a few centimetres long. Therefore, the area of the eraser will be measured in sq. cm.

Did You Know?

Example 5: Find the area of a square of side 5 cm.

Here, s = 5 cm

Area of a square = s × s = 5 cm × 5 cm = 25 sq. cm

The smallest country in the world is Vatican City. It is surrounded by the city of Rome. It has a total area of about 0.44 sq. km.

Example 6: Find the length of the side of a square with an area measuring 81 sq. cm.

Here, Area = 81 cm

Area of a square = s × s

81 = s × s = 9 × 9

s = 9 cm

So, side length = 9 cm

Find the area of the photo frame.

Here, s = 8 cm

Area of a square = s × s

The area of the photo frame = cm × cm = sq. cm.

Area of a Rectangle

Let us consider a rectangle of length 5 cm and breadth 3 cm, as shown.

Area of the rectangle = number of squares in a row × number of squares in a column = 5 × 3 = 15 small squares

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, the area of a rectangle = l × b

Area of Compound Shapes

Compound shapes are formed by combining two or more basic shapes.

For example, this is a compound shape formed with two rectangles A and B, as shown in the given figure.

Now, let us find the area of the given compound shape.

We will follow these steps:

1. Identify the shapes that come together to form the compound shape.

2. Find the missing lengths.

3. Find the area of rectangle A and 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

4. Find the area of the whole shape.

Area of the whole shape = area of rectangle A + area of rectangle B = 15 sq. cm + 48 sq. cm = 63 sq. cm

Example 7: 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 = 300 sq. cm.

Therefore, the area of the rectangle is 300 sq. cm.

Example 8: Find the area of the given shape.

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 = (12 + 27 + 30) sq. cm = 69 sq. cm

Find the area of the shape.

The area of the big rectangle = 12 × 6 = sq. cm

The area of =

The area of the whole shape = .

Area of a Triangle

Look at the rectangular wall. It has been divided into two right-angled triangles. 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.

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 = 1 2 of the area of the rectangle

Area of the rectangle = l × b = 10 × 5 = 50 sq. m.

So, the area of each triangle = 1 2 × 50 = 25 sq. m.

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: Draw the triangle on the square grid and find the base. Base = 6 cm.

Step 2: Draw a rectangle around the triangle and split the triangle into 2 triangles.

Step 3: Find the area of the 2 triangles.

Rectangle 1: Length = 2 cm; Breadth = 6 cm

Area of rectangle 1 = 2 × 6 = 12 sq. cm

Area of triangle 1 = Half of 12 = 6 sq. cm

Rectangle 2: Length = 4 cm; Breadth = 6 cm

Area of rectangle 2 = 4 × 6 = 24 sq. cm

Area of triangle 2 = Half of 24 = 12 sq. cm

Area of whole triangle = 6 + 12 = 18 sq. cm

Example 9: What is the area of the red triangle?

Area of triangle 1 = Half of area of rectangle 1 = 1 2 × 6 = 3 sq. units

Area of triangle 2 = Half of area of rectangle 2 = 1 2 × 6= 3 sq. units

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

Example 10: 5 identical squares are placed side by side to form a rectangle. What is the area of each square if the length of the rectangle formed is 22.5 cm?

Length of rectangle = 22.5 cm

Since the 5 squares are identical, side length of 5 squares = 22.5 ÷ 5 = 4.5 cm

Did You Know?

The ancient Egyptians used the concepts of perimeter and area to measure and plan fields for agriculture.

Area of square = 4.5 × 4.5 = 20.25 sq. cm. 1 unit 1 2

Which unit would you prefer to find the area of the following?

a blackboard b a book c classroom d your school

e your city f lake g table h a farmer’s field

Find the area of the rectangles of the given measurements.

a l = 15 cm, b = 22 cm b l = 58 m, b = 70 m

Find the area of the squares of the given measurements.

a s = 56 m b s = 67 cm

Advita wants a new carpet for her room. If her room measures 12 m by 12 m, then how much carpet does she need to cover the entire floor?

Take each side of a square on the squared paper as 1 unit. Find the area of the triangles. a b

Find the area of the given compound shapes.

a 3

Use squared paper to show rectangles or squares with these measurements. Find the measurement that is not given for the following shapes.

Measure both the length and width of your notebook, then use these measurements to draw a rectangle on a square grid. Connect the opposite corners of the rectangle. Shade one side of the rectangle with your favourite colour. Can you find the area of this shaded region?

A rectangle has a length of 8 metres and a breadth that is one-fourth of its length. A square has a side length that is twice the breadth of the rectangle. Which shape has a larger area?

A rectangle is cut into 2 squares. The perimeter of each square is 36 cm. What is the perimeter of the rectangle?

Points to Remember

• The perimeter of a rectangle is 2l + 2b, and the perimeter of a square is 4 × s

• The area of a rectangle is l × b, and the area of a square is s × s.

• Area is always measured in square units.

• To find the area of compound shapes, we split the shapes into squares or rectangles.

• To find the area of a triangle, we draw a rectangle along the vertices of the triangle.

Math Lab

Setting: In groups of 3

Experiential Learning & Collaboration

Build a Paper House!

Materials Required: Cardboard sheets, a pair of scissors, glue or tape, a ruler, markers or coloured pencils, templates of squares, rectangles and triangles

Method: All 3 members of each group must make one cardboard house.

1 Collect templates of different shapes, such as squares, rectangles and triangles, from your teacher. These templates will serve as the bases for your cardboard houses.

2 Cut out the cardboard sheets using the templates. Decorate them to represent different parts of a house, such as walls, roofs, doors and windows.

3 Once the decoration and calculations are done, fold and assemble the shapes to create your house. You may use glue or tape to secure the different parts of the house together.

4 Measure the lengths and calculate the area that your house occupies.

Find the perimeter and area of the following rectangles.

and

Complete the given table.

Find the area of the following triangles.

Find the area of the following compound figures.

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:

Humayunʹs Tomb was built in memory of the second Mughal Emperor Humayun. It features a central platform that is square in shape and measures approximately 300 feet on each side. What is the area of the platform?

Find the total area of 8 square wooden panels if the length of side in each panel a 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.

Use 4 straws for the length and 2 straws for the width to make a rectangle. Now, with the same 12 straws, make a square. Which shape has a larger perimeter and area?

Three rectangular tiles having the same length and breadth are placed as shown. If the perimeter of each tile is 22 cm, find the perimeter of the shape formed.

If the length and width of a rectangle doubles, the area of the rectangle also doubles. Is this statement true? Explain your answer.

The School Map

Farah’s school is very big, with many rooms and a playground. One day, she and her friends noticed a large square-shaped map of the school on the bulletin board in the reception area. Intrigued, they studied the map, which showed the different sections of the campus. Here the size of the classrooms was the same. They looked closer, and they wondered about what the areas of the rooms in the building were. Inspired, they decided to calculate the perimeter and areas of different sections to better understand their school’s layout.

1 What is the total area covered by all four classrooms ?

2 Which area is bigger: the staff room or the art room?

3 If the layout is square in shape, what is the area of the school?

4 What is the area of the entire school except for the area of the square shaped ground?

5 What skills did Farah and her friends show by working together?

13 Capacity and Volume

Let’s Recall

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 with 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.

Letʹs Warm-up

Fill in the blanks.

1 Between a jug and a glass, has more capacity.

2 If 8 glasses of juice fill a jug, then is the capacity of the jug.

3 Four tin cans of paint fill a bucket of paint. If the capacity of 1 tin can is 2 L, then is the capacity of the bucket.

4 Half of a container fills the bucket completely. The capacity of the container is full buckets.

I scored out of 4.

Height of the vessels

Understanding Capacity

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 by looking at it. This is exactly what estimating capacity is. For example, look at these objects:

Did You Know?

The ancient system used different hour glass shaped measures like Chhataank, Pav and Seer to measure liquids.

The bottle of water holds about 1 L of water.

The medicine dropper holds about 2 mL of medicine.

We can estimate that 2 small 1 L bottles will fill the big 2 L bottle.

Example 1: Which of these objects can have a capacity of about a litre?

The capacity of the teapot can be about a litre. Circle the best estimate.

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?

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.

The figure below shows a set of measuring cups with capacities of 1 L, 500 mL and 300 mL. The jars are filled with different amount of water.

Think and Tell

How do we know how much water each cup holds?

Example 2: Look at the syringe. What is the amount of the liquid in it? What is its capacity?

The syringe contains 6 mL of liquid but its capacity is 10 mL.

Example 3: What is the capacity of the 2 cups? Write the amount of water in each of them.

The capacity of the 2 cups is 100 mL each.

Cup A contains 700 mL of water.

Cup B contains 400 mL of water.

What is the amount of juice in the jug? Write its capacity.

The smaller markings on the jug stand for 50 mL.

The jug contains mL of liquid. Its capacity is .

Units of Capacity

Litre (L) is the standard unit of capacity. It is used to measure big capacities. A millimetre (mL) is used for measuring smaller capacities.

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.

Remember!

A millilitre (mL) is less than a litre (L).

Now, look at the conversion chart.

Divide by 10 for every step as you move left

Multiply by 10 for every step as you move right

As Samidha bought 2 L of water, let us convert it to mL.

1 L = 1000 mL 2 L = 2 × 1000 = 2000 mL

Let us now convert 2500 mL to litres. 1000 mL = 1 L 1 mL = 1 1000 L

Example 4: Convert 6548 dL to L. 10 dL = 1 L

the

Example 5: 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.

Do It Yourself 13A

Circle the best estimate of the given objects.

Convert the given measurements.

a 658 mL into L b 8437 L into mL c 2567 dL into L

d 5054 cL into L e 821 L into daL f 136 L 80 dL into mL

Complete the given table.

Aman had a bucket in his bathroom which a capacity of 20 L. What is the capacity of the bucket in mL? Do you bathe everyday?

Ridhi pours 1050 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).

Challenge

Critical Thinking

Are there enough bowls to fill a jar of 2700 mL? Use the information given below to find the answer.

Adil and Mitali want to distribute sweets to their relatives on Diwali.

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

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.

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 Box 2

One layer of 8 milk cakes.

1 layer

Two layers of 8 milk cakes. There is no empty space left.

2 layers

So, Box 1 can hold 16 (8 × 2) milk cakes.

One layer of 16 milk cakes.

1 layer

Three layers of 16 milk cakes. The box was full.

3 layers

So, Box 2 can hold 48 (16 × 3) milk cakes.

Since Box 2 holds more, Box 2 has a greater volume than Box 1.

Volume is the amount of space a solid object occupies, or the amount of space enclosed within a container.

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 mm

1 mm

This is a millimetre cube (mm cube). It is about the size of a grain of sugar. 1 cm 1 cm 1 cm

This is a centimetre cube (cm cube). It is about the size of a dice. 1 m 1 m 1 m

This is a metre cube (m cube). It is about the size of a 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, the 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 has 3 such layers.

So, the 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), where the units can be ‘mm’, ‘cm’ or ‘m’.

Example 6: Which object has the greater volume?

The shoe box occupies more space than the tiffin box. So, the shoe box has a greater volume than the tiffin box.

Example 7: Find the volume of the given solids. Do they both have the same volume?

Solid A has 10 unit cubes. 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.

Find the volume of the solid. Do It Together

Layer 1 (blue) has unit cubes.

Layer 2 (yellow) has unit cubes.

Total number of unit cubes = Volume of the given solid = cu. units

tiffin box shoe box

Volume of Solids Using the Formula

Do you 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.

8 units height

Box A

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.

Box A

Length = 12; Breadth = 5

Thus, number of pethas (unit cubes) in 1 layer = 12 × 5 = 60

Box B

Length = 5; Breadth = 5

Thus, number of pethas (unit cubes) in 1 layer = 5 × 5 = 25

Height = 8

So, number of pethas (unit cubes) in box A = 60 × 8 = 480 pethas

Thus, volume of box A = 480 cu. units

Height = 5

So, number of pethas (unit cubes) in box B = 25 × 5 = 125 pethas. Thus, volume of box B = 125 cu. units

Volume = Number of unit cubes in a layer × Number of layers = (l × b) × h

What if we need to find the volume of objects that are not shaped like cubes?

The volume of such small objects can be found by using a simple technique.

Step 1

Take a measuring cup and fill it with water and note down the volume.

Volume = 50 mL

Step 2

Add the marble for which the volume has to be found.

Step 3

Note down the volume of water now.

Volume = 100 mL

Step 4

The difference in the mark is the volume of marble

Difference = 100 mL –50 mL = 50 mL

So, the volume of the marble is 50 mL.

Example 8: The water in a measuring cup was 10 mL. Naveen added a coin in the measuring cup and the water raised to 15 mL. What is the volume of 12 such coins?

Volume of 1 coin = final level of water – initial level of water = 15 mL – 10 mL = 5 mL

Volume of 12 coins = 12 × 5 mL = 60 mL.

Example 9: 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 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

Did You Know?

So, the volume of the container = 60 cu. cm

Look at the box. If the box is completely filled with unit cubes, what is the volume of the box?

Number of unit cubes in one layer =

So, volume of the box = × × h = × _____ × 3

Word Problems on Volume and Capacity

Sneha bought two containers, A and B, for packing food. What is the volume of the two containers?

Volume = l × b × h = 16 × 12 × 10 = 192 × 10 = 1920 cu. cm

Volume = l × b × h = 10 × 10 × 10 = 1000 cu. cm

Example 10: 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

20 . 1112 6 + 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.

Total quantity of petrol = 50 L 342 mL = 50.342 L

Quantity of petrol filled in the bus = 50.342 L – 33.805 L

So, the quantity of petrol in the bus is 16.537 L (or 16 L 537 mL).

Example 11: A 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?

Length = 15 cm, breadth = 10 cm, height = 8 cm

Volume of the container = l × b × h = 15 × 10 × 8 = 150 × 8 = 1200 cu. cm.

Ketan wants to rent two of the rooms in his house. He uses the storeroom 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

Find the volume of each solid. Circle the solid with the greater volume.

Rhea wants to fill an almirah that measures 3 m by 2 m by 1 m with cartons of old clothes. What is the volume of the almirah?

A big box that measures 10 cm by 15 cm by 6 cm is packed with smaller boxes. What is the volume of the bigger box?

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 its volume.

Vanita had 20 mL of milk, and a biscuit fell inside. Vanita’s milk increased by 10 mL.

a What is the volume of the biscuit?

b What is the volume of 3 such biscuits?

Rinku’s lunch box measures 16 cm by 8 cm by 3 cm, whereas Rita’s lunch box measures 10 cm by 7 cm by 3 cm. Whose lunch box is bigger in size? Do you finish the food in your lunch box without wasting it?

In an experiment Rekha prepares 16 L of sugar water. She pours them equally into some beakers. If each beaker can hold 2000 mL of the sugar water, how many beakers are needed?

Srishti has a jug filled with juice. She pours it into 2 glasses of different sizes. One glass has 350 mL of juice and the other has 890 mL of juice. If the jug contained 1.5 L of juice, how much juice was left in the jug?

1 Tina is making a building with cube-shaped blocks. The base of the building showing the number of cubes to be placed at the base is shown by the figure. The structure is 6 blocks high and each level has the same shape and number of blocks. What is the volume of Tina’s structure in cubic units?

Points to Remember

• Capacity of a container is the amount of liquid it can hold. Volume of the container is the amount of liquid it has.

• Volume of a solid is the amount of space an object occupies or the space enclosed within a 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

Setting: In groups of 5

Experiential Learning & Creativity

Cube Art Gallery

Materials Required: nets of cubes, cardboard, glue or tape, a pair of scissors, a ruler, markers or crayons or coloured pencils, decorative materials like stickers, glitter, etc. of your choice

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.

Chapter Checkup

Tick () the best estimate.

a Glue in

Sahil filled 500 mL of water in a jug. Sahil put 6 matchboxes in it. The water level rose to 800 mL. What is the volume of 1 matchbox?

Read and write the amount of coloured water in each of the jugs.

600 mL mL mL mL mL mL

Colour the beakers up to the amount of water written with each beaker. a

Find the length, breadth, and height of the solids by counting the number of cubes.

a b

Convert the measurements.

a 980 mL into L

d 3456 cL into L

Find the volume of the figures.

b 6869 L into mL

e 243 L into daL

c 9796 dL into L

f 907 L 56 dL into mL

a b c d

Juhi drinks 235 mL of juice from a pack of 1 L of juice. How much juice is left in the pack?

Ashish had two flasks. One flask can hold 850 mL of water, and the other flask can hold 1 L 250 mL of water. How much water can the flasks hold together? Do you drink 2 litres of water every day?

The containers are partly filled with unit cubes. Find the volume of each container.

a b c

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?

Seema’s bucket can hold six and a half litres of water. The water tank holds four times as much water as Seema’s bucket. So, how much water does the tank hold?

5 cups of water can fill a bottle, and 5 cups of water can fill a bowl. 5 bottles of 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?

Challenge

1 Jassi is fond of building blocks. He glues 8 cubes together to make a block as shown.

Now, he wants to make a block that is 6 small cubes long, 5 small cubes wide and 4 small cubes high. He wants to use the smallest number of cubes possible, by leaving the largest possible hollow space inside the block such that it looks like a box. How many blocks will he use?

2 Read the statements and choose the correct option.

Assertion: The volume of a cube will always be greater than the volume of a cuboid.

Reason: The volume of a cube is l × l × l, and the volume of a cuboid is l × b × h.

a Both assertion and reason are true.

b The assertion is true, and the reason is the correct explanation for the assertion.

c The assertion is true, but the reason is not the correct explanation for the assertion.

d The assertion is not true, but the reason is true.

Case Study

Value Development & Cross Curricular

Amul (Anand Milk Union Limited) is a dairy federation run by the Gujarat Cooperative Milk Marketing Federation Limited. Amul was founded on December 19, 1946. It is a major producer of diary products like milk, curd, paneer, different types of cheese, ice cream, chocolates and many more.

1 Write True or False.

a The volume written on the buttermilk packet will be less than 10 mL.

b The volume written on the pack of milk will be in mL.

2 The volume of the flavoured milk is 0.45 L. The volume of the flavoured milk in mL is

3 Amul produces paneer which is distributed across the country. It produces 15,000 kg of paneer from 45,000 litres of milk. How much milk will be required to produce 30,000 kg of paneer in a day?

4 Amul cheese is a famous vegetarian cheese worldwide. The dimension of a 400 g pack of cheese is 7 cm × 3 cm × 4 cm. If the company packs 10 such packs in a box, what should be the minimum height of the box?

5 Milk is one of the dairy products that gives us calcium. Calcium is very important for bones and teeth. Do you drink milk every day? If yes, how much mL do you have in a day?

14 3-D Shapes on Flat Surfaces

Letʼs 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.

With a little effort, they remember the names of these shapes as cube, cuboid, cylinder, cone and sphere.

Letʼs Warm-up

Fill in the blanks with the correct names of the shapes.

1 The is in the shape of a (cube/cuboid).

2 The is in a (cylindrical/cubical) shape.

3 The is in the shape of a (cuboid/cone).

4 The is in the shape of a (sphere/cone).

I scored out of 4.

Cuboid

Cube

Cylinder

Cone

Sphere

3-D Shapes

Isha and Misha are at the market. Both of them decide to eat something. Let us see what they have bought.

Isha: I see that my can of juice is long and round.

Misha: I see that my sandwich is triangular.

They both wondered at the shapes of the things that they had bought.

I have bought a sandwich. Misha, I have bought a can of juice.

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 (3-D) shapes.

The faces of 2-D shapes are joined together to form 3-D shapes.

Solid shapes like cubes, cuboids, cylinders, cones and spheres have faces, edges and corners.

Features of 3¯D Shapes

• The flat or curved surface of a solid shape is called a face.

• The line where two faces meet is an edge.

• 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

Sphere: 1 curved face, no edges, no vertices

Curved face

Cone: 1 flat face, 1 curved face, 1 curved edge, 1 vertex

Vertex

Curved face

Curved edge

Flat face

Rectangular pyramid 5 flat faces, 8 edges, 5 vertices

Vertex Edge Face

Let us read about two types of solid shapes: prisms and pyramids.

Prisms

A prism is a solid shape with two identical flat bases and rectangular lateral surfaces perpendicular to the bases. A cube is an example of a square prism.

Pyramids

A pyramid is a 3-D solid shape with a flat base and three or more than three triangular lateral surfaces that meet at the point called the vertex.

Example 1: Which of the given objects looks like a cone?

From the above objects, object 3 looks like a cone.

Example 2: Write the number of edges for the given shapes.

1 Sphere: 0 edges

2 Square pyramid: 8 edges

3 Triangular prism: 9 edges

Did You Know?

Astronomers in ancient Greece were the first to suggest that the Earth was spherical and not flat like most people believed.

Example 3: Janvi placed a number of ₹1 coins one over another. Name the solid shape that she formed. Write the number of faces, edges and vertices.

If a number of ₹1 coins is placed one over another, the shape that Janvi gets is a cylinder.

A cylinder has 3 faces, 2 edges and 0 vertices. Look at the objects and complete the table.

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 two squares of the same dimensions leaving one square unit from each vertex as shown.

Join the corresponding vertices to make a cube.

Example 4: Draw a cylinder with the help of rectangles on a square grid.

Draw a rectangle aligned properly with the grid lines as shown.

Draw a cuboid on the square grid. Do It Together

Join the vertices of the rectangle using curved lines to form a cylinder.

Do It Yourself 14A

1 Circle the shape that is a:

2 Write the number of vertices, edges and faces for the given shapes.

a Cone b Sphere

c Square Prism d Cylinder

3 The Skylon Tower is situated in Canada and is 160 m tall. Its construction was completed in 1965. What is the shape of the building?

4 What shape am I?

a I have 8 vertices, 6 flat faces and 12 edges

b I only have 1 face.

c I have edges but no vertex.

d I have a curved face and a vertex.

5 Sarika placed 4 cubes, one on top of the other. What shape did she get? Write the number of faces, edges and vertices.

6 Draw a cone on a square grid.

7 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?

1 A prism has two identical flat faces. Can a cylinder also be considered as a prism? Why or why not?

Nets and Views

Isha and Misha went to the bakery 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 as shown in the image.

Rohanʼs tent

Simiʼs tent

Remember!

A solid can have more than one net.

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

Let us look at the nets of a few more 3-D shapes.

Example 5: Which of these is not the net of a cube?

Hence, it is a cube.

As two faces are overlapping, it is not the net of a cube.

Example 6: Select the correct net of the given shape.

Did You Know?

Drawings of bridges, structures, homes, etc. are created using nets.

The shape has a pentagon-base with 5 triangular faces. Hence, 3 is the correct net.

Cuboid

Do It Yourself 14B

1 Identify the shape for which the net is drawn.

2 Look at the net and identify the solid shape it belongs to.

3 Draw the net of the given shapes.

4 Rishi and Megha made the net of a square-based pyramid. Who made the net correctly? Explain your answer with reasons.

a Rishi’s drawing

5 Draw the net of a hexagonal prism.

b Megha’s drawing

6 Show using nets how a rectangular prism is different from a rectangular pyramid.

Cuboid

1 A dice has P, Q, R, S written in a clockwise order on the adjacent faces and T and U at the top and bottom faces respectively. When R is on the top, what will be at the bottom?

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 she drew the side view. Which is the third view?

Drawing

The third view is the top view.

Example 7: Which is the top view of the shape?

1 is the side view of the shape. It is also the front view. 2 is the top view of the shape. Colour the square grids to make the top, front and side view of the object.

Misha’s

Isha’s Drawing

Which view is this?

Front view

Side view

Do It Yourself 14C

1 If you look at this object from the front, what will you see?

2 Draw the views of these figures.

a Right-side view of

b Top view of

3 Art Integration This table is made of many cubes joined together. Draw and label three different views of the table.

c Front view of

4 Mehar has made a shape using her building blocks. Count and write the number of building blocks she used to make the shape.

1 Dice are cubes with dots on each face. The opposite faces of a dice always have a total of 7 dots on them. If the front view shows 4 dots and the left side view shows 5 dots, how many dots will be shown on the back view and the right side view?

Maps and Floor Plans

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.

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 North, South, East and West.

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 size of the paper. This reducing of the actual size of a place to fit on a piece of paper is called scaling.

This is a map with the directions and scale marked on it.

In this map, the scale is 1 cm = 1 km.

Let us find the actual distance between National Stadium and India Gate with the help of the map.

Distance between National Stadium and India Gate on the map = 2 cm

Scale on the map = 1 cm = 1 km

So, the actual distance between National Stadium and India Gate = 2 × 1 km = 2 km.

We can also use a square grid to find and compare the areas of different places.

Do not miss the scale while reading a map in order to find the actual distance. Error Alert!

School Fire Station

Post Oﬃce House Park

Library Mall Pond

South East West North

Rajpath

Ashoka Road

Man Singh Road

Pandara road

Kasturba Gandhi

Example 8: Look at the map and answer the questions.

1 Which places are to the north and the east of the Little Town neighbourhood?

The hospital and the library are to the north and the Little Town Park is to the east of the Little Town neighbourhood.

2 Suhani is standing close to the Little Town Lake. In which direction is the post office from her location?

The post office is towards the south from Suhani's location.

Answer the questions and also find the area of different places given on the cm grid map.

1 The city park is of the highway.

2 The hotel is east of the highway.

3 The area of the city park is 12 cm2 on the grid.

4 The area of City Lake is on the grid.

5 The area of the highway is on the grid.

Do It Yourself 14D

1 Look at the map and answer the questions.

a The hospital is to the of the airport.

b The hotel is to the of the bank.

c The airport is to the of the school.

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?

3 What is the actual distance between the complex and the college?

= 12

4 What is the length and width of the cupboard, table and desk?

Map scale: 1 square = 2 m

1 cm = 12

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?

6 India shares its borders with several countries, known as neighbouring countries. Identify the nations situated to the far east, south, and west of India. (Look at the atlas for reference.)

1 Mihir goes to the stadium for football practice after school. On the map, his house is 2 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 = 3 km? Challenge

Critical Thinking

Floor Plans and Deep Drawings

On their way to the mall, the two sisters spotted some architects at a construction site, one of whom held a sheet of paper. 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.

It illustrates where the windows and doors of the house will be located.

The special way of drawing a house which shows 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.

Sometimes it is not possible to show all the windows and doors on a deep drawing. Remember!

Look at the drawings of a floor plan and a deep drawing of a house as given below.

Example 9: Compare the floor map and deep drawing and identify if the drawing is correct.

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.

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.

There are windows and door in the floor map.

Do It Yourself 14E

1 Look at the floor map of the house and label the doors and windows.

2 How many window(s) does the front of the house have in the floor map?

3 Draw the floor plan for the given deep drawing.

4 Which room will you be in if you enter through the back door?

5 Which are the window(s) that you cannot show in a deep drawing?

6 Draw a floor map of your classroom using symbols for different areas and objects.

1 Rakesh made the floor plan of his house which is shaped like a cuboid. He had 4 windows on each side of his house. How many windows can he not show in the deep drawing of his house? Draw a floor map and a deep drawing to show the same.

Points to Remember

• All 3-D objects have faces, edges, and/or vertices.

• All 3-D objects can be seen from different views—top, side 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

Setting: In groups of 5

Exploring Views and Deep Drawings

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 let the other groups draw the floor plan of this tower.

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.

2 A globe is like a round map that shows us how the Earth looks from space, with all the continents, countries and oceans. How many faces, vertices and edges does a globe have?

3 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 .

b A special way of drawing a house to show its length, width and height is called a .

c The 2-D representation of the map of a house is called a

4 Identify the shapes and write the number of faces, edges and vertices of each shape.

5 Draw the net of these shapes.

6 Draw the top, front and side views of the figures.

7 Match the distance on the map with the actual distance, if the scale of the map is 1 cm = 8 km.

8 The floor plan of a library is ready. Draw the doors and windows on the deep drawing of the library.

a Is there any window which you could not show on the deep drawing? Circle them.

b How many windows were you able to show on the deep drawing?

9 How many doors do you see in the floor plan?

11 Draw a floor plan of your house and show it to your friends/family.

10 Look at the layout of Ruchi’s apartment. How many times bigger is the bathroom than closet 1?

12 Ben goes from his house to May’s house and then to Jule’s house. What is the total distance covered by Ben?

Challenge

1 What is the minimum number of only flat faces a solid can have? Name the solid.

2 Draw the front, top and side view of a cone surmounted on a sphere.

Case Study

Tale of Three Viewpoints

A new building was made. It was taller than the buildings around it. Right in front of it was a small building. Three people were looking at the small building from different spots. Person 1 was looking at it from the top of the tall building. Person 2 was looking from the side of the building in the nearby park. Person 3 was looking from the front. This is shown in the picture below.

1 is Person ’s perspective. 2 is Person ’s perspective.

3 Write the views as seen by the 3 persons as front, side or top view.

4 What safety measures do you think Person 1 should take?

15 Time and Temperature

Letʼs Recall

Time is the measure of each moment in our lives. The basic unit of time is the second. We also use various units, such as minutes, hours, days, weeks, months and years to measure longer periods of time.

For example, brushing our teeth takes a few minutes; tying shoelaces requires just a few seconds; we spend several hours at school. We have a school timetable for a week, while the summer holidays last for months and birthdays occur once a year.

To read time, we rely on clocks. Clocks typically have three hands. The hour hand which is the shortest, points to the current hour, and the minute hand is the longest hand, which shows the current minute. Lastly, there is a thin hand that moves the fastest—it measures the seconds.

Letʼs Warm-up

Time

Kiran and Sahil are in the school cricket team. They are talking about the amount of time they spend on 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 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

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 equal to 60 minutes.

So, 3 hours = 3 × 60 = 180 minutes

Hence, they played for 180 minutes.

Did You Know?

1 millennium is 1000 years.

Example 1: Neethu takes 6 minutes to walk to her school bus stop from her home. For how many seconds does she walk?

Time spent walking = 6 minutes

1 minute = 60 seconds

6 minutes = 6 × 60 seconds = 360 seconds

Neethu walks for 360 seconds.

Think and Tell

How many seconds are there in a day?

Example 2: 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

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.

Kiran practised playing badminton for 75 minutes in the morning and 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 total time by 60.

Total number of minutes of practice = 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: How many hours and minutes are there in 220 minutes?

Total minutes = 220 60 minutes = 1 hour

Quotient = 3

Remainder = 40

So, 220 minutes is equal to 3 hours and 40 minutes.

Kiran and Rohit timed 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 = 1 = 250 × 250

Thus, Shreya juggled the tennis balls for minutes and seconds.

Do It Yourself 15A

Convert into minutes.

a 3 hours b 4 hours c 2 hours 20 minutes

d 3 hours 10 minutes e 3 hours 50 minutes f 4 hours 30 minutes

2 Convert into seconds.

a 5 minutes b 8 minutes c 9 minutes 10 seconds

d 10 minutes 20 seconds e 12 minutes 40 seconds f 15 minutes 50 seconds

3 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

f 550 seconds 8 minutes

4 A flight from Delhi to London takes around 8 hours and 30 minutes. How many minutes is the journey?

5 Maya reads a book for 50 minutes in the morning and 45 minutes in the afternoon. How much time does she spend reading the book?

6 Create a word problem to convert hours into seconds.

Sarah’s teacher asked her to practise on the piano for 1 hour and 50 minutes a day. She practised for 55 minutes in the morning and 50 minutes in the evening.

Which statement is true regarding this situation?

Statement 1: Sarah practised for the same duration as asked by the teacher.

Statement 2: Sarah should practise 5 more minutes to complete the duration requested by the teacher.

Calculating Time and 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.

Let us find out how much older Manish is than Sahil.

Step 1

Write the information.

Sahil’s age = 10 years 5 months

Manish’s age = 12 years 3 months

Step 3

Write the data vertically and perform the operation.

1 year 10 months11 3 + 12 = 15

12 years 3 months 10 years 5 months

Step 2

Apply the correct operation.

We need to find out how much older Manish is than Sahil, so we will subtract their ages.

Error

Alert! 1

Hence, Manish is 1 year 10 months older than Sahil.

Example 4: Subtract 4 hours 45 minutes from 5 hours 25 minutes.

5 hours 25 minutes - 4 hours 45 minutes = 0 hours 40 minutes or 40 minutes.

Example 5: Add: 7 weeks 4 days + 3 weeks 5 days 7 weeks 4 days + 3 weeks 5 days = 11 weeks 2 days

5 hours 25 Minutes 4 hours 45 Minutes 4 25 + 60 = 850 hours 40 Minutes 4 hours 85 Minutes 4 hours 45 Minutes7 weeks 4 days 3

Example 6: On November 20, Raj and Tina began counting the number of days left for their school's annual sports day. The annual sports day is on December 15. Find out the number of days they need to count down.

We know November has 30 days.

Number of days left in November from November 20 to November 30 = 10 days

Number of days in December starting from December 1 to December 15 = 15 days

So, adding the number of days = 15 + 10 = 25

Therefore, they are counting down 25 days until the sports day.

Example 7: Mahi started preparing for his match on June 23. The match is scheduled for 25 days later. On which date was the match scheduled?

Starting date of the preparation = June 23; Preparation time = 25 days

Date on which the match was scheduled = ?

We can find the date of the match by counting forward.

23 June to 30 June = 30 - 23 = 7

Days left after the month of June = 25 - 7 = 18

Therefore, Mahi’s match is on 18 July.

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 for which the experiment lasted = hours minutes

We need to find the time at which the experiment ended, hence we will add the data.

The experiment ended at hours minutes = p.m.

Do It Yourself 15B

What is the duration between the times shown on the two clocks?

c 3 weeks 10 days + 6 weeks 5 days

2 years 6 months + 1 year 5 months e 7 weeks 20 days − 2 weeks 5 days f 3 years 1 month − 7 months

3 If it’s 2:40 p.m. now, what time will it be 35 minutes later? Show the time on a clock.

4 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?

Solve.

5 A new museum opened on 25 October 2020, and closed on 15 February 2022. For how many months and days was the museum open?

6 A school was closed for 1 month 25 days during the summer vacation and for 20 days during the winter vacation. For how many months was the school closed? (Consider 30 days = 1 month)

7 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?

8 The Vande Bharat Express from Delhi to Katra departs from New Delhi at 6:00 a.m. and arrives at Katra at 2:00 p.m.. It halts at Ambala Cantt (8:10 a.m. to 8:12 a.m.), Ludhiana (9:19 a.m. to 9:21 a.m.) and Jammu Tawi (12:43 p.m. to 12:45 p.m.). For what duration does the train run from Delhi to Katra?

9 Create a word problem on time duration.

Meera’s school bus arrives at her bus stop at 8:10 a.m. every morning. Meera takes 15 minutes to get ready and needs to reach the bus stop at least 5 minutes before the bus arrives to maintain social distancing at the bus stop during the COVID-19 pandemic. It takes her 10 minutes to walk to the bus stop. At what time should she wake up to ensure she catches the bus on time?

Temperature

Riya and Aarav were spending their summer holiday 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 to her 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?

Measuring Temperature

Temperature is the measurement of how much heat an object or place has. 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)

2 Fahrenheit (°F)

However, when we talk about or write the temperature of something, we also use the degree along with the unit. For example, the temperature shown in the thermometer is 30℃.

Did You Know?

Aryabhata, in the 5th century, proposed a mathematical model for calculating time and studying the Earth's rotation, which influenced the development of calendars and timekeeping systems in India.

Remember!

Freezing point of water = 0℃

Water changes to steam = 100℃

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 or in the armpit 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

Temperature Range (°C) Weather

Below 0

Did You Know?

Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.

Above

Example 8: 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 = 37°C + 2.3°C = 39.3°C

So, Aliya’s body temperature was 39.3°C.

Example 9: 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.

Write the readings of the given thermometers.

Converting Between Units of Temperature

Let us look at the formulas that help us convert between Celsius (°C) and Fahrenheit (°F).

Celsius (°C) to Fahrenheit (°F) Fahrenheit (°F) to Celsius (°C)

For example: Converting 20 °C to °F.

Example 10:

(°F – 32) ()°−×=° 5 F 32 C. 9

For example: Converting 68 °F to °C.

(68 – 32) ()−×=° 5 68 32 20C. 9

So, 68°F is 20°C.

So, 20 °C is 68 °F.

a Convert 50°C to Fahrenheit (°F).

Putting in the formula,

9 50 32 5 122°F b Convert 86°F to Celsius (°C).

Putting in the formula, (86 – 32) ()−×=° 5 86 32 30C 9

If the temperature early in the morning was 12°C and warmed up by 9°C due to the sun, what was the temperature during the day? Also, convert the new temperature into Fahrenheit.

Initial temperature = 12°C

Increase in temperature =

New temperature = = Do It Together

Converting °C into °F:

9 C 32 F. 5

×+=°

So, °C is °F.

Do It Yourself 15C Chapter 15 • Time and Temperature

Choose the correct temperature.

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 Convert.

a 35 °C to °F b 80 °F to °C c 45 °C to °F d 149 °F to °C

4 The temperatures in 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 in Delhi and Shimla?

d How many degrees will the temperature need to rise in Bangalore to reach 25°C?

5 The highest temperature ever recorded in India was 51.0°C in Phalodi, Rajasthan while the lowest temperature ever recorded was –60.0°C in Drass, Jammu and Kashmir. What is the difference between the highest and the lowest temperatures in India? Give your answer in Fahrenheit.

In a city somewhere, something incredible happens with the temperature. On a particular day of the year, the temperature reads the same on both the Celsius and Fahrenheit scales. The relationship between Celsius (°C) and Fahrenheit (°F) is given by the formula:

Can you figure out at what temperature the values of °C and °F are the same?

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.

• The body temperature is around 37°C for a healthy individual.

• Different temperature ranges indicate different weather conditions.

Material Required:

Communication & Collaboration

• Prepare event cards with different scenarios (e.g., “Birthday,” “Sports Day”).

• A calendar.

Settings: Divide the class into small teams.

Method:

• Prepare time unit cards (years, months, weeks, days, hours, minutes and seconds).

Each member of the team draws an event card and a time unit card. In groups, 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.

Chapter Checkup

1 Convert the time into smaller units.

a 2 hours b 1 hour 30 minutes

c 2 hours 20 minutes d 6 minutes 40 seconds

2 Convert.

a 30 °C to °F b 175 °F to °C

3 A flight departs from New Delhi at 10:00 a.m. and takes 6 hours and 30 minutes, including a stopover, to reach Chennai. How many minutes does the flight take to reach its destination?

4 Shreya sleeps for 6 hours 15 minutes a day. For how many minutes does she sleep?

5 A soccer match started at 3:45 p.m. and continued for 1 hour and 15 minutes. When did the match end?

6 Prashant was 5 years 4 months old when he first went to school. Today he is 12 years 3 months old. For how long has he been going to school?

7 Sanya started to draw a picture at 1:32 p.m. She completed it at 5:15 p.m. How much time did she take to draw the picture?

8 Vivaan is fond of reading books. He starts reading a novel on 6 July and it takes him 40 days to finish the novel. On which day did he finish the novel? Why is reading books important for us?

9 The doctor advised Kunal to take his medicine every 75 minutes. How many times did Kunal take the medicine in 12 hours?

10 A train arrived at Jaipur at 10:45 a.m. It reached Jaipur 1 hour 15 minutes late. What was the scheduled time of arrival of the train at Jaipur?

11 Sara is in an air-conditioned room with a room temperature of 32°C. If she decreases the temperature by 2°C every minute, how many minutes will it take to reach 16°C?

Challenge

Critical Thinking & Cross Curricular

1 Read the assertion and reason, and choose your answer from the given options.

Assertion: Priya has taken 15 days leave from her office. Her holidays will start on Saturday, 25 June. The holidays end on Sunday, 10 July.

Reason: The number of days in the month of July is 31.

a Both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

b Both Assertion and Reason are true, but the Reason is not the correct explanation of the Assertion.

c The Assertion is true, but the Reason is false.

d The Assertion is false, but the Reason is true.

2 Vishesh was travelling from New Delhi to London. It takes 9 hours 35 minutes for a flight to reach London from New Delhi. The flight departs from Delhi airport at 6:00 a.m. (according to Indian standard time). What will be the time in London when the flight lands if London’s time is 5 hours 30 minutes behind Indian standard time?

Monitoring Climate Change

In a coastal town, scientists are studying the effects of climate change on the local environment. One aspect they are investigating is the change in temperature over the past few decades. Thus, for the past 30 years, researchers have been recording daily temperatures in the town. They have noticed a gradual increase in the average temperature over this period.

Data:

a In 1990, the average daily temperature was 20°C.

b In 2020, the average daily temperature was 24°C.

Answer the following questions:

1 What was the average daily temperature in 1990?

2 By how many degrees has the average daily temperature increased from 1990 to 2020?

3 The average daily temperature in 2020 was °C.

4 If the same trend follows, the temperature in 2050 would be °C.

5 The average daily temperature in the coastal town has decreased over the past 30 years. True / False

6 How do you think understanding changes in temperature over time can help us take better care of the environment?

16 Money

Letʼs Recall

In India, money is used in the form of rupees (₹) and paise (p). It is used 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 five rupees and fifty paise.”

The number to the left of the decimal point shows the rupee amount.

The decimal point separates the rupee amount from the paise.

How can we convert rupees to paise and paise to rupees?

The number to the right of the decimal point shows the amount in paise.

Remember that 1 rupee = 100 paise. Therefore, to convert rupees to paise, we multiply the given amount by 100, remove (₹) and (.) and write p for paise at the end.

For example, ₹7.25 = 7.25 × 100 = 725 p

To convert paise to rupees, we count 2 digits from the right, put a dot (.) and write the rupee symbol (₹) before the numbers. Otherwise, we can 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.

Working with Money

Mr Gupta owns a stationery store. He buys stationery items from a wholesaler and sells them to the customers at his shop for a 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

Find the cost of one pen.

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.

Step 2

Find the cost of 5 pens.

We know the cost of 1 pen = ₹6

Cost of required number of pens

= Cost of one pen × Number of pens

= ₹6 × 5 = ₹30

So, the cost of 5 pens is ₹30.

1. We divide to find the price of one unit.

2. We multiply to find the total amount for all units. Remember!

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.

Example 1: If 4 chocolates cost ₹40, the cost of one chocolate is .

Cost of 4 chocolates = ₹40

Cost of 1 chocolate = ₹40 4 = ₹10

Example 2: One dozen bananas cost ₹60. The cost of one banana is .

Cost of one dozen (12) bananas = ₹60

Cost of 1 banana = ₹60 12 = ₹5

Example 3: Aman buys 3 books 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

So, the cost of 1 book is ₹70.

Cost of 7 books = Cost of 1 book × Number of books wanted = ₹70 × 7 = ₹490

So, the cost of 7 books is ₹490.

Example 4: Which of these two will be a better buy?

Did You Know?

The Indian currency is called the Indian Rupee. Its short form is “INR”. It has a unique symbol '₹', which was adopted in 2010.

Rani pays ₹40 for 5 balls. Or Rani pays ₹56 for 8 balls.

When buying an item, a better buy is the one where we have to pay less money for the same number of 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

So, ₹56 for 8 balls will be a better buy.

Fill in the table using the unitary method.

S.No.

2 Rani pays ₹56 for 8 balls.

Cost of one ball = Total cost ÷ Number of balls = ₹56 ÷ 8 = ₹7

1 Cost of 3 pens = ₹60 Cost of one pen = ₹20 Cost of 5 pens = ₹100

2 Cost of 8 oranges = ₹80 Cost of one orange = ₹ Cost of 12 oranges = ₹120

3 Cost of 4 notebooks = ₹ Cost of one notebook = ₹50

Cost of 3 notebooks = ₹

It Yourself 16A

Fill in the blanks.

a If the cost of one candy is ₹5, then the cost of 7 candies will be ₹ .

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.

Find the cost of 20 candles at ₹15 per dozen. (1 dozen = 12 units)

Envelopes have been used in India for sending letters and documents. The cost of 35 envelopes is ₹630. What will be the cost of 57 such envelopes?

Suhani went to purchase cloth for her centre table. If 3 4 metres of cloth cost ₹54. Find the cost of 1 2 metre of cloth.

In 1947, at the time of India's independence, the price of gold was approximately ₹88.62 per 10 grams. What was the price of 100 grams of gold that time?

Dinesh has a spinal injury and he has been asked to take physiotherapy 3 days a week. A physiotherapist charges ₹550 for one sitting. How much money did Dinesh spend in 2 months?

Challenge

Critical Thinking

You have ₹50 to spend on school supplies. Pencils are priced at ʺBuy 2, Get 1 Freeʺ at ₹2 each. How many pencils can you buy without exceeding your budget?

Profit and 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 for ₹10, and one pen for ₹4.

Let us now understand the terms.

The price at which an item is bought. The price at which an item is sold.

Cost Price (CP) = ₹6

Selling Price (SP) = ₹10

Since SP > CP, a PROFIT was made.

Profit = SP – CP = ₹10 – ₹6 = ₹4.

Example 5: Find the missing value.

1 CP = ₹500, SP = ₹600, Profit = ?

Profit = SP − CP.

Profit = ₹600 − ₹500 = ₹100

3 CP = ₹1000, Profit = ₹100, SP = ?

Profit = SP − CP. So, SP = Profit + CP

SP = ₹100 + ₹1000 = ₹1100

Cost Price (CP) = ₹6

Selling Price (SP) = ₹4

Since SP < CP, a LOSS was made.

Loss = CP – SP = ₹6 – ₹4 = ₹2.

2 CP = ₹800, SP = ₹300, Loss = ?

Loss = CP − SP.

Loss = ₹800 − ₹300 = ₹500

4 SP = ₹250, Loss = ₹58, CP = ?

Loss = CP – SP. So, CP = SP + Loss

CP = ₹250 + ₹58 = ₹308

Example 6: The cost price of a toy is ₹1550. What will be the selling price if it is sold at a loss of ₹250?

Cost price (CP) of the toy = ₹1550

Loss made = ₹250

SP = CP – Loss = ₹1550 – ₹250 = ₹1300

Thus, the selling price (SP) of the toy = ₹1300

Remember!

Do It Together

Complete the table.

Mary bought a necklace for ₹500 and sold it for ₹750.

A shopkeeper bought a football for ₹200 and sold it for ₹500. ₹500

Yash bought a pair of shoes for ₹800 and sold them to his friend for ₹540.

Do It Yourself 16B

Fill in the blanks.

a The amount of money that a seller spends to pay for an item is called the

b When the cost price is higher than the selling price, then there is a .

c When the selling price is than the cost price, then the seller is said to have gained a profit.

d A flower vase is sold at ₹725 for a profit of ₹225. Its cost price is .

Raj set up his shop at the book fair. He sold a book for ₹540 and earned ₹30 on the book. Was my cost price more or less than the selling price?

Find the missing value.

a CP = ₹525, SP = ₹575, Profit = ?

b CP = ₹200, SP = ₹155, Loss = ?

c Loss = ₹550, SP = ₹7850, CP = ? d CP = ₹1200, Profit = ₹85, SP = ?

Find the missing element.

a SP = ₹1100

Profit/Loss =

c SP = ₹940

Loss = CP = ₹1100

b Profit = ₹240

SP =

The Gulf War in 1990-91 caused crude oil prices to rise. They rose from ₹262.50 per barrel in July 1990 to ₹612.50 per barrel by October 1990. This rise led to a significant rise in the cost of petroleum products in India. If a person purchased a barrel of oil in July and sold it in October then how much profit did he earn?

Create a problem in which CP and loss are given and SP needs to be calculated.

Challenge

Critical Thinking

A shopkeeper buys 3 kg of oranges for ₹510 and 4 kg of apples for ₹456. He sells the 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 on Money

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?

Total profit gained on each toy

CP of one toy = Total cost ÷ Number of units = ₹80 ÷ 2 = ₹40

CP = ₹40, SP = ₹100; Profit = SP – CP = ₹100 – ₹40 = ₹60

Thus, the profit gained on each toy is ₹60.

Thus, the profit gained on 70 toys = ₹60 × 70 = ₹4200.

Did You Know?

Earlier, people used to follow a barter system where goods and services were directly exchanged without using money.

Seema buys items from the wholesaler regularly to run her company.

The wholesaler makes a bill for the items she bought. He notes down the cost of each item and the number of items bought. Let us see what the bill looks like:

Bill No. 001/1

Darsheel’s Store Bill Date: 04/02/2023

Seema bought items worth ₹3350 from the wholesaler for her company.

To expand her business, Seema wants to set up a factory to make toys. She decides to take out a loan. A loan is money you borrow and have to pay back later. It is like getting help to buy something now and paying it back bit by bit. The bank also charges an amount over and above the loan. This extra amount that the bank charges, is called the interest.

Seema takes a ₹40,000 loan from the bank and pays ₹4000 every month. Let us calculate how many months it will take her to pay the money back if the interest amount is ₹4000.

No. of months = ₹44,000 ₹4000 = 11

Seema will be able to pay back the entire loan in 11 months.

Example 7: A shopkeeper lost ₹1750 on selling a refrigerator for ₹18,500. What was the cost price of the refrigerator?

SP of the refrigerator = ₹18,500

Loss = ₹1750

Loss = CP – SP

⇒ CP = SP + Loss

CP = ₹18500 + ₹1750 = ₹20,250

So, the cost price of the refrigerator is ₹20,250.

Did You Know?

Brahmagupta was an ancient Indian mathematician. He wrote important rules for arithmetic, including how to add and subtract numbers, which are also used when dealing with money.

Example 8: At the local fish shop, Mrs Patel bought 3 kg of fresh fish at ₹120 per kg. She also purchased 2 kg of prawns at ₹550 per kg, and some hooks and nets for a total of ₹600. Make a bill and calculate the total cost of Mrs Patel’s purchase.

The Fish Shop Bill Date: 05/01/2024

Thus, the total cost of Mrs Patel’s purchase is ₹2060.

Example 9: A worker takes a ₹25,000 loan from the bank to repair his house. He pays back the loan in 25 months. He paid an interest of ₹2000. How much does he pay each month if the payments are equal for each month?

Loan amount = ₹25,000; Interest amount = ₹2000;

Total Amount to be paid = ₹25,000 + ₹2000 = ₹27,000.

Number of months = 25

Amount paid each month = ₹27,000 25 = ₹1080.

Thus, the worker has to pay ₹1000 every month.

Example 10: Anita has ₹200 and wants to buy some chocolates that cost ₹25 each. How many chocolates can she buy with her money, and how much money will she have left with her?

Cost of 1 chocolate = ₹25

Number of chocolates she can buy with ₹200 = 200 25 = 8

Thus, Anita can buy 8 chocolates.

Since 8 × ₹25 = ₹200, Anita will not have any money left with her.

Example 11: Amy bought two shirts for ₹850 and sold both of them for ₹300 each. Calculate her profit/loss for:

1 One shirt:

CP of one shirt = Total cost ÷ Number of units = ₹850 ÷ 2 = ₹425

CP = ₹425, SP = ₹300

Now, since CP > SP, this means Amy incurred a loss.

Loss = CP – SP = ₹425 – ₹300 = ₹125

Thus, the loss incurred on one shirt is ₹125.

2 Two shirts:

Total CP of two shirts = ₹850

Total SP of two shirts = ₹300 × 2 = ₹600

Now, since CP > SP, this means Amy incurred a loss.

Total Loss = Total CP – Total SP = ₹850 – ₹600 = ₹250

Thus, the total loss incurred on two shirts is ₹250.

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 for:

1 One chocolate

CP of each chocolate = Total cost ÷ Number of units = ÷ 20 = ₹10

SP of each chocolate =

Since, SP > CP, Vansh gained a profit.

Profit = SP – CP = –

Profit gained on one chocolate is .

2 All chocolates

Total CP = ₹200

Total SP =

Since, SP > CP, Vansh gained a profit.

Total Profit = Total SP – Total CP = –

Thus, the total profit gained is .

Do It Yourself 16C

Sahil sold his TV for ₹30,000 at a profit of ₹1563. Find the CP of the TV.

A shopkeeper bought 20 pencil boxes for ₹480. If he sold each pencil box for ₹50, calculate the profit or loss on each pencil box.

At a trade fair, a merchant bought 15 kg of spices for ₹3000 and sold them for ₹250 per kg. Did he make a profit or a loss? How much was the profit or loss on each kilogram?

Ayan bought an old smartphone for ₹5620. He spent ₹530 to repair it. He then sold the smartphone to his friend for ₹6150. Did Ayan make a profit or a loss, and how much?

Bicycles produce no emissions, making them an eco-friendly mode of transport. Aryan bought a new bicycle for ₹7830 and spent ₹270 on its transportation. He then sold the bicycle for ₹8000. Find the profit or loss. What are the other benefits of cycling?

Rita goes to a toy shop and buys a doll for ₹150, 2 puzzles for ₹100 each and 3 balls for ₹50 each. Make a bill showing Rita’s purchases.

Rina took out a loan of ₹40,000 to buy a bike. She paid back ₹3500 every month for one year. How much money did she pay back to the bank? What was the interest amount?

The exchange rate between Singapore and India was: 1 SGD = ₹62.5.

Tanya takes a loan of 2000 SGD from a friend living in Singapore. In how many months will Tanya pay off the loan if she pays ₹5000 per month to her friend?

1 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): The amount of money you spend to buy an item.

• Selling Price (SP): The 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.

• 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.

• Loans are like borrowed money that you have to give back later, usually with extra money called “interest.”

• Bills shows you how much you owe for things, like groceries, electricity or services.

Ship Shop Shap!

Collaboration & Experiential Learning

Setting: In 2 groups

Materials Required: Pencils, pens, 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 cost price for the customers and the customers will buy items using play money and the shopkeepers will write down the selling prices. Ensure that students take turns in buying things.

4 Calculate the profit or loss after each transaction and discuss the results.

5 Make a bill for the customers for the items bought.

Chapter Checkup

State True or False.

a Selling price is the amount of money a seller pays to buy an item for his store.

b When CP < SP, the seller makes a profit.

c Priya bought a bag for ₹350 and sold it for ₹400. She made a profit.

d If the cost of 5 pencils is ₹30, then the cost of one pencil will be ₹8.

Math Lab

Calculate the profit or loss.

a CP = ₹167 and SP = ₹185 b CP = ₹36 and SP = ₹29 c CP = ₹147 and SP = ₹125

Kanika sold 20 bulbs for ₹15 each at a profit of ₹5 on all the bulbs. What was the cost price of each bulb?

Amit purchased two tea sets for ₹780 and ₹675. He sold both of them for ₹1450. Did he make a profit or a loss? How much was it?

A retailer bought 30 shirts for ₹200 each and 20 shirts for ₹100 each. He sold all of them for ₹300 each. Calculate the total profit or loss made by the retailer.

Which of these will be a better buy?

4 pencils for ₹24 Or 10 pencils for ₹50

Which is not a better deal: 36 chairs at ₹550 each or 22 chairs for ₹14,300?

A store bought 7 phones at ₹2100 each. They want to sell them at a total profit of ₹700. What should be the selling price of each phone?

Sharmin visited a nearby fruit shop and purchased 2 kg of apples at ₹50 per kg, along with 1 kg of bananas at ₹30 per kg and 3 kg of grapes at ₹40 per kg. Create a bill showing Sharmin's purchases.

In villages, farmers need new farming tools so that they can produce more food crops in a faster way. They borrow ₹5000 each from the local cooperative to buy seeds and equipment. They have to pay back the loan in 10 monthly payments without any extra money added. How much does each farmer need to pay back every month?

month.

Challenge

2 Diya spent 1 5 of her salary on rent, 1 4 of what was remaining on food. Then she spent 1 3 of what was left on clothes and transport. She saved the remaining ₹5000. What was Diya’s salary? Critical Thinking

1 Read the question below. Which of the statements is true?

3 workers can paint a house in 6 days for ₹9000. They divide the amount equally among themselves.

Statement 1: Each worker gets paid ₹500 every day.

Statement 2: In 1 day each worker will get 1 3 of the total amount.

Case Study

Shreya's Stamp Collection

Shreya works at a busy shopping centre. In her first year, she earned 2 valuable stamps. In her second year, she worked harder and earned twice as many stamps as she did in the first year. Shreya decided to sell all her stamps. Each stamp was worth ₹450, and she sold them at a profit of ₹50 per stamp.

Answer the questions given below:

1 Shreya collected stamps in her second year.

2 If Shreya sold all her stamps, how much money did she make?

a ₹1000 b ₹2250 c ₹3000 d ₹4500

3 How much would she have sold her stamps for, if she had sold each of them at a loss of ₹25?

a ₹100 b ₹225 c ₹300 d ₹425

4 Why do you think it is important to work hard and try to improve like Shreya did in her second year?

17 Data Handling

Letʼs Recall

Tally marks are a form of numerals that are used for counting and recording numbers. They are used primarily for small or quick calculations.

When a survey was conducted to find the favourite fruit of people in a colony, it was found that 35 people like apples, 30 people like oranges, 10 people like bananas and 25 people like kiwi.

A tally marks table was drawn to show this data.

Fruit

Letʼs Warm-up

The number of stamps that Kavya, Jishu and Rehan have is shown below in the form of a tally marks table. Read the table and answer some questions.

Fruit

1 Who has the highest number of stamps in their collection?

2 Who has the same number of stamps as Jishu?

3 Who has half of the total number of stamps?

4 What is the total number of stamps that the three children have?

5 How many more stamps does Kavya have than Rehan?

I scored out of 5.

Tally Marks

Bar Graphs

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.

Single and Double 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.

1 How many students chose football as their favourite sport?

The number of students who chose football is 10.

2 What is the favourite sport of most of the students?

The longest bar is that of cricket (green line), hence the favourite sport of most of the students is cricket.

The same data can be shown using a horizontal bar graph.

Think and Tell

drawing the bars horizontally make any

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.

Double Bar Graphs

We read double bar graphs in the same way we read single bar graphs.

Let us read the double bar graph shown above and answer the question.

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.

Example 1: The marks of a student in different subjects are shown in the bar graph. Read the graph and answer the questions.

1 How many marks did the student score in Science? 65

2 What is the difference between the marks scored in French, and the marks scored in SSc.? 75 – 70 = 5 marks

3 In which subject were the highest marks scored? Maths

Example 2: The number of laptops sold by two different brands in 5 months is shown using a double bar graph. Read the graph and answer the given questions.

1 How many total laptops were sold by Brand 1?

Number of laptops sold by Brand 1 in 5 months = 50 + 70 + 60 + 90 + 80 = 350 laptops

So, the total number of laptops sold by Brand 1 = 350 laptops

2 Which brand sold more laptops and how many more?

Number of laptops sold by Brand 1 = 350 laptops

Number of laptops sold by Brand 2 = 70 + 70 + 80 + 80 + 60 = 360 laptops

Difference between the number of laptops sold by both brands = 360 – 350 = 10 laptops

Thus, Brand 2 sold 10 more laptops then Brand 1.

Read this bar graph. It shows the runs scored by two players in different matches. Then, answer the questions.

Drawing Bar Graphs

Rohan now tries to find the favourite subject of his classmates. He first writes the data in tabular form.

Look at the correct bar while answering the question. Remember!

Let us help him draw a bar graph for the data.

Step 1: Draw a vertical and horizontal line as shown below. Choose a scale as 1 unit = 2 students. The vertical line is the y-axis and the horizontal line is the x-axis.

Step 3: Draw rectangular bars for all subjects adjacent to each other. Keep the width of the bars the same. Remember, the height of the bar should match the number given in the table.

Step 2: Mark the horizontal line with subjects and vertical line with numbers at fixed intervals so that all the readings can be marked on the graph.

The same graph can be drawn by interchanging the data on the x and y axes as shown below.

Example 3: Read the table showing the favourite fruit of 57 children. Draw a horizontal bar graph using the data.

Fruits No. of Children

Grapes 20 Bananas 12

The number of trees planted by an organisation is given below. Complete the bar graph.

The bar graph represents the number of students in different sections of Grade 5 of a school. Which section has the least and the most students?

Recycling is the process of taking old or used materials and turning them into new products instead of throwing them away. 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.

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 books. Here is the data they collected:

Create a bar graph to represent this data.

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 between the tickets sold to children on Sunday and those sold on Friday?

b What is the total number of tickets sold throughout the four days?

c How many more tickets were sold on Sunday as compared to Saturday?

Create two more questions on the bar graph given in Q4.

Refer to the bar graph in Q4 and answer the given question.

On Monday, Tuesday and Wednesday, the total number of tickets sold to adults was two-thirds of the total number of tickets sold to adults on Saturday and Sunday. The total number of tickets sold to children was four-fifths of the total number of tickets sold to children on Friday and Saturday. How many total tickets were sold in the three days?

Jaya and Shreya wanted to treat their mother on Mother’s Day. So, they decided to prepare apple pie for the party. Their mother was delighted by the surprise. She decided to divide each pie into 16 equal parts.

Reading and Drawing Pie Charts

Let us now learn to read and draw pie charts.

Interpreting Pie Charts

Jaya, Shreya and their mother shared one 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

2 How many more slices of apple pie did Shreya eat as compared to Jaya?

Slices of apple pie eaten by Shreya = 1 2 of 16 = 1 2 × 16 = 8slices

Slices of apple pie ate by Jaya = 3 8 of 16 = 3 8 × 16 = 6 slices

We multiply the numerator of the fraction with the whole number and divide the product by the denominator.

Difference between the number of slices eaten by Shreya and Jaya = 8 – 6 = 2 slices.

Example 4: Look at the pie chart. Rhea and her classmates were asked about their favourite subjects. If 72 students were surveyed, how many students voted for art?

The pie chart shows that 1 4 of the students voted for art as their favourite subject.

Hence, the number of students who voted for art = 1 4 of 72 = 1 4 × 72 = 18 students.

The pie chart below shows how a family spends its monthly income. Use the chart to answer the questions.

1 What is the largest expense of the family?

On comparing the fractions, 3 _____ _____ 10 >>

Hence, is the largest expense in the family.

2 If the family decides to cut their entertainment expenses by half, what fraction will they be spending on entertainment?

Representing Data on a Pie Chart

Do you remember Jaya, Shreya and their mother sharing an apple pie? After some days, they prepared some laddoos to distribute among needy people. Father also helped them. Below is a table showing the number of laddoo boxes filled by each member.

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.

Step 1

Find the fraction and angle for each category by multiplying 360° with the fraction of each category.

Step 2

Draw a circle with any radius and mark the radius. Mark the first angle, keeping the radius as the reference line.

Step 3

Mark the next angle, keeping the previous line as the reference. Continue until all the angles are marked.

Step 4

Colour the categories and write the data inside the pie chart.

Error Alert!

The sum of the angles inside a pie chart is not necessarily 360°.

The sum of the angles inside a pie chart is always 360°.

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:

Draw a pie chart to show the given data of students across these activities during the event.

Do It Yourself 17B

Riya gets a fixed amount of pocket money from her parents, and she uses it responsibly. The pie chart shows the uses of pocket money managed by Riya.

a What activity consumes the largest portion of her pocket money?

b If she receives ₹100 as pocket money, how much money goes towards savings?

c If she decides to spend only 1 10 of her pocket money on snacks/food, how much money is that approximately?

Rohan recorded the time he spends on different activities during 12 hours over the weekend. Here is the data he collected:

Playing: 3 hours

Studying: 4 hours

Watching TV: 2 hours

Reading: 3 hours

a Find the fractions and angles for the data. b Draw a pie chart and label the data.

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.

Read the pie chart showing the data for 200 students and answer the questions.

a How many students are interested in Maths?

b If 40 students are interested in Science, what fraction of the total students does this represent?

c If 10 students are interested in EVS, what fraction of the students are interested in EVS?

The pie chart shows the global energy consumption. If the total global energy consumption is 13,000 million tonnes then how much total energy is consumed by nuclear and renewables combined?

a 1625 million tonnes b 1040 million tonnes

c 1560 million tonnes d 520 million tonnes

Line Graphs

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.

Reading Line Graphs

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.

1 In which year did the siblings read the most books?

To find the greatest number of books, look at the peak (highest point) of the given line graph. Now look at the year corresponding to the peak.

Hence, 45 is the greatest number of books read by the siblings in the year 2020.

2 How many books did the siblings read in 2021?

and y axes.

Looking at the corresponding data for 2021, the siblings read 40 books in this year.

3 How many books did the siblings read in total for these 5 years?

To find the number of books read in each year, look at the corresponding data for each year.

Hence, the total number of books read across 5 years = 25 + 30 + 45 + 40 + 40 = 180 books. Given below is a line graph that shows the annual food grain production (in tonnes) from 1992 to 1997. Read the graph and fill in the blanks.

1 The production of food grain in the year 1992 was 20 tonnes.

2 The highest production of food grain was in the year .

3 The difference in production between the year 1995 and the year 1993 was .

4 The total production across the years = .

The graph shows Kavya’s internet usage across five days. On which days did she spend the maximum number of hours on the internet?

The line graph shows the number of persons who visited different cities on a certain day. Name the cities visited by the most and the least number of people.

Gulmarg is a beautiful hill station and popular tourist destination located in the Baramulla district of Jammu and Kashmir, India. The line graph represents the weekly minimum temperature readings (in °C) of Gulmarg for five weeks. Study the graph carefully and answer the questions.

a What was the temperature recorded during the second week?

b In which week was the highest temperature recorded?

c What was the temperature difference between the first and the last week?

Rahul drew a line graph to show the sales of guitars over the course of six years. Read the graph and answer the questions.

a In which year did Rahul sell 700 guitars?

b How many guitars were sold in the year 2021?

c In which year was the least number of guitars sold?

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 sales cloth double of that in May?

b What were the average sales over the period of five months?

Pune Kolkata Delhi

City

Points to Remember

• A bar graph is a visual representation of 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

Let’s Do Plotting

Aim: Students will learn how to create, interpret and analyse simple bar graphs and pie charts using real-time data.

Setting: In groups of 2

Materials Required: Chart paper, coloured pencils or crayons, ruler, coloured rectangular strips, coloured circular cutouts of the same-sized circles, glue Method:

The students from each group will collect real-time data of any type and record it in the form of a table.

Next, the group will analyse the data and represent it using bar graphs and pie charts.

For the bar graph, paste the coloured rectangular strips to represent the bar graphs.

For the pie charts, cut out the appropriate sectors from the coloured circular cutouts and paste these adjacent to each other to form the pie chart.

The group that accurately represents all the data in both types of graphs wins!

Chapter Checkup

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 graph and fill in the blanks.

a families like cats.

b families prefer birds and rabbits as their pets.

c The number of families who like dogs as pets is more than the number of families who like fish as pets.

The graph shows the number of pages read by boys and girls for five days of a week. Read the graph and answer the questions.

a On which day did boys and girls read an equal number of pages?

b Who reads more pages on Wednesday?

c What is the difference in the total number of pages read by girls and by boys?

The data represents the sale of refrigerators by a showroom in the last six months. Draw a horizontal bar graph.

The total area of the oceans on the Earth is about 362 million sq. km. The pie chart shows the area of the 5 oceans on Earth .

a What is the area of the Pacific Ocean?

b What is the total area of the Indian and the Atlantic Oceans?

c How much more is the area of the Antarctic Ocean than the Arctic Ocean?

a On which day did the fruit seller sell 15 kg of fruit?

The data shows the number of hours spent by Kunal on different activities in a day. Draw a pie chart for the given data.

b How many kilograms did he sell together on Saturday and Sunday?

c What is the difference between the sales of the initial 4 days and the last 3 days?

Read the graph showing the temperature of different cities. Answer the questions.

a Which is the hottest city?

b Which is the coldest city?

c What is the temperature in Chennai?

d Which city has a temperature of 10℃?

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.

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 Which ice cream is the most popular?

b Which is the least popular ice cream?

c How many children like chocolate ice cream?

d How many children like either vanilla or strawberry ice cream?

Create a question based on the graph in Q9.

Challenge

1 A fertiliser is a substance added to soil or plants to help them grow better and stronger. The bar graph shows the production of fertilisers by a company over the years.

If the production in 2003 was twice of that of 1998, and in 2004, it was 8 6 of 1997, then what is the total production of fertilisers over the span of 10 years?

2 600 students were surveyed on their favourite foods. Read the pie chart and identify the options that are not true.

a 375 students have Indian food as their favourite food.

b 60 students have Mexican food as their favourite food.

c 90 students have Italian food as their favourite food.

d 85 students have Chinese food as their favourite food.

Wildlife in India

Ruhaan is a wildlife biologist. He studies animals and other wildlife and how they interact with their ecosystems. Ruhaan collected data on the number of tigers and elephants for the year 2022 in different states of India and drew a bar graph on it. Read the bar graph and answer the questions.

1 What is the total number of tigers in the 4 states?

a 198 b 202 c 204 d 200

2 What is the difference between the number of tigers and elephants in the 4 states altogether?

a 30 b 35 c 38 d 43

3 Draw a horizontal bar graph to represent the same data.

4 Which state had 12 more elephants than tigers?

5 How can we conserve endangered (a species that is at serious risk of extinction) species like tigers and elephants? Cross Curricular &

Answers

Chapter 1

Let’s Warm-up 1. 8,00,000 2. four lakh thirty-six thousand five hundred thirty-six 3. 4 4. thousands

Do It Yourself 1A

1. a. 9000; 9 b. 50000; 5 c. 90000000; 9 d. 20000000; 2

e. 60000000; 6 f. 800; 8 g. 3000000; 3 h. 100000; 1

2. a. Indian System: 2,16,43,332; Two Crore Sixteen Lakh FortyThree 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; International System: 1,200,621

c. Indian system: 4,62,07,219; International System: 46,207,219

d. Indian system: 9,59,10,158

3. a. 4,00,00,000 + 10,00,000 + 90,000 + 800 + 80 + 7

Four crore forty-one lac ninety thousand eight hundred eighty-seven

b. 1,00,00,000 + 90,00,000 + 80,000 + 1000 + 700 + 2

One crore ninety lac eighty-one thousand seven hundred two

c. 80,000,000 + 1,000,000 + 80,000 + 5000 + 400 + 30 + 2

Eighty-one million eighty-five thousand four hundred thirty-two

d. 10,000,000 + 9,000,000 + 800,000 + 50,000 + 4000 + 4

Nineteen million eight hundred fifty-four thousand four

4. a. 60,08,098 b. 20,000,569 c. 4,090,000

d. 8,00,01,002 5. a. 1 b. 10 c. 10,000 d. 7

6. Indian number system- 27,22,000

International number system- 2,722,000

7. a. 91,19,199 b. 170,70,707 8. 353,000,000

Challenge 1. Option c

Do It Yourself 1B

1. 76,24,578

2. a. < b. = c. > d. < e. < f. <

3. a. 1,00,00,000 b. 2,90,52,468 c. 10,00,000 d. 3,00,52,468

4. a. Increasing order: 93,12,820 < 1,00,36,782 < 5,00,00,367 < 8,87,21,460

Decreasing order: 8,87,21,460 > 5,00,00,367 > 1,00,36,782 > 93,12,820

b. Increasing order: 36,81,910 < 92,56,890 < 6,92,10,350 < 8,26,00,031

Decreasing order: 8,26,00,031 > 6,92,10,350 > 92,56,890 > 36,81,910

c. Increasing order: 5,00,21,138 < 6,04,50,821 < 6,50,24,567 < 9,45,21,823

Decreasing order: 9,45,21,823 > 6,50,24,567 > 6,04,50,821 > 5,00,21,138

5. a. 98,54,310; 10,34,589 b. 87,65,321; 12,35,678

c. 65,43,210; 10,23,456

6. 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

7. 9,88,75,420 and 2,04,57,889

8. a. 9,99,99,998; 10,00,000 b. 9,99,99,876; 10,00,023

c. 9,99,98,765; 10,00,234

9. Descending Order = Russia, China, Australia, India

10. Answers will vary. Sample answer.

The government of a country has allocated its national budget for the upcoming fiscal year. The budget for the education sector was ₹45,678,912 and for healthcare was ₹45,345,678.

Which sector had a greater budget?

Challenge 1. 97653210, 10235679, 9765321, 1023567

2. 39,87,654

Do It Yourself 1C

1. a. 85,48,750 b. 89,05,460 c. 6,07,85,890

d. 1,56,48,950

2. a. 1,25,89,200 b. 87,52,400 c. 68,67,800

d. 77,59,900

3. a. 8,97,00,000 b. 53,12,000 c. 8,21,59,000

d. 5,89,90,000 4. 239,000 approximately

5. The municipal corporation spent ₹65,95,000 on planting trees.

6. a. 99,63,210; 99,63,000 b. 96,32,100; 96,32,000

Challenge 1. 3,23,45,500 and 3,23,46,449

Chapter Checkup

1. a. Indian system: 35,07,681; Thirty-Five Lakh 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 Eighty-One; 3,000,000 + 500,000 + 7000 + 600 + 80 + 1

b. Indian system: 4,20,87,950; Four Crore Twenty Lakh 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 Crore Thirty-Five Lakh 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 Forty-Two; 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 EightyFour; 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

2. a. 60,715,239; 6,07,15,239 b. 8,09,50,002; 80,950,002

c. 1,100,039; 11,00,039

3. a. 6,45,87,120; 6,45,87,100; 6,45,87,000

b. 89,09,010; 89,09,000; 89,09,000

4. a. > b. < c. = d. <

5. 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

6. 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

7. 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

8. 3,151,324

Indian number system - 31,51,324 = 30,00,000 + 1,00,000 + 50,000 + 1,000 + 300 + 20 + 4

International number system- 3,000,000 + 100,000 + 50,000 + 1,000 + 300 + 20 + 4

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 Seventy-Seven Thousand Five Hundred Fifty-five b. 1,99,77,560; 1,99,77,600; 1,99,78,000

Challenge 1. 12,15,696 2. 99887765

Case Study

1. Option c 2. Option d 3. Germany

4. Poland, Italy, France, United Kingdom, Germany

5. Germany

Chapter 2

Let’s Warm-up 1. 90 2. 5500 3. 2350 4. 280

Do It Yourself 2A

1. a. True 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. 7,28,944 b. 4,62,917 c. 7,71,214 d. 7,24,168

5. a. 81,898 b. 9,31,609 c. 2,47,880 d. 5,31,550

6. 9,03,550 7. 2,72,950 blazers 8. 22,448 kilometres

9. ₹3,81,030

Challenge 1. 47,865 to 99,999

Do It Yourself 2B

1. a. False b. False c. False d. True

2. a. 5,65,670; 56,56,700; 5,65,67,000

b. 4,78,520; 47,85,200; 4,78,52,000

c. 8,25,870; 82,58,700; 8,25,87,000

d. 19,84,540; 1,98,45,400; 19,84,54,000

3. a. 2500; 250; 25 b. 35,400; 3540; 354 c. 89,500; 8950; 895 d. 98,700; 9870; 987

4. a. 61,525 b. 27,36,600 c. 78,71,525 d. 1,46,55,870 e. 3,03,57,504 f. 4,45,49,588 g. 4,72,86,288 h. 5,35,25,760

5. a. Quotient = 183; Remainder = 134 b. Quotient = 124; Remainder = 228 c. Quotient = 159; Remainder = 280 d. Quotient = 161; Remainder = 219 e. Quotient = 233; Remainder = 366 f. Quotient = 135; Remainder = 365 g. Quotient = 136; Remainder = 15 h. Quotient = 102; Remainder = 219

6. ₹28,28,232 7. 159 pages 8. ₹680

Challenge 1. 9,60,000 chairs

Do It Yourself 2C

1. a. (ii) b. (ii) c. (ii) d. (ii)

2. a. False b. True c. True d. False

3. a. 266 b. 44 c. 12 d. 40 e. 15 f. 16

4. a. 168 b. 272 c. 3171

5. ₹78 6. 48 stickers 7. 7 pencils 8. ₹74,750

Challenge 1. So, the resultant number is 348.

Chapter Checkup

1. a. 80,245 b. 10,33,428 c. 12,41,211 d. 3,02,572

2. a. 20,101 b. 7,68,289 c. 8,03,271 d. 1,09,241

3. a. 8960; 89,600; 8,96,000 b. 45,460; 4,54,600; 45,46,000 c. 64,570; 6,45,700; 64,57,000 d. 98,760; 9,87,600; 98,76,000

4. a. Quotient = 210; Quotient = 21; Quotient =2, Remainder = 100 b. Quotient = 408, Remainder = 6; Quotient = 40, Remainder = 86; Quotient = 4, Remainder = 86 c. Quotient = 5120; Quotient = 512; Quotient = 51, Remainder = 200 d. Quotient = 7456, Remainder = 2; Quotient = 745, Remainder = 62; Quotient = 74, Remainder = 562

5. a. 29,32,848 b. 17,29,200 c. 1,28,80,901 d. 40,83,256

6. a. Quotient = 237; Remainder = 20 b. Quotient = 180; Remainder = 265 c. Quotient = 115; Remainder = 437 d. Quotient = 101; Remainder = 684

7. a. 27 b. 76 c. 43 d. 179 8. 45

9. 8, 99, 999 10. 7,03,66,258 people 11. 50 days 12. 38,400 flowers 13. 149 rows 14. 65 marbles

15. ₹4,87,800 16. 6,00,000 17. 2226 oranges in each carton

18. ₹1,20,500 19. Answer may vary. Sample answer: A company manufactures 1,86,320 pencils and 98,346 erasers. The company packs 2 pencils and 1 eraser in a packet. How many pencils and erasers will be left unpacked?

Challenge 1. Both statements are true. 2. Option a

Case Study

1. 12,10,111 kg 2. a. False b. True

3. 172 kg 4. 210 kg

Chapter 3

Let’s Warm-up 1. 1, 2, 4 2. 1, 3, 9 3. 1, 3, 5

4. 1, 23 5. 1, 5, 11

Do It Yourself 3A

1. a. 1, 2, 7 and 14 b. 1, 2, 17 and 34

c. 1 and 37 d. 1, 2 , 3, 4, 6, 8, 12, 16, 24 and 48

e. 1, 5, 11 and 55 2. a. 1, 3, 5 and 15 b. 1 and 41

c. 1, 3, 7, 9, 21 and 63 d. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72

e. 1, 3, 9, 27 and 81

3. Prime numbers: a, c, g, i

Composite Number: b, d, e, f, h, j

4. 4 prime numbers - 11, 13, 17 and 19 5. 2 factors

6. The chairs will be arranged in 13 rows with 13 chairs in each row.

7. Answer may vary. Sample answer

Alika has a ribbon of length 27 m. In how many ways she can cut the ribbon equally?

Challenge 1. True. 2 is a factor of 10.

Do It Yourself 3B

1. b, c, e 2. d, e 3. a, b, d 4. a, b 5. 660

6. pencils, erasers and notebooks

Challenge 1. Yes, because both 324 and 135 are divisible by 9.

Do It Yourself 3C

1. b 2. c 3. a. 8 = 2 × 2 × 2 b. 20 = 2 × 2 × 5

c. 24 = 2 × 2 × 2 × 3 d. 33 = 3 × 11 e. 63 = 3 × 3 × 7

f. 72 = 2 × 2 × 2 × 3 × 3 g. 90 = 2 × 3 × 3 × 5

h. 112 = 2 × 2 × 2 × 2 × 7

4. a. 16 = 2 × 2 × 2 × 2 b. 22 = 2 × 11 c. 30 = 2 × 3 × 5

d. 45 = 3 × 3 × 5 e. 51 = 3 × 17 f. 60 = 2 × 2 × 3 × 5

g. 100 = 2 × 2 × 5 × 5 h. 148 = 2 × 2 × 37 5. 1, 5, 73, 365

Challenge 1. 11

Do It Yourself 3D

1. a. 1 b. 1 c. 1, 7 d. 1, 3 e. 1, 3, 5, 15

f. 1, 2, 3, 6 g. 1, 2 h. 1, 5

2. a. 8 b. 15 c. 5 d. 1 e. 27 f. 14 g. 10

h. 11 3. a. 3 b. 13 c. 18 d. 8 4. a. 6

b. 14 c. 20 d. 27 5. 5 cm 6. 30 days

Challenge 1. 9 litres

Chapter Checkup

1. Prime Numbers: b and c

Composite number: a and d

2. a. 1, 2, 4, 8 and 16 b. 1, 2, 4, 5, 10 and 20

c. 1, 2, 3, 4, 6, 8, 12 and 24 d. 1, 2, 3, 6, 7, 14, 21 and 42

3. a. 1, 2, 4, 7, 14 and 28 b. 1, 2, 3, 4, 6, 9, 12, 18 and 36

c. 1, 2, 4, 7, 8, 14, 28 and 56

d. 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80

4. a. divisible by 5 b. divisible by 3

c. divisible by 2, 3, 5, 9 and 10 d. divisible by 2, 5 and 10

5. a. 88 = 2 × 2 × 2 × 11 b. 102 = 2 × 3 × 17

c. 112 = 2 × 2 × 2 × 2 × 7 d. 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

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 answer: 120

12. 25 litres 13. 8 bunches 14. 6 inches; 44

15. 80; 3 cupcakes; 2 sandwiches 16. 6 containers

Challenge 1. 10 is the greatest number that divides 178 and 128 leaving remainder as 8.

2. HCF is 4. Twice the HCF is 4 × 2 = 8. So, the number is 8. Case Study

1. d 2. b 3. 32

4. 12 necklaces

Chapter 4

Do It Yourself 4A

1. a. 8, 16, 24, 32, 40 b. 11, 22, 33, 44, 55 c. 21, 42, 63, 84, 105 d. 25, 50, 75, 100, 125 e. 50, 100, 150, 200, 250

2. Yes, 7209 is completely divisible by 9.

3. 2025 4. Answers may vary. Sample answer 10, 20, 30, 40, 50, 60, 70, 80, 90, 100; Even. 5. No

Challenge 1. Yes A can also be a multiple of B. For example, if B = 8, the factors are 1, 2, 4, 8. If A = 1, 2 or 4, it cannot be B’s multiple but of A = 8, it becomes a multiple of B which is also 8.

Do It Yourself 4B

1. d 2. a. 60 b. 63 c. 55 d. 40 e. 18 f. 30

g. 60 h. 150 3. c 4. 2036 5. 36 days 6. 10:30 a.m.

7. Answers may vary. Sample answer.

A shopkeeper sells candles in packet of 12 and candle stands in packets of 8. what is the least number of candles and candle stands Rita should buy so that there will be one candle for each candle stand?

Challenge 1. 12

Do It Yourself 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. a. 40 b. 16 c. 80 d. 50

3. a. 48 b. 175 c. 180 d. 315 e. 360 f. 864 g. 1680 h. 180

4. No, LCM of 12, 24, and 56 = 2 × 2 × 2 × 3 × 7 = 168

So, Rohan is not right.

5. 4 times

6. Answers may vary. Sample answer.

Find the least length of a rope which can be cut into whole number of pieces of length 45 cm and 75 cm.

Challenge 1. 63 years

Do It Yourself 4D

1. a. 168 b. 150 c. 96 d. 300 e. 168 f. 300 g. 315 h. 540

2. No, Rohit’s calculation is incorrect, as he didn’t consider all the prime factors.

3. A B

LCM of 42, 70 210

LCM of 63, 105 315

LCM of 30, 45 90

LCM of 9, 15 45

LCM of 12, 48 48

4. 60

Challenge 1. a. Only conclusion 1 is true.

Do It Yourself 4E

1. August 7, 14, 21, 28. 2. 6 groups

3. 15 litres 4. 10:31 a.m. 5. 24 balls

6. 48 students 7. 30 weeks 8. 3360 stacks

9. Answers may vary. sample answer.

Three pieces of ribbons 42 cm, 49 cm and 63 cm long need to be divided into the same length. What is the greatest possible length of each ribbon?

Challenge 1. a. 10 visits b. 10th visit

Chapter Checkup

1. 9, 18 and 27 2. a 3. a. 63 b. 81 c. 99 d. 117 4. b 5. a. 630 b. 90 c. 1386 d. 6930 e. 180 6. a. 54 b. 70 c. 45 d. 600 7. d 8. a. 480 b. 3720 c. 360 d. 630 9. 39 10. 180 minutes 11 2041

Challenge 1. c. Rahul and Sahil are correct. 2. a

Case Study

1. b. Every 60 minutes 2. 9:00 a.m. 3. 10 4. 4:00 p.m. 5. Answers may vary.

Chapter 5

Let’s Warm-up 1. one-third 2. a half 3. one-fourth

4. one-eighth 5. two-fourths

Do It Yourself 5A

1. a. Unlike fractions b. Like fractions c. Unlike fractions

d. Like fractions e. Like fractions

2. Proper fractions: 41212 ,,. 51529

Improper fractions: 658179 ,,,. 25113

Mixed numbers: 157 4,5,6. 789

3. a. 61 15 b. 14 20 15 c. 79 15 d. 15 e. 44

4. a. 2 4 3 b. 2 8 7 c. 2 5 5 d. 3 7 6 e. 8 7 9 f. 9 2 g. 18 7 h. 55 9 i. 35 3 j. 135 11

5. 2 17 5

6. Mixed Number 2 3 6 Improper Fraction 15 6

Challenge 1. 22 3

Do It Yourself 5B

1. Answers may vary, sample answers:

a. 1218 , 1421 b. 1218 , 3045 c. 2233 , 3451 d. 2436 , 4060

2. Answers may vary. Sample answers:

a. 63 , 105 b. 72 , 288 c. 1510 , 2114 d. 168 , 2412

3. a. 2 5 b. 3 7 c. 1 3 d. 4 5

4. a. Not equivalent b. Equivalent

c. Equivalent d. Equivalent

5. a. 28 b. 56 c. 2 d. 70 6. d.

7. Answer may vary. Sample answer = =

Challenge 1. Green- 1 6, Pink - 3 18 or 1 6 , Blue - 1 4 or 3 12, Purple - 1 6, Yellow- 1 4 or 3 12 ; Green, pink and purple are equivalent. Yellow and blue are equivalent. Do It Yourself 5C

1. a. < b. < c. > d. <

2. a.  b.  c.  d. 

3. a. 132 674 << b. 1125 5236 <<<

c. 31575 13412127 <<<< d. 5487 128158 <<<

4. a. 3311 5846 >>> b. 8131 9283 >>>

c. 1111 691114 >>> d. 42431 5311156 >>>>

5. Answer may vary. Sample answer.

6. Reshma 7. Answer may vary. Sample answer.

Shalini studied for 14 9 hours, Priya studied for 8 3 hours.

Who studied for more hours?

Challenge 1. Statement II

Chapter Checkup

1. a. 15 2 b. 17 3 c. 103 8 d. 51

3. Answers may vary. Sample answers: a. 468 ,, 121824 b. 234 ,, 182736

c. 101520 ,, 162432 d. 223344 ,, 284256

4. Answers may vary. Sample answers: a. 842 ,, 1263 b. 1284 ,, 21147 c. 25155 ,, 653913

d. 543618 ,, 664422

5. a. 3

7. a. 1514 61225 <<< b. 1313 5825 <<<

c. 2126 5237 <<< d. 14311 27512 <<<

8. Least: Alex and Casey; Greatest: Bailey

9. Australia < South America < North America < Europe < Africa < Asia

Challenge 1. 3 8 2. Option a

Case Study

1. b 2. a 3. 8 10 4. False 5. Answer will vary.

Chapter

Do It Yourself 6A

9. Answers may vary. Sample answer.

Neeta

Challenge 1. b Do It Yourself 6B

8. 20 L 9. 19 bottles 10. Answers may vary. Sample answer. Roy ran 4 5 of a km each day for 6 days. How many kms did he run in all?

Challenge 1. No Chapter Checkup

1. a. 5 4

4. a. 13 1 20

6. a. 1 12 2 b. 16 c. 11 15 d. 5 162

7. a. Sum of 3 and 1 4 1 2 4 b. Product of 5 and 4 5 c. 2 5

8. 71 100 9. 11 17km 28

10. 357 km; Phase 1: 65 357 ; Phase 2: 125 357 ; Phase 3: 167 357

11. 88 cars 12. 9 10litres 28 13. 23 4m 30

14. Answers may vary. Sample answer.

A glass of water holds cups of water. How many 1 4 cups will it take to fill the glass?

Challenge 1. 1 3 cup; 4 cheese cubes 2. b

Case Study

1. c 2. b 3. 1 13 20 hours

4. Students will create something using recyclable materials.

Chapter 7

Let’s Warm-up 1. 3 10 2. 1 100 or hundredths

3. 1 10 or tenths 4. 1 5. 3 10

Do It Yourself 7A

1. a. forty-five point five or forty-five and five tenths

b. twelve point three or twelve and three tenths

c. forty-two and fourteen-hundredths or forty-two point one four

d. one hundred seventy-eight point six four or one hundred seventy-eight and sixty-four hundredths

e. four hundred fifty-seven point one eight or four hundred fiftyseven and eighteen hundredths

f. twenty-five and one hundred twenty-seven thousandths or twenty-five point one two seven

g. one hundred seventy-four and two hundred one thousandth or one hundred seventy-four and two zero one

h. seven hundred sixty-five and four thousandths or seven hundred sixty-five point zero zero four

2. a. 413.2 b. 72.33 c. 361.04 d. 52.402

3. a. 1.6

One and six-tenths.

1.72

one point seven two or one and seventy-two hundredths

4. a. 40 + 5 + 1 10 or 40 + 5 + 0.1

b. 700 + 80 + 9 + 4 10 or 700 + 80 + 9 + 0.4

c. 40 + 5 + 1 10 + 3 100 or 40 + 5 + 0.1 + 0.03

d. 400 + 80 + 6 + 7 10 + 2 100 or 400 + 80 + 6 + 0.7 + 0.02

e. 900 + 70 + 6 + 7 100 or 900 + 70 + 6 + 0.07

f. 10 + 5 + 6 100 + 2 1000 or 10 + 5 + 0.06 + 0.002

g. 70 + 8 + 6 10 + 2 1000 or 70 + 8 + 0.6 + 0.002

h. 100 + 50 + 4 + 3 100 + 1 1000 or 100 + 50 + 4 + 0.03 + 0.001

5. a. 4.1 b. 867.17 c. 16.106 d. 489.982

6. a. 0.9, 90% b. 0.32, 32% c. 0.34, 34%

d. 0.475, 47.5%

7. a. 13 10 , 130% b. 627 50 = 1254%

c. 803 23 = 3212% d. 1151 500 = 230.2%

8. Answers may vary. Sample answers.

a. 187.2 = 187.20 = 187.200

b. 87.02 = 87.020 = 87.0200

c. 963.14 = 963.140 = 963.1400

d. 189.221 = 189.2210 = 189.22100

9. a. 21 10 , 2.1, 210% b. 7 20 , 0.35, 35%

c. 131 250 , 0.524, 52.4% d. 78 25 , 3.12, 312%

10. a. > b. < c. = d. <

11. 400 + 50 + 5 + 8 10 + 9 100 or 400 + 50 + 5 + 0.8 + 0.09

12. 0.625 13. 60%

Challenge 1. 0.9978 and 0.9987 Do It Yourself 7B

1. None

2. a.  b.  c.  d. 

3. a. 1 b. 2 c. 1 d. 3

4. a. 13.15, 1.20 b. 3.48, 1.20 c. , 1.526 d. 1.400, 47.584

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

6. 0.540, 60.900 and 83.394

7. Answers may vary. Sample answer. Mahika bough 1.5 L of milk and 3.245 L of curd. Convert both the decimals into like decimals.

Challenge 1. Option c Do It Yourself 7C

b. 0.6

1. a. 0.9 > 0.89 0.5

2. a. True b. False c. True

3. a. 4.2 b. 15.67 c. 87.654 d. 294.98

4. a. 3 b. 8 c. 48 d. 15

5. 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

6. a. < b. = c. = d. <

7. 206 8. 24.68 > 24.54 > 24.36.

9. Telangana, Rajasthan, Bihar, Maharashtra, Uttar Pradesh

Challenge 1. No

Chapter Checkup

1. a.

1.2 = 1 + 0.2 = 1 + 2 10

1.48 = 1 + 0.4 + 0.08 = 1 + 48 10100 +

2. a. fifteen and two-tenths or fifteen point two

b. seventy-one and sixty-five tenths or Seventy-one point six five

c. eight hundred fourteen and thirty-six hundredths or eight hundred fourteen point three six

d. one hundred seventy-six and eight hundred one thousandths or one hundred seventy-six point eight zero one

3. a. 40.101 b. 103.05 c. 420.011 d. 31.005

4. a. 10 + 1 + 1 10 or 10 + 1 + 0.1

b. 10 + 6 + 54 10100 + or 10 + 6 + 0.5 + 0.04

c. 400 + 90 + 2 200 or 400 + 90 + 0.02

d. 800 + 40 + 3 + 33 1001000 + or 800 + 40 + 3 + 0.03 + 0.003

5.a. 0.125, 12.5% b. 0.8, 80% c. 0.6, 60% d. 0.9, 90%

6. a. 63 10; 5 ; 1260% b. 261 10; 5 ; 5220%

c. 104 10; 5 ; 2080% d. 5063 1000; 200 ; 2531.5%

7. a. 12.4 12.3 < b. < 14.55 14.5

c. > 222.22 222.02 d. < 3.003 3.033

8. a. 1 b. 100 c. 100 d. 1000

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. Answers may vary. Sample answer.

5. a. < b. > c. = d. > 6. 4.85 litres 7. 1.47 m 8. 3.005 kg 9. 5.862 km

10. a. 70.88 kg b. 83.735 kg 11. 3.45 km 12. ₹4939

Challenge 1. 9

Do It Yourself 8B

1. a. ;0.8 b. ;2.0

2. a. 456 b. 1547.8 c. 63,157 d. 10 e. 179.34 f. 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. ₹89,190 6. 27.6 7. ₹7516.7

8. Suresh’s Father, ₹17,458 9. ₹11,870.4; Answer may vary

10. ₹1344.89

Challenge 1. 3; 3.75

Do It Yourself 8C

1. a. 0.2

11. Utkarsh

12. 12.9 < 12.99 < 13.01 < 13.1 Challenge 1. 0.97 2. Option c Case Study

1. Option c 2. Option d 3. Jin Mao Tower, Wuhan Greenland Centre, CITIC Tower, Lotte World Tower, Shanghai Tower, Merdeka 118

4. 679 m; 528 m; 421 m; 599 m; 476 m; 632 m

5. 678.9 = 600 + 70 + 8 + 9 10

527.7 = 500 + 20 + 7 + 7 10

420.5 = 400 + 20 + 5 10

599.1 = 500 + 90 + 9 + 1 10

475.6 = 400 + 70 + 5 + 6 10

632.0 = 600 + 30 + 2

Chapter 8

Let’s Warm-up 1.

Do It Yourself 8A

b. 0.2

2. a. 1.56 b. 0.5123 c. 0.012 d. 0.032561 e. 0.0231 f. 0.2

3. a. 20.1 b. 9.13 c. 5.415 d. 4.725 e. 21.02 f. 42.05

4. a. = b. > c. < d. = e. > f. < 5. 1000 pounds 6. 4.2 kg 7. Yes

8. Answer will vary. Sample answer.

A community centre received a donation of 47.5 litres of hand sanitizer to be distributed equally among 5 local schools. How much hand sanitizer will each school receive to ensure an equal distribution?

Challenge 1. 1166.72 Euros

Chapter Checkup

1. a. 0.7 b. 0.78 2. a. 1.2

Do It Yourself 9B

1. Answers may vary. Sample answer.

3. a. 63.123 b. 46.34 c. 33.48 d. 51.792

e. 518.755 f. 114.921 4. a. 10 b. 1000 c. 100

d. 10 e. 1000 f. 1000 5. a. 4.701 b. 100 c. 1000

d. 1000 e. 0.0142 f. 100 6. a. 18.48 b. 58.518

c. 289.26 d. 25.632 e. 516.675 f. 1,449.612

7. a. 18.4 b. 12.35 c. 30.02 d. 46.39 e. 23.65

f. 12.142 8. a. 33.07 b. 85.971 c. 55.33

d. 37.399 e. 50.605 f. 24.512 9. a. 25.23 b. 24.85

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. 7:30 am: Acute angle 1:30 pm: Obtuse angle

6. Acute angles and obtuse angles

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 12. ₹4.2 13. $1.22 14. ₹14,259.2 15. 5 days 16. Answer may vary. Sample answer. John’s car consumes 0.08 gallons of fuel per mile. If he drives 45.7 miles, how many gallons of fuel does he use?

Challenge 1. 25.543 kg 2. option d

Case Study

1. Option c 2. Option a 3. 1444.6 mm 4. 790 mm 5. 1288.78 mm 6. Answers may vary

Chapter 9

Let’s Warm-up 1. Slanting Line 2. Sleeping Line

3. Standing Line 4. Sleeping Line 5. Sleeping Line

Do It Yourself 9A

1. a. Parallel lines b. Ray c. Intersecting lines d. Line 2. a. False b. False c. True

d. True 3. Line segment, Parallel lines, Point, Ray

4. a. M b. N M

c. Q P d. T S

e. W O X V U

5. a. A, B, C, D, E, F; ,, ACCBAB

b. Answers may vary. Sample answer. FD

c. 1 ray, .BE

6. Intersecting lines

Challenge

1. True Q P Q P

2. a. Intersecting lines b. Parallel lines

Challenge 1. a. obtuse angle b. straight angle c. 2 acute angles

Do It Yourself 9C

1. a. Acute angle b. Obtuse angle c. Right angle

d. Acute angle 2. a. 74° b. 115° c. 152°

d. 120° 3. a. 53° b. 90° c. 123° d. 32°

e. 57° f. 127°

4. a.

5. 30° angles

Challenge 1. students will draw a straight angle

Do It Yourself 9D

1. a. Quadrilateral b. Nonagon c. Hexagon

d. Decagon 2. a. nine b. three c. seven

d. five 3. a. True b. True c. True d. False

4. octagon 5. Hexagon 6. PQ, QR and PR.

Challenge 1. Triangles -15 Quadrilaterals -5

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. 65° b. 115° c. 140° d. 155°

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.

125° O N M 10 cm

11. Parallel lines 12. Hexagon

13. Answers may vary. Sample image.

Challenge 1. Option c 2. Case Study

1. option c 2. option b 3. option c

4. Chapter 10 Let’s Warm-up 1. 2.

Do It Yourself 10A

1. c and d 2. 3.

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. d

6. a. b.

Challenge 1.

Do It Yourself 10B

4. b; Answers may vary. Sample answer.

5. b

Challenge 1.

Do It Yourself 10C

1. a. Adding 4 to each term. b. Adding 2 to each term.

c. The pattern has squares of natural numbers.

2. a. 53453 b. 89012 c. 14523

3. 5 red and 5 yellow flowers. 4. 84

5. Answer may vary. Sample answer:

Challenge 1. 14

Do It Yourself 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 b.

c. no line of symmetry d.

3. a. A b. H c. T d. Asymmetrical.

4. a. False b. False c. True d. True

5. Answer may vary. Sample answer:

6. a. b. c.

Challenge 1.

Chapter Checkup

1. d, h.

2 a. b.

6. 53 m 16 cm 7. 25 km 794 m 8. 53 dm

9. Answer may vary. Sample answer. The length of pipe A is 67 m 25 cm and the length of pipe B is 87 m 30 cm. How much more is the length of pipe B than the length of pipe A?

Challenge 1. 14 cm

Do It Yourself 11B

1. a. 30 g b. 85 g c. 70 kg d. 100 g 2. a. 0.079 kg

4. 2 1 4 16 8

3. a. b. d.

Case Study

b. 9,75,000 mg c. 46,77,000 g d. 165.5 g e. 68.76 g

f. 8,40,000 mg 3. a. 6.01 kg; 6010g b. 16.08 g; 16,080 mg

c. 547.006 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 6. 4 kg 56 g 7. 6 kg 133 g 8. 6 kg 100 g

9. a. 1 kg 490 g b. 1 kg 80 g 10. 0.0051 kg

11. Answer may vary. Sample answer.

The weight of a watermelon is 11 kg and the weight of a jackfruit is 5,500 g. What is the weight of both the fruits together?

Challenge 1. 10 quintals is 1 ton

Chapter Checkup

1. a. m; kg b. cm; g c. m; kg 2. a. 12.5 cm b. 17.9 cm

c. 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

5. a. b. c.

6. a. b. c.

8. 1,25,000 mg 9. No, because their overall weight is 293 kg which is greater than the carrying limit of the lift.

10. (Answers may vary) Sample answer

a. One 4 cm and one 8 cm b. One 4 cm and two 8 cm

c. Three 8 cm 11. a. 36 cm b. No

12. 20 kg flour, 7.5 kg butter and 10 kg sugar.

Challenge 1. 41,000 g 2. Option c

Case Study

1. a. cm b. mm

7. 256, 241, 226; 15 is subtracted each time.

8. a. 1, 3, 9, 27, 81 b. 15, 18, 23, 26, 31

9. a. 57, 64, 72, 81, 91 b. 75, 69, 62, 54, 45 10. No 11. a. Yes b. Yes

2. option d

3. Length of the play area = 6,682 m

4. The van can shift 4 tigers or 1 tiger and 1 lion in a trip

5. Answer will vary.

Chapter 12

Let’s Warm-up Perimeter: 10 cm; 12 cm; 12 cm; 14 cm

12. 128, 256, 512; The number is getting doubled each time. 13. ₹2560 14. 300 steps

Challenge 1. 28 2. Option 3- A is true but R is false.

1. a. Rectangle 2. False 3. reflectional symmetry

4. No 5. Dispose of trash in designated bins, and avoid bringing items that can cause litter.

Chapter 11

Let’s Warm-up 1 cm 2. m 3. g 4. Kg 5. cm

Do It Yourself 11A

1. Answers may vary, Sample answer.

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. 0.547 cm c. 1.056 km d. 689.2 m e. 203.4 m f. 0.7698 km

4. a. 45.1 m b. 16,080 mm c. 280 m 5 dm 5. 0.0001 km

Area: 6 sq. cm; 6 sq. cm; 5 sq. cm; 6 sq. cm

Do It Yourself 12A

1. a. 80 cm b. 300 cm c. 60 cm d. 100 cm

2. a. 128 m b. 212 cm c. 108 m d. 312 cm

3. a. 54 cm b. 280 m c. 76 m d. 80 cm

4. a. 24 mm b. 37 mm 5. 180 m 6. ₹1400

7. 1800 sq. m. 8. 8 cm

9. Answers may vary. Sample answers. 10 units 2 units 8 units 4 units

Challenge 1. 9 m

Do It Yourself 12B

1. a. sq. cm b. sq. cm c. sq. m d. sq. m e. sq. km f. sq. m

g. sq. cm h. sq. m 2. a. 330 sq. cm b. 4060 sq. m

3. a. 3136 sq. m b. 4489 sq. cm 4. 144 sq. m

Do It Yourself 14C 1. a

3. Front view Side view Top view 4. 15 building blocks Front view Side view Top view

Challenge 1. 3 and 2

Do It Yourself 14D 1. a. south b. east

4. Cupboard = 2 m, 8 m; table = 6 m, 4 m; desk = 2 m, 6 m

5. 12.5 km 6. Myanmar, Sri Lanka and Pakistan Challenge 1. 12 km

Do It Yourself 14E

5. Windows at the back and either left or right side.

Cylinder Cuboid

2. Faces = 1, Vertices = 0, Edges = 0

3. a. Net b. Deep drawing c. Floor plan

4. a. Cylinder: Faces = 3; Edges = 2; Vertices = 0

b. Cube: Faces = 6; Edges = 12; Vertices = 8

c. Cone: Faces = 2;Edges = 1; Vertices = 1

7. a. 32 km b. 64 km c. 120 km d. 200 km e. 256 km

8. Answer may vary. Sample answers.

Yes

9. 4 doors 10. 3 times.

11. Answer may vary. Sample answer

4. a. 35 tickets b. 715 tickets c. 35 tickets

5. Answer may vary. Sample answer.

Q1. What is the number of adult tickets sold in the 4 days?; Q2. What is the difference in the number of children tickets sold on Thursday to that on Sunday.

Challenge 1. 280 tickets

Do It Yourself 17B

50 students b.

5. Answer may vary. Sample answer. How many more students are interested in sports than Maths?

Challenge 1. 1560 million tonnes Do It Yourself 17C

1. Wednesday and Thursday 2. Most = Pune; Least = Goa

3. a. 8°C b. Week 1 c. 8°C 4. a. 2017 b. 800 guitars

c. 2018 5. Answers will vary. Sample answer.

Q1. What is the total numbers of guitars sold during the given years?

Q2. In which year(s) did Rahul sell more than 600 guitars?

Challenge 1. a. April b. 6800

Chapter Checkup

1. a. 15 b. 19 c. 2

2. a. Thursday b. Boys c. 3 pages 3.

6.a. Friday b. 65 kg c. 5 kg

7.a. Jaipur b. Ooty c. 15°C d. Bangalore 8.

Case Study

1. Option b 2. Option c

Andra Pradesh Bihar Telangana State

Rajasthan

9. a. Chocolate b. Blueberry c. 14 children d. 22 children

10. Answers will vary. Sample answer. What is the difference in the number of students who like mango flavour to the one who like blueberry flavour? Challenge 1. 6,10,000 tonnes 2. Option d

3. 010203040 50607080

Number of Elephants/Tigers Elephants Tiger

4. Andhra Pradesh

5. Answers may vary.

About the Book

Imagine Mathematics seamlessly bridges the gap between abstract mathematics and real-world relevance, offering engaging narratives, examples and illustrations that inspire young minds to explore the beauty and power of mathematical thinking. Aligned with the NEP 2020, this book is tailored to make mathematics anxiety-free, encouraging learners to envision mathematical concepts rather than memorize them. The ultimate objective is to cultivate in learners a lifelong appreciation for this vital discipline.

Key Features

• Let’s Recall: Introductory page with a quick recall of concepts learnt in previous grades

• Real Life Connect: Introduction to a new concept related to real-life situations

• Examples: Solved problems showing the correct method and complete solution

• Do It Together: Guided practice for learners with clues and hints to help solve problems

• 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

• Challenge: Critical thinking questions to enhance problem-solving and analytical thinking skills

• Real-Life Maths: Scenario-based questions to apply theory to real-life situations

• QR Codes: Digital integration through the app to promote self-learning and practice

About Uolo

Uolo partners with K-12 schools to provide technology-enabled learning programs. We believe that 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, Southeast Asia and the Middle East.

ISBN 978-81-979482-3-7

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