Master Mathematical Thinking MATHEMATICS
6
MATHEMATICS
Master Mathematical Thinking
Grade
Acknowledgements
Academic Authors: Animesh Mittal, Muskan Panjwani, Anjana AR, Gitanjali Lal, Anuj Gupta, 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.
© Uolo EdTech Private Limited
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Book Title: Imagine Mathematics 6
ISBN: 978-81-979482-8-2
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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.
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.
The temperature can change with different weather conditions. On a sunny day, the air feels warm, and the thermometer shows a higher temperature. On a cloudy or rainy day, the air feels cooler, and the temperature on the thermometer drops. Weather conditions like rain, wind and sunlight can all affect the temperature of the air.
Temperature Range (°C)
Below 0 Very Cold 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.
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.
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 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.
2
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
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.
GRR Steps Unit Component
Do It Yourself Challenge
You do
Chapter Checkup Challenge
Case Study
Do It Yourself 14B
1 Identify the shape for which the net is drawn.
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.
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
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
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.
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.
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.
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.
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.
d The face value of the digit in the thousands place is 5.
Thus, Australia was donated more vaccine doses.
Remember!
e The digit in the tens place is the biggest 1-digit number. What is the secret code?
e The digit in the tens place is the biggest 1-digit number. 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
Case Study
How much longer is the red straw than the blue straw?
Cross Curricular
Cross Curricular
Population of Countries
Population of Countries
The population of different countries is shown using a table. Read the data carefully and answer the questions.
Country
Convert the lengths.
Convert the weights.
The population of different countries is shown using a table. Read the data carefully and answer the questions.
United
United
a 5 kg into g b 4 g 64 cg into g c 5487 g into mg d 43 kg 7 dag into kg
A baby koala is called a joey. A young joey weighs about 0.38 kg. How much is that in g?
58,870,762
A candle weighs 125 g. How much is it in mg?
1 Which country has the least population?
1 Which country has the least population?
a Italy b Germany c Poland d United Kingdom
a Italy b Germany c Poland d United Kingdom
2 Which country has the greatest population?
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?
3 Which country has approximately double the population than that of Poland?
4 Arrange the countries in ascending order as per their population.
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?
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.
NCF Page 218
Curricular Goals
CG-1
Understands numbers and sets of numbers (whole numbers, fractions, integers, rational numbers, and real numbers), looks for patterns, and appreciates relationships between numbers
CG-2
Understands the concepts of variable, constant, coefficient, expression, and (one-variable) equation, and uses these concepts to solve meaningful daily-life problems with procedural fluency
Competencies
C-1.1 Develops a sense for and an ability to manipulate (e.g., read, write, form, compare, estimate, and apply operations) and name (in words) large whole numbers of up to 20 digits, and expresses them in scientific notation using exponents and powers
C-1.2 Discovers, identifies, and explores patterns in numbers and describes rules for their formation (e.g., multiples of 7, powers of 3, prime numbers), and explains relations between different patterns.
C-1.3 Learns about the inclusion of zero and negative quantities as numbers, and the arithmetic operations on them, as given by Brahmagupta
C-1.4 Explores and understands sets of numbers, such as whole numbers, fractions, integers, rational numbers, and real numbers, and their properties, and visualises them on the number line
C-1.5 Explores the idea of percentage and applies it to solve problems
C-1.6 Explores and applies fractions (both as ratios and in decimal form) in daily-life situations
C-2.1 Understands equality between numerical expressions and learns to check arithmetical equations
C-2.2 Extends the representation of a number in the form of a variable or an algebraic expression using a variable
C-2.3 Forms algebraic expressions using variables, coefficients, and constants and manipulates them through basic operations
C-2.4 Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems
C-2.5 Develops own methods to solve puzzles and problems using algebraic thinking
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 Develops a sense for and an ability to manipulate (e.g., read, write, form, compare, estimate, and apply operations) and name (in words) large whole numbers of up to 20 digits, and expresses them in scientific notation using exponents and powers
C-1.2 Discovers, identifies, and explores patterns in numbers and describes rules for their formation (e.g., multiples of 7, powers of 3, prime numbers), and explains relations between different patterns.
C-1.3 Learns about the inclusion of zero and negative quantities as numbers, and the arithmetic operations on them, as given by Brahmagupta
C-1.4 Explores and understands sets of numbers, such as whole numbers, fractions, integers, rational numbers, and real numbers, and their properties, and visualises them on the number line
C-1.5 Explores the idea of percentage and applies it to solve problems
C-1.6 Explores and applies fractions (both as ratios and in decimal form) in daily-life situations
C-2.1 Understands equality between numerical expressions and learns to check arithmetical equations
C-2.2 Extends the representation of a number in the form of a variable or an algebraic expression using a variable
C-2.3 Forms algebraic expressions using variables, coefficients, and constants and manipulates them through basic operations
C-2.4 Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems
C-2.5 Develops own methods to solve puzzles and problems using algebraic thinking
C-3.1 Describes, classifies, and understands relationships among different types of twoand three-dimensional shapes using their defining properties/attributes
C-3.2 Outlines the properties of lines, angles, triangles, quadrilaterals, and polygons and applies them to solve related problems
C-3.3 Identifies attributes of three-dimensional shapes (cubes, parallelepipeds, cylinders, cones), works hands-on with material to construct these shapes, and also uses two-dimensional representations of three-dimensional objects to visualise and solve problems
C-3.4 Draws and constructs geometric shapes, such as lines, parallel lines, perpendicular lines, angles, and simple triangles, with specified properties using a compass and straightedge
C-3.5 Understands congruence and similarity as it applies to geometric shapes and identifies similar and congruent triangles
CG-1
Understands numbers and sets of numbers (whole numbers, fractions, integers, rational numbers, and real numbers), looks for patterns, and appreciates relationships between numbers
CG-2
Understands the concepts of variable, constant, coefficient, expression, and (one-variable) equation, and uses these concepts to solve meaningful daily-life problems with procedural fluency
CG-3
Understands, formulates, and applies properties and theorems regarding simple geometric shapes (2D and 3D)
C-4.1 Discovers, understands, and uses formulae to determine the area of a square, triangle, parallelogram, and trapezium and develops strategies to find the areas of composite 2D shapes
C-4.2 Learns the Baudhayana-Pythagoras theorem on the lengths of the sides of a right-angled triangle, and discovers a geometric proof using areas of squares erected on the sides of the triangle, and other related geometric constructions from the Sulba-Sutras
C-4.3 Constructs various designs (using tiling) on a plane surface using different 2D shapes and appreciates their appearances in art in India and around the world
C-4.4 Develops familiarity with the notion of fractal and identifies and appreciates the appearances of fractals in nature and art in India and around the world
C-5.1 Collects, organises, and interprets the data using measures of central tendencies such as average/mean, mode, and median
C-5.2 Selects, creates, and uses appropriate graphical representations (e.g., pictographs, bar graphs, histograms, line graphs, and pie charts) of data to make interpretations
C-6.1 Applies both inductive and deductive logic to formulate definitions and conjectures, evaluate and produce convincing arguments/proofs to turn these definitions and conjectures into theorems or correct statements, particularly in the areas of algebra, elementary number theory, and geometry
C-7.1 Demonstrates creativity in discovering one’s own solutions to puzzles and other problems, and appreciates the work of others in finding their own, possibly different, solutions
C-7.2 Engages in and appreciates the artistry and aesthetics of puzzle-making and puzzle-solving
C-8.1 Approaches problems using programmatic thinking techniques such as iteration, symbolic representation, and logical operations and reformulates problems into series of ordered steps (i.e., algorithmic thinking)
C-8.2 Learns systematic counting and listing, systematic reasoning about counts and iterative patterns, and multiple data representations; learns to devise and follow algorithms, with an eye towards understanding correctness, effectiveness, and efficiency of algorithms
C-9.1 Recognises how concepts (like counting numbers, whole numbers, negative numbers, rational numbers, zero, concepts of algebra, geometry) evolved over a period of time in different civilisations.
C-9.2 Knows and appreciates the contributions of specific Indian mathematicians (such as Baudhayana, Pingala, Aryabhata, Brahmagupta, Virahanka, Bhaskara, and Ramanujan)
Recognises interaction of Mathematics with multiple subjects across Science, Social Science, Visual Arts, Music, Vocational Education, and Sports
CG-4
Develops understanding of perimeter and area for 2D shapes and uses them to solve day-to-day life problems
CG-5
Collects, organises, represents (graphically and in tables), and interprets data/information from daily-life experiences
CG-6 Develops mathematical thinking and the ability to communicate mathematical ideas logically and precisely
CG-7
Engages with puzzles and mathematical problems and develops own creative methods and strategies to solve them
CG-8 Develops basic skills and capacities of computational thinking, namely, decomposition, pattern recognition, data representation, generalisation, abstraction, and algorithms in order to solve problems where such techniques of computational thinking are effective
CG-9 Knows and appreciates the development of mathematical ideas over a period of time and the contributions of past and modern mathematicians from India and across the world
CG-10 Knows about and appreciates the interaction of Mathematics with each of their other school subjects
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.
Let’s Recall: Introductory page with a quick recall of concepts learnt in the previous grade
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.
Expanded form:
Number name: Two lakh
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.
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.
Expanded form: 2,00,000 + 1000 + 300 + 1 Number
1
All About 7-Digit and 8-Digit Numbers
I scored out of 4.
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.
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
Temperature Range (°C) Weather
3 The place value of the
Explanation:
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.
The temperature can change with different weather conditions. On a sunny day, the air feels warm, and the thermometer shows a higher temperature. On a cloudy or rainy day, the air feels cooler, and the temperature on the thermometer drops. Weather conditions like rain, wind and sunlight can all affect the temperature of the air.
Temperature Range (°C) Weather
Below 0 Very Cold
0–10 Cold
11–20 Mild
21–30 Warm
31–40 Hot
Numbers 8-digit Numbers
Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.
Place Value, Face Value and Expanded Form
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?
Normal body temperature = 37°C.
Above 40 Very 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.
Aliya’s body temperature = 37°C + 2.3°C = 39.3°C
Normal body temperature = 37°C.
Drass in Jammu and Kashmir is the second coldest inhabited region on Earth.
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?
Aliya’s body temperature = 37°C + 2.3°C = 39.3°C
So, Aliya’s body temperature was 39.3°C.
All About 7-Digit and 8-Digit Numbers
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?
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?
37.5°C – 21.2°C = 16.3°C
The temperature difference is 16.3°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.
Write the readings of the given thermometers.
The temperature difference is 16.3°C. Write the readings of the given thermometers. Temperature = 45°C
Do It Together: Guided practice for learners with partially solved questions
Converting Between Units of Temperature
Place Value, Face Value and Expanded Form
2 4
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.
Converting Between Units of Temperature
Now, if we add 1 to this number, we get 1000000.
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.
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. Did You Know?
1000000 is the smallest 7-digit number and is read
15 • Time and Temperature
We saw in the news article that the number of vaccinations 1360477. Let us try to place this 7-digit number in
Let
1
1
Fill
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 by Sana =
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. a b c d
2 Look at the net and identify the object it belongs to. a b c d
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 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
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.
1 Sanya wants to solve a 7-digit secret code in a safe. Use the given clues to help Sanya solve the secret code.
b The digit in the lakhs place is 4 less than the digit in the ones place.
a The digit in the hundreds and ones place is 6.
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.
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.
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
2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice. Critical Thinking
e The digit in the tens place is the biggest 1-digit number. What is 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.
2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.
c The digit in the ten lakhs and ten thousands place is the smallest odd number.
Case Study
Case Study
Thinking Case Study
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. What is the secret code?
Population of Countries
2 Write the greatest 8-digit odd number using only 5 digits. Do not repeat any digit more than twice.
Cross Curricular
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.
Curricular
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
1 Which country has the least population?
2 Which country has the greatest population?
a Poland b United Kingdom c Italy d Germany
2 Which country has the greatest population?
3 Which country has approximately double the population than that of Poland?
4 Arrange the countries in ascending order as per their population.
1 Which country has the least population?
3 Which country has approximately double the population than that of Poland?
5 If all the digits in the population of each country is rearranged to form the greatest
Cross Curricular
Large Numbers and Operations
Letʼs Recall
The numbers used in our daily lives are written using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We can form various numbers using these digits. These numbers are written using commas after every period, starting from the ones period.
For example, let us say the population of a country is 2347709. This number can be written using commas by representing it in a place value chart.
Each digit of a number has a place value and a face value. The place values of all its digits are used to write its expanded form and number name. For example, the population of a country can be represented as shown below.
Face Value
Place Value
9 ones = 9 × 1 = 9
0 tens = 0 × 10 = 0
7 hundreds = 7 × 100 = 700
7 thousands = 7 × 1000 = 7000
4 ten thousands = 4 × 10000 = 40000
3 lakhs = 3 × 100000 = 300000
2 ten lakhs = 2 × 1000000 = 2000000
Expanded form: 2000000 + 300000 + 40000 + 7000 + 700 + 9
Letʼs Warm-up
Write True or False.
1 The number 586592 can be rewritten using commas as 5,865,952.
2 The place value and face value of 0 in 9,80,564 is 0.
3 The expanded form of 3,83,452 is 3,00,000 + 80,000 + 3000 + 400 + 50 + 2.
4 The number name of 7,03,478 is seven lakh three hundred four thousand seventy-eight.
5 Between 40,73,548 and 40,72,187, 40,73,548 is greater. 2347709
Number name: Twenty-three lakh forty-seven thousand seven hundred nine I scored out of 5.
Understanding Large Numbers
The data on the population of a few countries is given.
277534122 144444359 339996563 123294513 Real
While reading the article, Ritu was confused and could not read the numbers given in the data.
Indian Number System
The population of each of the countries that was given in the data is a 9-digit number. Indonesia Russia United
Japan 277534122 144444359 339996563 123294513
9-digit Numbers
Let us learn how to read and write large numbers using place value.
Face Value and Place Value
We know that the greatest 8-digit number is 99999999. If we add 1 to this number, we get 100000000. This is the smallest 9-digit number with a new place called the ‘Ten Crores’ place.
Crores Lakhs Thousands Ones
Ten Crores (TC) Crores (C) Ten Lakhs (TL) Lakhs (L) Ten
new place
Ten Crores (TC) place
We saw in the article that the population of Indonesia is 277534122. We can write it as:
Crores Lakhs Thousands Ones
Ten Crores (TC) Crores (C) Ten Lakhs (TL) Lakhs (L) Ten Thousands (TTh) Thousands (Th)
(T) Ones (O) 2 7 7 5 3 4
(H)
Every digit in a number has a fixed position according to which the digit has its value. The fixed position is called the place of a digit, and the value of the digit is called the place value of the digit. The face value is the digit itself.
To find the place value of any digit, we multiply the digit by its value in a place of the number, whereas the face value does not depend on the position of the digit in the number.
For example, 277534122 can be represented as:
Place value of a digit = Digit or face value × Value of the place
Face Value
277534122
Place Value
2 ones = 2 × 1 = 2
2 tens = 2 × 10 = 20
1 hundreds = 1 × 100 = 100
4 thousands = 4 × 1000 = 4000
3 ten thousands = 3 × 10000 = 30000
5 lakhs = 5 × 100000 = 500000
7 ten lakhs = 7 × 1000000 = 7000000
7 crores = 7 × 10000000 = 70000000
2 ten crores = 2 × 100000000 = 200000000
Remember!
The place value of zero is always 0. It may hold any place in a number; its value is always 0.
Example 1: Write the face value and the place value of the coloured digits in the given numbers.
1 43,07,64,936 — 0 is in the ten lakhs place. Both the place value and the face value of 0 are 0.
2 85,50,99,225 — 8 is in the ten crores place. So, the place value of 8 is 80,00,00,000 and the face value of 8 is 8.
Example 2: Find the sum of the place value and the face value of 4 in 23,43,08,970.
The place value of 4 = 40,00,000
The face value of 4 = 4
Their sum = 40,00,000 + 4 = 40,00,004
Write the place value and the face value of 3 in 13,98,70,465.
3 is in the place.
The place value of 3 is
The face value of 3 is .
Expanded Form and Number Names
Did You Know?
Godfrey Harold Hardy is an English mathematician known for his achievements in number theory and mathematical analysis.
When the values of the digits of a number are written according to their place values, separated by a plus sign, it is called the expanded form of that number. For example, the expanded form of 27,75,34,122 is 20,00,00,000 + 7,00,00,000 + 70,00,000 + 5,00,000 + 30,000 + 4000 + 100 + 20 + 2 = 27,75,34,122
Expanded Form
Standard Form
Number Names
To read and write 9-digit numbers in figures and words, we group the digits according to the periods and put commas after each period, starting with the ones. This helps us read the numbers easily. For example, 277534122 can be read in the following way.
Crores period Lakhs period Ones period Thousands period
20,00,00,000 + 7,00,00,000 + + + 70,00,000 + 5,00,000 30,000 + 4000 100 + 20 + 2
seventy-five lakh twenty-seven crore
Example 3: Write the number for the number name.
Ninety crore twenty-one lakh eleven thousand two hundred two
We can write it as 90,21,11,202, using commas.
one hundred twenty-two thirty-four thousand
Think and Tell
Can you now read and write the population of the United States, Russia and Japan using the Indian number system?
Do It Together
Write the number in its expanded form and as a number name.
56,70,34,219 = 56,00,00,000 +
Number name: Fifty-six crore
Do It Yourself 1A
Write the face value and the place value of the coloured digits in the numbers written in the Indian number system.
a 38,57,64,999
d 81,87,50,658
b 27,89,73,653
e 58,76,03,653
Write the expanded form of the numbers.
c 18,90,37,890
f 20,56,78,450
a 24,85,67,835 b 56,42,86,604 c 93,27,59,191
Write the number names for the numbers.
a 23,34,26,578 b 74,36,54,679 c 46,57,68,979
Write numbers for the number names.
a Sixteen crore twenty-six lakh ninety-two thousand one hundred forty-three
b Sixty-five crore twenty-two lakh forty-four thousand nine hundred
c Eighty crore ninety-nine lakh forty thousand two hundred one
In a number 76,47,56,229, the digits 5 and 4 are interchanged to form a new number. Answer the questions.
a Will there be a change in the face values of 5 and 4?
b What will be the new place values of 5 and 4?
c What is the difference of the place values of 5 in the new number and the original number?
The first state-controlled lottery emerged in Kerala in 1967. Mary won a lottery with a ticket bearing the 9-digit number 272545619. Rewrite the number using commas. Write its number name and expanded form.
The distance from Jupiter to Saturn is 64,62,70,000 km. Write the distance in words.
Form the smallest 9-digit numbers using all odd digits at least once. Write the number as a number name.
International Number System
Like the Indian number system, there is a globally recognised system of numbers called the international number system. For 9-digit numbers, there are the ones, the thousands and the millions period. Each period has three places, and the commas are put after each period, starting from the right.
1 periods place 3 places 3 places 3 places
Face Value and Place Value
Like the Indian number system, each digit of a number has a place value and a face value. We find the place value and the face value in the same way as in the Indian number system.
For example, let us take the population of Indonesia (277534122). Its face value and place value can be shown in this way.
277534122
Face Value 2 ones
Place Value
Example 4: Write the face value and the place value of the coloured digits in the numbers.
1 203,565,768 — 5 is in the hundred thousands place.
The place value of 5 is 500,000. Its face value is 5.
2 909,358,767 — 9 is in the hundred millions place.
The place value of 9 is 900,000,000. Its face value is 9.
Example 5: Find the difference of the place value and the face value of 8 in 682,344,590.
The place value of 8 = 80,000,000
The face value of 8 = 8
The difference of the place and the face value = 80,000,000 − 8 = 79,999,992
The face value of 4 is . Do It Together
Write the place value and the face value of 4 in 134,870,965.
4 is in the millions place.
So, the place value of 4 is .
Expanded Form and Number Names
The international number system differs from the Indian number system mainly in the grouping of digits. The numbers are grouped in sets of three digits, separated by commas, starting from the right. For example, let us consider the population of Indonesia (277534122).
Expanded form:
Number form:
277,534,122 five hundred thirty-four thousand one hundred twenty-two two hundred seventy-seven million 200,000,000 + 70,000,000 + 70,000,00 + + 500,000 + 30,000 + 4000 100 + 20 + 2 Millions period Ones period Thousands period
Remember!
In every number in each system, the value of each digit is 10 times the value of the digit on its right.
Example 6: What is the expanded form of 112,086,785? Also, write its number name.
Expanded form: 112,086,785 = 100,000,000 + 10,000,000 + 2,000,000 + 80,000 + 6000 + 700 + 80 + 5
Number name: One hundred twelve million eighty-six thousand seven hundred eighty-five
Number name: Do It Together
Write the expanded form and the number name for the given number.
902,564,875 = 900,000,000 + 2,000,000 +
Write the face value and the place value of the coloured digits in the numbers written in the international number system.
a 179,043,456 b 205,145,450 c 385,876,243
d 580,398,653
Write the expanded form of the numbers.
a 248,567,465
Write the number names of the numbers.
a 345,768,979
Write numbers for the number names.
a Eight hundred twenty-six million one hundred twenty thousand sixty-six
b One hundred ninety-two million five hundred four thousand sixteen
c Seven hundred eighteen million seven hundred ten thousand one hundred fifty-six
There were 4824238 teachers working in government schools in India in 2021-22. Write the number using commas and number name in the international number system.
The speed of light is 299,792,458 metres per second. Write the number in words.
I am a 9-digit number. All my digits are non-zero and distinct.
- My ones digit is four times the tens digit. The tens digit is 1.
- The sum of the ones and the tens digits is equal to the sum of the hundred thousands and millions digits.
- The sum of the ten thousands and the thousands digits is equal to double the hundreds digit.
- The millions digit is greater than the hundred thousands digit, and the ten thousands digit is 2 more than the thousands digit.
- The digit at the hundred millions place is the largest digit.
- The digit at the ten millions place is the remaining digit. Which number am I?
Comparing and Ordering Numbers
Do you remember how Ritu read and wrote the population of the countries in both the Indian and international number systems?
Let us now compare the populations of various countries and arrange them.
Comparing Numbers
To compare two numbers, let us say 21,04,76,827 and 21,98,70,539, we follow the steps.
Step 1
Arrange the numbers in the place value chart and check the number of digits.
Both the numbers have 9 digits.
Step 2
Start comparing the digits from the highest place. The number with the greater digit is greater.
Here, the digits are the same up to the crores place. Compare the digits in the ten lakhs place.
We see that, 0 < 9.
So, 21,04,76,827 < 21,98,70,539.
Remember!
A number with more digits is always greater.
Ordering Numbers
Think and Tell
Can you compare the populations of Russia and Indonesia and tell which country’s population is less?
To arrange the populations of the countries in ascending order, we follow the steps.
Step 1
Write the smallest number
On comparing, we notice that all the numbers have 9 digits and 1 < 2 < 3.
So, let us compare the numbers starting with the digit 1. On comparing the digits in the next place we get, 2 < 4.
So, 12,32,94,513 is the smallest of all.
Step 2
Compare the remaining numbers. Write the next smallest number.
As 1 < 2 < 3, the next smallest number is 14,44,44,359.
Step 3
In the same way, compare the remaining numbers, if any, and write the next smallest number or write the greatest number.
As 2 < 3, the next smaller number is 27,75,34,122.
Therefore, 33,99,96,563 is the greatest number of all.
Think and Tell
Can you now arrange the populations of the countries in descending order?
Thus, 12,32,94,513 < 14,44,44,359 < 27,75,34,122 < 33,99,96,563 is the required ascending order.
Forming Numbers
Let us now understand this using an example. Suppose, we are given the digits 9, 5, 1, 0, 6, 7, 2, 8 and 3.
Let us try forming numbers so that each digit appears exactly once.
Think and Tell
Can you form more such numbers?
Now what if we want to form the greatest and smallest 9-digit numbers, using these digits exactly once?
To form the greatest number, we write the digits in descending order, 98,76,53,210.
This becomes the greatest 9-digit number that can be formed using each of the digits exactly once!
To form the smallest 9-digit number, we write the digits in ascending order, 10,23,56,789.
Remember!
0 cannot appear in the leftmost place because then it will be an 8-digit number.
This becomes the smallest 9-digit number that can be formed using each of the digits exactly once!
What if the repetition of 1 digit is allowed and we want to form 9-digit numbers using 9, 5, 1, 0, 6, 7, 2 and 3?
To write the greatest number, we choose to repeat the greatest digit.
9 9 7 6 5 3 2 1 0
This becomes the greatest 9-digit number that can be formed while repeating only 1 digit once.
To write the smallest number, we repeat the smallest digit once. Remember not to put a 0 in the highest place.
1 0 0 2 3 5 6 7 9
This number becomes the smallest 9-digit number that can be formed while repeating only 1 digit once.
Example 7: Arrange 100,000,000; 318,675,080; 702,354,758 and 450,500,809 in descending order.
Start comparing the digits in the highest place and find the number with the biggest digit. Continue finding the largest number from the remaining numbers.
Here, 702,354,758 > 450,500,809 > 318,675,080 > 100,000,000 is the descending order.
Do It Together
Example 8: Form the greatest and smallest 9-digit numbers, using the digits 1, 8, 6, 7, 0, 9, 2, 5 and 4. Repetition of digits is not allowed.
The greatest number is 98,76,54,210, and the smallest number is 10,24,56,789.
Arrange the following numbers in both ascending and descending order. 126,856,384; 860,778,653; 400,897,234; 165,121,349
Ascending Order: ; 165,121,349; ;
Descending Order:
Understanding Roman Numerals
The clock with unusual numbers shows another way of writing digits up to 9 except 0. Such numbers are called Roman numbers.
Think and Tell
How can we make the greatest and smallest number using only 2 digits?
This is an ancient system of writing numbers and is called the Roman number system. It is still in use in some cases, like in numbered lists and on clock faces.
In this system, we use seven basic symbols to write different numbers. This system does not have the concept of place value, so there is no representation for 0.
Numbers in the Indian System 1 5
There are several rules for writing Roman numbers.
Rule 1: A Roman number symbol cannot be repeated more than three times. Repeating it means adding its value as many times as it is repeated.
XXX = 10 + 10 + 10 = 30 CC = 100 + 100 = 200
Rule 2: If a symbol of a smaller value is written to the right of a symbol of greater value, the resulting number is equal to the sum of the symbols.
VI = 5 + 1 = 6 CI = 100 + 1 = 101
Rule 3: If a symbol of a smaller value is written to the left of (before) a symbol of greater value, the resulting number is equal to the difference of the symbols.
IX = 10 – 1 = 9 and XC = 100 – 10 = 90
Rule 4: If a smaller number is placed between two larger numbers, it is always subtracted from the larger number to the right.
XIV = 10 + (5 – 1) = 14 and MCD = 1000 + (500 – 100) = 1400
Example 9: Write the number form of the Roman numbers.
1 CDXXXIX = CD + X + X + X + IX
= (500 – 100) + 10 + 10 + 10 + (10 – 1)
= 400 + 10 + 10 + 10 + 9 = 439
Remember!
The symbols V, L and D can never be repeated or subtracted, whereas the symbols I, X, C and M can be repeated up to three times.
2 MCMLXXXIV = M + CM + L + X + X + X + IV
= 1000 + (1000 – 100) + 50 + 10 + 10 + 10 + (5 – 1)
= 1000 + 900 + 50 + 10 + 10 + 10 + 4 = 1984
Example 10: Write the Roman numbers for the numbers.
1 97 = 90 + 7 = (100 – 10) + 7 = XC + VII = XCVII
Error Alert!
215 = 100 + 100 + 5 + 5 + 5 = CCVVV
215 = 100 + 100 + 10 + 5 = CCXV
Match the numbers with their Roman numbers.
2 1567 = 1000 + 500 + (50 + 10) + 7
M + D + LX + VII = MDLXVII
Remember!
The symbol I can be subtracted only from V and X, whereas the symbol X can be subtracted only from L and C. Similarly, the symbol C can be subtracted only from D and M.
Write Roman numbers for the numbers.
a 99 b 184 c 285 d 438 e 1564 f 1287
Write the number form for the given Roman numbers.
Arrange the following numbers in ascending order.
a 100,356,782; 500,040,367; 887,210,460; 931,124,820
b 927,516,890; 360,841,910; 692,180,350; 826,020,031
c 500,216,138; 604,503,821; 650,241,567; 945,241,823
Arrange the following numbers in descending order.
a 826,374,510; 871,926,345; 670,814,256; 450,070,921
b 423,516,789; 801,210,450; 962,115,108; 678,203,001
c 543,343,867; 967,208,891; 788,216,134; 578,206,010
Write the greatest 9-digit number and the smallest 9-digit number, using all the digits only once.
a 5, 3,
Write the greatest 9-digit number and the smallest 9-digit number, using all the digits by repeating the greatest digit only once.
Chapter 1 • Large Numbers and Operations
Write the greatest 9-digit number, using: a two digits b five digits c four digits
The amounts of cotton produced by the US, India, China and Brazil are 3,180,410 tonnes, 6,131,050 tonnes, 5,910,500 tonnes and 2,757,353 tonnes respectively. Arrange the names of the countries in increasing order of cotton production.
Draw a clock using Roman numerals to show the time 4:30.
Sana's year of birth is MMXII, and Yana's year of birth is MCMXCIV in Roman numerals. Who is older and by how much?
Estimation in Large Numbers
Disha was reading a newspaper. A news clipping in the newspaper read:
The concert raised ₹52,00,67,890 as a donation for the frontline workers of the Covid-19 battle in 2020.
Using Rounding Off for Estimation
In the news article, the number 52,00,67,890 is inconvenient to read and speak out loud.
We can also just read it as:
More than ₹52,00,00,000 has been raised for the frontline workers of the COVID-19 battle through a concert.
This gives a fair idea of how much money has been donated for the frontline workers. This is exactly what rounding off a number is! Let us learn more about rounding off numbers.
Rounding Off
To show rounded-off numbers, terms like 'about', 'over', 'more than' and 'approximately' are used to convey that the number is close to the actual number.
We can round off numbers to various places, such as tens, hundreds and thousands.
Rounding off a number to the nearest ten
Look at the digit in the ones place of the number.
If the digit is less than 5, the digit in the tens place remains the same, and the digit in the ones place is replaced by 0.
If the digit is 5 or greater than 5, the digit in the tens place is increased by 1, and the digit in the ones place is replaced by 0. For example, look at the numbers given below.
Remember!
Rounded-off numbers are the nearest values of the exact numbers.
Rounding off a number to the nearest hundred
Look at the digit in the tens place.
If the digit is less than 5, the digit in the hundreds place remains the same, and the digits at the tens and the ones places are replaced by 0.
If the digit is 5 or greater than 5, the digit in the hundreds place is increased by 1, and the digits at the tens and ones places are replaced by 0. For example, look at the numbers given below.
Rounding off a number to the nearest thousand
Look at the digit in the hundreds place.
If the digit is less than 5, the digit in the thousands place remains the same, and the digits in the hundreds, tens and ones places are replaced by 0.
How do we round off an 8-digit number to the largest place of a number? 0 < 5 5 = 5
Think and Tell
If the digit is 5 or greater than 5, the digit in the thousands place is increased by 1, and the digits in the hundreds, tens and ones places are replaced by 0. For example, look at the numbers given below.
Example 11: Round off 86,45,00,679 to the nearest ten.
86,45,00,679 has 9 in the ones place. 9 > 5. So, add 1 to the digit in the tens place and write 0 in ones place.
So, 86,45,00,679 rounded off to the nearest ten is 86,45,00,680.
Example 12: Which of the following is the rounded-off number for 32,75,98,065, if it is rounded off to the nearest thousand?
a 32,75,98,060 b 32,75,98,070 c 32,75,98,000
On rounding off 32,75,98,065 to the nearest thousand, we get 32,75,98,000. So, option c is the correct answer.
Round off 73,26,63,547 to the nearest ten, hundred and thousand.
So, the number 73,26,63,547 rounded off to the nearest 10 is 73,26,63,550.
So, the number 73,26,63,547 rounded off to the nearest 100 is .
So, the number 73,26,63,547 rounded off to the nearest 1000 is
Estimation in Operations
Did You Know?
The population of India was 1,43,16,60,000, as of September 24, 2023.
Sometimes we need to solve a problem quickly where we need to find the estimated sum, difference, product or quotient. In such cases, we first need to round off the numbers and perform the operation. This helps us easily find the answers.
Estimating the Sum and Difference
To find the estimated sum and difference, we round off each of the numbers and then add or subtract.
For example, we need to find the estimated sum and difference of 6584 and 2431 to the nearest hundred.
6584 rounded off to the nearest hundred = 6600
2431 rounded off to the nearest hundred = 2400
6 6 0 0 + 2 4 0 0
9 0 0 0
So, the estimated sum of 6584 and 2431 = 9000.
Estimating the Product
The estimated difference of 6584 and 2431 = 4200.
Like the estimated sum and difference, we can also find the estimated product of two numbers.
For example, we need to find the estimated product of 84 and 28.
84 rounded off to the nearest ten = 80
28 rounded off to the nearest ten = 30
80 × 30 = 2400
So, the estimated product of 84 and 28 is 2400.
Now, what if we have to find the estimated product of 568 × 44?
To find the estimated product in such cases, we round off each number to its greatest place, and then multiply the rounded-off numbers.
Hence, the estimated product is 600 × 40 = 24,000.
Estimating the Quotient
Like the estimated product, we can also find the estimated quotient. Say we need to find the estimated quotient of 96 and 9 to the nearest ten.
96 rounded off to the nearest ten = 100
9 rounded off to the nearest ten = 10
100 ÷ 10 = 100 10 = 10
Thus, the estimated quotient of 96 and 9 is 10.
Example 13: Find the estimated sum of 79,643 and 10,056 by rounding off to the nearest thousand.
79,643 rounded off to the nearest thousand = 80,000
10,056 rounded off to the nearest thousand = 10,000
Thus, the estimated sum of 79,643 and 10,056 when rounded off to the nearest thousand is 90,000.
Example 14: What is the estimated difference of 42,431 and 20,564 if they are rounded off to the nearest thousand?
42,431 rounded off to the nearest thousand = 42,000
20,564 rounded off to the nearest thousand = 20,000
Thus, the estimated difference of 42,431 and 20,564 when they are rounded off to the nearest thousand is 22,000.
Example 15: Find the estimated product of 834 and 25 by rounding off each number to its greatest place.
834 rounded off to the nearest hundred is 800.
25 rounded off to the nearest ten is 30.
800 × 30 = 24,000
Thus, the estimated product of 834 and 25 is 24,000.
Example 16: A company produced 5634 gadgets in total. If they need to pack them into boxes with each box containing 231 gadgets, about how many boxes will they need? (Round off to the nearest hundreds)
5634 rounded off to the nearest hundred = 5600
231 rounded off to the nearest hundred = 200
5600 ÷ 200 = 5600 200 = 28
Thus, the company will need around 28 boxes.
Estimate the answers for given operations. Do It Together
rounded off to 10 1 765 + 213 (to the nearest hundred) 3 758 × 510 (to the nearest hundred)
So, the estimated sum of 765 and 213 = .
rounded off to 100
rounded off to 100
So, the estimated product of 758 and 510 is
332 ÷ 25 (to the nearest ten)
rounded off to 10
______ = So, the estimated quotient of 332 ÷ 25 is .
So, the estimated difference of 984 and 356 = . 2 984 – 356 (to the nearest ten) Do as directed.
b Round off to the nearest 100 i
c Round off to the nearest 1000
Write all the possible numbers that can be rounded off to the nearest hundred as 4300.
What are the greatest and smallest numbers that can be rounded off to the nearest thousand to give 7000 as the answer?
Solve to find the estimated answer by rounding off to the nearest 100.
4356 + 9120
+ 23,589
Reeti bought a suit for ₹3659, a saree for ₹6342. Find the estimated amount she paid for the two items (Round off to the highest place). If she also bought a bedsheet and paid around ₹12,000 to the shopkeeper, what was the aproximate price of the bedsheet?
There are 8869 apple trees in an orchard. If there are 303 rows of apple trees, then find the estimated number of trees in each row. Do you eat any fruit every day? (Round off to the highest place)
The first commercial portable calculators appeared in Japan in 1970. Chirag purchased 45 calculators for his stationery shop. If each calculator costs ₹595, then estimate the total cost of the calculators. (Round off to the nearest tens)
Challenge
Critical Thinking
A number is formed by interchanging the digits 6 and 1 in 4,65,271. On rounding it off to the nearest ten, we get 4,15,280. If the original number is rounded off to the nearest hundred, then what is the difference of the original rounded-off number and the new rounded-off number?
Srikant estimated a number to the nearest ten and got the answer 5,67,790. What are the smallest and greatest possible numbers that can make Srikant’s answer correct? 1 2
Operations on Large Numbers
A factory produces 15,000 garments every 3 days.
First day: 4876 garments
Second day: 5823 garments
It produced the rest of the garments on the third day. How can we find the number of garments the factory produced on the third day?
Addition and Subtraction
Let us now understand how we can find the number of garments produced by the factory on the third day.
Total number of garments produced by the factory in 3 days = 15,000
Number of garments produced on the first day = 4876
Number of garments produced on the second day = 5823
We can find the number of garments produced by the factory on the third day by subtracting the number of garments produced on the first day and the second day from the total number of garments.
Thus, the number of garments produced on the third day is 4301.
Example 17: A printer prints 47,489 copies of a newspaper on Monday. The same printer prints 45,784 copies of a newspaper on Tuesday. How many copies does it print in two days?
Number of copies of the newspaper printed on Monday = 47,489
Number of copies of the newspaper printed on Tuesday = 45,784
Total number of copies printed in two days = 47,489 + 45,784 1 1 1 1 4 7 4 8 9
So, the total number of copies printed in two days is 93,273.
Example 18: The weight of a cylinder filled with gas is 30 kg 100 g, and the weight of the empty cylinder is 14 kg 500 g. What is the weight of the gas in the cylinder in grams?
Weight of the cylinder filled with gas = 30 kg 100 g = 30,100 g
Weight of the empty cylinder = 14 kg 500 g = 14,500 g
Weight of the gas in the cylinder = 30,100 g – 14,500 g 2 9 11 3 0 1 0 0
Think and Tell
Do you think 30 kg 100 g and 14 kg 500 g can be written as 30.100 kg and 14.500 kg, respectively?
So, the weight of the gas in the cylinder is 15,600 g or 15 kg 600 g.
Example 19: Write a word problem on adding two 6-digit numbers.
There are 2,56,689 people in city A, and there are 5,75,908 people in city B. How many people are there in both cities in total?
After a survey, Misha found that the total population of a city in India was 37,65,392, of which, 18,90,898 were adult males, 16,54,976 were adult females and the rest were children. How many children were there in the city?
Total population of the city in India = 37,65,392
Number of males in the city = 18,90,898
Number of females in the city = 16,54,976
Number of children in the city = – ( + 16,54,976)
So, the number of children in the city was .
Multiplication and Division
When the factory produces 15,000 garments in every 3 days, it is clear that a large number of garments are produced in a month. Now, let us see how we can find the number of garments produced in a month. 1 month has 30 days.
Number of garments produced every 3 days = 15,000
Number of garments produced in 1 day = 15,000 ÷ 3 = 5000
Number of garments produced in 30 days = 5000 × 30 = 1,50,000
So, the number of garments produced in a month is 1,50,000.
Example 20: A car can cover 85 km in 1 hour. How much distance (in metres) will it cover in 32 hours?
Distance covered by a car in 1 hour = 85 km
Distance covered by the car in 32 hours = 32 × 85 km
1 km = 1000 m
2720 km = 2720 × 1000 = 27,20,000 m
So, the distance covered by the car in 32 hours is 27,20,000 m.
Example 21: To stitch a pair of trousers, 2 m 35 cm cloth is needed. Out of 52 m 25 cm, how many pairs of trousers can be stitched and how much cloth will be left?
Length of cloth available = 52 m 25 cm = 5225 cm
Length of cloth required for 1 pair of trousers = 2 m 35 cm = 235 cm
Number of trousers that can be made from the given cloth = 5225 ÷ 235
Thus, 22 pairs of trousers can be made and 55 cm of cloth will be left. 235 5225
55 525 22
A school in a city has 5678 students. If the annual fee per student is ₹16,000, what is the total fee collected by the school?
Total number of students in a school = 5678
Annual fee per student = ₹16,000
Total fee collected by the school = × =
Parentheses and Order of Operations
We know that in the first month, the number of garments produced by the factory was 1,50,000. What if the production of garments in the factory increases by 3500 each month for the subsequent 6 months? Let us find the number of garments produced in 7 months.
Number of garments produced in the first month = 1,50,000
In the next month, production increased by 3500 and then remained constant for 5 months after that.
Number of garments produced in each of the 6 months = 1,50,000 + 3500
Number of garments produced in the 7 months = 1,50,000 + [(1,50,000 + 3500) × 6]
Now, we can see that the above expression has more than 1 operation and more than 1 set of brackets.
2 operations
Addition and Multiplication
1,50,000 + [(1,50,000 + 3500) × 6]
Now, let us learn how to solve problems with multiple operations.
2 brackets small brackets (parentheses) and square brackets
Each operation and each set of brackets has an order for solving them. Let us see how the order is followed.
Small brackets
Curly brackets
Square brackets
So, the expression 1,50,000 + [(1,50,000 + 3500) × 6] as per the given order can be solved in this way.
Step 1: Solve the brackets in the order of (), {} and []. There are no curly brackets, so we only solve the small and square brackets.
Step 2: Next, we have to solve the operations in the order of ÷, ×, + and −. We do not have the ‘÷’ and ‘×’ signs. So, we skip these operations. Next, we solve for the ‘+’ sign.
1,50,000 + [(1,50,000 + 3500) × 6] = 1,50,000 + [1,53,500 × 6] = 1,50,000 + 9,21,000
= 10,71,000
As we do not have any other operation in the expression, we will stop here.
Thus, 1,50,000 + [(1,50,000 + 3500) × 6] = 10,71,000
Using Brackets in Solving Problems
When we use brackets, it is easy to interpret the problem and simplify the process of solving it without any confusion.
For example, Sanya bought 3 bouquets of 8 roses each, and Aakash bought 5 bouquets of 8 roses each. Let us see how we can find the total number of roses they have. There are two ways of solving this problem.
Method 1 (without the use of brackets):
Number of roses purchased by Sanya = 3 × 8 = 24
Number of roses purchased by Aakash = 5 × 8 = 40
Together, the number of roses they purchased = 24 + 40 = 64
Method 2 (with the use of brackets):
Number of roses Sanya and Aakash purchased = (3 × 8) + (5 × 8) = 24 + 40 = 64
In each case, the answer is the same, but the methods are different. Method 2 was much faster and easier than method 1 because we used a pair of brackets to put the number of bouquets together.
So, using brackets appropriately is important to make the mathematical calculations easier and faster.
Expanding Brackets
We can also use brackets to expand a number.
For example, to solve 5 × 102, we use brackets to expand the second number and then use the expansion to solve it. Let us see how.
5 × 102 = 5 × (100 + 2) = 5 × 100 + 5 × 2 = 500 + 10 = 510
Example 22: Simplify [10 + (15 ÷ 5) × 4 – (2 × 2)]
= 41 × 3 = 123
= 36 + 15 = 51
[10 + (15 ÷ 5) × 4 – (2 × 2)] = [10 + 3 × 4 – 4] (solve small brackets) = [10 + 12 – 4] (solve square brackets through DMAS) = 18
Example 23: John ate 6 cookies from a packet of 26 cookies. Carl ate 5 cookies from the rest. How many cookies are left? Use brackets to find your answer.
Number of cookies eaten by John = 6
Number of cookies eaten by Carl = 5
Total number of cookies = 26
Total number of cookies left = (26 – 6) – 5 = 20 – 5 = 15
Solve: 8 × 105
8 × 105 = 8 × ( + 5) = 8 × + 8 × = + =
Error Alert!
Do It Yourself 1E
Solve the problems.
a 8,79,436 – 3,24,255 b 56,47,658 – 21,432
c 4,37,548 + 2345 + 7,65,675 d 76,98,608 + 23,13,542 + 2313 + 498
e 6,58,769 – 23,143 + 34,546 f 9,80,76,576 + 32,435 – 7,65,878
Solve.
a 54,76,587 × 40 b 3,24,543 × 120 c 65,78,598 ÷ 3
d 9,87,98,706 ÷ 9 e (79,69,889 × 6) ÷ 2 f (3,24,45,670 ÷ 5) × 13
Solve to find the answer. a 402 – 118 +
d 35 ÷ 5 × 9 + 7 × 30 ÷ 3 e 250 + [24 – {4 × 3 + (8 – 5)}]
Which number when added to 46,98,799 gives 95,34,657?
Find the missing digits. a 2 2 7 5 0 2 + 7 8 1 2 8 0 5 0 9
The product of two numbers is 5,87,69,880. If one of the numbers is 30, then find the other.
The quotient of two numbers is 9,04,35,460. If one of the numbers is 20, what is the other number?
Ajay travels a distance of 18 km 685 m from his house to his office. He travels in the metro two times a day. How much distance in metres will he cover in 5 days?
The distance at a particular time between Mercury and Earth is 9,63,77,000 km, and the distance between Earth and Jupiter is 6,56,96,000 km. Assuming that Mercury, Earth and Jupiter formed a straight line, find the total distance between Mercury and Jupiter.
A vessel contains 8 L 650 mL of water. How many glasses of capacity 500 mL each will be filled with the total quantity?
Sara gets a scholarship from the central government of ₹3408 each year. How much money will she receive in 5 years? How can you help to spread information about the scholarship schemes among your family and friends?
Protective gloves are made from natural rubber latex or nitrile rubber. A clinic orders 78 sets of protective gloves. If there are 546 gloves in total, how many gloves are there in one set?
Red carnation is the national flower of Spain. Megha buys 8 sets of carnations, with 5 carnations each decorating her house. The next day, she buys 12 sets of carnations with 5 carnations each. How many carnations does she buy in total?
Write a word problem on subtracting two 6-digit numbers.
The numbers are following a sequence. Find the sequence and fill in the next numbers.
4,03,804; 4,02,804; 4,03,304; ; 4,02,804; ; 4,02,304
Insert brackets to make the calculations true.
Points to Remember
• In the Indian number system, 9-digit numbers begin with the ten crores place in the place value chart (from the left).
• In the international number system, for 9-digit numbers, we have the ones period, thousands period and the millions period.
• The place value of a digit is the value of the digit in a number based on its position in the number.
• The face value of a digit for any place in the given number is the digit itself.
• In the Roman number system, we use seven basic symbols to write various numbers.
• Rounded-off numbers are the nearest values of the exact numbers.
• While rounding off a number to the nearest ten, hundred or thousand, we always check the digit in the place to the right. If the digit is equal to or greater than 5, we round up. We round down if the digit is less than 5.
• To find the estimated sum, difference, product or quotient, we round off each of the numbers and then perform the required operation.
• BDMAS stands for Brackets, Division, Multiplication, Addition and Subtraction.
Math Lab
Setting: In groups of 4
Experiential Learning & Collaboration
Mystery Number Math
Materials Required: Flash cards with maths riddles written on them, pencils, an A4-size sheet, erasers, chalk and duster and one dice
Method: All 4 members of each group must follow these steps.
A number table is drawn up to 20 like a hopscotch game (as shown below) before the game begins. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Flash cards with riddles will be kept at random numbers to solve. One example of a riddle is given below.
I am a 9-digit number. If you divide me by 85, then add 7, and finally multiply by 6, you will get 16,422,990. What number am I?
Choose your captain, and he/she rolls the dice.
Move as many steps forward on the number table as the number you get on the dice.
If you find a flash card at any number while moving through the table, solve it. You may use sheets of paper, a pencil and an eraser to solve the riddle.
If you get the answer right, you are allowed to move two extra steps forward as a reward. If you get the answer wrong, you need to move 5 steps back as a penalty from the number that you get on the dice.
(Every time a riddle is solved, it is replaced by a new one by the teacher.)
The group that completes 1 round from 1 to 20 first wins a small reward.
Chapter Checkup
Rewrite the numbers in figures and words, using both the Indian and the international number systems. Also, write their expanded forms.
a 350427681 b 420879502 c 635658421 d 901500084
Write the numbers for the number names.
a Four hundred sixty million seven hundred twenty-two thousand two hundred thirty-nine
b Sixty-three crore twelve lakh fifty-eight thousand one hundred forty-three
c Eighty crore nine lakh fifty thousand sixty-two
d One hundred million one hundred thousand thirty-nine
Fill in the blanks using <, > or =.
a 656,502,567 648,900,650 b 314,572,879 314,527,879
c 900,760,518 900,768,757 d 113,005,885 113,005,885
Arrange the following numbers in ascending order.
a 67,23,56,475; 19,08,04,365; 68,91,63,896; 76,90,87,687
b 676,162,895; 676,817,980; 435,406,576; 324,335,678
c 87,12,63,256; 65,45,12,845; 97,12,36,125; 65,78,15,325
Write the Roman numbers for the numbers given.
a 635 b 2015 c 2087 d 2750
Write the number forms of the following Roman numbers.
a CCCXVII b MCDIX c DCCXCVII d CCXVIII
Round off the numbers to the nearest ten, hundred or thousand.
a Ten 7,65,378 65,408 57,67,883
b Hundred 2,34,568 43,336 1,23,35,032
c Thousand 9,03,426 83,211 47,69,865
Simplify the problems.
a [72 – 36 ÷ 6 × 2] + (30 – 22) + 5 b 9 [8{12 – 7 + 5}] × 8
c 888 × [370 ÷ {65 + (18 ÷ 2)}] d 60 + 13 – {(5 × 1 2 × 10) – 75 ÷ (17 – 2)}
Solve and find the answers.
a 19,87,654 + 8,23,901 b 98,74,320 – 43,71,001
c 49,26,115 × 50 d 6,81,279 ÷ 3
e 67,589 – 23,543 + 13,678 f 90,800 + 32,552 – 45,765
A certain 9-digit number has only eights in the ones period, only sevens in the thousands period and only fours in the millions period. Write the number in figures and words, using both number systems.
The first Indian bank after independence was the State Bank of India in 1955. Junaid opened a bank account. He was given a customer ID: 750635389. Rewrite this number in figures and words using both number systems.
An e-commerce company sold 76,578 books in the first month of the year 2000. Estimate the number of books that the company will sell in the whole year if the company sells an equal number of books each month.
The Thanjavur doll is a traditional bobbled toy made of terracotta material. A factory manufactured 7,65,875 Thanjavur dolls in January and 4,36,586 Thanjavur dolls in February. Find the estimated difference of the dolls manufactured in February and January.
Write any two possible number sentences with all the operations that result in value of 100.
Form the greatest and smallest 9-digit numbers, using the digits 1, 3, 6, 8, 4, 9, 0, 2 and 5 only once. Also, form the greatest and the smallest 9-digit numbers using all of these digits except 0. Arrange the numbers in both ascending and descending order.
A shipping company has a fleet of 35 cargo ships, each capable of carrying 27,850 tonnes of cargo. These ships are divided into three fleets: Fleet A, Fleet B and Fleet C. Fleet A has 9 ships, Fleet B has 12 ships and Fleet C has the remaining ships. Find:
a the total cargo capacity of Fleet A and Fleet B combined.
b the total cargo capacity of Fleet C.
c the total cargo capacity of all 3 fleets.
1 In 15,87,12,528 how many times is the value of 5 in the crores place to that of the value of 5 in the hundreds place?
2 Read both the statements given below and choose the correct option.
Assertion (A): The number 6468 rounded off to the nearest hundred is 6500.
Reason (R): To round off a number, if the digit at the tens place is 5 or more, then the number is rounded off to the nearest 100 by increasing the hundreds digit by 1 and replacing each digit on its right by 0.
a A is true, but R is false.
b A is false, but R is true.
c Both A and R are true, and R is the correct explanation for A.
d Both A and R are true, but R is not correct explanation for A.
Case Study
Cross Curricular
Health Benefits of Grapes
Grapes are delicious fruits that offer health benefits like lowering cholesterol and blood sugar levels. They are a rich source of copper and vitamin K. The table shows the grapes production in different countries in the year 2022.
Country Production (in tonnes)
Italy 84,37,970
France 61,99,950
Türkiye 41,65,000
India 34,01,000
Chile 24,02,686
1 The quantity of grapes produced in Italy rounded off to the nearest thousand is .
2 Write True or False.
a The total quantity of grapes produced in India and Chile together is about 58,00,000 tonnes.
b France has produced about 20,00,000 tonnes more than Türkiye.
3 What is the total amount of grapes produced in Türkiye and Chile?
4 How much more grapes are produced in France than in India?
5 Write the number name of the quantity of grapes produced by Türkiye in the international number system.
2 Patterns in Whole Numbers
Letʼs Recall
Imagine we are in an apple orchard.
The number of ripe apples on the first tree is 1, on the second is 2, on the third is 3 and so on. The numbers 1, 2, 3... are called natural numbers.
Natural numbers are the counting numbers that start from 1 and go on infinitely, without any end.
Let us look at natural numbers on a number line.
Natural Numbers
1 2 3 4 5 6 7 8 9
We can skip count by numbers
Letʼs Warm-up
Help the squirrel jump by using skip counting.
Real Life Connect
Whole Numbers and Natural Numbers
Robin and Riya have flowers of different colours in their backyard. They were counting the number of flowers of each colour.
Robin: There are 10 red flowers and 19 yellow flowers, but there are no pink flowers.
Whole Numbers on a Number Line
The numbers 1, 2, 3, 4… are called natural numbers. All natural numbers along with 0 are called whole numbers
Predecessor and Successor
Did You Know?
Aryabhata, a renowned astronomer from ancient India, is credited with inventing the number ʺ0ʺ. However, later recognition for this invention was also given to Brahmagupta, another Indian mathematician who lived about a century after Aryabhata.
Predecessor comes before a given number, i.e., given number – 1.
Successor comes after a given number, i.e., given number + 1.
Predecessor of 10 9 10 11
Successor of 10
Example 1: Write the successor and predecessor of 12,499. Write the next 3 consecutive numbers after 12,499.
The successor of 12,499 = 12,499 + 1 = 12,500
The predecessor of 12,499 = 12,499 – 1 = 12,498
The three consecutive numbers after 12,499 are 12,500, 12,501 and 12,502.
Write down three consecutive numbers just preceding 87,453.
The predecessor of 87,453 = 87,453 – 1 = 87,452
The predecessor of = – 1 =
The predecessor of = _______ –_______ =
Whole Numbers
Think and Tell
Are all natural numbers whole numbers?
Remember!
Every whole number except zero has an immediate whole number predecessor.
Therefore, the three consecutive numbers before 87,453 are , and .
Representing Numbers on the Number Line
The distance between two consecutive points on a number line is called a unit distance.
Do It Together
Example 2: Represent 7 on the number line.
1 0 2 3 4 5 6 7 8 9 10
Think and Tell
What is the largest whole number?
Example 3: What is the distance between the points 3 and 8 on the number line?
1 0 2 3 4 5 6 7 8 9 10
The distance between 3 and 8 is 5.
Imagine you are at a railway station with a platform divided into sections, each labelled with whole numbers.
1 0 2 3 4 5 6 7 8 9 10
1 If you are standing in section 3, the whole number that represents the section just before the one you are in is .
2 If you take a step to the right and move to section 5, the whole number that represents the section you were previously in is .
Operations on a Number Line
Add: 5 + 3
Start at 5. Jump 3 steps of the unit distance forward. 1
Thus, 5 + 3 = 8
+1 +1 +1
Multiply: 4 × 3
Start at zero and jump 3 units forward, 4 times.
Subtract: 9 – 4
Start at 9. Jump 4 steps of the unit distance backwards.
–1 –1 –1 –1
Thus, 9 – 4 = 5
Divide: 15 ÷ 3
Start at 15 and jump 3 units backwards until we reach 0. –3 –3 –3
Therefore, 4 × 3 =12
Therefore, 15 ÷ 3 = 5
Example 4: Each student in a class has 3 pencils. How many pencils in total do 6 students have?
Number of pencils each student has = 3; Total number of students = 6
Do It Together
Total number of pencils = 6 × 3 pencils. Therefore, 6 students have 18 pencils.
Vansh bought 4 candies for ₹20. How much did one candy cost? Show your answer on the number line.
1
2 3
Cost of 4 candies = ` ; Cost of 1 candy = ________ ÷
Therefore, the cost of 1 candy is ` .
Properties of Whole Numbers
Properties of Addition of Whole Numbers
1. Closure property
Sum of 2 whole numbers = Whole number
For example: If 4 and 3 are two whole numbers, then 4 + 3 = 7 is also a whole number.
3. Additive property of zero
Adding 0 to any number results in the number itself. Zero is the identity element and is referred to as the additive identity.
For example: 3 + 0 = 0 + 3 = 3
Example 5: Find the sum of 137 and 295 in two ways.
137 + 295 = 432; 295 + 137 = 432
⇒ 137 + 295 = 432 = 295 + 137
Properties of Subtraction of Whole Numbers
1. Closure property
2. Commutative property
Two whole numbers can be added in any order.
For example: If 4 and 3 are two whole numbers, then 4 + 3 = 3 + 4 = 7.
4. Associative property
When three or more whole numbers are added, the sum remains the same regardless of the grouping of the addends.
For example: (4 + 3) + 5 = 4 + (3 + 5) = 12
The difference of two whole numbers may not always be a whole number. For example, If a and b are two whole numbers, then:
Case 1: a > b
As 14 is greater than 7, 14 – 7 = 7 is a whole number.
Case 2: a = b
As 7 and 7 are equal, then 7 – 7 is a whole number.
Therefore, the closure property does not hold true for subtraction.
2. Commutative property
If we change the order of subtraction, then the difference of the two whole numbers will not be the same.
As 7 and 4 are whole numbers, then 7 – 4 ≠ 4 – 7
Therefore, the commutative property does not hold true for subtraction.
4. Associative property
Two whole numbers cannot be regrouped and subtracted first. Hence, the order of subtraction is an important factor.
Let 14, 7 and 4 be three whole numbers.
14 – (7 – 4) π (14 – 7) – 4
Therefore, the associative law does not hold true for subtraction.
Case 3: a < b
As 7 is less than 14, then 7 – 14 is not a whole number, as subtraction is not possible if the subtrahend is smaller than the minuend.
3. Property of zero
When 0 is subtracted from a whole number, then the difference is the number itself.
Then, 7 – 0 = 7
Therefore, the property of zero holds true for subtraction.
5. Inverse Operations
Inverse operations are pairs of mathematical operations in which one operation undoes the other operation.
Addition and subtraction are inverse operations. We can say that addition undoes subtraction and vice versa.
Let 7, 4 and 3 be three whole numbers.
If 7 – 4 = 3; then 4 + 3 = 7.
Example 6: Use a suitable property to solve (958 − 445) − 510. Also, check if the associative law exists for subtraction.
(958 – 445) – 510 ⇒ (513) – 510 ⇒ 3
Using the associative law, 958 – (445 – 510); but 445 < 510
Therefore, the associative law does not hold true for subtraction.
Properties of Multiplication of Whole Numbers
Let us look at the properties of multiplication of whole numbers.
1. Closure property
When we multiply two whole numbers, the product is always a whole number.
For example: If 3 and 5 are two whole numbers, 3 × 5 = 15 is also a whole number.
3. Multiplicative property of zero
When a whole number is multiplied by zero, the resulting product is always zero.
For example: 3 × 0 = 0 × 3 = 0
5. Associative property
When three or more whole numbers are multiplied, the product remains the same regardless of the grouping of the numbers.
For example:
If 1, 2 and 3 are three whole numbers, then (3 × 2 ) × 1 = 3 × (2 × 1 ) = 6
Example 7: Solve and state the properties used.
1 544 × 6 + 544 × 9
= 544 × (6 + 9) = 544 × 15 ⇒ 8160
The property used is the distributive property of multiplication over addition.
Properties of Division
Let us look at the properties of division of whole numbers.
1. Closure property
When one whole number is divided by another whole number, then the quotient may not always be a whole number.
For example: 7 ÷ 2 = 3 1 2 , which is not a whole number
2. Commutative property
Two whole numbers can be multiplied in any order.
For example: If 3 and 5 are two whole numbers, 5 × 3 = 3 × 5.
4. Existence of multiplicative identity
When a whole number is multiplied by one, the product is equal to the number itself.
For example: 3 × 1 = 1 × 3 = 3
6. Distributive property of multiplication over addition
Multiplying the sum of two addends by a whole number gives the same result as multiplying each addend individually by the number and then adding the products together.
For example:
1 × (2 + 3) = 1 × 5 = 5 (1 × 2) + (1 × 3) = 2 + 3 = 5
So, 1 × (2 + 3) = (1 × 2) + (1 × 3)
2 823 × 5 – 823 × 3
= 823 × (5 – 3)
= 823 × (2) ⇒ 1646
The property used is the distributive property of multiplication over subtraction.
2. Commutative property
If the order of the dividend and divisor is changed, the quotient will not be the same.
For example: 6 ÷ 3 = 2 and 3 6 = 1 2
So, 6 ÷ 3 ≠ 3 ÷ 6
3.
Associative property
When three or more whole numbers are divided, the result changes if the order of the grouped numbers is changed.
For example:
(18 ÷ 3) ÷ 6 = 6 ÷ 6 = 1
18 ÷ (3 ÷ 6) = 18 ÷ 1 2 = 36
(18 ÷ 3) ÷ 6 ≠ 18 ÷ (3 ÷ 6)
5. Division by a number itself
When a non–zero whole number is divided by itself, it gives the quotient 1.
For example: 6 ÷ 6 = 1
7. Division by zero
A non–zero whole number cannot be divided by zero. Division by zero is not defined. 6 ÷ 0 is not defined.
For example: Division of a whole number by zero is not possible.
4.
Division by 1
When a non–zero whole number is divided by one, the quotient is the whole number itself.
For example: If 6 is a whole number, then 6 ÷ 1 = 6
6. Zero divided by a whole number
When zero is divided by a non–zero whole number, it gives the quotient zero.
For example: 0 ÷ 6 = 0
8. Division Algorithm
According to the division algorithm,
Dividend = (Divisor × Quotient) + Remainder
For example: When we divide 21 by 2, then the quotient and the remainder are 10 and 1, respectively. This can be written using the division algorithm as: 21 = 2 × 10 + 1
Example 8: Divide 75,850 by 450 and check the result by the division algorithm.
We are given that, divisor = 450 and dividend = 75,850.
On dividing 75,850 by 450, we get:
So, the quotient = 168 and the remainder = 250.
According to the division algorithm:
Dividend = (Divisor × Quotient) + Remainder
= (450 × 168) + 250 = 75,600 + 250 = 75,850 → Dividend
Example 9: Find the largest 5–digit number that is exactly divisible by 340.
The largest 5–digit number is 99,999.
On dividing 99,999 by 340, we get:
According to the division algorithm:
Dividend = (Divisor × Quotient) + Remainder
99,999 = (340 × 294) + 39
So, the required number = 340 × 294 = 99,999 – 39
Therefore, the required number is 99,960.
Fill in the blanks using the appropriate properties.
Write the next 5 whole numbers after 13,214.
Find the predecessor and successor of the numbers.
a 66,778 b 54,557 c 86,457 d 89,665 e 97,564
New Delhi is the one of the largest cities in India, covering an area of approximately 1484 square kilometres. What are the predecessor and successor of 1484?
Fill in the blanks with a whole number to make the following statements true.
a 452 × 0 = b 6567 × 1 =
c (4 × 7) × 10 = 4 × (7 × ) d 785 × 45 = 785 × (________ – 5)
e 14 × (10 + 12) = (14 × ) + (________ × 12) f 9875 ÷ 1=
g ÷ 789 = 0 h 448 × (224 + 28) = (448 × ) + (________ × 28)
i 999 + (45 ÷ 45) = + (1) = j (14,500 ÷ 50) – (99 ÷ 3) =
Locate the missing points on the number line below.
Write the results of the number line representation in each case.
Evaluate using the number line.
a 15 + 7 b 14 – 9 c 7 + 5 d 18 ÷
Replace the letter by a suitable whole number to make the statements true: a 41 ÷ 41 = x b 0 ÷ y = 0 c (36 − 25) ÷ p = 11 d (144 ÷ 12) × r = 12
The total surface area of the Earth is 510 million square km. Of this total surface, 357 million square km is covered by water, and the rest is covered by land. Use the distributive property to calculate the area covered by land.
What is the distance between the points on the number line?
a 0 and 12 b 6 and 16 c 3 and 20 d 4 and 18 e 12 and 18 f 2 and 14
Adi was baking cookies. Out of 15 cookies that he baked, 7 got burnt. Find the number of cookies that were not burnt using the number line.
A class of 20 students decides to contribute ₹5 each for a donation camp. Find the total amount of money contributed using the number line.
What is the smallest whole number? How many whole numbers are there between 67 and 75?
Find the product of the successor and predecessor of the largest six-digit number.
On dividing a whole number by 5, the quotient and the remainder are the same. Find such a whole number.
Patterns
Jiya and her friends were climbing a staircase.
She climbed one step at a time. Her friend skipped one step each time. Her sister skipped two steps every time she climbed. She noticed they were forming a pattern while climbing.
Patterns in Numbers
1, 2, 3, 4, 5,…
2, 4, 6, 8, 10,… Counting by threes Counting by tens
3, 6, 9, 12, 15, 18,….
Even and odd whole numbers
Even numbers are numbers that form pairs with no number left. Even numbers end with 0, 2, 4, 6 or 8. Some examples of even numbers are 432, 134, 456, 978 and 590
Odd numbers are numbers that have one number left after forming pairs. Odd numbers end with 1, 3, 5, 7 or 9. Some examples of odd numbers are 521, 133, 945, 707 and 959
10, 20, 30, 40,…
Did You Know?
Pythagoras was a Greek philosopher and a mathematician. He studied and emphasized the properties of numbers, including odd and even numbers.
Whole numbers can also be represented in the form of shapes based on their arrangement.
Every number that is greater than 1 can be represented in the form of a line.
The number 2 can be shown as:
The number 3 can be shown as:
The number 4 can be shown as:
The number 5 can be shown as:
Let us see the pictorial arrangement of some more number sequences.
Let us now look at the arrangement of dots to see patterns in whole numbers.
Triangular Numbers
The number 1 can be shown as:
Square Numbers
The
Rectangle Numbers
The number 6 can be shown as:
The number 10 can be shown as:
The numbers 1, 2, 4, 8, 16, 32, 64…are power of 2 numbers. 1 1 1 × 2 2
1 × 2 × 2 4
1 × 2 × 2 × 2 8
1 × 2 × 2 × 2 × 2 16
1 × 2 × 2 × 2 × 2 × 2 32 1 × 2 × 2 × 2 × 2 × 2 × 2 64
The numbers 1, 3, 9, 27, 81, 243…are power of 3 numbers.
× 3 × 3 × 3 × 3 × 3 × 3 729
Can you think of ways of representing these numbers using dots? Let us see one of the ways to represent the power of 2 numbers.
We can also simplify the mathematical calculations by just observing certain patterns, which include addition, subtraction or multiplication of certain numbers.
Addition of 9, 99, 999, etc., to a whole number
100 + 9 = 100 + 10 – 1 = 110 – 1 = 109
100 + 99 = 100 + 100 – 1 = 200 – 1 = 199
100 + 999 = 100 + 1000 – 1 = 1100 – 1 = 1099
Subtraction of 9, 99, 999, etc., from a whole number
Multiplication of a whole number by 9, 99, 999, etc.
1000 – 9 = 1000 – (10 – 1) = 1000 – 10 + 1 = 990 + 1 = 991 112 × 9 = 112 × (10 – 1) = 112 × 10 – 112 × 1 = 1120 – 112 = 1008
1000 – 99 = 1000 – (100 – 1) = 1000 – 100 + 1 = 900 + 1 = 901
1000 – 999 = 1000 – (1000 – 1) = 1000 – 1000 + 1 = 1
112 × 99 = 112 × (100 – 1) = 112 × 100 – 112 × 1 = 11,200 – 112 = 11,088
112 × 999 = 112 × (1000 – 1) = 112 × 1000 – 112 × 1 = 1,12,000 – 112 = 1,11,888
Example 10: For an event, 4569 invitations have to be sent out by Friday. If on Wednesday, 999 invitations have been sent out, then how many more invitations have to be sent out?
Number of invitations to be sent out = 4569; Number of invitations sent out = 999
Number of invitations that are yet to be sent out = 4569 – 999 = 4569 – (1000 – 1) = 4569 – 1000 + 1 = 3569 + 1 = 3570
Therefore, 3570 more invitations need to be sent out.
Example 11: Study the pattern and write the next two steps.
We can see that the whole numbers are represented in the form of a pentagon.
So, the next two terms will be:
Find the missing numbers. a 3, 9, 27, 81, b 5, 16, 49, 148, 445 c 1, 2, 6, 24, 120,
Patterns in Shapes
We can form patterns using different shapes. Let us look at some examples below. Shape Sequences
Let us now look at patterns in shapes and what is the basis of the order in which they are placed.
Can you think of the shape that will come next in the pattern above?
The shape sequence given above is given by counting the number of sides, starting with 3, 4, 5, 6 and 7. The next shape in the sequence should have 8 sides, so the next shape in the sequence will be an octagon. Now, look at another pattern. What is the rule of the pattern?
The sequence above shows the formation of shapes by joining the dots. The rule in the sequence is the increasing number of dots starting from 2, 3, 4, 5, 6 and so on.
Example 12: Look at the shape sequence given. Write the rule and the number of squares in the next shape in the sequence.
The number of boxes in each shape is 1, 4, 9, 16, 25.
These are squares of numbers: 1 × 1 = 1,
2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25
The next shape in the sequence will have 36 boxes which is 6 × 6 = 36.
Example 13: Look at the shape sequence and write the rule. How many sides will the next shape in the pattern have?
An inverted equilateral triangle is added in the middle of each shape in the sequence.
So, the number of sides increases to 4 times the number of sides.
The first triangle has 3 sides; the next shape in the sequence has 3 × 4 = 12 sides. The next shape has 12 × 4 = 48 sides, the next shape has 48 × 4 = 192.
The next shape in the sequence will have 192 × 4 = 768 sides.
So, the next shape in the sequence will have 768 sides.
Look at the given shape sequence. Write the rule followed by the sequence. How many triangles will the next shape in the sequence have?
Rule followed by the sequence .
The number of triangles in the next shape in the sequence is
Do
It Yourself 2B
Look at the given pattern. Draw the next term in the pattern .
Draw the missing term in the pattern .
Study the pattern and write the next two steps. 1 × 1 = 1 11 × 11 = 121 111 × 111 = 12321
Study the patterns and write the next 3 steps.
1 × 8 + 1 = 9 1 × 9 + 1 = 10 1 × 10 + 1 = 11
Aman gets ₹99 for 1 task. For 3 tasks, he gets ₹999, for 5 tasks, he gets ₹9999 and for 7 tasks, he gets ₹99,999. How much will he get for 13 tasks?
Look at the pattern. Write the rule followed in the pattern. How many rectangles will the fifth shape in the pattern have?
Solve and establish a pattern.
The Fibonacci series was introduced to the West by the Italian mathematician Leonardo of Pisa. It is a sequence in which each number is the sum of the two preceding numbers. What are the next two numbers in the Fibonacci series: 1, 1, 2, 3, 5, 8, , ?
Create your own pattern using shapes.
There are 7 students standing in a line. Each student has some pencils, so that the number of pencils each student has is double the number of pencils the student in front has. How many pencils does the 7th student have if the first student has 1 pencil?
Points to Remember
• Numbers 1, 2, 3, 4 and so on are called natural numbers.
• Whole numbers are a set of numbers that includes all the natural numbers (positive counting numbers) along with zero.
• Numbers can be represented in the form of a line, a triangle, a square and a rectangle.
Math Lab Communication &
Setting: In pairs
Material: stickers, squared paper, glue, scissors, pen
Method:
1 Cut out the square grid to form squares with side lengths as 1 unit, 2 units, 3 units and so on.
Experiential Learning
3 Paste the squares on chart paper.
2 Draw circles inside the squares and colour them to show square numbers.
4 Write the number of circles below each square of paper.
5 Write the rule of the pattern below.
6 Repeat with rectangle numbers and cube numbers.
Chapter Checkup
Write the successor and predecessor of the numbers.
Study the patterns and write the next two steps.
Solve on a number line.
What will be 6th shape in the patterns shown below?
Look at the given pattern. Write the rule for the given pattern. How many dots will the fourth shape in the pattern have?
How many dots will come in the fifth pattern?
Create your own pattern using different shapes.
According to the People's Linguistic Survey of India, India has the second highest number of languages, totalling 780, after Papua New Guinea, which has 840. What is the difference in the number of languages available in Papua as compared to India?
During Diwali, Riya and her friends decided to decorate an orphanage and give gifts to the children there. They have already collected 1245 gifts for the children. Riya and her friends want to add 999 more gifts to their collection. Help them calculate the total number of gifts they will have after adding 999 using the short method of addition. What value do Riya and her friends demonstrate by helping the children with gifts?
The budget for an event is `82,360. Of this `15,500 was spent on the decorations, `10,000 was paid to the music band and `6500 was spent on refreshments. How much money is left after these expenses?
Find the number which when divided by 65 gives 9 as a quotient and 5 as a remainder. Did you use any property? Name it.
A librarian purchased 45 English books and 45 Hindi books. If the cost of one English book is `100 and one Hindi book is `75, find the total amount he has to pay.
A school ordered 96 chairs and 48 tables. The cost of each chair is `200 and the cost of each table is `150. If the school gave `3500 as an advance, what amount is yet to be paid?
Solve using the distributive property.
a 550 × 45 – 550 × 15 b 865 × 12 + 865 × 45
c 420 × 36 – 420 × 23
d 630 × 25 - 630 × 15
A rectangular playground is 324 m long and 220 m wide. How much distance will Vicky cover going 4 times around it? Which property did you use to solve the problem?
Read the statements given below and choose the correct option.
Assertion (A): The sum of 100, 150 and 100 is 15,00,000.
Reason (R): If a, b and c are three whole numbers then, (a + b) + c = a + (b + c).
Choose the correct option:
a Both A and R are true, and R is the correct explanation for A.
b Both A and Rare true, but R is not the correct explanation for A.
c A is true, but R is false.
d A is false, but R is true.
Look at the pattern. Draw the next term of the pattern.
Case Study
Warli Art
Cross Curricular & Creativity
Warli art is a tribal art mostly created by tribal people from the North Sahyadri Range in Maharashtra, India. These paintings use a set of basic geometric shapes such as a circle, triangle and square. These shapes are symbolic of different elements of nature.
1 The shapes used in the pattern above are
2 Choose the correct number of shapes in the first figure in the pattern shown above.
a 1 triangle and 2 circles
c 2 triangles and 1 circle
b 3 circles and 2 triangles
d 2 triangles and 2 circles
3 Write the rule followed by the pattern shown above.
4 What will be the next term in the pattern shown above?
5 Create your own Warli pattern using squares, triangles and circles.
3 Playing with Numbers: Factors and Multiples
Letʼs Recall
Shreya arranges 16 tiles in different ways, as shown. She arranges them in rows so that each row has the same number of tiles in it. She writes the multiplication sentence for each arrangement and finds that 18 can be written as the product of two numbers in different ways.
1 and 18, 2 and 9 and 3 and 6 are the numbers that when multiplied give the same product. They are the factors of 18. We can divide 18 by any of these factors without a remainder.
Letʼs Warm-up
Fill in the blanks with any two factors of the numbers given below:
Reviewing Factors and Multiples
Neerja and her mom went to the nearby grocery store to purchase fruits.
Mother: Neerja, we need to buy 12 apples. I see that they are arranged in different ways in boxes.
Neerja: Yes. I can see boxes having 1 row with 12 apples each, 2 rows with 6 apples each and 3 rows with 4 apples each.
Mother: Yes, very good. Are there other ways in which the 12 apples can be arranged? Let us find out!
= 1 ×
12 = 2 × 6
12 = 3 × 4
12 = 4 × 3
12 = 6 × 2
12 = 12 × 1
1 × 12, 2 × 6, 3 × 4 are called factor pairs. 1, 2, 3, 4, 6 and 12 are factors of 12.
Factors are whole numbers that can be multiplied to give another number as a result.
Two factors of a number that multiply to give the number are called factor pairs. We can find these pairs by identifying numbers that divide 12 completely without leaving a remainder.
Multiples are numbers that can be obtained by multiplying a number by other whole numbers. For example, we can multiply 4 by different numbers to get the multiples of 4 as 4, 8, 12, 16 and so on.
Let us now learn about the properties of factors and multiples.
Properties of Factors
Every number is a factor of itself, and 1 is a factor of every number.
Properties of Multiples
Every number is a multiple of 1 and itself.
Division of a number by its factor leaves no remainder. A multiple can be expressed as a product of its factors. A factor of a number is always less than or equal to the number.
The number of factors of a number is fixed.
Multiples of a number are always equal to or more than the number.
The number of multiples of a number is infinite.
A number can be a factor of more than one number. A number can be a multiple of many numbers.
Example 1: Write all the factors of 48.
1 × 48 = 48 2 × 24 = 48 3 × 16 = 48 4 × 12 = 48
6 × 8 = 48 (We stop here as 8 and 6 are already covered.)
So, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Factor Pairs
Example 2: Find two numbers whose difference is 3 and whose product is 54.
The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54.
Clearly, 6 and 9 are the required numbers, as their difference is 3 and their product is 54.
List all the factors and any three multiples of the numbers given below.
Factors = 1, 2, 4, 8, 16
Factors = Multiples = 16, 32, 48 Multiples
Factors
Factors = Multiples = Multiples =
ll the factors of the given numbers.
Write the first six multiples of each of the given numbers.
The product of two numbers is 24. Their sum is 14. Find the numbers.
Find two numbers of which the difference is 1 and the product is 72.
Shade the square grids to show factor pairs of the given numbers.
1 Raj claims to have 100 rupees, all in 2-rupee coins. Roy claims to have 67 rupees, all in 5-rupee coins. Who is wrong? Give a reason for your answer.
Types of Numbers
Ankita got a bag of blocks from her mother. The bag had 12 red blocks and 13 blue blocks. When Ankita and Naman tried to share the blocks equally, they found that they could split the red blocks evenly but not the blue ones. They were left with one blue block, which made them think about how they could divide the red blocks evenly but not the blue ones.
Let us understand the mystery behind these numbers.
Even and Odd Numbers
Even Number
Any number that can be divided exactly by 2 is called an even number. These numbers always end with the digits 0, 2, 4, 6 or 8.
Example: 26, 30, 18, 42, 64, etc.
Example 3: Which of the numbers is even?
Odd Number
Odd numbers are numbers that are not divisible by 2, meaning they cannot be divided exactly by 2 and always leave a remainder of 1 when divided by 2. These numbers always end with the digits 1, 3, 5, 7 or 9.
Example: 15, 19, 27, etc.
Even numbers end with 0, 2, 4, 6 or 8. Hence, 62 is an even number.
Example 4: Which of the numbers is odd?
Odd numbers end with 1, 3, 5, 7 or 9. Hence, 41 is an odd number.
Classify the numbers as even or odd.
Perfect Number
A perfect number is a number that is the sum of all its factors (excluding itself).
For example, 6: The factors of 6 are 1, 2 and 3.
1 + 2 + 3 = 6
Example 5: Check whether 496 is a perfect number or not.
Factors of 496 = 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
Sum of factors (excluding itself) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
Therefore, 496 is a perfect number.
Check whether 8128 is a perfect number.
The factors of 8128 are 1, 2, 4, 8, , 32, , 127,
Sum of factors (excluding itself) = 1 + 2 + 4 + 8 + ______________________________________________
Therefore, 8128 (is/is not) a perfect number.
Prime and Composite Numbers
Prime Number
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19 and so on.
Here 2 has 2 factors; 1 and 2; 17 has 2 factors 1 and 17.
Composite Number
A composite number is a natural number greater than 1 that has more than two factors.
Examples of composite numbers are 4, 6, 9 and so on.
Factors of 4 — 1, 2, 4 (more than 2 factors)
Factors of 9 — 1, 3, 9 (more than 2 factors)
Factors of 15 — 1, 3, 5, 15 (more than 2 factors)
Note:
• 1 is neither a prime number nor a composite number.
• 2 is the smallest prime number.
• 2 is the only even prime number.
Twin primes are pairs of prime numbers that have a difference of 2 between them.
For example: (3, 5), (5, 7) and (11, 13).
Think and Tell
How many even prime numbers are there?
Prime triplets are sets of three prime numbers that have a difference of 2 between each consecutive pair.
For example: (3, 5, 7) and (17, 19, 23)
Co-primes, also known as relatively prime or mutually prime numbers, are two or more numbers that share no common factors other than 1.
For example: (8, 9), (15, 28), (7, 11)
Twin Primes
Prime Triplets
Co-primes (Relatively Prime)
Co-prime Art
Look at these thread art diagrams. In the first picture, there are 18 pegs, and the thread is wrapped around every sixth peg (this is called a thread gap of 6). In the second picture, there are 8 pegs, and the thread is wrapped around every third peg (a thread gap of 3).
Did you notice that in the first picture, the thread is not touching all the pegs, whereas in the second picture, it is touching all the pegs? There is a relation between the number of pegs and the thread gap. When the thread is touching all the pegs, the number of pegs and thread gap are co-prime (meaning they don’t have any common factors except 1). Here 8 and 3 are co-prime numbers since they have no common factors except 1. 18 and 6 are not co-prime numbers.
Example 6: Complete the thread diagrams. Write if the numbers are co-primes. 1 20 pegs, Thread gap of 11
10 pegs, Thread gap of 4
20 and 11 are
Example 7: Which of the numbers below is a prime number?
and 4 are not co-prime
Here options (2), (3) and (4) have 2 as a factor since these numbers end with 2, 6, and 8 respectively. On the other hand, 37 has only two factors, 1 and itself. So, 37 is a prime number.
Example 8: Which of the given pairs are co-prime?
1 12 and 15
Solution:
5 and 15
10 and 15
1 12 and 15: The common factors are 1 and 3, so they are not co-prime.
2 5 and 15: The common factors are 1 and 5, so they are not co-prime.
3 10 and 15: The common factors are 1 and 5, so they are not co-prime.
4 17 and 15: The only common factor is 1, so they are co-prime. So, 17 and 15 are co-prime.
17 and 15
Determine if the given pairs of numbers are co-prime (relatively prime) or not. Justify your answers.
1 15 and 28 2 21 and 40
Factors of 15: 1, 3, 5, 15
Factors of 28: 1, 2, 7, 14, 28
Only 1 is the common factor.
Hence, 15 and 28 are .
Divsibility Rules
Factors of 21:
Factors of 40:
Hence, 21 and 40 are .
Let us look at some rules that will help us figure out whether a number is divisible without dividing it.
Divisible by
Divisibility Rules
Examples
2 The digit in the ones place of the number is 0, 2, 4, 6 or 8. 10, 32, 54, 76, 98 and so on
5 The digit in the ones place of the number is 0 or 5. 25, 70, 135, 160 and so on
10 The digit in the ones place of the number is 0. 20, 50, 70, 100 and so on
3 The sum of the digits of the number should be divisible by 3.
36: Sum of digits = 3 + 6 = 9 (divisible by 3)
315: Sum of digits = 3 + 1 + 5 = 9 (divisible by 3)
6 The number should be divisible by both 2 and 3. 18: divisible by both 2 and 3.
4 The number formed by the last two digits should be divisible by 4.
8 The number formed by the last three digits should be divisible by 8.
9 The sum of the digits of a number should be divisible by 9.
11 The difference of the sum of the digits in the odd places and the sum of the digits in the even places is 0 or a multiple of 11.
Example 9: Which of the numbers is divisible by 3?
A number is divisible by 3 if the sum of its digits is divisible by 3. Let us calculate the sums:
For 572: 5 + 7 + 2 = 14 (not divisible by 3);
For 648: 6 + 4 + 8 = 18 (divisible by 3)
For 793: 7 + 9 + 3 = 19 (not divisible by 3);
For 921: 9 + 2 + 1 = 12 (divisible by 3)
72: divisible by both 2 and 3.
424: 24 is divisible by 4
736: 36 is divisible by 4
2016: 16 is divisible by 8
4136: 136 is divisible by 8
108:Sum of digits = 1 + 0 + 8 = 9 (divisible by 9)
2169: Sum of digits = 2 + 1 + 6 + 9 = 18 (divisible by 9)
847:Sum of digits in odd places – sum of digits in even places
= (7 + 8) – 4 = 11 (divisible by 11)
935: Sum of digits in odd places – sum of digits in even places
= (5 + 9) – 3 = 11 (divisible by 11)
Did You Know?
Lakshit Pusri from India made a world record by solving 100 division sums of 3-digit by 1-digit numbers in just 1 minute 54 seconds.
Find out which of these numbers are divisible by 4, 6, 9 or 11.
1 5628
Divisibility by 4: 28 is divisible by 4.
So, 5628 is divisible by
Divisibility by 6: Digits at ones place is 8 which is divisible by .
5 + 6 + 2 + 8 = 21, which is divisible by .
So, 5628 is divisible by both 2 and 3, hence it is divisible by .
Divisibility by 9:
Divisibility by 11:
So, 5628 is divisible by .
2 4356
Divisibility by 4:
Divisibility by 6:
Divisibility by 9:
Divisibility by 11:
So, 4356 is divisible by .
Determine whether the given numbers are even or odd. Write “E” for even and “O” for odd next to each number.
a 42
Identify if there are any perfect numbers among the numbers given below. a 57
Write True or False for the given statements.
a 25 is an even number.
c 6 is a composite number.
b 17 is a prime number.
d 13 and 17 are twin prime numbers.
Write all the prime numbers within each range given below. Create a co-prime pair from each range.
a 40 to 50 b 60 to 70
c 10 to 30
Write all pairs of twin primes between 60 and 90.
d 0 to 10
Check the divisibility of the given numbers by 2, 3, 4, 5, 6, 8, 9 or 10.
990; 2484; 41,870 and 9,12,048
Give a number divisible by:
a 2 but not 6
c both 11 and 9
b both 6 and 8
d 3 but not 2
Draw square grids to show all the factor pairs of 14. Write if 14 is a prime or composite number. Give a reason for your answer.
A maths teacher asks students to organise themselves into two groups based on whether their roll numbers are prime or composite. How many students are there in each group if there are a total of 36 students in the class having roll numbers 1 to 36?
Write a question on your own using the divisibility test for 11.
Use thread art to find if 20 and 8 are co-primes or not.
Challenge
Critical Thinking
Find two prime numbers which can be expressed as the sum of 4 different prime numbers. 1
Prime Factorisation
Maya runs a factory that produces cardboard boxes. The factory uses prime factorisation to figure out how many different-sized boxes can be created from a certain size of cardboard sheet to minimise waste.
Let us see how prime factorisation helps us!
Prime factor:
When a prime number is a factor of a given number, it is referred to as a prime factor. Here is one of the methods for finding the factors and prime factors of the number 36.
Every composite number can undergo further factorisation. This factorisation process continues until all the factors derived from it are prime, and these prime factors are highlighted.
36 = 2 × 2 × 3 × 3
Hence, 36 has been expressed as a product of prime factors. This is the prime factorisation of 36.
The prime factorisation of a number can be done by these methods.
1 Factor tree 2 Repeated division
Let us factorise 75 using both the methods. Remember!
The prime factors may be presented in varying orders, yet they remain the same.
75 = 3 × 5 ×
= 3 × 5 × 5
Example 10: Express each number as the product of its prime factors by using a factor tree.
Example 11: Express each number as the product of its prime factors by using repeated division.
Example 12: Write the largest 4-digit number and express it in terms of its prime factors. The largest 4-digit number is 9999.
Example 13: Find the prime factors of 252. Arrange them in ascending order. Determine the difference of two consecutive prime factors.
252 = 2 × 2 × 3 × 3 × 7
The prime factors of 252 in ascending order are 2, 3 and 7.
The difference of 2 and 3 is 1. The
of 3 and 7 is 4.
Express each number as the product of its factors by using both factor trees and repeated division.
Prime factors of 76 Factor tree of 76
Prime factors of 99 Factor tree of 99
Do It Yourself 3C
Show the prime factorisation of each of the given numbers.
a 126 b 882 c 6241 d 192
State whether these are prime factorisations or not.
a 189 = 3 × 3 × 21 b 252 = 2 × 6 × 3 × 7
Find the prime factors of 345. Arrange them in ascending order.
Express each number as the product of its prime factors by using a factor tree.
Find the prime factors of 560. Arrange them in ascending order. Calculate the product of the two smallest prime factors. Write the smallest 5-digit number and express it in terms of its prime factors.
A biologist, a scientist who studies living organisms and their interactions with the environment, wants to organise several samples of bacteria into rectangular petri dishes for an experiment. Should the biologist have 56 samples or 63 samples of bacteria, if the samples are to be arranged into 4 rows?
Challenge
Find the smallest positive integer that can be expressed as the product of four distinct prime numbers and is greater than 100.
Working with LCM and HCF
Maya: Ansh, I have 36 red balloons, and 48 blue balloons. Can you tell me what is the maximum number of identical groups that I can make so that each group has both types of balloons?
Ansh: We can find the common factors of 36 and 48. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, while the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
The common factors are 1, 2, 3, 4, 6 and 12. 12 is the highest common factor (HCF).
So we can make 12 identical groups, each having 3 red balloons and 4 blue balloons.
Let us revise common factors and common multiples and learn how finding the HCF and LCM can help us solve such problems in our daily lives.
Revisiting HCF and LCM
Common Factors and Common Multiples
We will first revise the concept of common factors and multiples to get started.
Common Factors
Common factors are numbers that can exactly divide two or more numbers without leaving a remainder.
Let us find the common factors of 12, 20 and 24.
Factors of 12: 1, 2, 3, 4, 6 and 12
Factors of 20: 1, 2, 4, 5, 10 and 20
Factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24
The common factors of 12, 20 and 24 are 1, 2 and 4.
Common Multiples
Common multiples are numbers that are multiples of two or more numbers.
Let us find the common multiples of 12, 20 and 24
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120,...
Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180,...
Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240,...
The common multiples of 12, 20 and 24 are 120, 240, 360,…
Example 14: List the common factors of 48, 36 and 56.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
So, the common factors of 48, 36 and 56 are 1, 2 and 4.
Example 15: List two common multiples of 6, 12 and 18.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72,...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...
Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180...
So, two common multiples of 6, 12 and 18 are 36 and 72.
Factors of 12 Factors of 20
3, 6, 12
5, 10, 20 8, 24 1, 2 4
Factors of 24
Multiples of 12 Multiples of 20
12, 36, 60, 84, 108 24,48,72,96144,168, 192,216
20, 40, 60, 80, 100, 140, 160 180 120
Multiples of 24
Draw Venn diagrams to show the factors of the given numbers. List their common factors.
1 12, 18, 24: 6, 2 18, 27, 36:
Factors of 12 Factors of 18
4, 12 9, 18
8, 24
Factors of 24
Highest Common Factor
HCF stands for Highest Common Factor, and it refers to the largest number that evenly divides two or more numbers without leaving a remainder.
Let us learn how to find the HCF of two numbers using three different methods.
By Listing Factors
Step 1: List the factors of 24 and 36.
Step 2: Identify the common factors.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors of 24 and 36: 1, 2, 3, 4, 6, 12
Step 3: Determine the largest common factor. The largest common factor is 12.
So, the HCF of 24 and 36 is 12.
Now, let us find the HCF of 324 and 540 using the prime factorisation and long division method.
By Prime Factorisation
Step 1: Find the prime factorisation of each number.
Step 2: Identify the common prime factors.
Step 3: Select one prime factor from each group of shared prime factors and multiply these selected prime factors to obtain the HCF.
So, the HCF of 324 and 540 is 2 × 2 × 3 × 3 × 3 = 108.
By Long Division
Step 1: Start by dividing the larger number by the smaller number. If the remainder is zero, the divisor is the HCF.
Step 2: If the remainder is not zero, divide the divisor by the remainder.
Step 3: Repeat this process until the remainder becomes zero. The last divisor used is the HCF.
So, the HCF of 216 and 324 is 108.
Example 16: Find the HCF of the numbers using prime factorisation. 168 and 252 168 = 2 ×
252 = 2 × 2 × 3 × 3 × 7
Hence, HCF of 168 and 252 = 2 × 2 × 3 × 7 = 84
Example 17: Find the HCF of the numbers using repeated division: 126, 198 and 315
Hence, the HCF of 126, 198, 315 is 9.
Find the HCF of the given numbers using prime factorisation. 72, 108, 96 Do It Together
72 = 108 = 96 =
Hence, the HCF of 72, 108 and 96 = 2
Lowest Common Multiple
The lowest common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the given numbers without leaving a remainder.
Let us learn the different methods to find the LCM.
By
Listing Multiples
Find the LCM of 24 and 36 by listing the multiples.
Multiples of 24: 24, 48, 72, 96, 120, 144...
Multiples of 36: 36, 72, 108, 144,...
The common multiples of 24 and 36 are 72, 144…
Chapter 3 • Playing with Numbers: Factors and Multiples
The lowest common multiple of 24 and 36 is 72.
So, the LCM of 24 and 36 is 72.
Find the LCM of 16, 24, 30 and 36 using the following methods:
By Prime Factorisation
Step 1: Find the prime factorisation of each number.
Step 2: Identify common prime factors among all numbers.
Step 3: Select one shared prime factor from each set of prime factors and multiply them with all the other common prime factors that are not common to find the LCM.
Hence, the LCM of 16, 24, 30 and 36 is 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720.
By Division Method
Step 1: Arrange the provided numbers in a line, separated by commas.
Step 2: Find a number that divides at least one of them exactly.
Step 3: Divide the divisible numbers and note the quotients below these numbers. Write the numbers that cannot be divided as they are.
Step 4: Repeat until you have co-prime numbers.
Step 5: Multiply divisors and co-prime numbers to get the LCM.
Hence, the LCM of 16, 24, 30 and 36 is 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720.
Properties of LCM and HCF
16 = 2 × 2 × 2 × 2
24 = 2 × 2 × 2 × 3
30 = 2 × 3 × 5
36 = 2 × 2 × 3 × 3
2 16, 24, 30, 36
2 8, 12, 15, 18
2 4, 6, 15, 9
2 2, 3, 15, 9
3 1, 3, 15, 9
3 1, 1, 5, 3
5 1, 1, 5, 1 1, 1, 1, 1
1 The highest common factor (HCF) of the numbers is always smaller than or equal to the smallest number.
2 The least common multiple (LCM) of the numbers is never less than any of the numbers.
3 Two numbers that are co-prime have an HCF of 1.
4 For two or more co-prime numbers, their LCM is equal to the product of those numbers.
5 The HCF of the numbers is always a factor of their LCM.
6 For any two numbers, their product is equal to the product of their HCF and LCM. Let us understand the relationship between the HCF and LCM through an example.
Consider the numbers 12 and 18. Let us find their HCF and LCM and compare their product with the product of their HCF and LCM.
HCF of 12 and 18 = 2 × 3 = 6
Product of numbers = 12 × 18 = 216
HCF × LCM = 6 × 36 = 216
Product of Numbers = HCF × LCM
LCM of 12 and 18 = 2 × 2 × 3 × 3 = 36
Example 18: Find the LCM of 135 and 246 using prime factorisation:
Example 19: Find the LCM of 40, 48 and 45 using the division method.
2 40, 48, 45
2 20, 24, 45
2 10, 12, 45
2 5, 6, 45
3 5, 3, 45
3 5, 1, 15
5 5, 1, 5 1, 1, 1
Hence, the LCM of 40, 48 and 45 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
Example 20: The HCF of two numbers is 9, and their LCM is 108. One of the numbers is 27. What is the other number?
Let us denote the two numbers as A and B.
Given: HCF = 9; LCM = 108
Let us say one of the numbers A = 27
We can use the relationship between HCF, LCM and the numbers: HCF × LCM = A × B
Substitute the values:
9 × 108 = 27 × B
B = 9 × 108 27 = 36
So, the other number B is 36.
Example 21: Is it possible for two numbers to have 25 as their highest common factor (HCF), and 600 as their least common multiple (LCM)? Explain your answer.
The HCF of the given numbers is always a factor of their LCM. Since 600 is divisible by 25, it is possible to have two numbers whose LCM and HCF are 600 and 25, respectively.
Example 22: Find the greatest number that divides 247 and 1031 leaves a remainder 7 in each case.
Subtract the remainder from each number:
247 − 7 = 240; 1031 − 7 = 1024
The HCF of 240 and 1024 is 16.
The greatest number that divides these numbers leaving a remainder of 7 is 16.
2 126, 154, 189: Do It Together
Fill in the blank with the LCM of the numbers below:
1 60, 84: 2 × 2 × × × =
Chapter 3 • Playing with Numbers: Factors and Multiples
Do It Yourself 3D
Find the HCF of the numbers.
a By listing factors.
i 48, 84 ii 30, 45 iii 18, 27 iv 36, 72, 108
b By prime factorisation.
i 20, 30 ii 30, 45 iii 45, 75 iv 54, 90, 126
c By repeated division.
i 48, 60 ii 63, 105 iii 72, 96, 120 iv 65, 132
Find the LCM of the numbers.
a By listing their multiples.
i 60, 20 ii 15, 35 iii 4, 8, 12 iv 12, 16
b By the prime factorisation method.
i 16, 30, 42 ii 28, 44, 132 iii 16, 28, 40, 77 iv 20, 25, 30, 50
c By the division method.
i 11, 22, 36 ii 96, 128, 240 iii 9, 12, 36, 54 iv 102, 170, 136
The highest common factor (HCF) of two numbers is 18, and their least common multiple (LCM) is 126. One of the numbers is 54. What is the other number?
If 25 is the HCF of two numbers, can the LCM of those numbers be 780? Explain.
What is the maximum number of Sundays that can occur in the first 57 days of the year?
Find the greatest number that can divide 655 and 889 leaving a remainder of 7 in each case.
Find the greatest number that can divide 77, 263 and 462 leaving remainders 5, 7 and 6, respectively.
The product of two numbers is 2028 and their H.C.F. is 13. How many such pairs of numbers are possible?
Word Problems on LCM and HCF
The knowledge of HCF and LCM can come in handy in many real-life situations. Let us explore some word problems!
Amy is planning to distribute chocolates (24), balloons (30) and pencils (18) equally among her friends for a party. What is the maximum number of identical gift bags she can create without any items left over?
As we need to find the maximum number of gift bags without any items left over, we will find the HCF of the given numbers.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24
Factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30
Factors of 18 = 1, 2, 3, 6, 9 and 18
So, the highest common factor (HCF) of the numbers is 6.
Hence, Amy can create a maximum of 6 gift bags without any items left.
Remember!
LCM is always greater than HCF.
Example 23: At a bakery, there are three ovens used for baking different types of bread. Oven A bakes a batch in 15 minutes, Oven B in 20 minutes, and Oven C in 25 minutes. At what time intervals will all three ovens finish baking their batches simultaneously?
To find out how often all three ovens will finish baking their batches simultaneously and the minimum time it will take for them to do so again, we need to determine the least common multiple (LCM) of the baking times for each oven.
The baking times for the ovens are:
Oven A = 15 minutes; Oven B = 20 minutes; Oven C = 25 minutes
LCM (15, 20, 25) = 2 × 2 × 3 × 5 × 5 = 300
So, all three ovens will finish baking their batches simultaneously every 300 minutes.
2 15, 20, 25
2 15, 10, 25
3 15, 5, 25
5 5, 5, 25
5 1, 1, 5 1, 1, 1
Emily is organising a bake sale fundraiser for her school. She has 24 chocolate chip cookies, 36 oatmeal cookies and 48 peanut butter cookies. She wants to make cookie packs with the same number of each type in each pack, with no cookies left. How many packs of cookies will Emily have in total?
To find the number of packs of cookies, we need to calculate the HCF of 24, 36 and 48.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 = 1, , , , 6, , , ,
Factors of 48 = , , , , , 8, , , , 48
So, the HCF of the numbers is . Hence, Emily will make packs.
Shaini is making gift bags for a party. She has 64 toy cars and 80 action figures. She wants to make identical gift bags with the same numbers of toy cars and action figures in each. If she wants to make the greatest number of gift bags possible without having any items left over, then how many gift bags can she make?
A toy store has three toy-making machines. Machine X produces toys every 25 minutes, Machine Y every 35 minutes and Machine Z every 40 minutes. After how much time will all three machines complete their work simultaneously?
Cattle farming involves raising cows primarily for milk production, ensuring proper nutrition and health to maximise milk yield. A farmer has 72 cows and 36 bales of grass. He wants to make identical groups such that each group has an equal number of cows and an equal number of bales of grass. What is the maximum number of groups he can make?
Different plants have varying watering needs which is crucial for their health. Sansevieria requires watering every 18 days, while Euphorbia milii needs it every 12 days. If the gardener waters all the plants today, after how many days will he need to water them together again?
A carpenter has wooden planks of lengths 54 cm, 72 cm and 90 cm. What is the maximum length he can cut these planks into, with no wastage?
Three different buses depart from a bus station. Bus X departs every 8 minutes, Bus Y every 12 minutes, and Bus Z every 16 minutes. If all three buses leave the station at 8:00 a.m., when is the next time all three buses will leave the station at the same time?
Simran volunteers at an old age home every 6th day, and Shiva visits the same old age home every 7th day. How many times will they both meet at the old age home during the months of December and January if we start counting from 1st December?
Earth takes 365 days to orbit the Sun, while Mars takes 687 days. If Earth and Mars are perfectly aligned with the Sun at a given time, how long will it take for this alignment to occur again?
Read the statements and choose the correct option.
Assertion (A): If three cyclists start at the same point and ride around a circular track with different lap times, they will all meet at the starting point together after the LCM of their lap times.
Reason (R): The LCM of the lap times represents the first time all events will coincide.
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.
Points to Remember
• A factor of a number is a number that can divide the number and leave no remainder.
• A multiple of a number is a number that can be obtained by multiplying that number by another number.
• A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.
• Co-prime numbers (or relatively prime numbers) are two or more numbers that have no common positive integer divisors other than 1.
• Twin primes are pairs of prime numbers that have a difference of 2.
• The LCM of two or more numbers is the smallest multiple that is divisible by each of them.
• The HCF of two or more numbers is the largest positive integer that evenly divides each of them without leaving a remainder.
Math Lab
Experiential Learning & Collaboration
The Sieve of Eratosthenes!
Setting: Each student should attempt it on their own.
Materials needed: Large sheet of grid paper with numbers from 1 to 100 pre-drawn, markers or coloured pencils, copies of the grid paper for each student
Methods:
Take a copy of the grid paper, which has numbers from 1 to 100.
Discuss that 1 is neither a prime number nor a composite number.
Circle a prime number like 2 and cross out its multiples, like 4, 6, 8, etc.
Continue this process for the numbers 3, 5, 7, 11 and so on. Encircle each prime number and cross out its multiples.
Continue the process until all the numbers in the grid are either encircled (prime) or crossed out (composite).
After completing the activity, discuss the prime numbers found.
Chapter Checkup
List all the factors of the numbers.
Write each number as the product of its factors by using a factor tree.
If two numbers are divisible by a certain natural number, prove that both their sum and their difference are also divisible by that same number.
a 54 and 36 are divisible by 9. b 546 and 364 are divisible by 13.
The product of two numbers is 36, and their sum is 13. Find the numbers. Are they co-primes? If so, use thread art to show this.
Find the HCF of the numbers. a By prime factorisation.
b By repeated division.
i 54, 72 ii 77, 121
Find the LCM of the numbers.
a By the prime factorisation method.
i 18, 35, 49 ii 24, 36, 120
b By the division method.
i 14, 28, 42 ii 80, 112, 200
Find the greatest number of five digits which is exactly divisible by 7, 14, 21, 28 and 35.
Determine the largest three-digit number divisible by 8, 12, 18, 24 and 32.
A library has 30 history books, 45 science books and 60 literature books. They want to arrange these books on shelves so that each shelf contains an equal number of each type of book. What is the maximum number of books they can place on each shelf?
The traffic lights at three different road crossings change after every 40 seconds, 60 seconds and 80 seconds, respectively. If they change simultaneously at 7 am, then at what time will they change simultaneously again?
Sarah has 18 erasers, 24 pencils and 30 pens. She wants to distribute them equally among her friends from a children's welfare organisation. What is the largest number of friends among whom she can distribute these supplies, with each friend receiving the same number of each item?
Create a five-digit number using the digits 1, 2, 3, 4 and 5 exactly once, so that the number is divisible by 6.
A garden has three sprinklers that operate at intervals of 20 minutes, 25 minutes and 30 minutes, respectively. What is the minimum amount of time one would have to wait to observe all three sprinklers operating simultaneously in the garden?
Rita wants to memorise the names of all 106 national parks in India. She wants to memorise the same number of names each day. How many names of national parks can she memorise each day: 2, 3, 4 or 5?
Find the smallest number from which if 15 is subtracted, the result is exactly divisible by 9, 12, 18, 21, 27 and 36.
Read the question and decide which of the statements are required to answer it.
The LCM and HCF of two numbers are 360 and 18 respectively. What is the second number?
Statement 1: The first number is 72.
Statement 2: The second number is 72 more than the HCF.
a Statement 1 alone is sufficient, but Statement 2 alone is not sufficient.
b Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
c Either of the statements are sufficient.
d Both statements together are not sufficient.
Geography Time!
Geographers studying transportation routes between cities need to calculate the optimal distances for road networks. Understanding factors and multiples helps them determine efficient distances and connections. In a region with three major cities—City A, City B and City C—urban planners are designing a highway system to connect these cities efficiently.
Answer the following questions:
1 The distance between City A and City C is 150 kilometres. If a new railway line is built, connecting City A and City C with a station at every 25 kilometres, how many railway stations will be needed along the route?
2 A cargo shipment departs from city A every 30 minutes and a parcel shipment departs from there every 40 minutes. If both shipments depart at 10 a.m., then when will they next depart at the same time?
3 The highway system is designed such that the distance between City A and City B is a multiple of the distance between City B and City C.
4 What is one step one can take to make the city greener?
4 Lines, Angles and Shapes
Letʼs Recall
Every day we come across various types of lines, angles and shapes. Let us look at some examples.
Parallel lines are like best friends who walk side by side but never cross each other. They stay the same distance apart everywhere they go, just like railroad tracks that run alongside each other without ever meeting.
Angles are formed when two lines meet at a point like the two handles of the scissors, and they can be measured in degrees.
Shapes are figures with specific forms, like circles, triangles and quadrilaterals, each with their own number of sides and angles, like the front of the birdhouse which has 5 sides.
Letʼs Warm-up
Fill in the blanks with the number of sides in the shapes.
Rays, Lines and Line Segments
One day, Anju and her dad wanted more space to dry their clothes. Anju took some rope and showed it to her father.
Anju: Dad, can we make a clothesline using this rope?
Dad: Of course, Anju. Did you know that the poles of a clothesline are examples of the geometrical concept known as a ‘line segment’?
Anju: Really?
Dad: Yes! Whether they are simple poles or massive pyramids, they all rely on simple geometrical ideas that help us understand different shapes.
Measuring and Drawing Line Segments
Let us recall key geometric concepts like points, rays, lines and line segments.
Key Geometrical Concepts
Point: A point denotes a location. A point has no dimensions like length, breadth or height. It can be denoted as X, Y, Z or N.
Line Segment: A line segment has a fixed length. The distance between two points is the length. It can be denoted as XY or YX
A ray is obtained by extending a line segment indefinitely in one direction. It does not have a fixed length. It can be represented as XY, where X is the initial point. Ray XY is not the same as ray YX.
A plane is a surface on which any two given points can be connected by a line so that the line lies wholly on the surface itself. It is a smooth, flat surface extending indefinitely in all directions. It has length and breadth but no height or thickness.
The walls of a room, a football field, a sheet of paper and a tabletop are some examples of parts of a plane that extends indefinitely in all directions.
Figure
Real‐life example
A line is obtained by indefinitely extending a line segment in both directions. It has no endpoints and no fixed length. It is denoted by small letters like l, m, n, etc. or also simply as XY
Points — A, B, P, M, Z
Lines — AB
Line segments — PM
Rays — AZ
Types of Lines
Parallel lines are two or more straight lines that do not intersect each other, no matter how far they are extended. They maintain a constant distance from each other.
Example 1: Look at the figure. List some points, lines, line segments and rays.
Think and Tell
Perpendicular lines are two lines that intersect at a right angle. Point P is the point of intersection here.
Is the surface of a volleyball part of a plane? What about a volleyball court?
Try relating the real‐life examples with their corresponding geometrical elements. Name them and draw to show. Do
Football field Stars in the sky Straw pipe Light emitted by the sun
Geometry Plane
Tabletop
Football ground
Walls in a room
Sheet of paper
Measuring and Drawing Line Segments
A ruler is split into 15 units. Each of these sections is exactly 1 cm long. Each centimetre is further split into 10 smaller units, called mm.
Drawing a line: Place one end of the ruler at the paperʼs edge, line up the ‵0′ mark. Draw along the rulerʼs edge.
Correct way of taking a reading from the scale
For precise measurements, align your vision directly above the mark in a vertical line. Viewing from an angle can lead to an error in the measurement due to the viewing angle.
Other Units for Measuring Lengths
Units like millimetre and centimetre are suitable for measuring shorter lengths only. For longer distances, we need larger units like metre and kilometre.
Bigger unit to smaller unit Multiply by 10
Smaller unit to a bigger unit divide by 10
Example 2: Measure the length of the line segments.
Draw a line segment
Comparison of Line Segments
two
The lengths of line segments can be compared by first measuring the line segments and then comparing them. There are multiple ways to compare the lengths of line segments.
Chapter 4 • Lines, Angles and Shapes
Comparison by Observation
By observation, XY > MN > AB. Comparison by observation is useful when differences in length are quite obvious.
It is difficult to compare the lengths of line segments PQ, SR and DC just by looking at them.
Comparison by Tracing
We can also use tracing paper to compare line segments. Let us say you want to compare CD and RS. Trace RS using tracing paper and place it on CD in alignment. You would be able to tell which line segment is longer.
Comparison using a ruler and divider
To measure a line segment using a ruler and divider, follow these steps:
1 Open the divider and place its arms on points A and B.
2 Carefully lift the divider put it on the ruler, with one end at zero.
3 Check where the other divider end point is on the ruler.
4 Read the mark on the ruler to know the length.
Example 3: Name the line segment with the shortest length in the figure shown below.
By observation, we can conclude that CD is the shortest line segment.
Measure the line segments using a ruler and divider. Write the lengths. Do It Together
Which of these represents a line?
a H b GH c OP d KL
Which of these is true about rays?
a They lack thickness. b They have a fixed length.
c They only extend in one direction. d They have only two end points.
Check whether the statements are true or false.
a A line extends in two directions.
c A line segment is part of a ray.
Write the length of the given line segments.
b A plane has a definite size.
d A ray has more endpoints than a line.
a b c
Draw line segments of the lengths given.
a 5.2 cm b 7.7 cm c 61 mm d 35 mm
Which of the following is the shortest length? a 2000
A road of length 23 km was constructed. What is the length of the road in these units? a metres b centimetres c decametres d hectometres
A constellation is a group of stars forming a pattern. How many stars do you see in the Leo constellation that form the pattern? The Great Wall of China is around 20,000 km long. How many cm will it be on a map if you use the scale 1 cm = 5000 km?
Look at the map with airports. If there are direct flights between every single pair of airports, how many total flight routes would be shown on the map if we draw a line segment to represent each possible flight connection?
Koi is learning about perpendicular lines. The teacher says that all perpendicular lines are also intersecting lines. Koi asks him whether all intersecting lines are perpendicular too. What do you think? 1 2
Angles
Alex and Emma are gathered with their toy cars in their study. They had built a makeshift ramp from wooden planks to have a race.
Emma: Alex, I wonder why our cars go faster when the ramp is steeper.
Alex: I think it is because of the angle, Emma.
Emma: What are angles, Alex?
Basics of Angles
To help understand how much a line segment, line or ray is inclined towards another, we use the geometrical idea of angles.
An angle:
• Is formed when two rays (or lines) meet at a common point called the vertex.
• Has the symbol: ∠
• Has the representation: ∠PQR, ∠RQP, ∠Q
Interior and Exterior of an Angle
As you can see in the figure, if the arms of the angle were extended indefinitely, they would divide the plane into two regions — interior and exterior. Points A, B and C lie on the exterior. Points D, E and F lie in the interior.
Adjacent Angles
Adjacent angles are a pair of angles that share a common vertex and a common arm. ∠XOY and ∠ZOY are adjacent angles.
Example 4: Observe the diagram given below and list the points that lie in the interior and exterior of angles ∠PON and ∠QON.
Interior Points
∠PON: E, F, D, G, Q
∠QON: D, G
Exterior Points
∠PON: A, B, C
∠QON: A, B, C, E, F, P
Example 5: Draw two adjacent angles to the angle, ∠POQ.
Adjacent angles drawn to ∠POQ are ∠MOP and ∠NOQ.
Draw an adjacent angle to ∠NMO so that:
1 Point B lies in the interior of the new angle, but not A.
2 Both points A and B lie in the interior region of the new adjacent angle.
Types of Angles
Angles can be classified on the basis of their measures.
More than 0º and less than 90º 90º
than 90º and less than 180º
The 90° angle is also called a perpendicular. It forms an ‘L’ shape, like when a vertical and a horizontal line meet. We see perpendiculars in the corners of a sheet of paper, windows and doors, or buildings.
Angles and Direction
Imagine you are standing with your arms extended, facing east. Your back would be to the west, your left arm indicating north, and your right arm indicating south. In the diagram, you can see that the lines joining the opposite directions are perpendicular to each other.
The child is facing east. What direction will he face if he:
1 turns clockwise by one right angle?
2 turns anticlockwise by one straight angle?
3 turns clockwise by two right angles?
4 turns anticlockwise by two straight angles?
Lines, Angles and Shapes
To be able to correctly predict the directions when you turn, you need to take care of two things:
• Amount of turn, such as the number of right angles or straight angles
• Sense of rotation or turn (clockwise or anticlockwise)
Notice the changes in the direction depending on the sense of rotation and the amount of turn. Turning in terms of right angles can also be expressed in terms of turning or rotation by a portion of a circle. Let us see how.
1 On turning clockwise by one right angle or one quarter circle, the child will face south.
2 On turning anticlockwise by one straight angle or one half circle, the child will face west.
3 On turning clockwise by two right angles or two quarter circles, the child will face west.
4 On turning anticlockwise by two straight angles or two half circles, the child will come back to his original position and face east.
Angles in a Clock
Now, let us understand angles using the turns that the hands of a clock make. As the hour hand moves 3 numbers, it makes 1 4 of the turn. On completing every 6 hours, it takes a 1 2 turn.
or 2 straight angles or one full turn
Example 6: Identify the type of angles for the given measures.
Example 7: What fraction of a full turn will the hands of a clock make if the hour hand moves from:
1 2 to 5 When the hour hand moves from 2 to 5, it covers a one‐fourth turn.
2 6 to 12 When the hour hand moves from 6 to 12, it covers a one‐half turn.
3 12 to 9 When the hour hand moves from 12 to 9, it covers a three‐fourths turn.
Do It Together
Example 8: Alice is standing facing north. She turns clockwise through 2 right angles. She then turns anti-clockwise through 3 right angles. What direction is Alice facing?
1 Clockwise through 2 right angles — New position: Faces south
2 Anticlockwise through 3 right angles — Final position: Faces west
Remember!
1 full rotation = 2 straight angles = 4 right angles
The starting position of Jiya is given below. She takes various turns clockwise and anticlockwise. Find the final position in each case.
Be cautious with rotation direction in problem solving; mixing up clockwise and anti‐clockwise can cause mistakes. Error Alert!
Measuring and Drawing Angles
Let us learn to measure the angles and construct them.
Measuring Angles
Do you remember how Emma and Alex were racing cars on the ramps? We can find out how steep the ramp is by measuring the angle it makes with the ground.
The steps to measure an angle using a protractor:
1 Put the centre of the protractor right over the vertex of the angle (the point where the two lines of the angle meet).
2 Make sure that the baseline of the protractor (the straight line at the bottom) is aligned with one of the angle's arms.
3 Look at where the other arm of the angle crosses the markings on the protractor. The number where the arm crosses is the measure of the angle in degrees.
Drawing Angles
Let us draw an angle using a protractor. Draw ∠XYZ of measure 56°.
1 Draw ray YZ. Place the protractor so that Y coincides with the midpoint of the baseline, and the baseline (0-180° line) aligns with ray YZ.
2 Start with the starting point at Z and mark point X at 56°.
Example 9: Use a protractor to draw ∠DEF measuring 294º.
3 Draw ray YX. Thus, ∠XYZ = 56°.
To begin drawing 294º, we will first subtract it from 360º. Hence, 360º − 294º = 66º. Now we will draw 66º.
Example 10: Measure the given angles. Write their measurements.
Remember!
Always align the centre and the baseline of the protractor with the vertex and arm of the angle while measuring the angle.
Measure the angles and write which one has the smallest measure.
Which of the following is an incorrect representation of an angle?
Classify the following angles as acute, obtuse or reflex angles.
a 36° b 93° c 192° d 89°
Which of the following objects is perpendicular to the ground?
a Bat kept on the floor b Clock hands at 3:00
c A boy leaning against the wall d A book kept on a table
Measure the given angles. Write their measure.
In the figure shown here, recognise the adjacent angles.
P
Write the difference between the measures of the angles formed by the blue arm and red arm with the black arm in each case.
a b
c d What fraction of a full rotation should the minute hand of a clock travel to go from the numbers:
a 5 to 8? b 3 to 7? c 11 to 5? d 9 to 6?
Draw adjacent angles to ∠NMZ so that:
a B lies in the interior of the new angle, but not A and D.
b Both A and B lie in the interior region of the new adjacent angle.
c Only C lies in the interior region of the new adjacent angle.
Adho Mukha Svana Yogasana commonly known as downward-facing dog, is a yoga pose that helps in strengthening the arms, shoulders and back muscles. In the Adho Mukha Svana Yogasana, your hands and feet create angles. Estimate how many degrees are in the angle made by the body in the given figure. Is it closer to a right angle or away from it?
Which direction would you be facing if your initial position is:
a north and you take a 1 2 turn, anticlockwise?
b south and you complete 1 4 of a turn, clockwise?
c east and you turn three‐fourths, clockwise?
Draw two angles ∠XOY and ∠AOB on point O. Mark four points such that:
a They are interior to ∠XOY only.
b They are interior to ∠AOB only.
Cycling is an activity that strengthens the heart muscle and improves blood circulation. If Reena's bicycle makes four and a half turns then how many right angles does it turn?
Suppose you were initially facing south. You were made to turn anticlockwise by three right angles. In which direction are you facing now? Draw to show the change in direction.
Write 3 letters of the English alphabet that have adjacent angles.
Rohit has a job interview scheduled at 3:00 p.m. He wants to make sure that he leaves two hours early. What angle will the minute and hour hands make on the clock when he should be leaving his house? 1 2 3
If a circular compass has eight cardinal directions evenly spaced around it, what is the angle between any two consecutive directions at the centre of the compass?
Understanding Polygons
Rita’s father bought her a beautiful pendant. Rita’s eyes lit up on looking at the gift. Rita called her friends from the neighbourhood and showed it to them.
Rita: Hey everyone, guess what? My father just gave me this amazing pendant.
Ravi: Wow! I’ve never seen such a shape before.
Let us see the features of triangles, quadrilaterals and other polygons one by one.
Triangles and Their Features
Triangles can be seen everywhere around us, in historical buildings, pyramids, etc. It is amazing how geometry is intertwined with architecture. Let us look into the details of triangles and their features.
Parts of a Triangle
A triangle has 3 vertices, 3 sides and 3 angles.
Vertices: P, Q, R
Sides: PQ, QR, RP
Did You Know?
The pyramids of Giza are around 4500 years old.
Angles: ∠PQR or ∠Q, ∠QRP or ∠R, ∠RPQ or ∠P.
Naming a Triangle
A triangle is denoted by the uppercase delta symbol followed by the names of all its vertices in any order. For example, ∆PQR, ∆QRP, etc.
The part of the plane enclosed by the triangle is called the interior of the triangle. The part of the plane outside the area enclosed by the triangle is called the exterior of the triangle.
Classification of Triangles on the Basis of Sides
Exterior of a triangle
Exterior of a triangle
Interior of a triangle
Exterior of a triangle
Triangles can be classified on the basis of their sides by measuring the lengths of their sides.
Equilateral Triangle Isosceles Triangle Scalene Triangle
All the sides are equal in length.
AB = BC = CA
The angles of an equilateral triangle are also equal.
Any two out of three sides are equal. Here, XY = XZ.
The angles opposite to the equal sides are also equal. Here, ∠XYZ = ∠XZY.
Classification of Triangles on the Basis of Angles
All the sides are unequal. MN ≠ NO ≠ OM
All the angles are unequal.
MNO ≠ ∠NOM ≠ ∠OMN.
Triangles can be classified on the basis of their angles by measuring the angles.
Acute-angled Triangle Obtuse-angled Triangle
The side opposite to the obtuse angle is the largest side of the triangle.
All the angles measure less than 90°, i.e., all three angles are acute angles.
One of the angles measures more than 90°. The other 2 angles are acute angles.
Example 11: List the vertices, sides and angles of the triangle given below. Identify the points lying on the interior and exterior of the given triangle.
Vertices: M, N and O
Angles: ∠MNO, ∠NOM and ∠OMN
Sides: MN, NO and OM
Interior: E, F and G
Exterior: A, B, C and D
Right-angled Triangle
A C 90°
The side opposite to the right angle is the hypotenuse. It is the largest side in the right-angled triangle.
One of the angles measures 90°. The other 2 angles are acute angles.
Did You Know?
Aryabhata, an ancient Indian mathematician explained how to calculate the area of a triangle and provided methods to solve equations related to triangles and polygons.
Do It Together
Example 12: Label the triangles as equilateral, scalene or isosceles.
4
As two sides of the triangle are equal, it is an isosceles triangle.
As all the sides of the triangle are equal, it is an equilateral triangle.
As none of the sides of the triangle are equal, it is a scalene triangle.
Example 13: One of the angles in a triangle is more than 90°. Which of the angles given below CANNOT be the value of one of the other two angles?
1 67° 2 11° 3 111° 4 1°
Solution: 111° cannot be the value of any of the other two angles because there can be only one obtuse angle in a given triangle.
1 Measure the lengths of the sides of the given triangles. Label them as equilateral, isosceles or scalene.
2 Measure the angles in the triangles. Label them as acute-angled, obtuse-angled, or right-angled. a b c
2
Do It Yourself 4C
List the vertices, sides and angles in the triangle.
Classify the triangles as equilateral, isosceles or scalene, given the lengths of their sides. a 6 cm, 7 cm, 9 cm b 15 cm, 15 cm, 15 cm c 19 cm, 23 cm, 29 cm d 9 cm, 11 cm, 9 cm e 18 cm, 31 cm, 27 cm f 3 mm, 3 mm, 3 mm
Classify the triangles according to the measures of the angles.
a 60°, 40°, 80° b 65°, 85°, 30° c 130°, 20°, 20° d 90°, 40°, 50° e 110°, 25°, 45° f 60°, 60°, 60°
Rani drew a rough map of 5 cities in Indian states using triangles as given below. Colour the scalene (red), equilateral (blue) and isosceles (yellow) triangles in the figure.
Madhya Pradesh
3.5 cm
Maharastra
6 cm
Sam and Maria work in an old-age home that has 2 triangular parks for the elderly to walk in. The first garden measures 7 metres, 7 metres and 10 metres and the second measures 5 metres, 6 metres and 7 metres. Identify the types of triangle formed by their gardens. Do you help old people?
Consider the following statements:
a Every equilateral triangle is necessarily an isosceles triangle.
b Every right-angled triangle is necessarily an isosceles triangle.
The correct statements are –i Only a ii Only b iii Both a and b iv Neither a and b
Classification of Quadrilaterals
A quadrilateral is a plane figure that has 4 sides or edges, 4 angles and 4 corners or vertices.
Parts of a Quadrilateral
Adjacent Sides: Adjacent sides of a quadrilateral are sides that share a common vertex (corner). They are next to each other in the sequence of sides.
E.g. — (PS, PQ), (SR, RQ)
Opposite sides: Opposite sides of a quadrilateral are sides that do not share a common vertex. They are located on the opposite sides of the quadrilateral.
E.g. — (PQ, SR), (PS, QR)
Diagonals: These are the line segments joining opposite vertices of a quadrilateral.
E.g. — PR, QS
Opposite angles: These are the angles in a quadrilateral with no common arm.
Adjacent angles: Adjacent angles of a quadrilateral are the angles that share a common side.
E.g. — (∠P, ∠R), (∠Q, ∠S)
E.g. — (∠P, ∠Q), (∠Q, ∠R), (∠R, ∠S), (∠S, ∠P)
Do It Together
Interior of a Quadrilateral: The interior of a quadrilateral refers to the region enclosed by the four sides of the quadrilateral.
Exterior of a Quadrilateral: The exterior of a quadrilateral refers to the region outside the four sides of the quadrilateral.
Exterior of a quadrilateral
Exterior of a quadrilateral
Interior
Exterior of a quadrilateral
Exterior of a quadrilateral
Example 14: List the diagonals, adjacent sides, adjacent angles and opposite angles in the given quadrilateral.
Adjacent Sides: (RU, UT), (UT, TS), (TS, SR), (SR, RU)
Adjacent Angles: (∠R, ∠U) , (∠T, ∠U), (∠T, ∠S), (∠S, ∠R)
Opposite Angles: (∠R, ∠T) , (∠S, ∠U)
Diagonals: RT, SU
Draw points P, Q, R, S, T and U as shown in the first figure. Draw quadrilaterals to show the points as per the given rules.
1 Only points P and R are in the interior.
2 Only points Q, R and S are in the interior.
3 Only points P and T are in the exterior.
Types of Quadrilaterals
Quadrilaterals can be found in many things around us, like windows and doors, kites and buildings. They can be classified based on the properties of their sides and angles.
Classification of quadrilateral based on their angles.
Convex Quadrilaterals
In these quadrilaterals, the measure of each angle is less than 180º. Both diagonals lie inside it.
Concave Quadrilaterals
In these quadrilaterals, the measure of one of the angles is more than 180º. One of the diagonals lies outside it.
Classification of quadrilaterals based on their sides.
All the sides are equal.
All the angles are right angles. Opposites sides are parallel.
Diagonals are equal and bisect each other at right angles.
All the angles are right angles.
Opposite sides and angles are equal.
Opposite sides are parallel.
Diagonals are equal and bisect each other.
Parallelogram
Opposite sides are equal and parallel.
Opposite angles are equal.
Diagonals bisect each other.
One pair of opposite sides is parallel. Here, AB CD.
A trapezium whose non-parallel sides are equal is known as isosceles trapezium. Here, PS = QR.
All the sides are equal.
Opposite sides are parallel.
Opposite angles are equal.
Diagonals bisect each other at right angles.
Kite Chapter 4 • Lines, Angles and Shapes
Opposite sides are unequal.
One pair of opposite angles are equal.
Two pairs of adjacent sides are equal.
One diagonal bisects the other.
Do It Together
Example 15: Which quadrilaterals satisfy the given conditions?
1 None of the sides may be equal to any other side: Trapezium
2 All the sides are equal but all its angles are not equal. Rhombus
PQRS is a parallelogram whose diagonals bisect at T. Label the parallelogram, and list the line segments that are equal in length.
PQ = SR;
Do It Yourself 4D
List the names of sides and vertices in the given quadrilaterals. Name all the adjacent and opposite sides in them.
Draw and name the diagonals of the quadrilaterals given below.
a only A, B and C are in the interior. b only B, D, E and F are in the exterior. 2 Art
Name the quadrilateral with the features given.
a All sides are of equal length and all angles measure 90°.
b Opposite sides are parallel and opposite angles are equal.
c Two pairs of adjacent sides are of equal length, and one pair of opposite angles is equal.
Draw two quadrilaterals and mark the points A, B, C, D, E and F in each quadrilateral such that:
In an archaeological survey, which is a methodical investigation to locate and document cultural artifacts and sites within a specific area, a mysterious patch of land was found with four statues at the corners. The archaeologist, curious about the shape of the land, measures the angles formed at each statue.
∠ ABC = 80°, ∠ BCD = 100°, ∠ CDA = 75°, and ∠ DAB = 105°.
a Is the patch of land a convex or a concave quadrilateral?
b Can the quadrilateral land be a rectangle or a square?
c Can the quadrilateral land be a parallelogram? Why?
Challenge
The quadrilateral formed by joining the midpoints of the sides PQ, QR, RS, SP of a quadrilateral PQRS is . Draw 2–3 quadrilaterals to find the answer.
a a trapezium but not a parallelogram b a quadrilateral but not a trapezium c a parallelogram d a rhombus
Polygons and Their Features
Let us look at the features of the types of polygons.
Parts of a Polygon
A polygon is a plane figure that has 3 or more straight sides. It has the same number of vertices and angles as the sides.
Adjacent Vertices: The end points of any given side are called adjacent vertices. (P, Q) and (P, T) are 2 examples.
Adjacent Sides: Adjacent sides of a polygon are sides that share a common vertex (corner). They are next to each other in the sequence of sides. (PQ, QR) and (SR, RQ) are 2 examples.
Types of Polygons
Regular and Irregular Polygons
Diagonals: These are the line segments joining two nonadjacent vertices of a polygon. For example, PR, QS, QT, RT, SP
Interior of a polygon: The interior of a polygon refers to the region enclosed by the sides of the polygon.
A regular polygon is a polygon in which all the sides are of equal length and all angles are of equal measure. Regular polygons are highly symmetrical and have a uniform appearance.
An irregular polygon is a polygon that does not have all the sides of equal length and all angles of equal measure. Irregular polygons lack the uniformity and symmetry of regular polygons.
Convex and Concave Polygons
Convex Polygons only angles measuring less than 180° . The diagonals lie inside the polygon.
Exterior of a polygon: The exterior of a polygon refers to the space outside the polygon's boundaries.
Concave Polygons have one angle measuring more than 180°. Some of the diagonals lie to the exterior.
Think and Tell
Is a right-angled triangle a regular polygon?
Let us classify polygons on the basis of their sides.
A polygon with equal sides alone may not be regular; it requires equal angles as well to be considered regular. Error Alert!
Example 16: Check whether the given figures are polygons or not.
Figure (d) is not a polygon as it does not have straight sides. All the others are polygons.
Example 17: List all the vertices and sides of the given polygon. Which line segments can be drawn to form its diagonals?
Vertices: A, B, C, D, E, F
Sides: AB, BC, CD, DE, EF, FA
Diagonals: AC, AD, AE, BD, BE, BF, CE, CF, DF
Example 18: Which of these options can be the angles of a regular polygon?
1 60°, 100°, 130°, 70°
3 90°, 100°, 120°, 90°
2 90°, 90°, 90°, 90°
4 45°, 60°, 75°, 90°
Since all the angles in a regular polygon are equal, option b is correct.
Diagonals: Do It Together
Name the polygon and list its vertices. Draw diagonals. Write the number of diagonals.
Draw 2 points A and B on the interior and 2 points X and Y on the exterior of the polygon.
Vertices:
Do It Yourself 4E
Which of these is a regular polygon?
a Square
c Scalene triangle
b Rectangle
d Right-angled triangle
List the names of all the sides and vertices of the polygon.
Draw an irregular polygon with the number of sides given.
Raj chose to build a logo in the shape of a polygon for his company. Read the clues and guess the type of polygon and the number of sides in it.
a All the sides and angles in his polygon are equal.
b He can draw 9 diagonals in his polygon.
Sarah and Alex are participating in a school project to study the effects of sunlight on plant growth. Sarah has a garden in the shape of a hexagon. She plants two sunflower plants on each side of her garden. Alex has an octagon-shaped garden. He plants one sunflower plant on each side of the garden. Who has more sunflower plants and how many more?
Six friends, A, B, C, D, E and F, are sitting around a hexagon-shaped table. Each friend sits at a corner of the hexagon and faces the centre. Who is sitting opposite to D?
Here are the clues to figure out who sits opposite D:
A is sitting two seats to the left of F. B is sitting next to both C and D. E is sitting two seats to the left of D.
Understanding Circles
On a sunny winter morning, Era and her son Laksha go to their favourite beach. Armed with sticks and surrounded by endless sand, they are ready to let their creativity flow.
Era: Hey Laksha, I’m drawing these wavy lines that flow like waves.
Laksha: Mom, look, I have drawn lines.
Era: Perfect! I’ll make circles as well, like the sun shining down on us.
Types of Curves
Era and Laksha create various types of curves with their drawings. Some form loops, others cross and a few are wavy. These distinctions allow curves to be grouped into various categories. Let us explore a few of these categories.
Simple and non-simple curves: A curve that does not cross itself is known as a simple curve.
Open and closed curves: Open curves donʼt begin and end at the same point, whereas closed curves do. Simple Curves Non-simple Curves Open Curves Closed Curves
Position with respect to a closed curve
Given below is a simple closed curve with points X, Y and Z marked at different places.
Think and Tell
Can you draw curves that are simple and closed?
Interior of the curve: The region inside the boundary of the curve.
Boundary of the curve Interior of the curve X Exterior of the curve Y Z
Exterior of the curve: The region outside the boundary of the curve.
Here, X lies in the interior, Y lies on the boundary and Z lies in the exterior of the curve.
Example 19: Which of these is an open curve?
Figure (d) is an open curve.
Circles: Parts and Construction
Circles are geometric shapes defined by a set of points equidistant from a central point. Circles appear in many shapes around us in nature.
Parts of a Circle
• A point within the circle that is equidistant from all points on the circle’s circumference. It is often denoted by the letter ’O’.
• The distance from the centre of a circle to any point on the circle.
• A line segment passing through the centre of a circle, connecting two points on the circle.
• It is the longest chord of the circle.
• The diameter is twice the length of the radius.
Chapter 4 • Lines, Angles and Shapes
• A line segment that connects two points on the circle.
• A straight line that intersects a circle at two distinct points and extends beyond the circumference of the circle. Centre Radius Chord
Did You Know?
The Parliament House in India has a central hall in a circular shape, representing the Ashoka Chakra.
Semicircle means half of a circle. When a circle is cut into two equal parts, you get two semicircles.
Circumference of a Circle
Concentric circles are circles that share the same centre but have different radii.
The circumference of a circle helps you understand the size of the circle. It is the length of its outer edge or boundary. It is equivalent to the perimeter of other shapes, but the term ‘circumference’ applies only to circles.
Circumference = 2πr, where r is the radius and π = 22 7 .
The symbol ‘π’ here is a constant whose value is 22 7 . We can also use π = 3.14 as a substitute to simplify calculations.
Did You Know?
Archimedes made significant contributions to the study of circles and their measurement, including the calculation of the value of pi and the development of formulas for the area and circumference of a circle.
Constructing Circles
Lets see how to contruct a circle using a compass
Step 1
Use your pencil to make a small dot on your paper. This will be the centre of your circle.
Step 3
Put the pointed end of the compass on the dot you made in step 1.
Centre of the circle
Think and Tell
Can there be a chord longer than the diameter of a circle?
Radius
Step 2
Place the needle of the compass on the zero mark of the ruler and open the arms of the compass to take the pencil to the required radius length.
Step 4
While holding the centre of the compass steady on the dot, carefully rotate the pencil end around the centre. This will draw the circle.
Example 20: In the circle shown below, draw and label the radius, diameter, secant and chord.
Remember!
Multiple circles within a circle may or may not all be concentric circles.
Example 21: What is the circumference of a circle with a radius of 6 cm? Use the value of π = 3.14.
Given, radius, r = 6 cm.
Using, C = 2πr = 2 × 3.14 × 6 cm = 37. 68 cm
Hence, the circumference of the circle is 37.68 cm.
Example 22: Find the radius of a circle whose circumference is 44 cm. Use π = 22 7 .
Given, circumference, C = 44 cm.
C = 2πr. So, 44 cm = 2 × 22 7 × r r = 7 cm
Example 23: Using a scale of 3 m on the ground to 1 cm on paper, illustrate how to make a scaled-down representation of a circular swimming pool, which has a diameter of 9 m. Label the diameter.
Scale is 3 m = 1 cm. So, 9 m = 3 cm
Diameter = 3 cm, so radius = 3 ÷ 2 = 1.5 cm. O 3 cm Q P
Fill in the blanks with the correct diameter or circumference. Use π = 22 7 .
Do It Together ) the simple curves.
6.3 m
cm
cm
Do It Yourself 4F
b c d e
Classify the curves as open or closed curves.
b c d e
Colour the minor segment in red and major segment in blue.
Chapter 4 • Lines, Angles and Shapes
Which of the following represents concentric circles?
a b c d
Which of these does not have a finite length?
a chord b arc c diameter d secant
Mark one point each on the interior, exterior and boundary of the curve.
Draw closed curves that cross themselves the number of times given.
Using a compass, construct circles with the given measures.
a radius = 3 cm b radius = 3.8 cm
Draw 2 simple and 2 non-simple curves.
Diameter = 8 cm
Find the circumference of the circles with the following radii. Take π = 22 7 .
7 cm
Find the value of the radius for the following circumference. Take π = 22 7 . a 66
Diameter = 7.2 cm
The Ashoka Chakra, a significant symbol in India, features prominently on the Indian national flag. The chakra has 24 spokes and is a perfect circle. The circumference of the Ashoka Chakra on a replica flag is 3.14 metres.
Calculate the radius of the chakra. (Take π = 3.14)
Read both the statements given below and choose the correct option.
Assertion (A): Chord, secant, sector, segment and radius are the parts of a circle.
Reason (R): The centre of the circle is always in the interior of the circle.
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.
Points to Remember
• A point is a mark of location. It has no dimensions.
• A line segment has a fixed length. It can be measured using a ruler.
• Angles are measured in degrees, using a protractor.
• A polygon is a 2-D shape with 3 or more sides. It has the same number of sides, vertices and angles. Triangles and quadrilaterals are also polygons.
• A polygon can be regular or irregular, concave or convex.
• A circle is a curve that starts and ends at the same point, and the distance from the centre to any point on the edge of the circle is the same.
• The circumference of a circle is the distance around its outer edge or boundary. Chapter 4 • Lines, Angles and Shapes
Setting: Individual
Creating Rangoli Designs with Circles
Materials needed: Large sheets of paper, pencils, markers or coloured pencils, compass
Steps:
Distribute sheets of paper and art supplies to each student.
Instruct the students to create rangoli designs by constructing circles using the compass.
Encourage the students to try constructing concentric circles to make the design. They can then colour the design.
After the drawings are complete, ask the students to share their rangoli designs with the rest of the class.
Chapter Checkup
Count the number of line segments in the figures.
Draw the line segments of these lengths. a 4.6 cm b 77 mm c 6.9 cm
Look at the points marked. Create new angles such that:
a Only A, E, D and F are in the interior of the angle.
b Only B, C, D and E are in the interior of the angle.
c Only C and D are in the interior of the angle.
d Only A, F and G are in the interior of the angle.
e Only B and E are in the interior of the angle.
Draw two adjacent angles to the given angle ∠XOY.
Classify the angles as acute, obtuse or reflex.
a 26° b 256° c 79° d 129°
165°
Draw an irregular polygon with the given number of sides. a 3 b 6
Identify simple curves among the figures.
a b c d
Identify open and closed curves among the figures.
a b c d
Classify the triangles as equilateral, isosceles or scalene.
a 7 cm, 7 cm, 10 cm b 14 cm, 15 cm, 16 cm c 29 cm, 29 cm, 29 cm
Classify the given triangles according to the measure of the angles.
a 70°, 30°, 80° b 45°, 85°, 50° c 120°, 20°, 40°
A clock shows the time as 12 o' clock. What is the angle that the hour hand moves through when the time is:
a 3:00 o' clock? b 6:00 o' clock? C 12:00 o' clock? d 9:00 o' clock?
What orientation will you have if you are initially facing
a West and making 1 2 of a revolution clockwise?
b North and completing 1 4 of a revolution anticlockwise?
RSTU is a square whose diagonals bisect at Q. Find as asked.
a Name the diagonals of this square. b Name any four right angles found.
c Is UQ = TQ? Give a reason.
d Name the pairs of sides which are parallel.
Construct 2 concentric circles with radii measuring 3 cm and 4 cm. Label a secant, a chord and a diameter of the outer circle.
Identify the scalene, equilateral and isosceles triangle in the figure.
Identify the points which are interior to:
a Both ΔDEF and ΔPQR
b Both ΔABC and ΔDEF
c Only ΔDEF
d Only ΔPQR
The figure MNOP is a kite. Answer the questions. Give reasons.
a Is MN equal to PM?
b Is ∠MNQ equal to ∠MPQ?
c Is ∠ONQ equal to ∠OPQ?
Challenge
How many quadrilaterals do you see in the figure? 1
Critical Thinking
2
Read the assertion and reason and choose the correct answer from the options.
Assertion (A): In an isosceles trapezium, the non-parallel sides are equal in length.
Reason (R): A trapezium is a quadrilateral with exactly one pair of parallel sides.
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.
Case Study
Tracking Time with Shadows
Sundials are ancient timekeeping devices that use the position of the sunʼs shadow to tell time. Throughout the day, the shadow of the gnomon moves around the dial, indicating different times.
In the morning, the shadow forms an acute angle with the dial markings. At noon, the shadow ideally forms a right angle or something close to it. In the afternoon, the shadow forms an obtuse angle.
Answer the following questions.
1 What type of angle is formed by the shadow of the gnomon in the morning? a Acute angle b Right angle c Obtuse angle d Straight angle
2 At what time of day does the shadow of the gnomon form a right angle with the dial? a Morning b Noon c Afternoon d Evening
3 In the the shadow of the gnomon forms an acute angle.
4 In the afternoon, the shadow of the gnomon forms an angle.
5 The angle of the gnomon affects the types of angles formed by the shadow throughout the day. (True/False)
Cross Curricular
Letʼs Recall
Rahul and Preeti are playing a game of shapes. They have to put the solid shapes inside the box, based on their faces.
Let us see if they can identify the shapes.
A dice can be put inside a box through a square-shaped hole of the same size.
A ball can be put inside a box through a circular hole of the same size.
This means that the face of the dice is a square and that of a sphere is a circle.
This means that the face of the dice is a square and that of a sphere is a circle.
Letʼs Warm-up
Match the 3-D shapes with the 2-D shapes that you see in it.
Understanding 3-D Shapes
Jimmy owns a gift shop and a bakery shop. The gifts are in different 3-D shapes. Let us see how these gifts are different from each other.
Features of 3-D Shapes
Jimmy looks at the gifts, and he can see a lot of 3-D shapes in them.
The three dimensions of a 3-D shape are length, breadth and height. Such shapes are also called solid shapes
Every solid has faces, edges and vertices.
The flat or curved surface of a solid is called a face. An edge is where the two faces meet, and the vertex is where two edges meet.
Let us look at various 3-D shapes in Jimmy’s gift.
Cylinder: 0 vertices, 2 edges, 2 plane faces + 1 circular face
Cube: 8 vertices, 12 edges, 6 plane faces
Cone: 1 vertex, 1 edge, 1 plane face + 1 circular face
Sphere: 0 vertices, 0 edges, 1 curved face
Cuboid: 8 vertices, 12 edges, 6 plane faces
3-D shapes with straight sides can be classified as prisms and pyramids.
Prisms: Prisms are 3-D shapes with straight sides and identical opposite bases that can be polygons. The lateral faces of prisms are rectangles, and the opposite faces are parallel.
Square Prism
Square opposite faces
Hexagonal Prism
Identical hexagonal bases
Rectangular lateral faces
Triangular Prism
Identical triangular bases
Octagonal Prism
Identical octagonal bases
Face
Vertex Edge
Pyramids: A pyramid is a 3-D shape with one apex and one polygonal base. It has the same number of lateral triangular faces as the number of sides in the polygonal base. The lateral triangular faces meet at the apex.
Triangular lateral faces
Triangular base Square base
Triangular pyramid Square pyramid
Rectangular base Pentagonal base
Rectangular pyramid Pentagonal pyramid
A triangular pyramid is also called a tetrahedron.
Remember!
Hexagonal base
Hexagonal pyramid
Cubes and cuboids are also prisms, as they have identical opposite faces that are squares in a cube, and squares or rectangles in a cuboid.
Think and Tell
Does a decagonal prism have the same number of faces as a decagonal pyramid?
Example 1: Identify the shape. Write the number of vertices, edges and faces in the shape.
The base of the shape has 8 sides, so this is an octagonal prism.
It has 16 vertices, 24 edges and 10 faces (2 octagonal and 8 rectangular).
Example 2: Name the 2 shapes. Write 1 difference and 1 similarity between the shapes.
Names: Shape A is a triangular pyramid, and Shape B is a triangular prism.
Difference: Shape A is a pyramid and Shape B is a prism.
Similarity: Both shapes have a triangular base.
Complete the table. Do It
Shape A Shape B
) the correct option.
a Which of the figures is not a solid figure?
i cube ii circle iii pyramid iv sphere
b How many faces does a triangular pyramid have?
i 3 ii 4 iii 5 iv 6
c What is the shape of a dice?
i cube ii cuboid iii cylinder iv cone
d How many vertices does a heptagonal pyramid have? i 7
Name the shape of the objects.
a b c d
Fill in the blanks.
a A has 2 circular edges.
b All the faces of a are identical.
c A pentagonal prism has more edges than vertices.
d A and a have square bases.
The Pyramids of Giza were royal tombs built for three different pharaohs. Their shape was that of a square pyramid. How many faces, edges and vertices do they have?
Draw two cubes side by side. Write the number of faces, edges, and vertices of the solid shape formed.
Match the shapes with their features.
a 6 faces
Octagonal pyramid
b 12 edges and 7 vertices Cone
c 1 vertex and 2 faces
d 16 edges
e 7 faces and 10 vertices
Cuboid
Pentagonal prism
Hexagonal pyramid
John is playing with his geometric shapes. He has an octagonal pyramid and an octagonal prism. Write 2 differences and 2 similarities between the 2 shapes.
How is a nonagonal pyramid different from a hexagonal pyramid? How are these shapes similar?
Pyramids generally have an equal number of faces and vertices, and their number of edges is two less than twice the number of faces or vertices. If a pyramid has 8 vertices, what is the name of this solid?
Nets of 3-D Shapes
Do you remember, Jimmy, who has a bakery and a gift shop? This time he gets an order for 4 cakes of all different shapes. The cakes have to be packed in boxes of the same shapes as those of the cakes. He folds the cardboard to form the boxes.
Let us look at some of the boxes.
Nets of Cubes and Cuboids
A net is a set of 2-D shapes drawn on a sheet of paper or cardboard that can be folded to form a 3-D shape.
Let us see the nets of some 3-D shapes.
Did You Know?
Aryabhata, the renowned Indian mathematician, also proposed that the Earth is a sphere. Additionally, it is the only shape that does not have a net.
Always look at the opposite faces in the net of a cuboid. They can never be different.
Remember!
The cube can be opened in 11 ways, whereas a cuboid can be opened in 54 different ways.
Nets of an open box
An open box does not have the top face of the box. Let us look at some examples.
Did You Know?
A solid shape with 6 or more plane faces is called a polyhedron.
Example 3: Which of the nets can form a cube whose opposite faces will have the same colour?
In Net B, the opposite faces will have the same colour.
Example 4: Which of the following nets will form a cube when folded?
The nets in option (a) and option (b) will form a cube when folded.
A net for an open box is shown. Where should the rectangles and squares be placed so that the figure becomes the net of a closed cuboid? Draw the new net.
Nets of Cones and Cylinders
Let us look at the nets of cones and cylinders.
Cone: 1 plane face, 1 curved face
Cylinder: 1 curved face, 2 plane faces
Example 5: How many faces does the net of a cone have?
The net of a cone is . It has 2 faces.
Example 6: Draw the net of a cylinder which is open at both ends.
The net of a cylinder which is open at both ends is a rectangle.
Draw any three nets for a cylinder. Do It Together
Nets of Pyramids and Prisms
Jimmy also packs some cakes in pyramid-shaped boxes. Let us see what the box looks like when we open it.
Square pyramid 1 square face, 4 triangular faces
Let us learn about the nets of pyramids and prisms.
Triangular Prism
2 triangular faces, 3 rectangular faces
Pentagonal Pyramid 1 pentagonal face, 5 triangular faces
Example 7: Draw the net of a heptagonal pyramid. A heptagon has 7 sides.
Heptagonal Pyramid
Example 8: Which 3-D shapes do these nets form?
Net of a Heptagonal Pyramid
Rectangular Pyramid Hexagonal Prism
Name the shape. Draw its net. Do It Together Name of the shape:
3-D Shapes
Do
Write the names of the solid shapes for which the nets are shown.
Fill in the blanks.
a A has no net.
b The net of a cylinder has faces.
c A cubical box is open from the top. The net of the box has faces.
d A square pyramid has square/s and triangles in its net.
Which of these is the net of an octagonal pyramid?
Draw the nets for the shapes given.
Draw the nets for the shapes given.
Draw an equilateral triangle. Locate the midpoints of all three sides and draw an inner triangle. Which shape does this net belong to?
A kaleidoscope is a tube containing loose, coloured glass or materials. When you peer through it, you'll observe dynamic and colourful patterns. It employs two triangular mirrors opposite each other and additional rectangular mirrors. Could you draw its shape and net?
It Yourself 5B Rajan placed a square pyramid on top of an open cube-shaped box with the same side length as the square base. How many faces, edges and vertices does this new shape have?
Points to Remember
• All 3-D objects have faces, edges and vertices.
• Two-dimensional shapes that are folded to make three-dimensional shapes are called nets.
Math Lab
Experiential Learning & Value Development
Nets of 3-D Shapes
Aim: To reinforce understanding of how faces of 3-D shapes can be used to draw their nets.
Setting: In pairs
Method:
Materials required: Different 3-D shapes, Old newspapers, scissors, glue, markers or pens and sample boxes
Distribute different 3-D shapes to the students.
Identify the shape of your box (e.g., rectangular prism, cube, pyramid, etc.).
Draw its net on a piece of old newspaper.
Cut out the net from the newspaper.
Fold the net to wrap the box and decorate it as desired.
What are the number of faces, edges and vertices in the shapes given? a
Fill in the blanks.
a A pentagonal pyramid has edges and vertices.
b The difference of the number of vertices and number of faces of a decagonal prism is
c The total number of edges in a heptagonal prism is
Which of these are NOT prisms?
Write the number of faces and their types in each of the shapes.
Given is the net of a cube. What colour face will be opposite the: a blue face? b orange face?
Jenna is participating in an origami competition. She wants to create a paper model of a triangular prism, using a net. Draw the net of a triangular prism.
David is a packaging designer. He is tasked with creating a unique gift box in the shape of a rectangular prism. Draw the net of a rectangular prism.
Choose the correct shape of the nets given.
Triangular pyramid Cone Cylinder Square pyramid
Name the 3-D shapes that these nets belong to.
a b c d
Draw the polygonal base for the nets of the 3-D shapes given.
a Nonagonal pyramid b Hexagonal pyramid
Draw the nets for the shapes given. a Octagonal pyramid b Cube c Cylinder d Triangular pyramid
1 Read the statements and choose the correct option.
Assertion (A): A cube and a cuboid are both examples of rectangular prisms.
Reason (R): A cube has all its faces as squares, whereas a cuboid has faces that are all rectangles but not necessarily squares.
Choose the correct option:
a Both A and R are correct, and R is the correct explanation for A.
b Both A and R are correct, but R is not the correct explanation for A.
c A is correct, but R is incorrect.
d A is incorrect, but R is correct.
2 The sum of the numbers on the opposite faces of a dice always equals 7. Draw the dots on the net of the dice.
Our Community Garden Shed
The local community centre is planning to build a garden shed to store tools and equipment. The shed will be constructed using various 3-D shapes to maximise space and functionality.
The community needs your help to understand the features and nets of these shapes to ensure the shed is well-designed.
1 The shed will be a rectangular prism with tin sheets all around to form the walls and the ceiling. How many tin sheets do they need?
2 Colour the walls pink and the ceiling red in the net of the shed. Cross out the part that does not form the walls and the ceiling.
3 On top of the shed, there will be a design in the shape of a rectangular pyramid. Which of the following statements correctly represent the shape?
a It has 5 faces, 5 edges and 8 vertices. b It has 5 faces, 8 edges and 8 vertices. c It has 8 faces, 5 edges and 5 vertices. d It has 5 faces, 8 edges and 5 vertices.
4 How are a rectangular prism and a rectangular pyramid different?
5 The community centre wants to make sure the shed is environmentally friendly. How can they do that?
6 Integers
Letʼs Recall
Natural numbers are the numbers that begin with 1 and go up to infinity, whereas whole numbers are numbers that begin with 0 and go up to infinity. They can be shown on a number line as:
The number to the left of a natural or whole number is its predecessor, and the number to the right of a natural or whole number is its successor.
For example, the predecessor and successor of 636 are 635 and 637, respectively.
Letʼs Warm-up
Fill in the blanks with the predecessor and successor of the following whole numbers.
Understanding Integers
Simran’s father bought her a new video game. Both Simran and her brother Jay are excited to play the game. While playing, they notice some strange numbers with ('+' and '−') signs appearing on the screen each time they achieve or miss their targets.
Round 1 points
Let us see what these numbers are!
Positive and Negative Integers
The numbers that the siblings see with the ʽ+ʼ and ʽ–ʼ signs are called integers. All negative and positive numbers including 0 are called integers. For example, −1, 2, −20 and 25 are all integers. Let us look at some real-life examples of integers.
We know how to represent natural numbers and whole numbers on a number line. Let us see how we can show integers on a number line.
Remember!
Positive integers can be represented without their sign as well.
Representing Integers
Now, we will learn how to represent integers on a number line. Let us mark –5, 9, –8 and 6 on a number line.
Step 1: Draw a line and mark numbers at equal intervals.
Step 2: Mark 0 at the centre point.
Step 3: Mark positive numbers to the right and negative numbers to the left of 0.
Step 4: Circle the integers to be shown on the number line.
The opposite of an integer is the number that is the same distance from zero but with the opposite sign. The integers –1, +1; –2, +2; –3, +3 and so on... are considered opposites of each other.
Some other opposite integers are:
Situation
Depositing ₹500 = +500
Rise by 100 points = +100
Opposite
Withdrawing ₹500 = −500
Fall by 100 points = −100
5 km towards the North = +5 5 km towards the South = −5
Successor and predecessor of an integer
Think and Tell
1 is the smallest positive integer. What are the smallest and largest negative integers?
The integer to the left of a given integer on a number line is called its predecessor, and the integer to the right of it is called its successor.
For example, the predecessor and successor of –2 and +2 can be given as:
Predecessor of –2 Successor of –2
Error Alert!
Predecessor of +2 Successor of +2
The predecessor of a negative integer is greater, and its successor is smaller than the integer in terms of its numerical value.
For integer –7
Predecessor = –6; Successor = –8
Example 1: Write the situations in the form of integers.
1 25 cm to the right = +25
3 25 m below ground level = –25
Example 2: Mark –3, +5, +7 and –6 on the number line.
The integers can be marked on the number line as:
Predecessor = –8; Successor = –6
2 A fall of 70 points = –70
4 15 km North = +15
Example 3: Write a situation for the integer 3.
One situation can be: The temperature of Srinagar is 3°C below 0°C.
Circle the integers from the given set of numbers. Represent the integers on the number line.
Comparing and Ordering Integers
We can compare and order the integers using a number line as shown below.
Smaller to greater Greater to smaller
• A positive number is always greater than a negative number.
• A positive integer with more digits is always greater than an integer with fewer digits.
• If two positive integers have the same number of digits, then compare the leftmost digits of the two numbers and continue until we come across unequal digits.
• A negative integer having a smaller numerical value is always greater than the one having a greater numerical value.
Absolute value of an integer
The absolute value of an integer is the actual distance of the integer from zero on the number line. In other words, the absolute value of an integer is the numerical value of the integer, regardless of its sign. The absolute value of a number is denoted by the sign ||.
Distance of 4 from 0
Example 4: Write the absolute value for –18 and 12.
The absolute values can be given as: |–18| = 18; |+12| = 12
Distance of 2 from 0
Example 5: Compare the given integers and arrange them in ascending order. –3, 5, 8, –7, –9, 3, 6
To arrange the integers in ascending order, we move from left to right on the number line.
–1 –2 –3 –4 –5 –6 –7 –8 –9 –10
Hence, the integers can be arranged as, –9 < –7 < –3 < 3 < 5 < 6 < 8
Example 6: Write all the integers between –4 and 3. The integers between –4 and 3 are –3, –2, –1, 0, 1, 2
Example 7: A diver is at a depth of 10 m below the sea level, and a balloon is at a height of 10 m above the sea level. Represent this using a number line.
Diver
Balloon
Rudra lives on the 5th floor, and Suhani lives on the 2nd floor of the same building. Rudra parks his car in basement 2, and Suhani parks her car in basement 3. Mark the floors and basements on a vertical number line.
Did You Know?
Brahmagupta, one of our greatest astronomers and mathematicians, was born in 598 AD. He was the first person to define zero, as well as positive and negative numbers.
Rudra’s car parking
Do
It Yourself 6A
Write the situations in terms of integers. Write their opposites as well.
a Going 10 km to the east
b Spending ₹1000
c Payal gets 5 apples
d 3 kg decrease in weight e Win by 3 runs f Be late by 10 minutes
Represent the given integers on a number line.
Given below is a number line representing integers. Write the integers corresponding to the points.
Write the predecessor and successor of the following integers.
If you move 4 numbers to the left of 1 on a number line, what integer would you reach?
Write all the integers between the integers given.
a –3 and 4 b –5 and 0 c 1 and 6 d –9 and –4 e –15 and –8
Write the absolute value of the integers.
a 8 b –10 c 25 d –35 e –65
Look at the pattern of integers. Identify the rule and fill in the blanks.
a –13, –16, –19, –22, b 18, 12, 6, c –30, –32, –34,
Compare using the >, < or = sign.
a –26 –33 b –15 –12 c 47 –47
d 14 18 e –51 –15
Write the integers in ascending order.
a –2, 5, 9, –8, –10 b –6, –11, 16, 20, –7 c 14, –9, 13, –15, 23
The Indian Meteorological Department is the primary agency for weather forecasting. A weather report showed the temperatures of certain cities as 5°C, –3°C, 12°C, –7°C, 18°C and 2°C. Arrange the temperatures in ascending order.
A submarine is at a depth of 15 km below sea level. A parachute is 35 km above sea level, and a helium balloon is 10 km above sea level. Which of these is between the other two? Represent this on a vertical number line.
Simran scored +4 points, and Jay scored –4 points in the first round. During the second round, Simran scored +2 points, and Jay scored +6 points.
They want to know who the final winner is after the two rounds. Let us help them out!
Addition of Integers
We already know about the addition of whole numbers. Let us now learn to add two or more integers using tokens.
Addition of Integers Using Tokens
Let us find the total scores earned by Jay and Simran using tokens.
Simran's scores were +4 and +2. Jay’s scores were −4 and +6.
We will use + to represent positive integers, and to represent negative integers.
Simran’s Total Score
+4 + (+2)
Step 1: Show the two scores using tokens.
Step 2: Count the total number of tokens if both integers are positive.
So, +4 + (+2) = +6
Example 8: Add the numbers using tokens.
Jay’s Total Score
−4 + (+6)
Step 1: Show the two scores using tokens. −4 will be shown using 4 red tokens. +6 will be shown using 6 green tokens.
Step 2: Cancel pairs of positive and negative tokens.
A positive token and a negative token cancel each other to make a ‘Zero Pair’.
Step 3: Count the tokens left since both numbers were of different signs.
We are left with 2 positive tokens.
So, −4 + (+6) = +2
We are left with 5 negative tokens.
So, +4 + ( –9) = −5
Draw and add +3 + (−8) using tokens.
We are left with 7 negative tokens.
So, –3 + (–4) = –7
+3 + (−8) =
Rules for Addition of Integers
Let us now see how we can add integers using the number line.
Addition of integers with the same sign
Let us add Simran’s scores during the two rounds, which are +4 and +2, using a number line.
Simran’s Scores: +4 and +2
Step 1: Start from 0.
Step 2: Move 4 steps to the right as we need to add +4.
Step 3: Again, move 2 steps to the right as the second digit is +2.
Hence, (+4) + (+2) = +6
What if Simran’s scores were –4 and –2?
Move 4 steps to the left of 0 reaching –4, then move 2 steps to the left.
Hence, (–4) + (–2) = –6 For two positive integers, move towards the right to add.
From the above discussion, we can conclude that:
To add two integers with the same sign:
• add their absolute values and
• place the common sign before the sum
Therefore, (+4) + (+2) = |+4| + |+2| = 4 + 2 = 6
And, (–4) + (–2), |–4| + |–2| = 4 + 2 = 6 (–4) + (–2) = –6
Addition of integers with different signs
For two negative integers, move towards the left to add.
Let us now add Jay’s scores, which are –4 and +6, using a number line.
Step 1: Start from 0.
Step 2: Move 4 steps to the left, as we need to add – 4.
Step 3: Move 6 steps to the right, as the second digit is +6.
Hence, (–4) + (+6) = +2
Move 6 steps to the right of 0 reaching +6, then move 4 steps to the left.
Hence, (+6) + (–4) = +2
From the above discussion, we can conclude that: To add two integers with different signs:
• find the difference of their absolute values and
• place the sign of the integer with the greater numerical value
Therefore, (–4) + (+6) = |+6| – |– 4| = 6 – 4 = 2
Example 9: Show 5 more than –4 on the number line.
5 more than –4 = –4 + 5
It can be shown on the number line as:
Hence, –4 + 5 = 1
Example 10: Add the given integers.
1 (–2345) and (1852)
As –2345 and 1852 are of opposite signs, |–2345| – |1852| = 493
Place the ʽ–ʼ symbol in front of 493 as 2345 has a greater numerical value than 1852.
So, –2345 + 1852 = –493
2 (– 430), (–360) and (1236) (–430) + (–360) = –790
Add the sum –790 with 1236 –790 + 1236 = |1236| + |–790| = 446
(–430) + (–360) + (1236) = 446
Example 11: Renu parked her car in the basement, 2 floors below ground level. She got in the elevator and went up 12 floors. At which floor did she get off the elevator?
2 floors below ground level = –2
As Renu went 12 floors up the basement, the floor she gets off at = –2 + 12 = 10
Hence, she left the elevator at the 10th floor.
A research team aboard an underwater research vessel descends 1200 feet beneath the surface of the water. They then rise 635 feet and descend 455 feet again. At which depth are they located?
1200 feet beneath the surface = –1200
Rise of 635 feet = +635
Descent of 455 feet =
Current location of vessel = (–1200) + (635) + _________ = _________
Hence, the vessel is feet the surface of the water.
Properties of Addition of Integers
Closure Property
If a and b are two integers, then a + b = c where c will always be an integer.
For example,
Commutative Property
For any two integers a and b, a + b = b + a
This means that the sum of two integers is the same regardless of the order in which they are added.
For example,
Associative Property
For any three integers a, b and c, (a + b) + c = a + (b + c)
This means that when adding three or more integers, the order in which additions are performed does not change the final sum.
For example,
1 (−5 + 3) + (−6) = −5 + (3 + (−6)) = −8 Do It Together
1 (+2) + (+8) = +10
2 (+5) + (–9) = – 4
1 (−5) + (+8) = (+8) + (−5) = +3
Existence of Additive Identity
For any integer a, we have:
a + 0 = 0 + a = a
This equation states that adding zero to any integer results in the same integer.
For example,
1 5 + 0 = 5 2 –5 + 0 = –5
Existence of Additive Inverse
For any integer, a, we have: a + (–a) = 0
For every integer, there exists an additive inverse such that the sum of the number and its additive inverse is zero.
For example, 1 10 + (–10) = 0 2 16 + (–16) = 0
Example 12: Which of these represent the additive inverse property?
1 18 + 3 = 3 + 18 2 30 + 0 = 30 3 15 + (–15) = 0
Option 3 represents the additive inverse property.
Example 13: Which property of the addition of integers is used in (–12 + 13) + 25 = –12 + (13 + 25)?
The associative property is used in the given addition sentence.
Fill in the blanks and name the property used.
3 39 + (−12) = Closure property 4 26 + (–13 + 9) = Do It Together
1 –25 + 0 = Existence of additive identity
2 –15 + 35 = 35 + (–15)
Do It Yourself 6B
Draw tokens to show the addition of the given integers. Write the answers. a +5 + (+6) b –1 + (+7) c –5 + (–9)
Draw number lines to show the addition of the given integers. Write the answer.
a 6 and 4 b –6 and –4 c –9, 3 and 5
Write the addition sentence shown by the number lines.
Fill in the blanks.
a 12 + (–14) = b –16 + (–18) = c –25 + 63 = d 52 + (–45) = e 125 + (–256) = f (–365) + (–145) =
Simplify.
a (–9) + 12 + (–15) b 18 + (–25) + 28 c –32 + (–14) + (–35)
d 36 + (–78) + (–14) + 63 e –175 + (–48) + 259 + 88 f –148 + 32 – 238 + 54
Add the successor of –1025 to the predecessor of 923. Write the additive inverse of the given integers. a –8 b 12 c –14
Find an integer a so that a 5 + a = 5 b 7 + a = 0 c 6 + a = 3 + 6
The temperature of a city was recorded at 42°C on Monday. The temperature fell by 15°C on Tuesday and fell by a further 7°C on Wednesday. Find the temperature recorded on Wednesday.
b Change one ‘−‘ to ‘+’ to get the answer as 11. Find three integers of different signs whose sum is –158. 1 2 Challenge
Look at the expression: 1 – 2 + 3 – 4 – 5 + 6 + 7 – 8 + 9
a Solve the above expression.
Critical Thinking
Subtraction of Integers
Subtraction of Integers Using Tokens
Let us now learn how to subtract integers using tokens. + is used to represent positive integers and is used to represent negative integers.
Positive Integer – Positive Integer
+5 – (+3)
Step 1: Draw positive tokens for the positive number.
+ + + + +
Negative Integer – Positive Integer
−5 – (+7)
Step 1: Draw negative tokens for the negative number.
Step 2: Take away +3 tokens from the given tokens.
+ + + + +
Step 3: Count the tokens left.
We are left with 2 positive tokens
Step 2: We have to take away +7 but since there are no positive tokens, we add 7 zero pairs.
+ + + + + +
Step 3: Take away +7 from the given tokens.
So, +5 – (+3) = +5 – 3 = +2
Example 14: Subtract the numbers using tokens. 1 −8 – (−6)
We are left with 2 negative tokens.
Step 4: Count the tokens left.
We are left with 12 negative tokens.
So, −5 – (+7) = −12
So, −8 – (−6) = − 2 2 +7 – (+4) + + + + + + +
We are left with 3 positive tokens. So, +7 – (+4) = +7 – 4 = +3
Do It Together
Subtract –2 – (+5) using tokens.
+ –2 – (+5) = –2 – =
Jay now wants to know the difference between the number of points he scored in the second round as compared to the first round. Let us help him!
Rules for Subtraction of Integers
Step 1: Take the additive inverse of the integer to be subtracted.
Additive inverse of −4 = +4
Step 2: Add it to the other integer
6 – (–4) = 6 + (+4)
Do It Together
Hence, Jay scored 6 – (–4) = 6 + (+4) = 10 more points in the second round compared to the first round.
From the above discussion, we can conclude that: To subtract integer b from a:
• change the sign of b and
• add to a
Example 15: Subtract.
1 85 – (–53) = 85 + (additive inverse of –53) = 85 + 53 = 138
Did You Know?
The highest temperature ever recorded on the Earth was 58°C in the Libyan desert. The coldest temperature ever measured was –88°C in Antarctica.
2 –269 – (136) = –269 + (additive inverse of 136) = –269 + (–136) = –405
3 –2445 – (–1196) = –2445 + (additive inverse of –1196) = –2445 + 1196 = –1249
Example 16: Subtract the sum of –215 and 635 from –856.
Sum of –215 and 635 = –215 + 635 = 420
Subtract 420 from –856 = –856 – 420 = –856 + (–420) = –1276
Example 17: The temperature of a city is 10°C on Monday. It drops by 15°C on Tuesday. What is the temperature of the city on Tuesday?
Temperature on Monday = 10°C
Drop in temperature on Tuesday = 15°C
Temperature on Tuesday = 10°C – (15°C) = –5°C
Subtract.
1 –36 from 58
= 58 – (–36)
= 58 + 36 =
3 –247 from 354
= 354 – ( )
= 354 + =
2 –145 from –365
= –365 – (–145)
= + =
4 3654 from –5214
= –5214 –
= + =
Properties of Subtraction of Integers
Closure Property
For any integer a and b a – b = c, where c will always be an integer.
For example,
1 (+12) – (+8) = +4
2 (+15) – (–9) = +24
Commutative Property
For any integers a and b, a – b π b – a
Subtraction is not commutative for integers. The order in which they are written is important.
For example,
1 (+22) – (+13) = +9; (+13) – (+22) = –9 (+22) – (+13) ≠ (+13) – (+22)
2 (+15) – (+10) = +5; (+10) – (+15) = –5 (+15) – (+10) ≠ (+10) – (15)
It Together
Associative Property
For any integers a, b and c, (a – b) – c π a – (b – c)
The integers are not associative under subtraction.
For example,
1 (5 – 3) – 4 = –2; 5 – (3 – 4) = +6 (5 – 3) – 4 ≠ 5 – (3 – 4)
2 (9 – 2) – 5 = +2; 9 – (2 – 5) = +12 (9 – 2) – 5 ≠ 9 – (2 – 5)
Subtraction Property of Zero
For an integer a, a – 0 = a
Subtracting zero from any integer leaves the integer unchanged. For example, 1 (–3) – 0 = –3
2 (–26) – 0 = –26
Example 18: Show that the integers 12, –18 and –34 are not associative under subtraction.
The associative property of subtraction of integers states that (a – b) – c = (12 – (–18)) – (–34) = 12 + 18 – (–34) = 30 + 34 = 64 a – (b – c) = 12 – (–18 – (–34)) = 12 – (–18 + 34) = 12 – (16) = –4 64 π –4, hence (a – b)
Fill in the blanks for the given properties.
1 Property Used: Commutative property
523 – = 365 – 523 =
Thus, 523 – 365 – 523
2 Property used: Associative property (125 – 523) − = 125 – ( 523 – 363) = Thus, (125 – 523) − 125 – ( 523 – 363)
Do It Yourself 6C
Draw tokens to show the subtraction of the given integers. Write the answers.
Subtract.
15 from –36
Subtract the integers and match the answers.
–52 from 69
Fill in the blanks using the properties of integers.
a –90 + 1 = b 25 – = 25
c –35 + 0 = d 32 – 15 =
Subtract the sum of 623 and –584 from the sum of –795 and 475.
The sum of two integers is –423. If one of them is 89, find the other.
Dev drove his car 125 km to the North on Monday. He drove 182 km South on Tuesday. How far is Dev from his initial point?
Mercury is a chemical element and the only metal that is liquid at room temperature. The melting point of mercury is –38.9°C and its boiling point is 356.7°C. How many degrees is the melting point below the boiling point?
The lowest point in Africa is Lake Asal, at 153 m below sea level, and the highest point in Africa is Mount Kilimanjaro, at 5895 m above sea level. What is the difference of these elevations?
Pooja had a balance of `–1250 in her bank account due to some penalties. If she wants to make the balance in her bank account `5000, how much money does she need to deposit in her account?
A submarine is 2514 feet below sea level. An aeroplane is 10,500 feet above sea level. What is the distance between the submarine and the aeroplane?
Challenge Critical Thinking
1
Anu and Piya are playing a game where they are going up and down the stairs. Anu goes up 4 steps and comes down 2 steps. Piya goes up 3 steps and comes down 1 step. Who reaches the top first if there are a total of 20 steps?
Points to Remember
• The numbers (…. −4, −3, −2, −1, 0, 1, 2, 3, 4, ….) are called integers.
• Negative integers lie to the left of 0 on the number line, whereas positive integers lie to the right of 0 on the number line.
• Integers to the right on the number line are greater than integers to the left on the number line.
• 0 is neither a positive nor a negative integer.
• The absolute value of an integer is its numerical value, regardless of its sign.
• To add two integers with like signs, we add their values and place the common sign before the total.
• To add integers with unlike signs, we take the difference of their absolute values and put the sign of the integer with the greater absolute value.
• To subtract an integer b from an integer a, we change the sign of b and add it to a.
Setting: In two groups
Experiential Learning & Communication
Expression Expedition
Materials Required: Paper and pencil, index cards with expressions written on them
Method:
1 Prepare index cards with various integer expressions written on them.
2 1 member from each team draws an index card and solves the expression.
3 After solving the expression, both players take turns explaining their solution to the class.
4 If both Player A and Player B provide the correct solution, they each earn a point for their team. If one or both provide incorrect solutions, they will earn −1 point.
5 Repeat the process for as many rounds as the time allows.
6 The team with the highest score wins!
Chapter Checkup
Solve the given problems using tokens. a +3 + (+8) b −2 + (+7) c −5 − (−7) d −4 − (+9)
Circle the integer which is to the left of the other on the number line.
a –2, 5 b 13, 18 c 8, –8 d –13, –17 e 25, 36
The table shows the temperatures of some places in India on a certain day.
Place
Chennai
Ladakh
Pune
Sonamarg
a Write the temperatures in the form of integers.
b Name the coldest and the hottest place.
Which point will be marked as −2 on the number line?
Tick () the integers that are given in their absolute values. a 9 b +12 c –36
Temperature
31° above 0°
5° below 0°
18° above 0°
8° below 0°
Add –164 and 98 Add –73 and –49
Write the predecessor and successor of the following integers.
a –8 b –26 c –38 d 45
Which of the statements is true?
a +8 is larger than –16
c –8 is larger than +16
b +16 is smaller than +8
d +16 is smaller than –8
How many positive integers lie between –5 and 0?
Which set of integers is written in ascending order?
a –25, –65, 28, 63, 100
c 29, 15, –9, –16, –32
Add the integers.
a –25 and 36
c –157 and 96
b –22, –18, –15, 16, 36
d –32, –17, –19, 42, 48
b –142 and –48
d 452 and –152
Subtract the predecessor of –526 from the successor of 658.
Write three integers with different signs whose sum is –48.
Write a number smaller than −858 but greater than −861 and is odd.
Thermometers are used to measure the degree of hotness or coldness of an object. The temperature of a liquid is 78°C. It dropped to –114°C. What is the difference between the two temperatures?
Muskan burnt 480 calories working out on a treadmill and an additional 260 calories by skipping. She indulged in a sweet that had a calorie count of 500. How many calories did she lose or gain?
On a hot summer day, the temperature in a desert was 132°F. It dropped to –250°F on a windy day. What is the difference between the two temperatures?
Priya uses an online wallet for shopping which allows her to shop for anything up to `750 even if she has no balance in the account. She wanted to purchase a dress for `1000, and her account balance was `–125. She added `1500 to her account. What is the balance in the account after the purchase of the dress?
In a test of 50 questions, +6 marks is given for each correct answer, –1 mark is given for each incorrect answer and 0 for not attempting. Manoj completed this test with 40 correct answers and 6 incorrect answers. Find the marks obtained by Manoj in the test.
The construction dates of the worldwide landmarks are given below. AD is represented by positive numbers and BC by negative numbers. Write the following differences in descending order.
Read the statements and choose the correct option.
Assertion (A): On a number line, 3 and −3 are equidistant from 0.
Reason (R): The absolute value of both 4 and −4 is the same.
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
Climate Zones: A Comparison
The average temperature varies across the countries. Some countries are warmer, whereas some countries are colder. The table shows the average temperatures of a few countries across the world.
1 The average temperature of Qatar is than the average temperature in Russia.
a 24.23°C more
c 31.81°C more
b 24.23°C less
d 31.81°C less
2 Arrange the countries in ascending order of the average temperature.
3 Which country among Russia and Greenland is warmer, and by how much?
4 Does it get colder or warmer when the temperature rises? Why?
Cross Curricular
Let’s Recall
A fraction is a number that represents a part of a whole. It has two parts, a numerator and a denominator.
Let us say Riyanshi has a strawberry pie, and she divides it into 7 equal parts. So, here a strawberry pie is considered a whole, and each piece of it is a part. She eats 3 parts out of the total.
To represent the pie eaten by her as a fraction, Riyanshi shades 3 parts out of 7. The shaded part represents 3 7 . Here, 3 is the numerator and 7 is the denominator.
Let’s Warm-up
Reviewing Fractions
Miya is baking an oat cake today. She divided the delicious oat cake into 10 equal parts.
Miya gave 2 slices of the cake to her father and 4 slices of the cake to her mother. Who got the bigger part of the cake? How do we write these as fractions?
Comparing and Ordering Fractions
First, let us learn how to represent fractions, and then how to compare them.
Representing Fractions
We know that Miya has divided her cake into 10 equal parts. The whole cake here represents a whole (1) and each piece of cake
represents a part of a whole cake.
Each piece of cake is 1 10 of the cake.
So, ten slices with each slice shown as 1 10 will make a whole or 1.
Let us now learn how to represent fractions on a number line. As we need to divide 1 in 10 equal parts; divide the space between 0 and 1 in 10 equal parts and note down the fractions.
Let us now understand different types of fractions.
Types of Fractions
Remember!
The gap between two whole numbers on a number line should be divided into the same number of equal parts as the denominator of the fraction.
Example 1: Classify the fractions as proper, improper or mixed numbers.
Did You Know?
The human body is primarily composed of water. On average, water makes up about 3 5 of an adult humanʹs body weight.
Example 2: Represent the fractions on a number line.
Write the fractions represented on the number line marked with black dots. Do It
Comparing Fractions
Remember, Miya was baking a delicious cake? She gave 2 10 of the cake to her father, and 4 10 of the cake to her mother.
We know that when two fractions have the same denominator, they are like fractions. Hence, 2 10 and 4 10 are like fractions.
To compare like fractions, we compare the numerators.
Greater the numerator, greater the fraction.
4 > 2
Thus, 4 10 > 2 10. So, her mother got bigger part of cake.
What if Miya gave 2 7 of the cake to her sister, and 2 8 to her brother? Who received the larger portion?
Here the 2 fractions are unlike. Unlike fractions are fractions that have different denominators. However, the two numerators here are the same. In this case, the numerators are the same. Here, we compare the denominators.
Different denominator
Do It Together
But what if both the numerators and the denominators are different? We can use either the LCM method or the cross-multiplication method to compare the fractions. Let us look at both methods in detail.
LCM Method: Make the fractions like, using the LCM method.
Let us compare 1 4 and 2 3 .
Step 1: Find the LCM of the denominators.
Step 2: Find the equivalent fractions of 1 4 and 2 3 such that their denominators are 12.
The LCM of 3 and 4 is 12. 1 4 × 3 3 = 3 12
LCM of 3 and 4 = 12
Step 3: Now that the fractions are like fractions, compare the numerators and identify which fraction is larger.
Cross-multiplication Method
Let us compare the same fractions 1 4 and 2 3 using the cross-multiplication method.
Cross multiply the denominators with the numerators.
1 × 3 = 3 and 4 × 2 = 8
3 < 8
Thus, 1 4 < 2 3 × 2 1 4 3
Example 3: Which fraction is bigger, 1 2 or 3 5 ?
Convert both the fractions into like fractions.
LCM of 2 and 5 = 10 3 5 = 3 × 2 5 × 2 = 6 10
1 2 = 1 × 5 2 × 5 = 5 10
6 > 5, so 6 10 > 5 10 . Thus, 3 5 > 1 2
Compare 5 7 and 3 6. Identify the larger fraction.
LCM of 7 and 6 = ______ 5 7 = 42 3 6 = 42
As > . Thus, the larger fraction is .
Ordering Fractions
To arrange fractions in ascending order (least to greatest) or descending order (greatest to least), remember to compare the fractions and then arrange.
Arrange 1 2 , 3 5 , 2 10 , 4 15 in ascending order.
Step 1
Convert all the fractions into like fractions.
LCM of 2, 5, 10 and 15 is 30.
1 2 = 1 2 × 15 15 = 15 30
3 5 = 3 5 × 6 6 = 18 30
2 10 = 2 10 × 3 3 = 6 30
4 15 = 4 15 × 2 2 = 8 30
Example 4: Arrange 1 2 , 5 6 , 5 12 , 2 3 in descending order.
LCM of 2, 6, 12, 3 = 12
1 2 = 1 × 6 2 ×
10 > 8 > 6 > 5
Thus, 5 6 > 2 3 > 1 2 > 5 12
Arrange 3 4, 4 5, 1 2, 7 8 in descending order.
LCM of 4, 5, 2, 8 = 3 4 = ; 4 5 = 1 2 = ; 7 8 =
Step 2
Since the denominators are the same, compare the numerators and order the fractions.
6 < 8 < 15 < 18
So, 6 30 < 8 30 < 15 30 < 18 30 Thus, 2 10 < 4 15 < 1 2 < 3 5
3 Represent the fractions on a number line. 4
Classify the fractions as proper fractions, improper fractions or mixed numbers.
Circle the larger fraction.
7 and 3 7
5 Compare the fractions using appropriate symbols (>, < or =).
and
and 12
6 Arrange the fractions in ascending order.
7
In school A, 480 students passed out of 500, whereas in school B, 550 students passed out of 600. Write the fraction of students who passed in both schools. In which school did the greater fraction of students pass?
The Earth’s surface is covered by both land and water. Approximately 3 parts of the Earthʹs surface are land, and the remaining 7 parts are water. Represent the fraction of the Earthʹs surface covered by land and water using a fraction circle.
Sana purchased different lengths of cloth. She purchased
and
Arrange the lengths of the cloth in descending order.
Jacky and Pritam go for a walk every day at the park. Jacky completes 5 8 rounds of the park in 10 minutes, whereas in the same time, Pritam completes 7 12 rounds of the park. Who covers the longer distance in 10 minutes? How is going for walks helpful?
You have a birthday brownie in a circular shape and have to divide it into 8 equal pieces by making 3 cuts only. How do you do it? What fraction of the whole brownie will each piece be?
Equivalence and Conversion
Let us learn how to convert improper fractions to mixed numbers and mixed numbers to improper fractions.
Conversion Between Mixed Numbers and Improper Fractions
Converting a mixed number into an improper fraction.
Convert 2 2 5 into an improper fraction.
Step 1
Multiply the denominator with the whole number.
Step 2
Add the result to the numerator.
Step 3
Keep the denominator the same, and write the result as the numerator.
Convert an improper fraction to a mixed number. Write 24 5 as a mixed number.
Step 1
Divide the numerator by the denominator.
Step 2
The quotient becomes the whole number part.
Step 3
The remainder becomes the numerator, and the divisor is the denominator.
Example 5: Convert the following mixed numbers into improper fractions.
Example 6: Convert the following improper fractions into mixed numbers.
The fractional part of a mixed number cannot have a numerator bigger than the denominator. Error Alert!
Convert. Do It
Equivalent Fractions and Simplest Form
While preparing, Miya added 3 4 tablespoon of baking soda, and 6 8 tablespoon of baking powder to the cake mixture
We can see from both figures that the amounts of baking soda and baking powder Miya used are equal in quantity.
Hence, 3 4 = 6 8 . Therefore, they are equivalent fractions.
Soda = 3 4
Powder = 6 8
Thus, equivalent fractions are fractions which may have different numerators or denominators, but represent the same value.
Let us find the equivalent fraction for 1 5 and 10 15 .
Hence, to find the equivalent fraction of any fraction, multiply or divide the numerator and denominator of the fraction by the same natural number.
What if we need to find whether two fractions are equivalent or not?
In such cases, we find the simplest form of the given fractions.
Simplest form of a fraction.
A fraction is in its simplest form if the HCF of the numerator and denominator is 1.
To find the simplest form of a fraction, divide the numerator and denominator by their HCF.
For example, for the fraction 15 25 ; the HCF is 5. Dividing both the numerator and denominator by 5; we get 3 5 = Simplest form of 15 25 .
If fractions have the same simplest forms then they are always equivalent.
Example 7: Convert 14 49 into its simplest form. HCF of 14 and 49 = 7
On dividing the numerator and denominator of the fraction by their HCF, we get 14 49 � 7 7 = 2 7
Thus, 2 7 is the simplest form.
Example 8: Check whether 3 7 and 9 21 are equivalent or not.
To find whether two fractions are equivalent or not, we need to find their simplest form.
In the fraction 3 7 , the HCF of 3 and 7 = 1, so it is in its simplest form.
In the fraction 9 21 , the HCF of 9 and 21 = 3
On dividing the numerator and denominator of 9 21 by their HCF, we get
As the simplest forms of both 3 7 and 9 21 are equal, the fractions are equivalent.
Write the equivalent fractions of 8 10 using multiplication and division.
The equivalent fractions are –
Do It Yourself 7B
Convert between improper fractions and mixed numbers.
Reduce the following fractions into their simplest forms.
Identify whether each pair of fractions is equivalent or not.
Write two equivalent fractions for the fractions. 4
Write the equivalent fraction for the given fractions.
The small intestine in an average human body is about 38 5 m and is actually the largest intestine based on length. Write this length as a mixed number.
Renu has baked 100 cookies and wants to divide them equally among her 3 siblings. What will be the share of each sibling? Express your answer as a mixed fraction. Do you think equal sharing is fair?
Riya was eating fruits. She had a total of 24 fruits. She gave 10 fruits to her friend and 10 fruits to her sister. What fraction of the fruit did she have left? Express your answer in the simplest form.
Eva and Sarah baked a chocolate and a vanilla cake, respectively. Eva’s chocolate cake was cut into 8 equal slices. Sarah’s vanilla cake was cut into 12 equal slices. If Eva gave away 3 slices of her cake and Sarah gave away 5 slices of her cake, which of them had the greater fraction of their cake left?
Minal asked her brother to guess the fraction based on the given clues:
a The numerator is three less than twice the denominator.
b When you add 1 to both the numerator and the denominator, the fraction becomes 2 3
How many more boxes need to be shaded to make the fraction shown by the figure equivalent to the other 2 fractions?
Operations on Fractions
Lisa and Maya run a fruit squash cafe. They are making a fruit punch. The list of ingredients used by them are shown in the table.
Addition and Subtraction of Fractions
Lisa wants to find the total fraction of blueberry and strawberry juice she used to make the fruit punch.
Addition and Subtraction of Unlike Fractions
Fraction of blueberry juice used by Lisa = 1 5 ; fraction of strawberry juice used by Lisa = 1 2
Remember!
Sum or difference of like fractions = Sum or difference of numerators Common denominator
Adding Unlike Fractions
As 1 5 and 1 2 are unlike fractions, we will use the below steps to add them.
Step 1: Find the LCM of 2 and 5.
of 5 and 2 = 10
Step 2: Find the equivalent fractions with denominators of 10.
Step 3: Add the fractions.
Thus, Lisa used a total of 7 10 of a cup of blueberry and strawberry juice in her recipe.
Subtracting Unlike Fractions
Let us see how much more strawberry juice than blueberry juice Lisa used by subtracting the unlike fractions. The fractions can be subtracted using the steps below.
Step 1: Find the LCM of 2 and 5.
Step 2: Find the equivalent fractions with denominators of 10.
It is clear that more strawberry juice was used than blueberry juice by 3 10 of a cup.
Example 9: Add 2 9 and 5 12 LCM of 9 and 12 = 36 Example 10: Find 7 8 –3 4
of 8 and 4 = 8
Step 3: Subtract the fractions.
Example 11: In Kaziranga National Park in Assam, India, the total area covered by different types of habitats is measured. If it has a grassland area of 3 8 of the total area and a forest area of 2 5 of the total area, what fraction is covered by both grassland and forest habitats?
Grassland area = 3 8
Forest area = 2 5
LCM of 8 and 5 = 40
Hence, 3 8 � 15 40 ; and 2 5 � 16 40
Fraction covered by both grassland and forest habitats = 15 40 � 16 40 � 31 40 .
Hence, the fraction covered by both grassland and forest habitats is 31 40.
Radha ate 1 9 of a cake, and her sister Riya ate 2 3 of it. Find the total fraction of the cake consumed and the fraction of the cake left.
Fraction of the cake eaten by Radha = 1 9 ; Fraction of the cake eaten by Riya = 2 3
LCM of 9 and 3 =
Fraction of cake eaten by Radha and Riya = ___________ + =
Fraction of cake remaining = – =
Addition and Subtraction of Mixed Numbers
While preparing fruit punch, Maya aimed to determine the total quantity of sugar used by both of them.
Adding Mixed Numbers
Fraction of sugar used by Lisa = 12 5 cups; fraction of sugar used by Maya = 31 3 cups
Total = 12 5 cups + 31 3 cups
To find the total, we will add = 12 5 + 31 3
Method 1:
Step 1
Add the whole numbers.
1 + 3 = 4
2
the fractional parts.
Step 3
Add the results of step 1 and step 2.
Step 1
Convert
Step 2
Find the LCM of the denominators.
Multiply to make the denominator the same as the LCM.
Step 3
Convert the improper fraction into a mixed number.
Subtracting Mixed Numbers
What if Maya had used 6 1 5 cups of sugar to make the same fruit punch instead of 3 1 3 cups? How much extra sugar would it have been then?
We subtract 3 1 3 from 6 1 5 .
Step 1
Convert the mixed numbers into improper fractions.
6
Example 12: Do as directed.
1. Add 1 1 7 and 2
2. Subtract
Step 2
Find the LCM of the denominators. Multiply to make the denominator the same as the LCM.
Step 3
Convert the improper fraction into a mixed number.
Did You Know?
Aryabhata (476–550 CE), an Indian mathematician used fractions in his astronomical calculations and gave rules for arithmetic operations with fractions.
Do It Together
Example 13: Jennifer went shopping for her birthday party and bought a total of 9 1 2 kg of candies. She purchased 2 3 4 kg of chocolate candies, 1 1 10 kg of gummy bears, 3 1 2 kg of jellybeans and some lollipops. What was the weight of the lollipops that she purchased?
Total weight of candies = 9 1 2 kg = 19 2 kg
Weight of chocolate candies = 2 3 4 kg = 11 4 kg
Weight of gummy bears = 1 1 10 kg = 11 10 kg
Weight of jellybeans = 3 1 2 kg = 7 2 kg
Weight of chocolate candies, gummy bears, and jellybeans together =
11 4 + 11 10 + 7 2
kg
LCM of 4, 10, 2 = 20
11 4 + 11 10 + 7 2
Weight of lollipops = 190 20 –147 20 = 43 20 = 2 3 20 kg
Subtract 5 2 6 from 7 1 4 .
LCM of 4, 6 =
On converting to improper fractions, we get:
=
Do It Yourself 7C
Add the fractions and reduce the fractions, if necessary.
Subtract the fractions and give the answer in simplified form.
What do we get if we subtract the difference of 4 1
?
While studying different types of trees present in the Himalayan foothill forest, researchers found that 1 4 of the trees are conifers (plants that are conically shaped to survive during snowfall), while 1 4 of the trees are deciduous (plants that shed their leaves during the winter to survive). Combine these fractions to describe the total proportion of trees that are either conifers or deciduous.
An empty box weighs 2 6 7 kg. An iron pan weighing 5 2 3 kg is placed inside it. What is the total weight of the box?
Sophia is baking cookies for a school fundraiser. She’s making two batches of cookies. In the first batch, she uses 3 5 cup of sugar, and in the second batch, she uses 1 5 cup of sugar. Find how much sugar Sophia used in total for both batches of cookies.
Priya bought some popcorn from the market. She generously gave three-fifths of them to her sister and distributed one-fourth of the popcorn among her neighbours and kept the rest for herself.
a What fraction of the popcorn did she give away?
b What fraction of the popcorn did Priya have left?
Challenge
In caffeine, a fraction of 1 12 constitutes oxygen atoms, 1 6 are nitrogen atoms and 5 12 are hydrogen atoms. The rest of the atoms consist of carbon. What fraction of caffeine’s atoms are oxygen, nitrogen or carbon?
Multiplication and Division of Fractions and Mixed Numbers
Lisa was thinking of trying a new fruit punch recipe which requires only 2 5 of the available quantity of apple juice. Refer to the table of ingredients given on page 136.
Multiplying Fractions or Mixed Numbers
Fraction of apple juice Lisa has = 3 4 of a cup.
Fraction of apple juice she wants to use in her new fruit juice recipe = 2 5 of the quantity that Lisa has.
Remember!
When the fraction is used with the word “of”, it acts as an operator. Ideally, “of” means multiplication.
Thus, Lisa will use only 3 10 of a cup of apple juice.
Do you remember Lisa used 1 2 5 cups of sugar for the fruit punch? What if she used 1 1 2 of what she was using earlier?
The fraction of sugar Lisa had = 1 2 5 cups; fraction of sugar Lisa used = 1 1 2 � 1 2 5 cups.
Step 1
Convert both mixed fractions into improper fractions.
Step 2
Multiply both improper fractions by multiplying the numerators and the denominators.
Remember!
Step 3
Convert 21 10 into a mixed fraction. 21 10 = 2 1 10
Thus, Lisa used 2 1 10 cups of sugar.
To multiply one fraction with another fraction, multiply the numerators, multiply the denominators and then write the answer in the lowest form.
Example 14: Multiply.
Think and Tell
What is the product of 5 8 and its reciprocal?
Dividing Fractions or Mixed Numbers
What if Lisa adds 1 3 cup of orange juice equally to 3 different glasses? How much juice would go in each cup?
Is there a different way to do this mathematically? Let us learn how.
Dividing 1 3 by 3 can also be written as 1 3 ÷ 3.
To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.
What
Example 15: Divide.
Remember!
Always convert a mixed fraction into an improper fraction when performing multiplication or division.
Do It Yourself 7D
Multiply the fractions and reduce the fractions, if necessary.
Write the reciprocal of the fractions.
Divide the fractions and reduce the fractions, if necessary.
Ananya ploughed 9 12 of a field, and Atif ploughed 1 3 of the area ploughed by Ananya. Find the fraction of the whole field ploughed by Atif.
Maya is visiting an old age home. She is making porridge to offer to the elderly residents as a kind gesture. She has 75 grams of raisins, but uses only 1 5 of them. How many grams of raisins does Maya use for the porridge?
How do you help the elderly at home and in your surroundings?
Emma is baking cream rolls and needs 2 3 cup of flour for each batch. She has 41 2 cups of flour in total. How many batches of cream rolls can Emma bake with the flour she has, and how much flour will be left over?
In ancient Egypt, land was often measured in units called “cubits”. If a farmer’s field is in the shape of an equilateral triangle with a perimeter 2 3 4 cubits, what is the length of each side?
Challenge
Examine this model and write a multiplication problem that involves fractions, based on its representation.
(Hint: When we mix the colours yellow and blue, we get the colour green.)
Points to Remember
Critical Thinking
• Fractions are numbers of the form a b , where a and b are whole numbers and b ≠ 0.
• Improper fractions are always equal to or greater than 1, whereas proper fractions are less than 1.
• When two like fractions are compared, then the fraction with the greater numerator will be greater.
• A fraction is said to be in its lowest form if the HCF of the numerator and denominator is 1.
• To add or subtract unlike fractions, find the LCM of the denominators, convert to like fractions, and then add or subtract.
• To multiply fractions, multiply the numerators, multiply the denominators, and simplify the answer.
• A reciprocal of a fraction can be obtained by interchanging the numerator and denominator.
• To divide one fraction by another, multiply the first fraction with the reciprocal of the other fraction.
Math Lab
Bingo!
Collaboration
Fraction bingo cards: Create bingo cards with different fractions (both proper and improper)
Fraction chip/markers: A set of fraction chips/markers (small tokens or stickers) to cover the fractions on their bingo cards
Caller’s cards: Prepare a deck of cards with multiplication problems, involving fractions.
Method:
Distribute fraction bingo cards and fraction chips/markers to each student. The goal is to cover five squares in a row (horizontally, vertically or diagonally) to win. 1 2
To cover a square, students must correctly solve the multiplication problem on the caller’s card. Students should call out ‘Bingo!’ when they have five covered squares in a row.
Start the game by having one student act as the caller.
Continue playing until a student calls out, ‘Bingo!’
Chapter Checkup
Convert the mixed numbers into improper fractions.
Convert the improper fractions into mixed numbers.
15 4
Find any three equivalent fractions.
5 7
Write the following fractions in their simplest form.
Check whether each pair of fractions is equivalent or not.
Solve and reduce to their simplest forms.
3 5 � 10 15
Answer the following questions.
a Which is more? The difference of 8 3 8 and 3 1 4 or the
b Which is less? The product of 1 2 3 and
c What is the sum of the product of 3 and 4 5 and the sum of 1 4 and 1 3 ?
d What is 1 5 of a dozen? What is 10 dozen?
A vitamin C tablet contains 1 4 gram of vitamin C. Sunny took 2 1 2 tablets today. How many grams of vitamin C did he take?
Shweta has 4 5 kg of sugar. She uses 3 4 kg to make a sweet dish. How much sugar does she have left?
Radha purchased 40 m of cloth. She used 2 5 m cloth for the curtains and 3 3 4 m cloth for the bed sheet. How much cloth does she have left?
A group of students created 2 different types of tote bags: X and Y.
For every 4 tote bags of type X sold, they would donate 5 12 of the proceeds to a blind school.
For every 2 tote bags of type Y sold, they would donate 2 5 of the proceeds to an old age home.
During the event, they sold
a 24 tote bags of type X for `400 each.
b 18 tote bags of type Y for `600 each.
Calculate the total amount donated from the proceeds of each type of tote bag sale.
Read both the statements given below and choose the correct option.
Assertion: 21 84 and 12 56 are equivalent fractions.
Reason: Two fractions a b and c d , (b, d = 0) are equivalent if their simplest form is the same.
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 Assertion is true, but Reason is false.
d Assertion is false, but Reason is true.
Case Study
Value Development
The Fitness Regime
Ravi is a dedicated athlete who spends 5 12 of the day training for physical fitness. He splits his training time among various art forms to improve his overall skills. Out of the 5 12 of the day, he spends 1 4 of his training time on martial arts, 1 3 of his training time on dance and the rest of the time on yoga.
Answer the following questions:
1 How much of the day does Ravi spend on martial arts?
2 What fraction of the day does Ravi spend on dance?
3 Ravi spends hours of his training time on yoga.
4 Arrange the activities in ascending order of their practice time.
5 What are the benefits of doing physical activities? Which physical activity do you do?
8 Decimals
Let’s Recall
Decimals are numbers that are used in real-life situations where we need to be more exact than whole numbers. For example, when we weigh ourselves, our weight might not be a whole number on the scale, so decimals are used to show the exact weight. Decimals can be used to write amounts, lengths, temperature, etc.
A decimal number is a number which consists of a whole and a decimal part. The decimal part is always less than 1.
The increasing order of the times is 15.45 seconds < 15.53 seconds < 15.6 seconds < 17.3 seconds. Thus, Sukant is the fastest followed by Yashu, Vyom and then Aman who secured the fourth position.
Letʼs Warm-up
Yashu takes part in a sprint race at his school. The time taken to finish the race is represented in decimal form. Yashu takes 15.53 seconds, Vyom takes 15.6 seconds, Aman takes 17.3 seconds and Sukant takes 15.45 seconds to finish the race. Fill in the blanks using >, < or =.
Understanding Decimals
Shraddha takes her cat to a vet clinic. The vet prescribes cough syrup for her cat.
0.8 mL of cough syrup is to be given to the cat every morning.
Representing and Converting Decimals
0.8 is a decimal number. The numbers which contain a decimal point (.) are called decimal numbers, or simply decimals. For example, 15.2, 1.3, 0.75 and 0.004 are all decimal numbers. Let us learn more about decimals with Shraddha.
Representing Decimals on a Number Line
Shraddha wonders how she can measure 0.8 mL and give this dosage to her cat.
Let us understand how to represent decimal numbers on a number line.
0.8 is more than 0 but less than 1.
So, 0.8 lies between 0 and 1.
The number of tenths between 0 and 1 = 10 tenths
The number of tenths in 0.8 = 8 tenths
To represent 0.8 on the number line, divide the unit length between 0 and 1 into 10 equal parts and represent 0.8 at the 8th division.
Let us now represent 0.54 on the number line.
1 one is divided into 10 tenths, and 1 tenth is further divided into 10 hundredths.
So, 0.54 lies between 0.5 and 0.6.
Example 1: Represent 1.5 on a number line.
1.5 lies between 1 and 2.
1.5 = 1 + 0.5 = 1 + 5 tenths
Divide the unit length into 10 equal divisions and then mark 1.5 five marks after 1.
Example 2: Represent 3.61 on a number line.
3.61 lies between 3.6 and 3.7.
Mark 12.6 on the number line. Do It
13 12
Place Values in Decimals
Shraddha took her cat to Dr Awasthi after a week. The dosage of cough syrup was increased to 0.925 mL.
Now, let us understand how to represent this number.
The place value increases by 10 times
The place value decreases by 10 times while moving from right to left. while moving from left to right.
Represent 1532.178 in a place value chart. Expanded form for 1532.178
Let us now see how to write the decimal number in words.
One thousand five hundred thirty-two
One thousand five hundred thirty-two
1532.178
one hundred seventyeight thousandths one seven eight point and
Example 3: Consider the decimal 213.018. Write it in words.
Did You Know?
The world record for the men's
213.018 is written as two hundred thirteen and eighteen thousandths or two hundred thirteen point zero one eight.
Expanded form = 200 + 50 + 6 + 0.003 or Number name = Do It Together
Consider the number 256.003. Write the number in expanded form and in words.
Conversion Between Fractions and Decimals
On the next visit to the vet, Shraddha’s cat had to be tested for FCoV. Dr Awasthi performed a blood test for it and drew 1 2 mL of blood using a syringe.
Converting fractions to decimals
Step 1: Multiply the denominator and the numerator so that we can get 10, 100 or 1000 in the denominator.
Converting decimals to fractions
Step 1: Write the decimal with the denominator of 1.
Step 2: Write a fraction with the denominator as 10, 100 or 1000.
Step 3: Insert a decimal point before as many places as the number of zeroes in the denominator from the rightmost side.
Step 2: Convert the denominator to 10, 100 or 1000 to eliminate the decimal point.
Example 4: Convert between fractions and decimals.
Step 3: Express the fraction in its simplest form.
Convert each of the decimals to a mixed fraction. Do It Together
Like and Unlike Decimals
Shraddha went to collect the FCoV reports of her cat.
The cat had to be given 1.2 mL of cough syrup in the morning, and 0.95 mL of cough syrup in the evening. Did you notice that the decimals 1.2 and 0.95 look different? Let us figure out how!
Decimal numbers that have the same number of digits after the decimal point are called like decimals
For example, 2.56 and 8.14 are like decimals.
Decimal numbers that have different numbers of digits after the decimal point are called unlike decimals.
For example, 1.2 and 0.95 are unlike decimals.
Same number of digits after the decimal point. 8.14 2.56
Example 5: Separate groups of like decimals from the decimal numbers.
4.48, 4.51, 1.1, 87.3, 148.484, 81.654, 781.925, 4.354, 15.68, 48.61, 14.3, 54.975
Group 1 (with 1 decimal place)
Group 2 (with 2 decimal places)
Different number of digits after the decimal point.
Group 3 (with 3 decimal places)
1.1, 87.3, 14.3 4.48, 4.51, 15.68, 48.61 148.484, 81.654, 781.925, 4.354, 54.975
Check whether 15.45 and 814.012 are unlike decimals or not.
Number of digits after the decimal point in 15.45 = 2
Number of digits after the decimal point in 814.012 =
So, the number of digits after the decimal point equal.
Thus, 15.45 and 814.012 are decimals.
Equivalent Decimals
Equivalent decimals are the decimal numbers which have the same value.
These types of decimals are also called equal decimals.
For example, 0.6 = 0.60 = 0.600 = 0.6000; 1.54 = 1.540 = 1.5400
Remember!
Adding 0 at the end of a decimal number does not change the value of the number.
The concept of equivalent decimals is used to convert unlike decimals to like decimals.
Let us see how to convert 2 unlike decimals 1.2 and 0.95 to like decimals.
To make 2 decimals like, we ensure that the number of decimals are the same in both decimals.
• Identify the decimal that has fewer digits after the decimal number.
• Add zeroes to that decimal number.
• Since 1.2 has 1 digit after the decimal point, we add a zero at the end.
So, 1.2 = 1.20; 0.95 = 0.95
Now, 1.20 and 0.95 are like decimals.
Example 6: Convert each of these to like decimals.
4.5, 6.63, 12.250, 14.02
4.5 = 4.500 6.63 = 6.630 12.250 = 12.250 14.02 = 14.020
Thus, 4.500, 6.630, 12.250, 14.020 are like decimals.
Example 7: Write any two equivalent decimals for the numbers given.
3.6 = 3.60 = 3.600
Write any two equivalent decimals of 18.3. Do It Together 18.3 =
Comparing Decimals
Compare 4.5 and 4.503.
Step 1: Convert the decimal numbers into like decimals.
Step 2: Start comparing the whole number parts. If the whole number part is the same then move to the tenths place. If the digits at the tenths place are the same, then compare the digits at the hundredths place.
Follow the same process until you find unequal terms.
Thus, 4.5 < 4.503
Example 8: Which is greater, 3.64 or 3.46?
Same number of digits after the decimal point.
Decimal point
The numbers are already like decimals so let us put these in a place value chart to compare the digits.
Remember!
Start comparing from the ones place; 3 = 3
Move to the tenths place. Compare the tenths place; 6 > 4
Thus, 3.64 > 3.46; Hence, 3.64 is the greater number.
Example 9: Arrange 1.23, 1.2, 1.235, 1.3 in ascending order.
We always start comparing numbers starting from the leftmost place.
1.23 = 1.230 1.2 = 1.200 1.235 = 1.235 1.3 = 1.300
Put the numbers in a place value chart.
On comparing, the ascending order for the numbers is:
1.200 < 1.230 < 1.235 < 1.300
Thus, 1.2 < 1.23 < 1.235 < 1.3
Arrange the numbers in descending order.
48.023, 48, 48.1, 48.02
On comparing, the descending order of the numbers is:
Do It Yourself 8A
Identify the decimal represented by the shaded part and represent it on a number line.
Write the given numbers in decimal form.
Write the given numbers in a place value table.
Express the decimals as fractions in their simplest form.
Write the decimal number represented by A, B, C and D. 5
Identify the number to multiply the fraction's denominator by to convert it into a decimal. Also write the decimal equivalent.
Write the decimals as mixed fractions.
Convert the given decimals into like decimals.
a 1.54 and 15.021 b 54 and 48.021 c 1.1 and 4.65 d 18.054 and 97
Identify the greatest decimal number.
a 15.4, 15, 15.14, 15.41 b 178.1, 178.2, 1.78, 178.15
c 12.810, 12.82, 12.815, 12.825
Arrange the given decimal numbers in ascending order.
d 15.151, 15.015, 15.51, 15.5
a 7.3, 8.37, 7.23, 8.32 b 58.37, 58.73, 58.45, 58.54
c 97.08, 97.18, 97.2, 97.8 d 18.1, 18.101, 18, 18.11
Express the amount using decimals.
a In kilograms – 3 kg 68 g, 14 kg 50 g, 5 kg 5 g, 725 g, 45 g
b In kilometres – 4 km 15 m, 8 km 40 m, 3 km 4 m, 750 m, 15 m
Reena bought 5 m 15 cm of cloth from the first shop and 5.25 m of cloth from the second shop. From which shop did she buy less cloth?
The temperature of Jaipur city for 5 days was recorded. Plot these temperatures on a number line.
Day Monday Tuesday Wednesday Thursday Friday
Temperature 34°C 33°C
Farhan's father runs a tailoring business. Farhan helps his father in his free time by purchasing cloth from the market. He purchased three lengths of cloth of 1.56 m, 21.5 m and 14.5 m. Write the numbers in words, and also write the expanded form of the numbers. Do you help your parents?
In a class of 60 students, 2 5 of the students participated in a discussion. What fraction of the students did not participate in the discussion? Express your answer as a decimal.
Kunal likes swimming. He swims 1.58 km on day one, 2.53 km on day two, 2.39 km on day three and 3.13 km on day four. Arrange the distances in ascending order and find the day Kunal swam the most distance. Do you do any physical exercise?
Renu has 4 bottles of water. The quantity of water in each bottle is shown.
Water bottle
Arrange the quantities in descending order.
1 Sejal, Vidhi, Ansh and Shiva each have a tablet. The table shows the screen sizes. Vidhiʼs tablet has the biggest screen. Anshʼs tablet screen is bigger than Sejalʼs but smaller than Shivaʼs. What is Anshʼs tablet screen size?
Operations on Decimals
Xavier had to transport some luggage from one place to another. He had an almirah weighing 49.75 kg and a box to pack it of 3.5 kg.
Addition and Subtraction of Decimals
Addition of decimals
What is the total weight of the luggage that Xavier has to transport? We find the total weight of luggage by adding 49.75 kg and 3.5 kg.
Thus, Xavier has to transport 53.25 kg of luggage.
Subtraction of decimals
Xavier can carry 65 kg of weight with him. If the weight of the luggage that he is carrying is 53.25 kg, then how much more weight can he carry?
Total weight that can be carried = 65 kg; Weight carried by Xavier = 53.25 kg
Extra weight that Xavier can carry = 65 kg – 53.25 kg
Thus, Xavier can carry 11.75 kg more in his truck.
Think and Tell
If Xavier carries a parcel of 5.2 kg more, then how much more weight can he carry in his truck?
10: Add 15.984 + 34.457 + 12.96
6 3 . 4 0 1
Thus, 15.984 + 34.457 + 12.96 = 63.401
11: Subtract: 51.47 – 12.072 Thus, 51.47 – 12.072 = 39.398
Example 12: Rakhil has 25 m 12 cm of cloth. He uses 12.56 m of the cloth. What length of cloth does he have left?
The total length of the cloth with Rakhil = 25 m 12 cm
= 25 m + 12 100 m = 25.12 cm
Length of the cloth used = 12.56 m
Length of the cloth left = 25.12 m – 12.56 m = 12.56 m
Example
Example 13: A warehouse stores sugar in its rooms. It has 584 kg 80 g, 888.045 kg and 748 kg of sugar in its 3 rooms. What is the total amount of sugar stored at the warehouse?
Weight of sugar in the different rooms of the warehouse
= 584 kg 80 g, 888.045 kg, 748 kg
584 kg 80 g = 584 kg + 80 1000 kg = 584.080 kg
The total weight of the sugar stored in the warehouse
= 584.080 kg + 888.045 kg + 748 kg
= 2220.125 kg
Example 14: Vedica walks for 1.65 km. She takes an auto and travels 3.4 km. If she has to travel a total distance of 11.6 km, how much more distance does she have to cover?
Distance walked = 1.65 km
Distance travelled by auto = 3.4 km = 3.40 km
Total distance covered on foot and by auto = 1.65 km + 3.40 km = 5.05 km
Total distance to be covered = 11.6 km = 11.60 km
Distance yet to be covered = 11.60 km – 5.05 km
Thus, Vedica has to travel 6.55 km more.
Thus, the difference of the heights of Aman and Rajneesh is . Do It
The heights of Aman and Rajneesh are 1.54 m and 1.71 m, respectively. What is the difference of their heights?
Height of Aman = 1.54 m
Height of Rajneesh = 1.71 m
Difference of the heights = 1.71 m – 1.54 m
Do It Yourself 8B
Add the decimals.
Subtract the decimals.
Compare using >, < or =.
Simplify the decimals.
It rained 4 mm on Monday, 3.5 mm on Tuesday, 3.75 mm on Wednesday and 4.2 mm on Thursday. How much did it rain in total on the 4 days? 5
Pawan goes to a jewellery store. He buys a 20 g 45 mg gold chain and a ring of 12.5 g in weight. What weight of jewellery did he buy? Write your answer in grams.
The distance between Ruhi’s home and the school is 6.98 km. She moved to a new place, so her new home is 9 km away from the school. How much extra distance does she have to travel each day?
Rajul had ₹587.2 in his bank account. He deposited ₹300 and ₹950 in his bank account. After a few days, he withdrew ₹450.5 from his account. What is the total amount of money he has left in his bank account? Do you save money?
Monica spent ₹148.6 on stationery, ₹1500.5 on clothes and ₹238.6 on transport. She had a total of ₹2000 with her. How much money does she have left?
A water molecule is made up of two hydrogen atoms and one oxygen atom. We know the mass (in amu) of one hydrogen atom and one oxygen atom. What is the mass of one water molecule (in amu)?
1 You mix 32.89 grams of citric acid and 11.889 grams of boric acid in a glass container for an experiment. You place the container on a balance and find the total weight to be 123.03 grams. What could be the reason?
Multiplication and Division of Decimals
Let us learn how to multiply and divide decimal numbers.
Multiplication of Decimals
Xavier charges ₹2 for every kilogram of luggage that he carries in his truck. How much will he charge for carrying 53.75 kg of luggage?
Total weight = 53.75 kg
Cost for carrying each kg = ₹2; Total cost = ₹2 × 53.75
Step 1: Ignore the decimal point in the decimal number and multiply the whole number.
Thus, Xavier charges ₹107.50 for transporting the luggage.
Step 2: Place the decimal point in the product to obtain as many decimal places as there are in the decimal number.
Example 15: Multiply: 36.56 × 9
3 6 5 6 × 9 3 2 9 0 4
Add a decimal point after 2 decimal places from the right.
Thus, 36.56 × 9 = 329.04
Example 16: Multiply: 15.2 × 12.24
Add the decimal point after 3 (1 decimal point + 2 decimal points) places from the right.
Thus, 15.2 × 12.24 = 186.048
Example 17: Virat earns ₹250.5 for each hour he works. If he works for 6.5 hours in a day, how much money does he earn?
Amount earned each hour = ₹250.5
Total hours worked = 6.5 hours
Total amount earned = ₹250.5 × 6.5 = ₹1628.25
Total cost of 18 books = Do It Together
Manu purchases 18 books of ₹156.56 each to distribute in a childrenʼs NGO. How much money does he have to pay in total?
Cost of each book = ₹156.56 Total number of books = 18
Division of Decimals
If 5 people equally contributed to a total of ₹107.50 and gave it to Xavier, how much money did each of them give to Xavier?
Total amount given = ₹107.50 = ₹107.5
Number of people = 5
Share of each person = ₹107.5 ÷ 5
Step 2: Divide the whole number part.
Step 1: Place the decimal point directly above the decimal point in the dividend.
Step 3: Divide the tenths.
As 107.5 ÷ 5 = 21.5, each person contributed ₹21.5.
Together
Dividing a decimal number by a decimal number.
Divide 48.15 by 1.5.
Step 1: Move the decimal point in the dividend towards the right until the divisor becomes a whole number, and write the dividend and the divisor. 1 decimal point
Step 2: Divide using long division.
Thus, 48.15 ÷ 1.5 = 32.1
Example 18: Divide 577.06 by 11.
Example 19: Divide 138.72 by 1.7.
÷ 1.7 = 1387.2 ÷ 17
577.06 ÷ 11 = 52.46
Example 20: Geetanjali is making a homemade mouthwash. She needs to mix water and mouthwash concentrate. She pours 2.5 litres of water into a container. She adds 0.83 litres of mouthwash concentrate to the water. If she decides to split the final mixture equally into 3 bottles, how much mixture will each bottle contain?
Volume of water: 2.5 litres
Volume of mouthwash concentrate: 0.83 litres
Total volume = 2.5 + 0.83 litres = 3.33 litres
There are 3 bottles, so we divide the total volume by 3:
Volume per bottle = 3.33 3 = 1.11 litres
Therefore, each bottle will contain approximately 1.11 litres of the mouthwash mixture.
Cost of each packet of paint = ₹3547.5 ÷ 15 = Do It
Rahul purchases 15 packets of paint for ₹3547.5. What is the cost of each packet of paint?
Cost of 15 packets of paint = ₹3547.5 Total number of packets = 15
Multiply the decimal number with the whole number.
51.2 × 4
12.6 × 9
Multiply the decimal number with the decimal number. a 1.2 × 5.3 b 2.6 × 23.5
Divide the decimal number by the whole number.
106.4 ÷ 7
Divide the decimal number by the decimal number.
2.76 ÷ 1.2
Answer the questions.
123.5 × 14
14.24 ÷ 1.6
a Find the quotient when 111.24 is divided by 9.
b Find the quotient when 466.44 is divided by 5.2.
Answer the questions.
a What is the product of 15.3 and 81.36? b Find the product of 87.64 and 23.6.
The length of a rectangular vegetable garden is 5.8 feet and the width is 3.6 feet. What is the area of the vegetable garden?
Anuj purchases 3.6 gallons of milk. How many quarts of milk does he purchase? (1 gallon = 4 quarts)
A chef has a recipe for 10 servings that calls for 0.75 cups of a special sauce. If she wants to make 35 servings, how many cups of the sauce does she need?
Vikas wants to donate books and stationery to the children at an orphanage. He buys fifty pencils at ₹2.5 each, twenty erasers at ₹5.6 each and thirty notebooks at ₹34.55 each for the children. How much does he pay the shopkeeper? What do you do to help the needy?
Criticial Thinking & Cross Curricular
1 Question: Is Suvarna getting enough protein each day? Statements:
1 Doctors recommend that sixth graders should eat about 5 ounces of protein each day. One serving of yogurt has 0.6 ounces of protein.
2 Suvarna, a sixth grader, eats 2.5 servings of yogurt each day.
Options:
a Statement 1 alone is enough to answer the question, but statement 2 alone is not enough.
b Statement 2 alone is enough to answer the question, but statement 1 alone is not enough.
c Both statements together are enough to answer the question, but neither statement alone is enough.
d Each statement alone is enough to answer the question.
e Statements 1 and 2 together are not enough to answer the question.
Points to Remember
• Numbers written in decimal form are called decimal numbers.
• The decimal numbers which have the same number of decimal places are called like decimals, and the decimal numbers with different numbers of decimal places are called unlike decimals.
• Unlike decimals can be converted to like decimals by adding 0’s as placeholders at the extreme right side of the decimal number.
• To add and subtract the decimal numbers, convert the decimal numbers into like decimals.
• To multiply the decimal numbers, remove the decimal points from the numbers to be multiplied.
• To divide the decimal numbers, place the decimal point directly above the decimal point in the dividend.
Math Lab
Game of Equivalent Fractions
Aim: To compare fractions and decimals
Collaboration & Communication
Materials Required: Pack of cards (20 cards) and an empty box
Setting: In pairs
Method:
Write the decimal numbers and their equivalent fractions on the cards.
Mix all the cards and give 5 cards to each student. Keep the rest of the cards in the empty box. Each student has to match one decimal number with its equivalent fraction.
The first student asks the other student if he/she has a particular card.
a If the second student has that card, then he/she gives it to the first student.
b If the second student does not have that card, then the student will take a card from the box.
Now, it’s the turn of the second student to ask for the card.
The game continues until all the cards have been used, the student who has the greatest number of matches wins.
Chapter Checkup
Mark the decimal numbers on a number line.
a 2.6 b 14.9 c 95.3
Identify the decimal number represented by the shaded part and represent it on a number line.
a b c
Write the numbers in decimal form.
a 40 + 3 + 1 10 b 900 + 50 + 4 10 + 3 100
Write the expanded form of the decimal numbers. a 5.23 b 12.056 c 507.633
Convert the fractions into decimals.
a 45 1 2 b 145 8 c 12 3 5
Express the decimals as fractions in their simplest form.
a 51.26 b 6.814 c 12.508
Arrange the decimal numbers in descending order.
a 15.1, 15, 15.05, 14.95 b 1.23, 1.2, 1.331, 1.303 c 153.32, 153.23, 152.1, 153.233
Solve.
a 1.5 + 1.555
d 492.36 ÷ 6
b 0.998 – 0.712 c 1.2 × 6.7
5.9 × 47.6
14.65 × 165.7
In a long-jump competition, Rachna jumped 3.6 m, Nina jumped 3.65 m and Rohini jumped 3.55 m. Who jumped the farthest?
Proper hydration is crucial for maintaining endurance and preventing dehydration. A cyclist is participating in a long-distance event. For this event, the cyclist needs to drink 0.75 litres of water for every 10 kilometres cycled, as recommended by sports health experts. The cyclist covered 15.486 kilometres in the morning and 8.75 kilometres in the afternoon. How much water should they drink after cycling the total distance covered?
Bhoomi wants to buy a laptop. The laptop costs ₹62,542.50, and she has ₹42,765.75 saved. How much more money does she need to buy the laptop. Why is saving money important?
Himani buys a shirt for ₹1998.50 and a pair of shoes for ₹6545.75. She pays ₹10,000. How much money should she receive as change?
A store has 65.75 kg of apples. After selling 42.5 kg, they restock with 28.325 kg of apples. What is the total weight of apples now?
Sarah earns ₹400.50 per hour and works for 25.50 hours in a week. How much does she earn in a week?
A smartphone plan costs ₹45 for the first 100 text messages in a month plus an additional ₹0.10 per text message. If you send 275 text messages in a month, what is the total cost?
A toy store sells two types of toys: Type A for ₹129.90 and Type B for ₹97.50. If the store sold a total of 60 toys and the total revenue was ₹7567.20, how many of each type of toy were sold?
Read the statements and choose the correct option.
Assertion (A): The sum of 23.56 and 78.4 is 101.96.
Reason (R): When adding decimal numbers, the decimal points do not need to be aligned; you simply add the numbers as if they were whole numbers.
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
Space Explorations!
In the vast expanse of space, astronomers use a special unit of measurement called an astronomical unit (AU) to describe distances within our solar system. One astronomical unit is the average distance between the Earth and the Sun, approximately 149.6 million kilometres. This makes it easier to compare the distances of planets from the Sun without dealing with extremely large numbers.
Let us look at the average distances of the planets from the Sun in AU:
Answer the following questions:
1 What is the average distance of Mars from the Sun in astronomical units (AU)? a 1.00 AU b 1.52 AU c 5.20 AU d 9.58 AU
2 If you add the distances of Mercury and Earth from the Sun, what is the result in AU? a 1.11 AU b 1.39 AU c 1.52 AU d 2.00 AU
3 The distance of Jupiter from the Earth is AU.
4 Arrange the following planets in ascending order of their distances from the Sun: Mars (1.52 AU), Venus (0.72 AU), Neptune (30.05 AU) and Uranus (19.22 AU).
9 Data Handling
Letʼs Recall
Shreya surveyed her class to know their favourite breakfast food. She listed them in her notebook. juice, toast, fruit, juice, milk, poha, toast, toast, toast, juice, milk, poha, poha, toast, fruit, milk, poha, toast, juice, juice, milk, juice, fruit, juice, milk, juice, milk, poha, juice, juice, milk
Let us draw a table to show the data.
Shreya concluded that:
Juice is liked by the greatest number of students which is 10.
Toast is liked by 6, fruit by 3, poha by 5 and milk by 7 students.
Letʼs Warm-up
Fill in the blanks based on the table above.
1 students like to have milk for breakfast.
2 Fruit is liked the ____________ (most/least).
3 Juice is liked the (most/least).
4 The total number of students who took the survey is .
5 more students like to have poha for breakfast than fruit. I scored out of 5.
Organising and Representing Data
Rajveer is fond of collecting data on various things. We know that data is a collection of information and numerical facts.
He collects data about various countries, such as their population, their areas and the number of states in each of them. When the data is collected in its original form, it is called raw data; and each item in the raw data is called an observation.
So, Rajveer conducted a survey and listed 4 countries. He wrote the number of states in each country.
Austria has 9 states, Malaysia has 13 states, Antigua and Barbuda has 2 states and South Sudan has 10 states.
He represented the data in tabular form.
This is also called the frequency distribution table
To find out how many times a given observation appears in this data, we use a frequency distribution table. Frequency is the number of times an observation occurs.
Organising Data as Tally Marks
Rajveer decides to represent the data using tally marks as shown.
Name of Country Tally Marks
Austria
Malaysia
Antigua and Barbuda
South Sudan
In tally marks, we use ‘ ’ for one item. So, for one state, we will draw one tally mark ‘ ’. For 4 states, we will draw ‘ ’. For 5 states, we will draw and not Note: For 5 items, we always use . To show 12 items, we will draw tally marks as
Now, let us look at another example of data and understand how to represent it.
The following data shows the marks obtained by 30 students.
5, 7, 6, 7, 8, 9, 5, 6, 7, 6, 6, 8, 8, 9, 5, 6, 7, 8, 8, 7, 6, 5, 7, 5, 7, 9, 9, 9, 6, 9
Let us learn how to draw a frequency distribution table.
Now, draw the frequency distribution table, as shown.
We can also arrange the raw data in ascending order as given below.
5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9
When the raw data is arranged in ascending order or descending order, then it is called an array.
Did You Know?
Statistics is the study of data. National Statistics Day is celebrated in India on June 29 every year to honor Prasanta Chandra Mahalanobis, revered as the father of Indian statistics.
Example 1: The table shows the number of children in various families in a colony. Answer the questions.
Number of Children 0
Tally Marks
1 How many families have 2 children? 7
2 For how many families is the data collected? 6 + 8 + 7 + 4 + 3 = 28
3 How many families have more than 1 child? 7 + 4 + 3 = 14
4 How many families have fewer than 4 children? 28 – 3 = 25
The height of 30 students in a class is given. Make a frequency distribution table for the data using tally marks. (All heights are in cm.)
135, 134, 136, 135, 132, 135, 136, 136, 133, 135, 134, 132, 135, 133, 133, 136, 134, 134, 135, 132, 132, 133, 136, 132, 134, 136, 135, 136, 132, 135 Do It Together
Height of Students Tally Marks Number of Students (Frequency)
Do It Yourself 9A
State True or False.
a Raw data is the organised form of data.
b The arrangement of numbers in ascending or descending order is called an array.
c Five as tally marks is represented as
d Frequency is always the same for every observation in data.
Data Handling
Draw tally marks for the given numbers.
a 20 b 27 c 31 d 42
The Mughal emperors were a succession of rulers who controlled significant regions of the Indian subcontinent from the early 16th to the 19th century. The prominent Mughal emperors included Babur, Humayun, Akbar, Jahangir, Shah Jahan and Aurangzeb. Each of them engaged in several major wars, with their totals being 3, 3, 5, 3, 4 and 6 wars, respectively. Create a frequency distribution table using tally marks to represent this data.
Fruits are naturally sweet and provide a quick source of energy, making them a great snack to keep us active throughout the day. Prepare a tally marks table for the favourite fruit of some children.
Orange, Apple, Banana, Guava, Litchi, Litchi, Orange, Apple, Banana, Banana, Apple, Guava, Guava, Apple, Litchi, Guava, Banana, Apple, Banana, Apple, Litchi, Orange, Apple, Orange, Litchi, Banana, Banana, Guava, Apple, Litchi, Banana, Orange, Banana, Apple, Litchi, Guava, Orange, Banana, Litchi, Apple, Apple, Apple, Orange, Banana, Litchi
The ages of 30 students at a school are shown below. 10, 11, 14, 12, 13, 11, 12, 13, 14, 10, 10, 12, 13, 14, 13, 12, 10, 11, 13, 14, 10, 12, 13, 14, 12, 10, 12, 13, 14, 10
Make a frequency distribution table for the data and answer the questions.
a How many students are 12 years old?
b How many students are 14 years old?
c How many students are less than 13 years old?
d How many students are more than 11 years old but less than 14 years old?
Challenge
1 The tally marks table shows the animals in five shelter homes.
a If the scale of the tally marks is changed from one tally mark representing one animal to one tally mark representing two animals, how will the new tally marks for the ‘Pet Rescue Centre’ be represented?
b If the number of animals in ‘Animal Haven’ increases to 15, how will this change be represented if one tally mark represents five animals?
Pictographs
Critical Thinking
Name of Shelter Home Number of Animals
Happy Tails Shelter
Paws Place
Furry Friends Home
Animal Haven
Pet Rescue Centre
Do you remember Rajveer who was collecting data for different countries? He drew a pictograph for the population of different countries.
But what is a pictograph?
Representation of data using images, icons or symbols is called a pictograph.
Let us look at the pictograph for the population of different countries.
Country Population
Ireland Nepal New
Cuba Key: = 2 million and = 1 million
The size of all pictures in a pictograph should always be the same. Error Alert!
Did You Know?
Pictographs are used in road signs so that people who speak different languages can understand them better. indicates a pedestrian crossing.
Example 2: The data shows the number of houses in 5 villages. Draw a pictograph to represent the data. Let us take = 200 houses.
Village A 1000
Village B
Village C
Village D
Village E
Village A
Village B
Village C
Key: = 200 houses
Example 3: The pictograph shows the number of various types of vehicles in a traffic survey. Answer the questions.
Key: = 600 vehicles
1 How many cars are there? 2400
2 How many more rickshaws are there than bicycles? 2100 – 1800 = 300
3 What is the total number of trucks and bikes/scooters? 2700 + 3300 = 6000.
4 How many vehicles are there in total? 2400 + 2700 + 2100 + 3300 + 1800 = 12,300
Car Truck
Rickshaw
Bike/Scooter
Bicycle
The number of bats sold by a shop in a week is shown using a pictograph. Answer the questions based on the pictograph. Do It Together
Key: = 80 bats
1 How many bats were sold on Thursday?
2 On which day were the most bats sold, and how many?
3 On which day were the least number of bats sold, and how many?
4 How many bats were sold in the entire week?
Do It Yourself 9B
The data shows different people’s favourite brands of soaps. Represent the data using a pictograph.
The pictograph shows the sale of electric bulbs by a shop on different days of a week.
Number of Bulbs Sold
Key: = 300 bulbs
a How many bulbs were sold on Tuesday?
b How many more bulbs were sold on Wednesday than on Thursday?
c How many bulbs were sold on Monday and Tuesday together?
d On which day were the least bulbs sold, and how many?
e How many bulbs were sold in total in the week?
Bar Graphs
Rajveer had collected data for the number of states and the population of some countries. Then, he collected data on the total area covered by the countries. Look at the data.
(The data is estimated and is in accordance with the year 2021).
Drawing Bar Graphs
Rajveer wanted to represent the data using a bar graph. Let us learn to present data using bar graphs.
Step 1: Draw the x-axis and y-axis.
Step 2: Write the title as Total Area of Countries.
Step 3: Label both the axes as Total Area and Country.
Step 4: Write the scale. Scale is the number that shows the units used.
Step 5: Draw the rectangular bars. The height of each bar shows the area of each country.
Example 4: The data shows the area of land (in acres) used by farmers for planting various types of vegetables. Draw a bar graph for the data.
The data shows the number of electronic appliances sold by various electronic companies in a month. Draw a bar graph to represent the data.
Interpreting Bar Graphs
The bar graph shows the marks obtained by Rajveer in various subjects.
What can be understood from the bar graph? Let us see!
Rajveer scored the highest marks in Maths and Social Science. He scored 90 marks each in both the subjects.
He scored the lowest marks in Hindi. He scored 75 marks in Hindi.
He scored 80 marks in Science and 85 marks in English.
Marks Obtained by Rajveer
Example 5: The bar graph shows the number of children in families. Study the bar graph and answer the questions.
1 How many families have only 2 children? 45
2 How many families have only 3 children? 25
3 How many families have more than 1 child?
45 + 25 + 20 = 90
4 How many families have less than 3 children?
50 + 45 = 95
Number of Children in Families
The bar graph shows the snowfall at a hill station in various months of a year. Study the bar graph and answer the questions.
1 In which month did the least amount of snow fall and how much? March,
2 How much snowfall occurred in the month of January?
3 How much more snowfall was there in December than in February?
Do It Yourself 9C
In a city, the average power supply from generators, instead of being constant, was fluctuating.
a The power supply from the generators was at its highest during the time .
b The power supply from the generators was at its lowest during the time
The data shows the number of members in various families of a colony. Draw a bar graph to represent the data.
Scale: 1 division = 10 families
The bar graph shows the expenditure of a family on various items in a month. Study the bar graph and answer the questions.
a What was the expenditure on food?
b On which item did the family spend the least, and how much?
c On which item did the family spend the most, and how much?
d What was the total expenditure of the family?
The bar graph shows the data on COVID vaccine doses administered in India. Read the bar graph and answer the following questions:
a What was the total number of COVID-19 vaccine doses administered in India as of June 1st?
b By how much did the number of vaccine doses administered increase from March to April?
c Which month saw the highest increase in the number of vaccine doses administered compared to the previous month?
Challenge
Critical Thinking
The bar graph shows the population of three nearby towns in the year 1975 and the year 2015. Read the statements and choose the correct option.
Assertion (A): Arsenal had the smaller increase in population from 1975 to 2015 compared to Cottage and Britannia.
Reason (R): The graph shows that Arsenal's population bar increased by only one division between 1975 and 2015.
a Both A and R are true, and R is the correct explanation for A.
b Both A and R are true, but R is not the correct explanation for A.
c A is true, but R is false.
d A is false, but R is true.
Points to Remember
• Numerical figures collected to convey information are called data.
• To get particular information from the collected data, it can be arranged in a tabular form, using tally marks. Such a table is called a frequency distribution table.
• The representation of organised data in the form of pictures or parts thereof is called a pictograph.
• A bar graph is a representation of data by a number of rectangular bars of uniform width erected vertically or horizontally, and there are equal spaces between two consecutive bars.
• The scale for making pictographs and bar graphs is chosen according to the need.
Math Lab
Experiential Learning & Collaboration
Favourite Ice-cream Flavours
Aim: To collect, organise, and analyse data about the favourite ice-cream flavours of the students in the class.
Settings: In groups of 5
Process:
Materials Needed: Chart paper, markers, a hat or box to draw from, ice-cream scoops and sticky notes or index cards
Distribute sticky notes or index cards to each student.
Ask them to write down their favourite ice-cream flavour on a sticky note or index card.
Collect the cards in a hat or box.
One by one, draw a card from the hat and read out the flavour.
Record each flavour on the chart paper.
Create a frequency table on the chart paper.
Using the data from the frequency table, create a bar graph on the chart paper.
Chapter Checkup
The data shows how people go to their office. Make a data frequency table for the data using tally marks.
The frequency data table shows the favourite subject of students. Read the table and answer the questions.
Subject Tally Marks
a Which subject is the favourite of the most students?
b How many students like English?
c Which is the least chosen subject?
d What fraction of the total students like Maths?
The bar graph shows the number of saplings planted by a farmer in different blocks of a farm. Study the bar graph and answer the questions.
a How many saplings were planted in block C?
b How many saplings were planted in block D?
c In which block were the most saplings planted?
d How many saplings in total were planted by the farmer?
India has eight Union Territories. Here is the bar graph representing the approximate area of 5 of the Union Territories. Read the bar graph and answer the following questions:
a Which Union Territory has the largest area?
b Which Union Territory has the smallest area?
c What is the area of the National Capital Territory of India?
d What is the total area covered by all the Union Territories?
Approximate Area (in sq. km)
Saplings Planted by a Farmer
Chandigarh Lakshadweep PuducherryDelhi DadraandNagarHaveli andDamanandDiu
Area of Union Territories in India
Create a new question on the graph shown in Q5.
The pictograph shows the number of ice creams sold by an ice-cream vendor in different months of the year. Read the pictograph and answer the questions.
a How many ice creams were sold in the month of March?
b In which month were the most ice creams sold and how many?
c What is the difference in the number of ice creams sold in January and May?
d What number of ice creams were sold in total?
The bar graph shows the number of books sold by a bookstore during 5 consecutive years. Read the bar graph and answer the questions.
a How many books were sold in the year 2019?
b How many books were sold in the year 2018?
c What fraction of the total number of books were sold in the year 2020?
d What is the ratio of the number of books sold in 2017 and 2021?
Key: = 150 ice creams
The bar graph shows the average temperature of a city in C for different months of a year. Read the bar graph and answer the questions.
a Which month has the highest temperature, and how much?
b Do any two months have the same temperature? If yes, which months and how much?
c How many months are warmer than September?
d Which month is colder than July but warmer than September?
The data shows the number of bedsheets manufactured by a power loom factory in different weeks of a month. Draw a bar graph to represent the data.
The data shows the number of mangoes sold by various shopkeepers. Draw a pictograph to represent the data.
Shopkeeper Sohail Ashraf Mahmood Raju Vivek
Critical Thinking
Read the bar graph and the statements given below. Choose the correct option.
Statement 1: Dealership 4 sold more cars than Dealership 1 but fewer than Dealership 2.
Statement 2: The total number of cars sold by the 4 dealerships is 140.
a Statement 1 is true but statement 2 is false.
b Statement 1 is false but statement 2 is true.
c Both statements 1 and 2 are true.
d Both statements 1 and 2 are false.
The pictograph represents the number of students participating in different extracurricular activities at school.
Key: = 10 students
a If 60 students participate in other activities, how many symbols should be used in the pictograph to represent this?
b If the key of the pictograph changes to each symbol representing 5 students, how would this affect the representation of chess and music?
c What will be the key if there were 35 students participating in drama and you need to use the same number of symbols as used in the given pictograph? 1 2
January
February
March
School Garden Project
Ms Kapoor, a teacher at Sunshine Elementary School, started a garden project with her class. Over five months, students planted different types of flowers—roses, sunflowers, tulips, daisies and marigolds. Each month, they recorded the number of flowers planted and represented the data using a pictograph. The pictograph uses a flower symbol, where each symbol represents 10 flowers.
April
May
1 In which month were the most sunflowers planted? a January b March c May d April
2 How many marigolds were planted in February? a 30 b 50 c 60 d 40
3 In which month were the most tulips planted? How many tulips were planted that month?.
4 Compare the number of daisies planted in February and May. Which month had more, and by how many?
5 Should we unnecessarily pluck flowers? Why or why not?
Month Roses Sunflowers Tulips Daisies Marigolds
10 Mensuration
Let’s Recall
The perimeter of a shape is defined as the total length of its boundary. We can find the perimeter of a shape using a square grid by counting the number of squares along the boundary of the figure.
For example, the perimeter of the shape below can be given as:
Perimeter = 9 + 3 + 3 + 5 + 6 + 8 = 34 units
The area of a shape is the amount of space it occupies. We can determine the area of the shape above by counting the number of squares it covers within the square grid.
The shape has 57 squares inside it.
So, the area of the shape is 57 square units.
Let’s Warm-up
Match the figures with their perimeters and areas.
Understanding Perimeter
Raman, a farmer, wants to fence his field to protect the crops from animals. He goes to buy fencing for his field.
Raman: I want to fence my field.
Shopkeeper: What length of fence do you need?
Raman shows a picture of his field with its dimensions.
Perimeter of Plane Figures
To find the length of the fencing, we need to find the total length of the sides of the field.
Perimeter of the field = Total length of the sides of the field
Perimeter of the field = 5 m + 7 m + 3 m + 10 m = 25 m
Hence, Raman buys 25 m of fencing.
Example 1: Which of the figures has a perimeter of 42 cm?
Perimeter of figure 1 = 12 cm + 7 cm + 4 cm + 2 cm + 8 cm + 9 cm = 42 cm
Perimeter of figure 2 = 6 cm + 5 cm + 7 cm + 6 cm + 13 cm + 11 cm = 48 cm
Hence, figure 1 has a perimeter of 42 cm.
Find the perimeter of the figures. Do It Together
= 7 cm + 5 cm + 6 cm + 2 cm =
Perimeter of Regular Shapes
The shapes in which the lengths of all the sides are equal are called regular shapes. We can find the perimeter of different regular shapes by using some formulas.
Equilateral Triangle
Perimeter
= 16
The lengths of all 3 sides of an equilateral triangle are equal.
Perimeter = 3 × side
Perimeter = 3 × 7 = 21 cm
The lengths of all four sides of a square are equal.
Perimeter = 4 × side
Perimeter = 4 × 4 cm = 16 cm
The lengths of all 5 sides of a regular pentagon are equal.
Perimeter = 5 × side
Perimeter = 5 × 5 = 25 cm
The lengths of all 6 sides of a regular hexagon are equal.
Perimeter = 6 × side
Perimeter = 6 × 4 = 24 cm
Remember!
Perimeter of regular polygons = Number of sides in the polygon × Length of each side
Think and Tell
What will be the perimeter of a regular octagon with sides of 7 cm?
Example 2: Find the length of the side of a square with a perimeter of 40 cm.
Perimeter of the square = 40 cm
Perimeter of the square = 4 × Length of the side = 40 cm
Length of the side = 40 ÷ 4 = 10 cm
Do It Together
Example 3: Find the perimeter of a regular polygon with 15 sides where each side is 30 cm.
Length of each side = 30 cm
Number of sides = 15
Perimeter = 15 × 30 cm = 450 cm.
Find the perimeter of the given shapes with sides of length 6 cm.
1 Regular Hexagon
Number of sides = 6
Perimeter = 6 × 6 cm =
Perimeter of Rectangles
2 Equilateral Triangle
Number of sides =
Perimeter = × 6 cm =
We know that the perimeter of a shape is the sum of the lengths of all sides.
Look at this rectangle. The length of this rectangle is 6 cm and the breadth is 4 cm.
So, its perimeter = 6 cm + 4 cm + 6 cm + 4 cm
We can write this as,
Perimeter = 2 × 6 cm + 2 × 4 cm = 2 × (6 cm + 4 cm)
Hence, we can say that the perimeter of a rectangle is twice the sum of length and breadth, or,
Perimeter = 2 × (l + b), where l is the length and b is the breadth of the rectangle.
Example 4: Find the cost of framing a rectangular photo of length 20 cm and breadth 17 cm at the rate of ₹12 per cm.
Length of the frame = 20 cm; Breadth of the frame = 17 cm
Perimeter = 2 × (length + breadth) = 2 × (20 + 17) = 74 cm
Rate of framing = ₹12 per cm
Cost of framing 74 cm = 74 × ₹12 = ₹888
Example 5: The length and breadth of a rectangular park are 41.5 m and 55.5 m, respectively. Find the perimeter of 12 such parks in km.
Length of the park = 41.5 m; Breadth of the park = 55.5 m
Perimeter of the park = 2 × (41.5 + 55.5) = 2 × 97 = 194 m
Perimeter of 12 parks = 12 × 194 = 2328 m
We know that 1000 m = 1 km
2328 m = 1 1000 × 2328 = 2.328 km
The perimeters of a regular heptagon and a rectangle are equal. If the side of the heptagon is 8 m and the length of the rectangle is 15 m, find the breadth of the rectangle.
Length of the side of heptagon =
Perimeter of heptagon = 7 × = 56 m
Length of rectangle =
Perimeter of rectangle = Perimeter of heptagon
2 × (l + b) = 56 b = 56 2 – ; b = 28 – = m
Hence the breadth of the rectangle is m.
Find the perimeter of the figures.
Find the perimeter of a rectangle of length 30 cm and breadth 12 cm.
Use a piece of thread to form a square figure with a perimeter of 164 cm. What is the length of the side of the square?
An isosceles triangle has a perimeter of 57 cm. If the measure of the unequal side is 17 cm, then what will be the measure of each of the two equal sides?
A piece of string is 42 cm long. What would be the length of each side if the string is used to form a an equilateral triangle? b a regular heptagon?
A rectangular piece of glass measures 3 m 25 cm by 1 m 42 cm. What is its perimeter?
On a square grid, draw 2 rectangles with the same perimeter.
The length and breadth of a rectangular football field are 77 m and 52 m, respectively. If a player takes two and a half rounds of the field, what is the distance covered by the player?
The perimeters of two squares are 40 cm and 60 cm respectively. What will be the side of the square whose perimeter is equal to the sum of the perimeters of these two squares?
A copper wire is bent to form a shape with each side of 4 cm as shown. If the same wire is bent again to form a regular decagon, what would be the length of each side of the decagon?
What is the cost of fencing a rectangular field of 25 m by 17 m at the rate of ₹28 per metre?
A square is cut out from a bigger square. Will the perimeter increase or decrease? Justify your answer with an example.
Understanding Area
Richa is working on her new craft in which she needs to cover a rectangular sheet of paper of length 8 cm by 10 cm with 1 cm by 1 cm pieces of paper. How many pieces of paper does she need to cover the entire sheet?
Area on Graph Paper
We have discussed the outer boundary (perimeter) of a figure. The region enclosed by a closed figure is called its area. We can find the region enclosed with the help of squared paper.
Did You Know?
Bhaskaracharya was a mathematician who, in his treatise “Lilavati,” advanced the field of mensuration by detailing methods to calculate the areas of shapes like triangles and circles, thereby shaping foundational geometry in ancient India.
Count the number of squares enclosed in the figure.
On counting, we found that there are 80 squares enclosed in the figure. So, the number of squares of paper required by Richa is 80.
Number of unit squares enclosed = Area of the figure.
Hence, the area of the rectangular sheet of paper = 80 sq. cm.
Remember!
The unit of area depends on the unit of the dimensions of the figure.
But the squares do not always fit into the area we want to measure. Can we still measure the area of such figures? Yes! Let us find out how.
Look at the figure below and the rules to estimate the area of such figures.
Count area = 0 sq. units if the portion is less than half a square.
Count area = 1 sq. unit if the portion is greater than half a square.
Area of 1 fully-filled square = 1 sq. unit.
Area = 1 2 sq. unit if exactly half the square is counted.
The area of the figure can be estimated as:
1 Fully-filled squares (Pink) = 12
2 Half-filled squares (Purple) = 0
3 More than half-filled squares (Blue) = 16
4 Less than half-filled squares (Orange) = 18
Area = 12 + 16 = 28 sq. units
Example 6: Find the area of the figures in sq. cm.
1
Number of squares enclosed by the figure = 36.
Area of the figure = 36 sq. cm.
Number of squares enclosed by the figure = 35.
Area of the figure = 35 sq. cm.
Number of squares enclosed by the figure = 57.
Area of the figure = 57 sq. cm.
Example 7: By counting the squares, estimate the area of the figure in sq. units.
1 Fully-filled squares = 14
2 Half-filled squares = 2
3 More than half-filled squares = 13
4 Less than half-filled squares = 13
Area = 14 + 2 × 1 2 + 13 = 14 + 1 + 13 = 28 sq. units.
Fully-filled squares = 33
Half-filled squares =
More than half-filled squares =
Less than half-filled squares = Area =
Fully-filled squares =
Half-filled squares = 0
More than half-filled squares =
Less than half-filled squares = Area =
Fully-filled squares =
Half-filled squares =
More than half-filled squares =
Less than half-filled squares = Area =
Area of Squares and Rectangles
Do you remember Richa who was covering her rectangular sheet of length 8 cm by 10 cm with 1 cm by 1 cm pieces of paper. Let us see her data along with the measures of a few more rectangles.
Number of squares = 80
Area = 80 sq. cm = 8 cm × 10 cm = 80 sq. cm
From the above data, we found that
Let us now find the area of a square with lengths given. 8 cm
Number of squares = 35 Area = 35 sq. cm = 5 cm × 7
Number of squares = 48
Area = 48 sq. cm = 6 cm × 8 cm = 48 sq. cm
Area of a rectangle = Length × Breadth
Number of squares = 16
Area = 16 sq. cm = 4 cm × 4 cm = 16 sq. cm
We can infer from the above data that, Area of a square = Side × Side
Example 8: The area of a rectangle is 252 sq. cm. If the length of the rectangle is 18 cm, then find its breadth.
Area = 252 sq. cm
Length of rectangle = 18 cm; Breadth = ?
Area of rectangle = Length × Breadth
252 = 18 × Breadth; Breadth = 252 18 = 14 cm
Hence, the breadth of the rectangle is 14 cm.
Example 9: Find the cost of polishing a table top with dimensions 556 cm by 2.75 m at the rate of ₹62 per sq. m.
Length of the table = 556 cm = 5.56 m; Breadth of the table = 2.75 m
Area of the table = Length × Breadth = 5.56 × 2.75 = 15.29 sq. m.
Cost of polishing the table = Area of the table × Rate per sq. m.
Number of squares = 36
Area = 36 sq. cm = 6 cm × 6 cm = 36 sq. cm
Did You Know?
Rajasthan is the largest state in terms of area. Its area is 3,42,239 sq. km.
= 15.29 sq. m × ` 62 = ` 947.98 Convert the dimensions into the
A room that is 5 m wide and 6 m long is to be covered with rectangular tiles. If each rectangular tile is 0.2 m wide and 0.3 m long, find the number of tiles required to cover the floor of the room.
Width of the room = 5 m; Length of the room = 6 m
Area of the room = Length × Breadth = × =
Width of the rectangular tile = ; Length of the rectangular tile = 0.3 m
Area of one tile = Length × Breadth = × =
Number of tiles required = Area of the room Area of one tile =
Thus, the number of tiles required is
Area of Triangles
Think and Tell
Can a shape be split in more than one way?
We know how to find the area of a rectangle. What if we draw a diagonal in the rectangle?
We get two triangles. What would be the area of each triangle?
Area of rectangle = l × b
Since the diagonal divides the rectangle into 2 equal triangles, the area of each triangle is half the area of the rectangle.
Area of triangle = 1 2 × l × b = 1 2 × 8 × 10 = 40 square cm. 10 cm
8 cm
Now let us verify this using squared paper.
We can find the area by finding the number of full, half and more than half filled squares.
Area of Triangle A
Number of fully-filled squares = 32
Number of half-filled squares = 0
Number of more than half-filled squares = 8
Total area = 32 + 8 = 40 square cm
Area of Triangle B
Number of fully-filled squares = 32
Number of half-filled squares = 0
Number of more than half-filled squares = 8
Total area = 32 + 8 = 40 square cm
We know that the area of the rectangle was 80 square cm. Did you notice that the area of each triangle formed using the rectangle is half of its area?
Hence, it is verified that:
Area of triangle = 1 2 × length × breadth or 1 2 × height × length of base
Now find the area of the triangle in the given figure. What do you notice?
Area of triangle = 1 2 × 10 × 8 = 40 sq. cm
Thus, the area of triangle in this case is also 40 sq. cm. 8
Example 10: Find the area of the given triangles ABC and DEF.
Area of triangle = 1 2 × height × length of base
Height of triangle ABC = 9 units
Length of base = 6 units
Area of ABC = 1 2 × 9 × 6 = 27 square units
Height of triangle DEF = 4 units
Length of base = 9 units
Area of DEF = 1 2 × 9 × 4 = 18 square units
Example 11: The area of a triangular field is 225 square cm. If the width of the field is 30 cm, what is its height?
Area of triangle = 1 2 × height × length of base
225 = 1 2 × height × 30
Height of the field = 225 × 2 30 = 450 30 = 15 cm
What is the area of the given triangle?
Height of the triangle = cm
Length of base = cm
Area = 1 2 × × = square cm
Area and Perimeter of Combined Shapes
Find the area and perimeter of the shape.
Step 1: Split the shape into rectangles or squares and name them.
Here the shape can be split into two parts named A and B.
Step 2: Find the missing dimensions.
Step 3: Find the area of each shape.
Area of part A = l × b = 60 × 25 = 1500 sq. cm
Area of part B = l × b = 20 × 25 = 500 sq. cm
Area of combined shape = 1500 + 500 = 2000 sq. cm
Step 4: Find the perimeter of the combined shape. Perimeter = 60 + 25 + 40 + 25 + 20 + 50 = 220 cm.
Example 12: Find the perimeter of the shape.
Find the missing dimensions.
Perimeter = 12 + 12 + 15 + 3 + 10 + 6 + 7 + 3 = 68 m.
Perimeter of the shape = 68 m.
Example 13: What is the area of the shape?
We split the shape into parts. Name them and find the missing dimensions.
Area of part A = Side × Side = 4 × 4 = 16 sq. cm
Area of part B = (l × b) = (14 × 5) = 70 sq. cm
Area of the shape = 16 + 70 = 86 sq. cm
Find the perimeter and area of the figure.
We split the shape into parts. Name them and find the missing dimensions.
Area of part A = l × b = 5 × = sq. cm
Area of part B = l × b = × 6 = sq. cm
Area of the figure = + =
Perimeter of the figure =
Find the area of a square of which the side is 7 cm.
Find the length of the rectangle with area 375 sq. cm and breadth 25 cm.
Find the area of the shapes in sq. cm.
The area of a triangular frame is 1125 sq. cm. If its height is 25 cm, what is the length of its base?
Draw a square of 10 cm length and then divide it into tiny squares of 2 cm length. What is the number of tiny squares that are created?
What is the cost of tiling a square plot with sides of 15 m at the rate of ₹15 per square m? If we have to tile a rectangular plot with length and breadth 45 m and 37 m respectively at the same rate, how much would it cost?
Draw a square that has a perimeter of 12 cm. Draw a rectangle with the same perimeter having a length of 4 cm. Which shape has a greater area?
Find the perimeter and area of the figures.
Compare the two plots: One is a rectangular plot of size 140 m by 30 m and the other is a square plot of side 6400 cm at the same price. Which one would you prefer to buy and why?
much will the area of a square increase if its perimeter is doubled?
Perimeter and Area Problems
Rohan wanted to sell his square plot. He was willing to buy a rectangular plot having the same area.
Word Problems on Perimeter and Area
The area of Rohan’s square plot was 484 sq. m. Rohan purchased a rectangular plot of breadth 11 m. He thought of fencing the plot. What would be the length of fencing that Rohan would require?
Area of the square plot = Area of the rectangular plot = 484 sq. m
Breadth of the rectangular plot = 11 m
plot
Length of fencing = Perimeter of the rectangle
We know that Perimeter of a rectangle = 2 × (l + b)
We also know that Area of a rectangle = Length × Breadth = 484 = (l × 11)
l = 484 11 = 44 m
Length of fencing required = 2 × (l + b) = 2 × (44 + 11) = 2 × 55 = 110 m
Hence, the length of fencing required is 110 m.
Example 14: Manisha wants to throw a birthday party for her sister in a square garden with a perimeter of 72 m. She plans to decorate one side of the garden with balloons and cover the ground with a carpet. What is the length of the side of the garden she needs to cover with balloons? Also, find the area of the carpet needed.
Perimeter of the square garden = Total length of the four sides = 72 m
Perimeter of the square = 4 × side = 72
Length of each side = 72 4 = 18 m
So, the length of side that needs to be covered with balloons is 18 m.
Area of the square = Side × Side = 18 × 18 = 324 sq. m
Hence, the area of the carpet required is 324 sq m.
Area = ?
Perimeter = 72 m
Example 15: Mahi has a square sheet of paper with a perimeter of 48 cm. The area of Shweta’s rectangular sheet is 18 sq. cm more than the area of Mahi’s sheet. If the length of the rectangular sheet of paper is 18 cm, find its breadth.
Perimeter of the square sheet = 48 cm
Area of the rectangular sheet = 18 sq. cm more than the area of the square sheet
Length of the rectangular sheet = 18 cm
Perimeter = 48 cm
Area = Area of square sheet + 18
Length = 18 cm
To find the breadth of the rectangular sheet of paper, we first need to find its area.
Perimeter of square sheet = 4 × side = 48 cm
Length of the side of the square sheet = 48 4 = 12 cm
Area of the square sheet = side × side = 12 × 12 = 144 sq. cm
Area of the rectangular sheet = 144 + 18 = 162 sq. cm
Also, the area of the rectangular sheet = Length × Breadth = 162 = 18 × Breadth
Breadth = 162 18 = 9 cm
Hence, the breadth of the rectangular sheet of paper is 9 cm.
Raghav plans to install borders around his crop field. The cost to install a border around a rectangular crop field with a width of 8 metres is ₹5200, with the installation rate being ₹130 per metre. Determine the length of the crop field, the total perimeter of the field with the borders, and the area of the crop field.
Cost of installing a border around the rectangular crop field = ₹5200
Width = 8 m; Rate of installing the border = ₹130 per metre
Total cost of installing the border = Perimeter × Rate of installing the border per metre
Perimeter = Total cost of installing the border ÷ Rate of installing the border per metre
Perimeter = ₹5200 ÷ ₹130 = metres
Perimeter = 2 × (Length + Breadth)
Length + Breadth = Perimeter ÷ 2
Length = (Perimeter ÷ 2) – Breadth
Length = – 8 = metres
Area = Length × Breadth = × 8 = square metres
The area of the floor of a rectangular kitchen is 130 sq. m. If the length of the kitchen is 10 m, then find its perimeter.
Yash wants to install strip lights around the edges of his square-shaped ceiling and paint it white. He needs 84 metres of strip lights. Calculate the area he needs to paint.
A triangular garden has two sides measuring 15 metres and 20 metres. The perimeter of the triangle is 60 metres. What is the length of the third side of the garden?
Pooja is planning to decorate the walls of her room with stickers. How many stickers with dimensions 8 cm and 13 cm are required to fit in a rectangular region of perimeter 552 cm and length 120 cm?
Seema was building a model of a house and formed the roof by placing two square rubber sheets, each with a side length of 10.5 cm, side by side. Find the perimeter and the area of the roof of this house model.
Kashish has a square room and she wants to make a decorative pattern along the length of one of the walls. She measures the perimeter of the room and finds it to be 32 metres. Can you help her find the length of each wall so she can plan her pattern design?
A rectangular park has an area of 3150 sq. m. The length of the park is 90 m. If Rahul walks around the park covering 6 km, how many times does he go round the park?
The Great Pyramid of Giza was built as a tomb for Pharaoh Khufu of ancient Egypt’s Fourth Dynasty and is the largest of the three pyramids on the Giza Plateau. The base of the pyramid is square-shaped with a perimeter of about 920 metres. Riya walked a distance equal to the length of one side of the pyramid’s base. How far did Riya walk?
finding the area of a rectangle when its perimeter and one of the sides are given.
A community hall consists of a square room and a triangular stage that shares its base with the square room. The side length of the square room is 10 metres. The stage is an isosceles triangle with its two equal sides being half the length of the base. Find the perimeter of the entire hall.
Points to Remember
• Perimeter is the distance around the outside of a shape.
Shape
Perimeter 4 × side 2 × (l + b) 3 × side 5 × side 6 × side
• The region enclosed by a closed figure is called its area.
• Area can be measured using a square grid.
• Area of a rectangle = length × breadth; Area of a square = side × side
• Area of a triangle = 1 2 × area of rectangle = 2 × base × height
Math Lab
Setting: In pairs
Experiential Learning & Collaboration
Exploring Area and Perimeter
Materials Required: Rectangular sheet of paper, ruler, coloured pencils, pencils and scissors
Method:
In pairs, measure the length of sides of the rectangular sheet and calculate the area.
Draw different shapes on the sheet and colour them in different colours.
Cut out the shapes carefully.
Measure the lengths and find the area of each of the shapes you cut out.
Determine the area of the sheet that is left after cutting out the shapes.
Discuss your results and observations with your classmates.
Find the perimeter of the given figures.
Find the area of the figures.
Find the area of a tabletop with length 21 cm and breadth 19 cm.
How many square tiles of length 12 cm will be needed to cover a square of length 3 m?
Suhani has a rectangular sheet of paper with a perimeter of 50 cm. The length of the sheet is 10 cm. She cuts the sheet along the diagonal to get two triangles. What is the area of each triangle so formed?
A piece of thread is 72 cm long. What will be the length of each side if the thread is used to form a square?
Sunita and Ramesh bought plots as shown. Whose plot has the greater area and perimeter?
A square of 144 sq. m area can be divided into how many rectangles of size 2 m × 1 m?
Sunita’s Plot Ramesh’s Plot
Find the cost of painting a rectangular piece of cardboard at the rate of 25 paise per sq. cm. The length and breadth of the cardboard are 80 cm and 1 4 m, respectively.
The length of one side of Mihir’s square plot is 25 m. He used 1 4 of his plot for farming. How much land is vacant?
The Chand Minar in Daulatabad is a historical monument that was constructed during the 14th century under the rule of Muhammad bin Tughluq. It features a square base, with each side measuring approximately 10 metres. Calculate the area of this square base.
Sailesh runs around a square park of side 88 m 3 times a day while Renu runs around a rectangular park of side 30 m by 70 m five times a day. Who runs the longer distance every day?
Kanika wants to place a carpet in her bedroom as shown. The area to be carpeted is shaded green. How many sq. m of carpet does she need for the shaded portion?
Use two pieces of wire of length 42 cm each and bend them to form a rectangle and a square. If the breadth of the rectangle formed is 5 cm, which of the two figures has the greater area and by how much?
Draw two rectangles that have different perimeters but the same area.
Challenge
Critical Thinking
1 Read the statements and choose the correct option.
Assertion (A): 1500 bricks of 40 cm by 40 cm can be used to construct a particular wall whose length and breadth are 16 m and 15 m, respectively.
Reason (R): Area of a rectangle is given as the product of its length and breadth.
a Both A and R are true, and R is the correct explanation for A.
b Both A and R are true, but R is not the correct explanation for A.
c A is true, but R is false.
d A is false, but R is true.
2 If the areas of two rectangles are 20 square cm and 12 square cm, respectively, what will be the perimeter of a rectangle whose area is equal to the sum of the areas of these two rectangles? Assume that the new rectangle has dimensions such that its length is twice its breadth.
Case Study
Cross Curricular & Value Development
A 2BHK apartment is a type of residential unit or flat that includes 2 bedrooms (B), 1 Hall (H) and 1 kitchen (K). Neha purchased a beautiful 2BHK apartment for her parents. The master plan of the apartment is as shown below. Look at the floor plan carefully and answer the questions.
1 What is the area of the flat?
a 80 square feet
c 150 square feet
b 160 square feet
d 1500 square feet
2 If the area of the hall is 480 square feet and its length is 40 feet, what is its perimeter?
3 If the width of Bedroom 1 is 16 feet, what is the width of the kitchen?
4 What would be the cost of tiling the floor of the kitchen if its length is 10 feet and the cost of tiling is ₹120 per square feet.
(Hint: Use the measures calculated in the above questions.)
5 What gift would you like to give to your parents when you grow older?
11 Introduction to Algebra
Let’s Recall
We know how to recognise patterns and extend them. Let us recall some patterns.
What shape comes after the circle? A triangle.
What shape comes before the triangle? A circle.
Extending this pattern in both directions, we get,
These are called repeating patterns.
In this pattern, the shape L is formed using dots.
Each figure has 2 more dots than the previous figure. This is the rule of the pattern.
Extending this pattern, the next figure in the sequence will have 9 dots.
These are called growing patterns
Let’s Warm-up
Match the following.
Number pattern
1 1, 2, 3, 4…
2 3, 6, 9…
3 1, 3, 5, 7…
4 2, 4, 6, 8…
5 4, 8, 12…
Next number in the pattern
I scored out of 5.
Algebra and Patterns
In a lecture hall, seats are arranged in such a way that the first row contains 3 seats, the second row has 6 seats, the third row has 9 seats and so on. Ravi is sitting in the 8th row. He wonders how many seats there are in his row.
Patterns
We can find the answer by making a table.
The number of seats in the 8th row is 24.
Did you analyse any pattern in the table?
Number Patterns
Let us learn how to find the rule in a number pattern and extend it.
The number of seats is three times the row number.
Write the letter ‘x’ to represent the row numbers.
Then, the number of seats = 3 × x or 3x.
This is the rule for the given pattern.
By substituting the row number in the rule, we get the total number of seats in that row.
For row 1, x = 1. Number of seats = 3x = 3 × 1 = 3.
For row 2, x = 2. Number of seats = 3x = 3 × 2 = 6.
For row 8, x = 8. Number of seats = 3x = 3 × 8 = 24.
Here, x is a variable, as x takes different values.
Think and Tell
A symbol that does not have a fixed value and takes various values is called a variable. In 3x, 3 is called the coefficient of x
A coefficient is a number that is written along with a variable, or it is multiplied by the variable.
Look at this matchstick pattern.
Shape 1 Shape 2
How many matchsticks are required to form shape 4, shape 5 and shape 20? Let us find the rule.
Do It Together
Shape number
Number of sticks required
1 1 × 3 + 1 = 4
2 2 × 3 + 1 = 7
3 3 × 3 + 1 = 10
Number of sticks required = (n × 3 + 1) or (3n + 1)
In the rule (3n + 1), n is a variable, 3 is the coefficient of n and 1 is the constant term.
The quantities with fixed numerical values are called constants.
Number of sticks in shape 4 = 4 × 3 + 1 = 13
Number of sticks in shape 5 = 5 × 3 + 1 = 16
Number of sticks in shape 20 = 20 × 3 + 1 = 61
Example 1: What are the next three numbers in the sequence 2, 5, 8, 11, …
First term of the sequence = (3 × 1) − 1 = 2
Second term of the sequence = (3 × 2) − 1 = 5
nth term of the sequence = (3 × n) – 1 = (3n – 1)
Fifth term = (3 × 5) − 1 = 14
Sixth term = (3 × 6) − 1 = 17
Remember!
A list of numbers which form a pattern is called a sequence.
Seventh term = (3 × 7) − 1 = 20
Example 2: While counting chairs in the lunch room, Madan noticed that there were 3 chairs in the first row, 7 chairs in the second row, 11 chairs in the third row, and 15 chairs in the fourth row. If this pattern continues, how many chairs will there be in the fifth row?
Number of chairs in the first row = (4 × 1) − 1 = 3
Number of chairs in the second row = (4 × 2) − 1 = 7
Number of chairs in the nth row = (4 × n) – 1 = (4n – 1)
Number of chairs in the fifth row = (4 × 5) − 1 = 19
Complete the table for the pattern of shapes. Use the rule to find the number of matchsticks needed for the shape in the 15th position.
Shape 1
Shape 2
Let us write the letter ‘n’ to represent the shape numbers.
Shape number
Number of sticks required 5 15
Rule
Rules in Patterns
Rules from Geometry
Perimeter = sum of the sides
P = a + a + a + a = 4 times a = 4a
Where P is the perimeter and a is each side of a square
Chapter 11 • Introduction to Algebra
Number of sticks required =
Shape 3
The opposite sides of a rectangle are equal in length
Perimeter = sum of the lengths of sides
)
Where P is the perimeter, l is the length and b is the breadth of the rectangle
Example 3: Write the general rule to find the perimeter of an equilateral triangle.
An equilateral triangle has all three sides of equal length.
The perimeter of an equilateral triangle = sum of the lengths of its three sides
So, P = a + a + a = 3 × a = 3a, where P is the perimeter and a is the side
Thus, we get the rule as P = 3a.
Example 4: Write a general rule to find the area of a square.
Area of a square = product of the length and width
So, A = a × a = a2 , where A is the area and a is the side
Thus, we get the rule for the area of a square as A = a2
Write the formula to find the perimeter of a regular pentagon using the variable a. Perimeter of a regular pentagon = sum of the lengths of all its sides
So, P = a + a + a + a + a = × a =
Thus, we get the rule for the perimeter of a regular pentagon as .
Do It Yourself 11A
Find the general rule for each of the number patterns. Then find the 18th term.
a 2, 4, 6, 8, 10, … b 3, 6, 9, 12, 15, … c 6, 11, 16, 21, ... d 3, 8, 13, 18, 23, …
Find the rule in terms of variables for the matchstick pattern and write the number of matchsticks used in the 50th shape. Draw the next shape for the pattern.
1
What is the rule for each of the number patterns?
a 3, 5, 7, 9, … b 5, 8, 11, 14, …
Complete the table, write the rule, and answer the question.
A bakery makes 10 cupcakes an hour.
Write the rule to work out the number of cupcakes this bakery produces within a certain amount of time. How many cupcakes will it make in 1 day?
Write the general rule to find the area of a rectangle.
Write the general rule to find the perimeter of the shapes. a a regular hexagon b a parallelogram c a regular nonagon d a regular octagon
Write the general rule to find the diameter of a circle using its radius.
The image shows the fifty pence coin used in Great Britain. Identify the shape and write the general rule to find the perimeter.
The perimeter of two pieces of land is written as 4a + 5a where a is the side of each shape. How will you write the perimeter of the new shape formed when they are joined on one side?
Algebraic Expressions and Equations
The charge for a taxi service includes a flat rate of ₹50 plus an additional ₹15 per km.
Algebraic Expressions
Let us try to express the total amount charged by the taxi service using variables. Let the total distance covered be x km.
So, the required expression is 15x + 50
The meanings and definitions of some of the important words used in the expression above are given in the table:
Words used
Meaning/definition
Variable A variable is an unknown quantity that may change.
Coefficient A numerical and constant quantity placed before the variable.
Constant A specific number that is assigned a fixed value.
Term A term can be a number, a variable, the product of two or more variables or the product of a number and a variable.
Algebraic expression An expression which is made up of terms having variables, along with algebraic operations (addition, subtraction, etc.).
Forming Algebraic Expressions
We can convert a verbal statement of fact into an algebraic expression.
For example, ‘3 added to a number’ can be converted into the expression ‘3 + x’, where x is a variable.
Some more examples of algebraic expressions:
Statement
Algebraic expression
Seven more than x x + 7
Four times n 4n
Twice of a added to thrice of b 2a + 3b
Five taken away from the product of x and y xy – 5
Six subtracted from y y – 6
Did You Know?
C. P. Ramanujam was an Indian mathematician who worked in the fields of number theory and algebraic geometry.
An expression with only operators and numbers is called an arithmetic expression.
An expression with operators, variables and constants is called an algebraic expression.
Example 5: Write the algebraic expression for each of the given statements.
a Decrease y by 10 = y – 10 b 3 times p = 3p
c y less than sum of x and 4 = (x + 4) – y d 12 divided by t = 12 t
Example 6: The base of a triangle is x cm. The height of the same triangle is 6 times the base of the triangle. What is the height of the triangle?
The base of the triangle is x.
Height of the triangle = 6 times the base of the triangle = 6x
Remember!
We do not use the '×' sign between a constant and a variable in an algebraic expression.
Example 7: If the length of a rectangular hall is 4 m more than 2 times the breadth of the hall, then what will its length be if the breadth is y metres?
Given, the breadth of the rectangular hall = y metres
Since the length of a rectangular hall is 4 m more than 2 times the breadth of the hall, length = (2y + 4) m.
Example 8: Rahul is n years old. Narendra is twice as old as Rahul, and Renu is 2 years younger than Narendra. Express Renu’s age in terms of the variable n.
Rahul’s age is n years. Narendra is 2 times as old as Rahul. So, he is 2n years old.
Renu is 2 years younger than Narendra.
So, Renu is (2n – 2) years old.
Example 9: A teacher gave 4 pens to each student. Find the number of pens the teacher had, if the number of students is n and she had 5 pens left.
Number of pens given to each student = 4
Number of students = n
Total number of pens given = 4 × n = 4n
Number of pens left = 5
Total number of pens the teacher had = 4n + 5
Example 10: The number of rooms on the ground floor of a building is 12 less than twice the number of rooms on its first floor. If the first floor has x rooms, how many rooms are on the ground floor?
Number of rooms on the first floor = x
Number of rooms on the ground floor = (2 × x) – 12 = 2x – 12
So, the age of Anil’s mother = years Do It Together
The age of Anil’s mother is 2 years less than 3 times Anil's age. Express the age of Anil’s mother in terms of Anil’s age.
Let Anil’s age be x years.
Solving Algebraic Expressions
To solve an algebraic expression, substitute the variable with its given value and simplify to find the answer.
For example, if x = 5 and y = 2, find x – y.
x – y = 5 – 2 = 3.
So, x – y = 3.
Example 11: Solve the given expressions if x = 9, y = 2 and z = 4.
1 8yz
8 × 2 × 4 = 64
2 2x – 4 (2 × 9) – 4 = 18 – 4 = 14
3 z2 + 3y – 5
42 + (3 × 2) – 5 = (4 × 4) + 6 – 5 = 16 + 6 – 5 = 17
Example 12: Rahul created an expression 3s + 4(t – v). Find the value of the expression if s = 10, t = 8 and v = 6.
Algebraic expression = 3s + 4(t – v)
s = 10, t = 8 and v = 6
3(10) + 4(8 – 6) = 30 + 4 × 2 = 30 + 8 = 38
Solve the given expression if a = 6, b = 1 and c = 3. Do It
Do It Yourself 11B
Sort these into arithmetic expressions or algebraic expressions.
a 7 × 2 + 3 − 2 b 7x + 5 c 12y d 10y – 6x
Write down the terms of the algebraic expressions. 2
3
Write algebraic expressions for the statements.
a 3 added to 6m b 10 subtracted from n c 15 times x
d Twice the product of x and y e 9 multiplied by y added to 1 f Thrice of y added to the difference of x and 3
Solve the algebraic expression for the value x = 2, y = 4 and z = 9.
a x(y + 8) b (x + z)(xy – z) c xy – 9zx d 10x – 3y 3xy + 8
A year has 12 months. How many months are there in u years. Write the algebraic expression and find the value of the expression for u = 13.
After sharing x pencils with your friend, you are still left with 7 pencils. What was the total number of pencils before sharing?
When a larger box of apples is emptied, the apples from it fill three smaller boxes, and 5 apples still remain outside. If the number of apples in a small box is x, what is the number of apples in the larger box?
Out of y students in grade 6, x are performing a group dance, and 6 are performing solo. Express y in an algebraic expression.
Mala is Rima’s younger sister. If Mala is five years younger than Rima, then write Mala’s age in terms of Rima’s age.
Karan had m chocolates. He gave his younger brother 5 chocolates and divided the remaining chocolates equally between his friend and himself. How many chocolates did his friend get in terms of m?
Rahul spends ₹y daily and saves ₹z per week. What is his income for three weeks?
Ashutosh purchased tomatoes at a price of ₹29 per kg and cucumbers at a price of ₹38 per kg. He purchased 4 kg less cucumbers than tomatoes. If he purchased x kilograms of tomatoes, write an algebraic expression for his total purchase.
Fill in the algebraic expression in the boxes and solve the expression using the given value of a.
a a – 6 × 3 for a = 5
b 15 + a – 8 for a = 10
c a × a + 2 for a = 7
Write a real
Challenge
expression that can be represented by 3x + 5.
The images show the expression for the parts and whole model. Fill in the missing parts of the expressions.
Algebraic Equations
We already know how to write an algebraic expression using variables. Let us now learn to write equations using one variable and solve them.
Forming Algebraic Equations
The expression for the amount to be paid for a cab can be given as 15x + 50 Gaurav paid ₹3800; hence the above expression would change to form an equation as:
Left-hand Side (LHS)
15x + 50 = 3800
Right-hand Side (RHS)
The equation formed above is a linear equation. A linear equation is an equation in which the highest power of the variables involved is 1.
Example 13: Write each of the following as equation.
1 7 less than a number is 16 b – 7 = 16 2 3 subtracted from the product of 4 and d is 15 4d – 3 = 15
3 One-fourth of a number p increased by 9 gives 41.
1 4 p + 9 = 41 Chapter 11 • Introduction to Algebra
Example 14: Translate the statement into an equation: Twice a number increased by 12 gives 24.
Let the number be x; Twice the number = 2x
Twice the number increased by 12 = 2x + 12.
Hence, 2x + 12 = 24 is the required equation.
Remember!
An equation remains the same if its LHS and RHS are interchanged. Thus, x + 25 = 47 is the same as 47 = x + 25.
Example 15: Ramesh is twice as old as his brother Rakesh. The sum of their ages is 18 years. Express the situation as an equation.
Let Rakesh’s age = x years.
Ramesh’s age is twice that of Rakesh = 2x years
Sum of their ages = 18 years
Hence, x + 2x = 18 is the required equation
Example 16: Let T represent the total number of plums. Let P represent how many each of 6 friends gets. What equation models the situation?
Total number of plums = T
Number of plums each friend gets = P
Number of friends = 6
Then, 6 × P = T ⇒ 6P = T
Example 17: Badri and Jai like to collect stamps. Badri has x stamps, and Jai has 29 more stamps than Badri. Together, they have a total of 73 stamps. Write an equation to describe this situation.
Number of stamps with Badri = x
Number of stamps with Jai = x + 29
Total stamps = x + x + 29 = 73 ⇒ 2x + 29 = 73
Example 18: The price of a muffin is ₹y. The price of a cake is 10 times the price of a muffin and the cost of a cookie is 7 less than 2 times the price of a muffin. Express the price of a cake and a cookie in terms of the price of the muffin.
Price of a muffin = ₹y
Price of a cake = 10 × ₹y = ₹10y
Price of a cookie = ₹(2y – 7)
Express the statement ‘Adding 3 to 4 times a number gives 21’ in the form of an equation.
Let the number be z. 'Adding 3 to 4 times a number' can be written algebraically as So, the required equation is .
Solving Linear Equations in One Variable
Gaurav used another taxi service that was not charging any flat fee. The taxi charges ₹15 per km. One day, he travelled with the taxi and paid ₹105, how many km did he travel?
Let the distance travelled by Gaurav = m km
Charge per km = ₹15; Charge for m km = 15 × m = 15m
As per the given condition, 15m = 105
The value of the variable for which LHS = RHS of an equation is called the solution or root of the equation. When we solve an equation, we find the value of the unknown variable.
Let us solve linear equations using different methods.
Trial and Error Method
Remember!
There can be only one value or solution to a given linear equation in one variable.
In this method, we substitute different values for the variable and check the equality of LHS with RHS. The value of the variable that satisfies the equation is the solution or root of the equation.
In the equation, 15m = 105
LHS = 15m and RHS = 105
If m = 1, LHS = 15 × 1 = 15 ≠ RHS; If m = 3, LHS = 15 × 3 = 45 ≠ RHS
If m = 5, LHS = 15 × 5 = 75 ≠ RHS; If m = 7, LHS = 15 × 7 = 105 = RHS
Therefore, for m = 7, LHS = RHS = 105
So, m = 7 is the solution of the equation 15m = 105.
Transposition Method
Transposing means shifting a number or variable from one side of the equality symbol to the other side by reversing the operation. The sign changes as shown below: + – – + × ÷ ÷ ×
Let us solve the equation 15m = 105 by the transposition method.
To isolate the variable m on the LHS, we transpose 15 from LHS to RHS by reversing the sign.
So, m = 105 15 = 7
Error Alert!
Do not forget to change the sign while transposing the digit.
Example 19: Solve the equation 8x + 2 = 50 by the trial-and-error method.
Here, LHS = 8x + 2 and RHS = 50
If x = 1, LHS = 8 × 1 + 2 = 8 + 2 = 10 ≠ RHS; If x = 2, LHS = 8 × 2 + 2 = 16 + 2 = 18 ≠ RHS
If x = 3, LHS = 8 × 3 + 2 = 24 + 2 = 26 ≠ RHS; If x = 4, LHS = 8 × 4 + 2 = 32 + 2 = 34 ≠ RHS
If x = 5, LHS = 8 × 5 + 2 = 40 + 2 = 42 ≠ RHS; If x = 6, LHS = 8 × 6 + 2 = 48 + 2 = 50 = RHS
Therefore, for x = 6, LHS = RHS = 50
So, x = 6 is the solution of the equation 8x + 2 = 50.
Chapter 11 • Introduction to Algebra
Example 20: Solve the given equations.
1 4x 6 = 8
4x = 8 × 6 = 48
x = 48 4 = 12
So, x = 12
= 13 2
Example 21: Solve 2m − 12 = 18 using the transposition method.
2m − 12 = 18 2m = 18 + 12 2m = 30 m = 30 2 = 15
So, m = 15 is the solution of the equation.
Example 22: Sahil’s age is 3 years more than 3 times the age of his daughter. If Sahil is 30 years old, find the age of his daughter.
Let the daughter’s age be x
3 times of daughter’s age = 3x
3 years more than 3 times = 3x + 3
Sahil’s age = 3x + 3 = 30
3x + 3 = 30
3x = 30 – 3
3x = 27
x = 27 ÷ 3
x = 9
So, Sahil’s daughter is 9 years old.
Example 23: The sum of two consecutive integers is 51. Find the numbers.
Let one number be x.
Let the other number be x + 1.
x + x + 1 = 51
2x = 51 – 1 = 50
2x = 50
x = 50 ÷ 2 = 25
x + 1 = 25 + 1 = 26
So, the 2 consecutive numbers are 25 and 26.
5x + 3 = 33 5x = 5x = x = 5 = Therefore, Sonal has friends. Do It Together
Sonal buys 33 balloons for a party. After distributing them among her friends, she has 3 balloons left. Each of her friends gets 5 balloons. Write a linear equation for this problem and find out how many friends Sonal has.
Let Sonal have x friends.
Number of balloons distributed among friends = 5x.
She has 3 balloons left. But the total number of balloons is 33.
So, the equation that satisfies the condition is 5x + 3 = 33.
Write the linear equations for each of the statements.
a When we add 3 to a number, we get 12. b When 7 is taken away from a number, it leaves 2.
c When 7 is taken away from twice a number, it leaves 3. d 2 added to the product of a number and 4 gives 26.
The denominator of a fraction exceeds the numerator by 5. If 3 is added to both, the fraction becomes 3 4 . Write an equation to represent this situation.
Seventy-two people signed up for the soccer league. After the players were evenly divided into teams, there were 6 teams in the league and x people in each team. Write an equation to represent this situation.
Tina thought of a number and then tripled it. After subtracting 7 from the result, she multiplied it by 2. If the product is 52, what equation represents the number Tina thought of ?
Write a statement for each of the equations.
a 4x + 5 = 9
Solve each of the equations by the trial-and-error method.
a 3x + 5 = 8 b 2(3x + 1) = 8 c 32 – 5x = 12
d 5x – 8 = 7 e 8y = 48 f 1 3 x – 3 = 0
One of the values in the brackets satisfies the equation. Find that value.
a 6a = 48 [10, 8, 7, 42]
3x + 2 = 23 [7, 8, 25, 9]
Verify if the value given is the solution of the equation. If not, find the solution.
a 4x + 2 = 10; x = 2 b 9y – 8 = 1; y = 3
y 4 = 3 [3, 7, 12, 8]
c 5x 2 = 25; x = 10
d 2n 3 – 6 = 2; n = 12 e 5z – 6 = 0; z = 1 f 3x + 2 = 5; x = 1
Solve the equations using the transposition method.
a x + 6 = 10 b 5 = a + 2 c m – 12 = 3
d −7 = x + 4 e 4 + k – 7 = 2 f 3 – 4x = −13
When two consecutive numbers are added, the sum is 25. Find the numbers.
Glass slides in biology labs are used to examine the specimens through a microscope. The length of the slide is 50 mm more than the breadth. If the perimeter is 200 mm, find the dimensions of the slide.
Rekha is 24 years older than her daughter, Ishita. After 8 years, she will be twice as old as Ishita. Find their present ages.
Sreeja scored x marks in English and 2x marks in mathematics. Her score in science was 70. If her score in the three subjects was 220, what was her score in mathematics and English?
Meenal has some money in her purse. If she adds ₹51 to it, it becomes four times the original amount. What is the original amount in her purse?
The sum of three consecutive multiples of 8 is 192. Find the numbers. 1
Points to Remember
• In algebra, we generally use letters of the English alphabet such as a, b, c, d, …, x, y and z to represent numbers. These letters are known as variables.
• Algebraic expressions are formed by meaningful combinations of variables, constants and operators.
• The parts of an algebraic expression joined by the ‘+’ or ‘−’ sign are called terms.
• An equation contains an equality sign ‘=’. The expression on the left-hand side of ‘=’ is called the LHS, while that on the right-hand side of ‘=’ is called the RHS.
• An equation in which the highest power of each variable is 1 is known as a linear equation.
• The solution or root of an equation is the value of the variable for which LHS = RHS.
• We can find the solution of an equation by trial-and-error or the transposition method.
Math Lab
Setting: In groups of 4
Experiential Learning & Collaboration
Patterns in Algebra
Materials Required: A4 size sheet, ice-cream sticks and glue stick Method:
Form different patterns by repeating letters, such as V, H and M using ice-cream sticks.
Write the number of ice-cream sticks used to form each shape in the sequence. Now, observe the number pattern and derive a general formula for each.
Chapter Checkup
Find the general rule which shows the number of matchsticks required to make the patterns. Use a variable to write the rule. Draw the shape for the next pattern.
If zero is added to a number or a number is added to zero, the result is the number itself. Generalise this property of numbers using a variable.
Write the algebraic expression for the situations.
a 4 added to 3 times x b 2 subtracted from 2 times y
c 4 less than the quotient of x by 3 d y is divided by 5 and the quotient is added to 6
State which of these are linear equations.
a 2x + 5 = 9 b 9x + 4 > 5
Solve the algebraic expression for x = 3, y = 7 and z = 5.
c 2y + 7 = 9
a 5x + 9y b 6xy – 2yz + 10 c 9xz –1 5 yz d 11yz + x + 4y
The lengths, in cm, of the sides of a triangle are 3x – 5, 2x – 1 and x + 1.
a Write down an expression, in terms of x, for the perimeter of the triangle.
b If the perimeter of the triangle is 31 cm, find the value of x.
Write your own situation based statements for the given equations. Solve the equations using the transposition method.
a x + 30 = 50 b 6x – 3 = 9 c 24 n = 4
4(3x + 7) = 40
Ajit has some marbles. Badri has twice as many marbles as Ajit. Charu has 5 more marbles than Badri. In total they have 55 marbles. How many marbles does Charu have?
Roller skating is done on roller rinks using roller skates. The length of the rink is ʻl ’ and the width is ʻw ’. What will be the perimeter of the rectangular rink?
If the present age of Rohit is z years, then express:
a his age 8 years from now.
b his grandmother’s age, who is 9 years more than five times his age.
Sahil has x drawing sheets. Rahul has 2 less than Sahil, while Aryan has 6 more than Sahil. Find out how many sheets Rahul and Aryan have in total. Also, find the total number of sheets that the three of them have.
The height of a cuboidal box is x cm. Its breadth is two times its height, and its length is 4 cm less than its breadth. Write the length and breadth of the box in terms of its height.
A sports teacher orders 42 tennis balls. Each package contains 3 tennis balls. Which of the equations represents the number of packages?
a x + 3 = 42 b 3x = 42 c x 3 = 42 d x = 3 42
The equation P = 2.5m + 35 represents the price P (in rupees) of a bracelet, where m is the cost of the materials (in rupees). The price of a bracelet is ₹115. What is the cost of the materials?
Lakhan is three times as old as Akash. Lakhan is 7 years older than Mala. The sum of their ages is 126. What is the age of Akash?
Challenge
1 Read the statements and choose the correct option.
Assertion (A): The equation for `3 subtracted from t to equal to 5’ is written as t – 3 = 5.
Reason (R): Any equation like the above, is a condition on a variable. It is satisfied only for a definite value of the variable.
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 false, but R is true.
d A is true, but R is false.
2 Sia bought a new multi-door refrigerator for ₹90,000. The value of the refrigerator decreases by ₹3000 per year. In how many years will the price of the refrigerator be half its original price?
Case Study
Economics is a social science that studies production, distribution and consumption of goods and services. It also deals with the demand and supply of products.
The price for supplying the product (P) is equal to the sum of the delivery charge (C) and the cost of making Q pieces.
1 The delivery charge is ₹300. The cost of making each piece is ₹30. Write the expression for the price for supplying the product.
a P = ₹300Q + ₹30
c P = ₹300 − ₹30Q
b P = ₹300 + ₹30Q
d ₹300Q + ₹30
2 If the value of Q is 4, find the price for supplying products.
3 The demand of the product is written as D, and the cost is written as C. Demand is half the cost. Write the expression if the cost rises 4 times.
4 What should be the value of the quantity in Q1 for the delivery charge to be equal to the product of quantity and cost of making one piece?
Cross Curricular
12 Introduction to Ratio and Proportion
Let’s Recall
A fraction is a representation of parts of a whole. For example, if we divide a circle into five equal parts and shade three parts of it, we say that 3 5 of the circle is shaded.
Fractions can be classified as like and unlike. Like fractions have the same denominator, whereas unlike fractions have different denominators.
For example, 3 8 and 7 8 are like fractions, whereas 1 4 and 2 5 are unlike fractions.
Fractions are represented in their simplest form to make them easier to work with.
For example, the fraction 4 8 can be simplified to 1 2
This shows that half of something is the same as four-eighths of it. Such fractions are also called equivalent fractions.
Equivalent fractions are fractions that represent the same portion of a whole, even though they may look different.
Let’s Warm-up
Fill in the blanks.
1 Simran has an apple cut into 6 equal slices. She eats 3 slices of the apple. The fraction of apples eaten is .
2 8 12 in its simplest form is
3 An equivalent fraction of 2 5 with a denominator of 10 is
4 4 7 and 6 7 are a pair of fractions.
5 Rohan has eaten 4 of 16 slices of cake. The fraction of cake left is .
Understanding Ratio
Shahid is fond of cooking and is preparing dosa batter for the next day. He knows that for 3 cups of rice, he requires 2 cups of dal
Ratios and their Applications
The ratio of the quantity of rice to the quantity of dal can be shown as 3 : 2.
This is known as the ratio of the two quantities.
Introducing Ratios
A ratio is the comparison between two quantities of the same kind and the same unit of measurement.
First term or antecedent
Second term or consequent
The order of terms in a ratio is important, and the ratio changes when the order of terms changes. That is a : b ≠ b : a
A ratio has no units as it is the comparison of quantities of the same unit.
Did You Know?
The ratio of land to water for the whole earth is 1 : 2.
Example 1: Ruhi made 7 parathas and 10 cupcakes. What is the ratio of the number of parathas and cupcakes?
Number of parathas = 7; Number of cupcakes = 10
The ratio of the number of parathas to the number of cupcakes = 7 : 10
Example 2: Suhani bought 12 carrots and 16 cucumbers from the market.
1 What is the ratio of the number of carrots to the number of cucumbers?
Number of carrots = 12; Number of cucumbers = 16
The ratio of the number of carrots to the number of cucumbers = 12 : 16
2 What is the ratio of the number of cucumbers to the total number of vegetables?
Number of cucumbers = 16; Total number of vegetables = 12 + 16 = 28
Required ratio = 16 : 28
Example 3: There are 25 boys and 22 girls in a classroom. What is the ratio of the number of boys to the total number of students in the class?
Number of boys = 25;
Total number of students = 25 + 22 = 47
Ratio of boys to total students = 25:47 a b a : b =
It
Example 4: In a box, there were 255 toffees and 375 chocolates. What is the ratio of the number of chocolates to the total number of sweets?
Number of chocolates = 375;
Total number of sweets = 255 + 375 = 630
Ratio of chocolates to total sweets = 375 : 630
Example 5: Sushma has 14 red marbles, 12 green marbles and 15 black marbles. Find the ratio of the number of red marbles to the marbles that are not black.
Number of red marbles = 14;
Marbles that are not black = Number of red marbles + Number of green marbles = 14 + 12 = 26
Ratio of red marbles to marbles not black = 14 : 26
A box contains 20 balls of which 4 are red, 5 are black, 9 are white and rest are blue. Find:
1 The ratio of the number of red balls to the number of white balls.
Number of red balls = 4; Number of white balls =
The ratio of number of red balls to the number of white balls = 4 :
2 What is the ratio of the number of black balls to the total number of balls?
Number of black balls = 5; Total number of balls =
The ratio of the number of black balls to the total number of balls = 5 :
Ratio in Simplest Form
Shahid took 9 cups of rice and 6 cups of urad dal to make the batter. The ratio of rice to dal taken to make the batter is 9 : 6. We can also say that for every 3 cups of rice, Shahid needs 2 cups of urad dal. This means that 9 : 6 can be simplified as 3 : 2.
If the HCF of both the terms of a ratio is 1, then we can say the ratio is in its simplest form.
Let us learn how to write the ratio in its simplest form.
Step 1
Find the HCF of both the quantities.
HCF of 9 and 6 = 3
Step 2
Divide both the quantities by their HCF.
9 : 6 = 9 6 = 9 ÷ 3 6 ÷ 3 = 3 2 = 3 : 2
Hence, the simplest form of 9 : 6 = 3 : 2
Example 6: Express the ratio 16 : 24 in its simplest form.
HCF of 16 and 24 is 8.
16 : 24 = 16 24 = 16 ÷ 8 24 ÷ 8 = 2 3 = 2 : 3
Thus, 16 : 24 in its simplest form is 2 : 3.
Remember!
HCF stands for the highest common factor, which is the largest number that divides both the numbers.
Example 7: Find the ratio of 175 litres to 325 litres.
1 HCF of 175 and 325 is 25.
175 : 325 = 175 325 = 175 ÷ 25
325 ÷ 25 = 7 13
The ratio of 175 litres to 325 litres is 7 : 13.
2 15 min : 300 sec
300 sec = 300 60 = 5 minutes
HCF of 15 and 5 = 5
15 5 = 15 ÷ 5 5 ÷ 5 = 3 1
Thus, 15 min:300 sec in its simplest form is 3 : 1
Example 8: Write the given ratios in their simplest form.
1 14 km : 21000 m
21000 m = 21 km
HCF of 14 and 21 = 7
14 21 = 14 ÷ 7 21 ÷ 7 = 2 3
Thus, 14 km:21000 m in its simplest form is 2:3
2 18 m : 25 cm
18 m = 1800 cm
HCF of 1800 and 25 = 25
1800 25 = 1800 ÷ 25 25 ÷ 25 = 72 1
Thus, 18 m:25 cm in its simplest form is 72:1
Example 9: If the HCF of two numbers is 7, and their ratio in the simplest form is 2 : 3, what are the two numbers?
HCF of two numbers = 7
Simplest form of the ratio = 2:3
The given numbers are, 2 × 7 = 14 and 3 × 7 = 21
Example 10: A rectangular field is 10 m long and 5 m wide. Find the ratio of its length to its perimeter in its simplest form.
Length of rectangular field = 10 m; Width of field = 5 m
Perimeter of field = 2 × (l + b) = 2 × (10 + 5) = 30 m
Ratio of length to perimeter = 10:30 = 1:3
The ratio of 1 year 8 months to 5 months is Do It Together
What is the ratio of 1 year 8 months to 5 months?
1 year 8 months = months + 8 months = months
The required ratio = months: 5 months = 5 =
Equivalent Ratios
We saw that the ratio of rice to dal was 3 : 2, and Shahid took the rice and dal in the ratio 9 : 6.
We also saw that on simplifying 9 : 6, we got 3 : 2. The ratios 9 : 6 and 3 : 2 are called equivalent ratios. Equivalent ratios are those that can be simplified to the same value.
Let us find the equivalent ratio of 2 : 5 using some steps.
Step 1
Write the ratio in the form of fractions.
2 : 5 = 2 5
Hence, 2 : 5 = 4 : 10 = 6 : 15 = 8 : 20
Step 2
Multiply the numerator and denominator with the same number.
2 × 2 5 × 2 = 4 10 Or 2 × 3 5 × 3 = 6 15 Or 2 × 4 5 × 4 = 8 20
Comparison of Ratios
We can compare two ratios by using some simple steps.
Let us compare 5 : 7 and 5 : 10
Step 1
Write the ratios in the form of fractions in their simplest form.
5 : 7 = 5 7 ; 5 : 10 = 5 10 = 1 2
Step 3
Compare the like fractions. 10 14 > 7 14
Hence, 5 7 > 5 10
So, 5 : 7 > 5 : 10
Step 2
Convert the fractions into like fractions.
LCM of 7 and 2 is 14.
5 7 = 5 × 2 7 × 2 = 10 14 and 1 2 = 1 × 7 2 × 7 = 7 14
Now, let us see how we can divide a certain quantity in a given ratio.
Suhani has ₹1500. She wants to divide the amount among her siblings, Sunita and Ravi, in the ratio 2 : 3. How much will each sibling get?
Step 1
Find the total number of parts in the ratio.
2 : 3 = 2 parts + 3 parts = 5 parts
Step 2
Find the number of parts given to each.
2 out of 5 parts are given to Sunita = 2 5
3 out of 5 parts are given to Ravi = 3 5
Step 3
Find the share by multiplying the parts by the amount to be shared.
Sunita will get 2 5 of 1500 = 2 5 × 1500 = ₹600
Ravi will get 3 5 of 1500 = 3 5 × 1500 = ₹900
Example 11: Find two equivalent ratios of 2 : 7.
2 : 7 = 2 7 = 2 × 2 7 × 2 = 4 14 = 4 : 14 or 2 × 3 7 × 3 = 6 21 = 6 : 21
Two equivalent ratios of 2 : 7 are 4 : 14 and 6 : 21.
Think and Tell
How many equivalent ratios can a ratio have?
Example 12: Compare the ratios 3 : 5 and 7 : 8.
As both the fractions are in their simplest form, convert them to like fractions.
LCM of 5 and 8 = 40
3 5 = 3 × 8 5 × 8 = 24 40 and 7 8 = 7 × 5 8 × 5 = 35 40
As, 24 40 < 35 40 3 5 < 7 8 or 3 : 5 < 7 : 8
12 • Introduction to Ratio and Proportion
Example 13: Two numbers are in the ratio 2 : 5. The larger number is 15 more than the smaller number. Find the numbers.
Let the numbers be 2x and 5x.
Given that the larger number is 15 more than the smaller number.
Larger number = 15 + smaller number
5x = 15 + 2x
5x – 2x = 15
3x = 15 x = 15 3 = 5
The numbers are 2 × 5 = 10 and 5 × 5 = 25.
Example 14: In a certain recipe, the ratio of flour to sugar is 3 : 2. If you need 600 grams of sugar, how many grams of flour do you need?
Ratio of flour to sugar = 3:2
Let the quantities of flour and sugar be 3x and 2x respectively.
Given that the quantity of sugar is 600 g.
So 2x = 600 x = 600 2 = 300
Amount of flour required = 3x = 3 × 300 = 900 grams.
Example 15: The ratio of kids to adults at a party is 5 : 3. If there are 18 adults at the party, how many kids are there?
Ratio of kids to adults = 5:3
Number of adults = 3x = 18 x = 18 3 = 6
Number of kids at the party = 5x = 5 × 6 = 30
The teacher gave the children 90 chocolates, and asked Manish to divide them in the ratio of 5 : 4 between Group A and Group B. How many chocolates will each group get?
Total number of parts in the given ratio = 5 + 4 = 9
Total number of chocolates = 90
Number of parts given to group A = out of 9 parts = 9
Number of parts given to group B = out of 9 parts = 9
Number of chocolates given to group A = 9 × 90 = chocolates
Number of chocolates given to group B = 9 × 90 = chocolates
Write True or False.
a A ratio can have units, depending on the types of quantities being compared.
b In 3 : 5, the consequent is 5.
c The terms of a ratio a : b are called the antecedent and consequent, respectively.
d 10 : 2 is the same as 2 : 10.
e 10 litres of water can be compared with 10 km.
Write the given ratios in their simplest form.
a 48 : 72
d 25 L : 750 mL
Find three equivalent ratios for the ratios.
b 400 cm : 6 m
c 1.5 kg : 750g
e 9 hours : 540 min f ₹100 : 10000 paise
a 5 : 7 b 4 : 9 c 12 : 16
d 6 : 15 e 8 : 11 f 9 : 17
Compare and fill in the blanks with <, > or =.
a 5 : 3 3 : 4 b 3 : 5
d 3 : 5 5 : 6 e 3 : 7
Match the ratios with their simplest forms.
a 480 : 32
b 378 : 105
c 138 : 111
d 42 : 280
46 : 37
3 : 20
18 : 5
15 : 1
Apoorv has a basket of apples. He has 16 red apples and 24 green apples. What is the ratio of the number of red apples to the number of green apples?
At a pet care centre, there are 30 dogs, 20 cats and 10 birds. What is the ratio of the number of cats to the total number of animals? How should we treat animals?
In Sarah's garden, there are 15 roses, 10 tulips and 5 lilies. Find the ratio of:
a The number of roses to the number of tulips.
b The number of lilies to the number of roses.
c The number of tulips to the number of roses and lilies.
Madhav visited the Mysore Zoo, officially known as Sri Chamarajendra Zoological Gardens in Karnataka, India, and observed various animals. On the information board, he read that the average weight of a male tiger is about 220 kg, and that of a one-horned male rhino is about 2200 kg. Find the ratio of the weight of a tiger to that of a rhino.
Express the ratio of 24 hours to 2 days in its simplest form.
If you have 18 litres of water and 12 litres of juice, what is the simplest form of the ratio of water to juice?
A rectangular field is 80 m long and 60 m wide. Find the ratio of its length to its perimeter.
If the HCF of two numbers is 5, and their ratio in the simplest form is 3 : 4, what are the two numbers?
If the scale of a floor plan is 1:300 (1 cm on the plan equals 300 cm (or 3 metres) in reality), determine the actual dimensions of a room that appears to be 3 cm by 2 cm on the plan.
A pole that is 20 metres tall casts a shadow of 25 meters. At the same time, a tree casts a shadow that is 18 metres long. What is the height of the tree?
An artist is mixing two colours of paint to create a new shade. She wants to mix them in the ratio of 1 : 2. If she uses 3 litres of the first colour, how much of the second colour should she use to maintain the ratio?
Sneha splits her pocket money in the ratio of 2 : 3 for spending and saving, respectively. If she saves ₹6000 every month, how much money does she spend each month?
A baker's grandma uses 3 cups of milk for every 6 cups of all-purpose flour in her famous cookies. The baker only has 2 cups of milk on hand. To maintain the same ratio of milk to flour, how much flour should they use?
Challenge
Critical Thinking
Adventure tourism is booming in India. As per a recent survey, the ratio of rafting rentals to bungee rentals at a store is 10:7 in Rishikesh. If the number of bungee rentals doubles and the number of rafting rentals stays the same, then the number of bungee rentals is how many times the number of rafting rentals?
Understanding Proportion
John and Fatima go cycling for 15 minutes on weekdays and cover half a kilometre.
John: Let us cycle for 30 minutes today, as it’s Sunday!
Fatima: Of course, John! How much distance can we cover in 30 minutes if we pedal at the same speed?
Let us see how we can find the distance covered!
Proportion and their Applications
Introducing Proportion
We saw that John and Fatima usually cover half a kilometre in 15 minutes. So, in the next 15 minutes they will cover the same distance.
Distance covered in 15 minutes = half a kilometre = 500 m
Distance covered in 30 minutes = 500 m + 500 m = 1000 m
Ratio of time of cycling = 15 : 30 = 1 : 2; Ratio of distance covered = 500 : 1000 = 1 : 2
Two ratios are said to be in proportion if they are equal. The symbol used to equate the two ratios is ': :' or ‘=’. Here we can say, 15 : 30 : : 500 : 1000.
Example 16: Are 11, 33, 22 and 66 in proportion?
11 : 33 = 11 33 = 11 ÷ 11 33 ÷ 11 = 1 3 = 1 : 3
22 : 66 = 22 66 = 22 ÷ 22 66 ÷ 22 = 1 3 = 1 : 3
11 : 33 = 22 : 66. Hence, 11, 33, 22 and 66 are in proportion.
Example 17: Are the ratios 25 g to 100 g and 5 kg to 25 kg in proportion?
25 g to 100 g = 25 : 100 = 25 100 = 1 4 = 1 : 4
5 kg to 25 kg = 5 : 25 = 5 25 = 1 5 = 1 : 5
1 : 4 ≠ 1 : 5
Did You Know?
Aryabhata was an ancient Indian mathematician who worked on the concepts of ratios while studying the motions of planets. He used ratios to explain astronomical calculations.
Therefore, the ratios 25 grams to 100 grams and 5 kg to 25 kg are not in proportion.
Ratio of Ramesh’s weight to father’s weight = = = Hence, the given weights are proportion. Do It Together
Mansi’s weight is 25 kg and her mother’s is 60 kg whereas Ramesh’s weight is 30 kg and his father’s is 75 kg. Are the weights in proportion?
Mansi’s weight = 25 kg; Mother’s weight = 60 kg
Ratio of Mansi’s weight to mother’s weight = 25 : 60 = 25 60 = Ramesh’s weight = ; Father’s weight =
Proportional Terms
Four numbers, a, b, c and d are said to be in proportion if a : b = c : d. The proportional terms can be given as: Middle terms or means
= c : d
If a, b, c and d are in proportion, then,
: b : : c : d
Extreme terms or extremes
of extremes
Do It Together
Example 18: If 32 : x : : 9 : 36, find the value of x
We know that, Product of means = Product of extremes
Therefore, x × 9 = 32 × 36
9x = 32 × 36
9x = 1152
x = 1152 9 = 128
Error Alert!
The operation changes when transposing the values.
5x = 25 x = 25 × 5 = 125
5x = 25 x = 25 ÷ 5 = 5
Example 19: If 4 pencils cost ₹26, how many pencils can you buy for ₹78?
Let the number of pencils be x.
Using the method of proportion, 4 : x : : 26 : 78
x × 26 = 4 × 78
26x = 312
x = 12
Hence, 12 pencils can be bought for ₹78.
A recipe requires 3 cups of flour to make 2 dozen cookies. How many cups of flour would be required to make 16 dozen cookies?
Let the number of cups of flour required be x.
Using proportion, 3 : : : 2 :
Therefore, × 2 = 3 × = =
Hence, cups of flour are required to make 15 dozen cookies.
Continued Proportion
Any three quantities a, b and c are said to be in continued proportion if,
First term
Example 20: If x, 15 and 25 are in continued proportion, then find the value of x.
As the given quantities are in continued proportion,
x : 15 : : 15 : 25
x × 25 = 15 × 15
25x = 225
x = 9
So, the value of x is 9.
Third term
Middle terms or Means
It Together
If 7, 14 and x are in continued proportion, then find x.
As the given quantities are in continued proportion,
: 14 : : : x
So, the third term is .
Do It Yourself 12B
Which sets of numbers are in proportion?
a 4, 5, 12, 15 b 8, 5, 9, 6 c 15, 5, 12, 4 d 9, 5, 18, 10
Check and state if true or false.
a 17 : 8 : : 1 : 3
b 5 : 15 : : 3 : 9
c 21 : 4 : : 84 : 16 d 7 : 12 : : 21 : 36
Create a proportion for each set. Use only 4 numbers from each set.
a 3, 13, 9, 16, 39 b 42, 5, 7, 3, 18 c 7, 21, 2, 4, 12 d 28, 14, 8, 56, 2
Find the missing numbers in the proportions given.
a 2 : x : : 5 : 10
Find the fourth proportional to:
b 3 : 6 : : 9 : x
c x : 16 : : 12 : 24 d 1 : 3 : : x : 12
a 12, 4, 9 b 2, 6, 1 c 9, 3, 6 d 32, 8, 24
Find the third term when the given terms are in continued proportion.
a 9, 3 b 2, 4 c 16, 36 d 36, 42
The second, third and fourth terms of a proportion are 21, 45 and 63. Find the first term.
If x, 9 and 3 are in continued proportion, then find the value of x.
If the ratio of the length and width of a rectangle is 5 : 3, and the length is 15 metres, what is the width of the rectangle?
A certain protein is required in an amount proportional to a person's weight. A person weighing 55 kg requires 132 grams of protein. What is the weight of the person who requires 156 grams of protein?
Rajiv was going to his village by boat. The boat can travel 126 miles on 18 gallons of gasoline. How much gasoline will it need to travel 175 miles?
In a chemical reaction, 3 moles of hydrogen react with 1 mole of nitrogen to produce ammonia. If 9 moles of hydrogen are used, how many moles of nitrogen are required?
To win a relay race, Riya must run 300 metres before her opponent runs 280 metres. Riya runs at a pace of 75 metres every minute. Her opponent runs at a pace of 15 metres every 12 seconds. Who wins the race?
Do It Together
Unitary Method
Do you remember John and Fatima, who cycle every day? After the cycling session, both of them had some juice. They finished 2 packs of juice in 6 days. Their father needs to stock the juice packs for 30 days. How can he do it?
This can be done with the help of the unitary method.
In this method we find the value of one unit first and then the value of the required number of units.
Step 1: Write the value of the units given.
Step 2: Find the value of 1 unit.
Step 3: Find the value of the required units.
Number of juice packs required for 6 days = 2
Number of juice packs for 1 day = 2 6
Number of juice packs required for 30 days = 2 6 × 30 = 10 packs
Example 21: A train takes 10 hours to cover a distance of 650 km. What distance will it cover in 4 hours?
Distance covered in 10 hours = 650 km
Distance covered in 1 hour = 650 10
Distance covered in 4 hours = 650 10 × 4 = 260 km
Example 22: 5 roses cost ₹320 and 7 lilies cost ₹595. Which flower is expensive?
Cost of 5 roses = ₹320
Cost of 1 rose = ₹320 ÷ 5 = 64
Cost of 5 roses = ₹595
Cost of 5 roses = ₹595 ÷ 7 = 85
So, the lily is more expensive than the rose.
Kunal went to buy some paint to paint his house. The shopkeeper showed him paint A that costs ₹750 for 15 litres, and paint B that costs ₹900 for 20 litres. Which paint will be cheaper for him?
Cost of 15 litres of paint A = ₹750
Cost of 1 litre of paint A = 750 = ₹
Cost of 20 litres of paint B = ₹
Cost of 1 litre of paint B = 20 = ₹
Hence, paint ___________ will be cheaper than paint
Do It Yourself 12C
If the cost of 12 pens is ₹108, what is the cost of 20 such pens?
Raj can cycle at a speed of 2 km/h. How long will it take him to cover 8 km? 2
The cost of 125 postcards is ₹375. How many postcards can be purchased for ₹180?
Amit is employed under the Mahatma Gandhi National Rural Employment Guarantee Act (MGNREGA), which guarantees 100 days of wage employment to rural households. He works for 5 days in a month and receives a salary of ₹1350 for his work. How many months will it take him to earn ₹12,150?
The temperature increased by 25℃ in the last 100 days. If the rate of temperature increase remains the same, how many degrees will the temperature increase in the next 16 days?
A school needs to transport its students on a field trip to an old age home, to teach students about empathy and compassion. 3 school buses can carry 120 students. How many school buses are required to transport 1000 students?
Manya pays ₹18,000 as interest for 1 year. How much interest does she have to pay for 7 months, if the interest per month remains the same?
Raju buys 24 oranges for ₹336 from shop A, and Kiran buys 21 oranges for ₹273 from shop B. Who has bought oranges at a cheaper price?
Shreya can type 660 words in half an hour. How many words will she be able to type in 10 minutes with the same efficiency?
A recipe requires 3 cups of flour to make 10 cookies. If you want to make 30 cookies, how many cups of flour are needed, considering that an extra 10% of flour is required due to adjustments in the recipe?
Points to Remember
• Ratio is the comparison between two quantities of the same kind and the same units.
• Ratios are expressed in their simplest form when the HCF of the terms is 1.
• Equivalent ratios are found by multiplying or dividing both terms of the ratio by the same non-zero number.
• Two ratios are in proportion if they are equal and are represented using ': :' or '='.
• Three numbers a, b, and c are in continued proportion if a : b and b : c are in proportion.
• Finding the value of one unit and then calculating the required number of units is called the unitary method.
Setting: In groups of 4
Materials Required:
Picture Challenge
Experiential Learning & Collaboration
A set of task cards with different scenarios (e.g., "Pizza Slices," "Toy Cars"), ratio cards (e.g., 1 : 3, 2 : 5, 3 : 4, etc.), sheets of paper, coloured pencils, markers
Method:
Each team randomly selects a task card with a scenario (e.g., "Pizza Slices") and a proportion card (e.g., 1 : 3).
Teams must use the proportion card to create a simple picture that represents the scenario. For example, if they have the scenario "Pizza Slices", and the ratio 1 : 3, they need to draw one pizza divided into four equal parts.
After creating the pictures, the teams can colour them and label the parts to indicate the ratio. (e.g., one red slice and three blue slices).
Each team presents their picture, explaining the scenario, the ratio used, and how they visually represented it.
Chapter Checkup
A supermarket has 45 packets of potato chips, 66 packets of banana chips and 32 packets of corn chips. What is the ratio of:
a Potato chips to corn chips? b Banana chips to the total number of chips?
Write the ratios in their simplest form.
a 8 m : 700 cm b 360 sec : 7 min c 5 L : 750 mL
Find four equivalent ratios for the ratios given.
a 3 : 8
Compare and fill in the blanks with <, > or =.
a 8 : 11 7 : 9
Which of the ratios are not in proportion? a 14 : 20 : : 28 : 40
If a box contains 24 red stamps and 36 blue stamps, what is the ratio of red to blue stamps in its simplest form?
If 12 : x : : 6 : 36, find the value of x
If 9, 12 and x are in continued proportion, then find the value of x.
Reshma went to market to buy eggs. The cost of 4 dozen eggs is ₹288. What is the cost of 15 such eggs?
If the ratio of pencils to pens in a box is 4 : 5, and there are 36 pencils, how many pens are there?
The sum of two numbers is 30, and their ratio is 3 : 5. What are the two numbers?
Ravi was going from Delhi to Jaipur. The distance between Delhi and Jaipur is around 180 miles. He travels at a constant speed of 60 miles per hour. How long will it take for Ravi to reach Jaipur from Delhi?
A machine in a soft-drink factory fills 750 bottles in 5 hours. How many bottles will it fill in the next 3 hours?
As part of the Swachh Bharat movement, a group of volunteers is working to paint a community building to promote cleanliness and beautify the area. If 18 volunteers can paint the building in 6 days, how many volunteers are needed to complete the painting in 4 days?
The ratio of John's age to Mary's age is 2 : 3. If John is 24 years old, how old is Mary?
An example of India's technological advancements in transportation, the Vande Bharat Express, India's semi-high speed train, can cover 1,200 km in 6 hours, how much time (in minutes) will it take to travel 800 km?
The present age of a mother is 36 years, and that of her daughter is 20 years less than the age of the mother. What is the ratio of the age of the mother after 12 years to the age of the daughter 10 years ago?
Critical Thinking
1 The ratio of first and second class fares between two stations is 3:2. The number of passengers travelling first and second class are in the ratio 1: 15. If ₹26,400 is collected as the total fare, then what is the amount collected from the first class passengers?
2 Read the statements and choose the correct option.
Assertion (A): If the ratio of the mass of zinc to copper in an alloy is 7:5 and the total mass of the alloy is 120 kg, then the mass of zinc in the alloy is 70 kg.
Reason (R): The mass of zinc in the alloy can be calculated using the formula part of zinc total parts × total mass
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.
Cross Curricular & Value Development
Wildlife Study!
India's rich wildlife is a national treasure. Dedicated conservation efforts, from establishing national parks to community-led initiatives, are steadily bringing back endangered species and restoring fragile ecosystems. Suppose you are studying two renowned wildlife sanctuaries in India to analyse the ratios and proportions of various species for a school project focused on conservation efforts.
Sanctuary 1: Ranthambore National Park
Tigers: 60
Leopards: 20
Sanctuary 2: Bandhavgarh National Park
Tigers: 80
Answer the following questions:
Leopards: 15
Deer: 200
Deer: 250
1 How many tigers are there in Ranthambore National Park per 100 deer? a 30 b 20 c 15 d 25
2 Bandhavgarh National Park has how many less leopards than Ranthambore National Park? a 5 b 10 c 15 d 20
3 In Ranthambore National Park, the ratio of tigers to leopards is .
4 True or False: Bandhavgarh National Park has a higher proportion of deer to total animals compared to Ranthambore National Park.
5 What should you keep in mind while visiting wildlife sanctuaries?
13 Symmetry
Let’s Recall
When a shape can be folded so that one half of it fits exactly on the other half along the fold line, the shape is said to be symmetrical.
For example, when we fold a sheet of paper in to two halves, the left side of the paper will overlap with the right side of the paper.
The shapes on the left can be called symmetrical.
The shapes that cannot be divided in two halves are called non-symmetrical shapes.
Let’s Warm-up
Look at the shapes. Write whether they are symmetrical or non-symmetrical.
Lines of Symmetry
Priya and Rishi visited different monuments in India during their summer holiday. They also went to see the Hawa Mahal in Jaipur.
Priya: Wow! This monument is so beautiful.
Rishi: Yes, Priya! Have you noticed something about the structure?
Priya: Yes! It looks the same on the left and the right when we look at it from the centre.
They noticed the same symmetry in other monuments as well.
Symmetry in Shapes
When a shape or figure is divided into halves that are identical so that one is a mirror image of the other, it is said to be symmetrical.
The line at which a shape or figure is found to be symmetrical is the line of symmetry.
Line of symmetry
Think and Tell
Where can you draw a line of symmetry in a line segment?
The two halves of a symmetrical shape, when folded along the line of symmetry, will perfectly superimpose on each other.
A shape or figure can have one or more than one line of symmetry. A shape with no line of symmetry is called an asymmetrical shape.
Let us look at some shapes and the lines of symmetry they have.
0 lines of symmetry 1 line of symmetry
A rectangle does not have diagonal lines of symmetry because when it is folded in half diagonally, there is no superimposition of one half on the other. Error Alert!
2 lines of symmetry 4 lines of symmetry
Example 1: How many lines of symmetry do equilateral, isosceles and scalene triangles have?
Equilateral Triangle (All sides equal in length)
Isosceles Triangle (2 sides equal in length)
Scalene Triangle (All sides unequal in length)
3 lines of symmetry
1 line of symmetry
No line of symmetry
Example 2: Which of the following shapes have infinite lines of symmetry?
Parallelogram Star Circle Ellipse
Let us draw the lines of symmetry for all the above shapes.
No lines of symmetry 5 lines of symmetry Infinite lines of symmetry 2 lines of symmetry
We can draw an infinite number of lines of symmetry in a circle. Draw and write the number of line(s) of symmetry for each shape. Do It Together
Think and Tell Why do you think a parallelogram is asymmetrical?
Symmetry in Designs or Figures
Similar to shapes, we can also find lines of symmetry in different figures and objects around us. Let us look at some figures and objects along with their lines of symmetry.
Did You Know?
The Taj Mahal is a masterpiece of symmetry, with its entire structure designed to be a perfectly mirrored reflection along a central axis.
Lines of symmetry = 5 Lines of symmetry = Lines of symmetry = Lines of symmetry = 1 line of symmetry 2 lines of symmetry 9 lines of symmetry
Example 3: Draw the lines of symmetry in the letters. How many lines of symmetry does each letter have?
Let us draw the lines of symmetry for the given figures.
0 lines of symmetry 1 line of symmetry 1 line of symmetry 2 lines of symmetry
Example 4: How many lines of symmetry does the figure have?
The colours of the design should also overlap along the line of symmetry. Error Alert!
The figure has 2 lines of symmetry.
Example 5: Which of the following figures has more than two lines of symmetry?
Let us draw the line of symmetry for the given figures.
Figure 2 has more than 2 lines of symmetry. It has 5 lines of symmetry.
Example 6: Draw the horizontal line of symmetry for the word ‘BOX’.
Draw and write the lines of symmetry for the figures.
1 line of symmetry lines of symmetry lines of symmetry _______ lines of symmetry
Do
It Yourself 13A
Draw an isosceles triangle. How many lines of symmetry does it have?
The Qutub Minar, located in Delhi, India, is the tallest brick minaret in the world, standing at 73 metres. Draw the possible lines of symmetry the monument has.
Natasha and Pihu are playing chess. Natasha says that the chessboard has 4 lines of symmetry, whereas Pihu says that it has only 2 lines of symmetry. Who is right and why?
Line 1 and line 2 are drawn to show the lines of symmetry in each figure. Write which is the correct line of symmetry in each figure.
Draw the lines of symmetry for each of
Write four letters with 2 or more lines of symmetry.
List and draw any 5 symmetrical shapes/objects in your surroundings. Also, draw their lines of symmetry.
Challenge
Check if the given figure is symmetrical. If not, then which part prevents the given figure from being symmetrical? How can you make this figure symmetrical without removing it?
Reflection and Rotational Symmetry
Kritika went to the hospital with her mother for an eye checkup. She saw an ambulance at the hospital. She noticed that the spelling of ‘AMBULANCE’ was written as .‘AMBULANCE’
Kritika: Look, Mom! The spelling on the ambulance is incorrect.
Mom: No, Kritika, the spelling is correct.
Reflection and Symmetry
Mom showed Kritika the image of the ambulance in a mirror.
Kritika was amazed to see that the spelling appeared correct in the mirror.
Think and Tell
After going home, Kritika wrote some letters in her notebook and started looking at them in a mirror.
S A E A E S Mirror
Mirror Image
Did You Know?
There are some mirrors that can reflect sound too. Such mirrors are called acoustic mirrors.
The image in the mirror appears due to reflection. This mirror image is called the reflection of the object. The concept of line symmetry is very closely related to the concept of mirror reflection. A line of symmetry is also called a mirror line. An object and its image in a mirror are symmetrical along the mirror line.
The figure below is shown with its reflection along the mirror line.
On comparing the lengths of the line segments AB and A’B’, and BC and B’C’, we can conclude that:
The image of an object or figure is exactly the same in shape and size as the original figure.
The corresponding line segments of the mirror image should be an equal unit distance from the mirror line as the original image.
Example 7: Which of the figures has an incorrect reflection symmetry along the mirror line?
Example 8: Which letter looks the same in its mirror image?
Figure 2 has an incorrect reflection symmetry. N N B B T T
The letter T looks the same in its mirror image.
Example 9: Which of the figures shows the correct reflection symmetry for the figure along the mirror lines m1 and m2?
The mirror image of Figure 1 is correct along mirror line m1 but there is no image along mirror line m2, so it is incorrect.
In Figure 2, the distance of the image from the mirror line is 3 units, while the distance of the mirror image from the mirror line is 4 units. Hence, it is not the correct answer.
Figure 3 has a symmetrical mirror image along both the mirror lines; hence, it is the correct answer.
Complete the mirror image of the figures along the mirror line. Do It
Rotational Symmetry
Remember Priya and Rishi who were out visiting the monuments. Their father bought them a packet of cookies.
Rishi and Priya found that the cookies looked the same when they were rotated. 120° 120° 120°
We can say that the cookie has rotational symmetry. There is a fixed point about which the object is rotated. The fixed point is called the centre of rotation.
An angle through which a figure can be rotated to look the same is called an angle of rotational symmetry or angle of symmetry.
The number of positions in which a figure can be rotated and appear to be the same as before the rotation is called the order of symmetry. Now let us look at this flower. If you rotate the flower at the centre by 90°, the flower will look the same.
The flower has 4 angles of rotation
Order of rotation of the flower is 4.
Hence, the flower has 4 angles of rotation or a 90° angle of rotational symmetry.
Let us now see rotational symmetry in figures that show rotational symmetry at different angles.
Example 10: Find the order of rotation for the given figures.
and Tell What is the angle of symmetry of a wheel?
The order of rotation is 4.
The order of rotation is 2.
Example 11: Write the angle of symmetry for the given figure. How many angles of symmetry does the figure have?
The figure has 2 angles of rotation. The angle of symmetry is 180°.
Chapter 13 • Symmetry
Look at the given figure. How many angles of symmetry does it have? Write all the angles of symmetry.
Number of angles of symmetry = Angles of symmetry = 60°, , , , 300°, 360°
2 Art Integration
3
4 Cross Curricular
6
Do It Yourself 13B
Which pair of figures are mirror images of each other?
Copy the figure in your notebooks. Draw the reflection of the figures along the mirror line.
Guess the word:
Suhani wrote a 3-letter word in her notebook. If you look at its mirror image along the horizontal mirror line, you will see the word ‘WOW’. What is the original word?
Large mirrors are placed at sharp turns on mountain roads so drivers can see around the corner and avoid accidents. Shubh saw a vehicle in the mirror with the following written on it:
What was actually written?
Write the mirror image of the following words along vertical lines of symmetry.
Draw the reflection symmetry of the following on a square grid.
Write the order of rotational symmetry for the given figures.
The logo shown means ‘Recycle’. Write the angle of symmetry for the given logo.
Challenge
Critical Thinking
Which colour is incorrectly shaded in the mirror image? 1
Points to Remember
• Symmetry is when a shape or figure is divided into halves by a line so that the two halves match exactly when folded along the line.
• The line along which a shape or figure is found to be symmetrical is the line of symmetry.
• A figure may have 0, 1, 2 or more than 2 lines of symmetry.
• A line of symmetry is also called the mirror line.
• An object and its image in a mirror are symmetrical along the mirror line.
• The image of an object or figure is exactly the same in shape and size as the original figure.
Setting: In pairs
Materials Required:
Collaboration & Experiential Learning
Exploring Reflection and Symmetry
Sheets of paper, paint, thread, markers or pens.
Method:
With your partner, take a sheet of paper and fold it in half.
Open the sheet so it lays flat.
96 � (–4) = ?
Dip a piece of thread into paint, ensuring it is well coated.
Randomly place the painted thread on one half of the sheet and move it around to form a pattern.
Fold the sheet in half again, pressing down gently.
Open the sheet to reveal the design created by the thread, then remove the thread.
Allow the paint to dry completely. Decorate the card further if desired, and write a message inside.
The Eiffel Tower, designed by Gustave Eiffel, was constructed for the 1889 World’s Fair in Paris, celebrating the 100th anniversary of the French Revolution. Does it have lines of symmetry? Draw to show.
Draw the lines of symmetry in the letters and numbers.
3 Z D 8 S U
Write True or False.
a There is 1 one line of symmetry in the letter T. b A regular pentagon has 5 lines of symmetry.
c All quadrilaterals have 4 lines of symmetry. d A semicircle has one line of symmetry.
Rahul saw a rainbow in the sky. He told his friend Vaibhav that the rainbow has infinite lines of symmetry. Is Rahul correct? Explain your answer.
Draw and write the number of all possible lines of symmetry for the figures.
Are the lines of symmetry marked on the given figures, correct? If not, draw the correct lines of symmetry.
Complete the figures to make them symmetrical along the line of symmetry.
Draw the mirror image of the figure.
c
In a figure, the angle of symmetry is 90°. What is the other angle of symmetry of this figure? What is the smallest angle of symmetry? 11
The given traffic sign board indicates ‘uphill ahead’. Name the shape of the outline of the sign board. What is the order of symmetry and what is the angle of symmetry of the outline shape? 12 Cross Curricular
Challenge
Read the given statements and choose the correct option.
Assertion (A): A regular hexagon has six lines of symmetry.
Reason (R): A line of symmetry divides a shape into two identical shapes.
a Both A and R are true, and R is the correct explanation of A.
Thinking
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.
A girl walks in a pattern on a map, starting from her house and walking north for 3 blocks, then west for 4 blocks, south for 3 blocks and finally east for 4 blocks, returning to her house. The path she traces forms a shape. How many lines of symmetry does it have?
A Nature Walk
Mrs Sharma is the class teacher of grade 6. The students in her class are on a nature walk in the nearby park to explore the concept of symmetry. The students are excited as they gather different natural objects like leaves, flowers, butterflies, and even take pictures of trees. They observe these items carefully to identify lines of symmetry and mirror images.
1 Shubham observed a butterfly with beautiful wings. How many lines of symmetry does this butterfly have?
2 Maya collects different leaves and folds them to find lines of symmetry. Which of the following leaves shows a line of symmetry?
3 A student found a flower broken in half. He places it in front of a mirror to see what it originally looked like. Help him complete the drawing of the flower he found.
4 Choose one of the symmetrical objects you find in nature. Draw it and indicate its line of symmetry.
314 Construction of Line Segments and Angles
Let’s Recall
Line Segments
A line segment is a part of a line with two end points. Imagine a piece of string streched between two points. That is a line segment! A line segment can be measured or drawn using a ruler.
We see line segments in different shapes and figures.
Angles
An angle is a geometric figure formed by two rays or line segments that share a common end point, called the vertex, and extend in different directions. We see angles in an inclined tree, pyramids, etc.
Angles can be measured and drawn using a protractor.
Let’s Warm-up
Fill in the blank with the count of the number of line segments and angles in the shapes given below.
Line Segments = Angles = Line Segments = Angles =
Construction of Line Segments
In a lively town, curious Lily loved to explore. When her handyman aunt, Aunt Sarah, visited, Lily couldn't resist asking about her toolbox.
Lily: Aunt Sarah, what is inside your toolbox?
Aunt Sarah: Well, Lily, it is filled with tools like measuring tapes and pencils for drawing different shapes. Do you want to learn how to use them?
Lily: Yes, Aunt Sarah!
Constructing Line Segments and Their Copy
Let us first learn how to draw line segments.
Constructing Line Segments
We can draw line segments using a ruler. We can also construct a line segment using the ruler and compass. Let us see how!
Draw a line segment of length 4.3 cm.
Using a Ruler
1 Mark a point and label it as X.
2 Place the 0 mark of the ruler on point X, and draw a line up to the 4.3 cm mark.
When measuring the length of an object, view the reading directly from above.
3 Label point Y to form a line segment XY of length 4.3 cm.
Using a Ruler and a Compass
1 Draw a line m. m
3 Position the compass' pointer at the zero mark of the ruler, and then adjust the opening until the pencil point aligns precisely with the 4.3 cm mark.
XY represents the line segment of length 4.3 cm.
2 Mark a point X on the line m. m x
4 Without altering the compass' width, position the pointer at X and draw an arc that intersects line m at point Y.
Example 1: Construct a line segment of length 5.6 cm using a ruler and a compass.
1 Draw a line x
2 Mark a point M on the line x.
3 Position the compass' pointer at the zero mark of the ruler, and then adjust the opening until the pencil point aligns precisely with the 5.6 cm mark.
4 With the same compass width, position the pointer at M and draw an arc that intersects line x at point N. MN is the required line segment.
Constructing a Copy of a Line Segment
Construct a copy of the line segment MN using a ruler and a compass.
Step 1: Begin by drawing a line called l, and mark a point X anywhere on it. X l
Step 2: Position the compass with its pointer at point M of the provided line segment MN and adjust the width so that the pencil tip aligns with point N.
Step 3: Without changing the compass' width, move the pointer to point X, and then draw an arc that intersects line l at point Y. This action results in line segment XY, which is an identical copy of MN in terms of length.
Example 2: Construct a line segment of length 5.6 cm. Extend its length to twice the original length using a ruler and compass. What is the total length?
1 Follow the steps to construct a line segment PQ of length 5.6 cm.
2 Keeping the compass width unchanged, position the pointer at Q, and draw an arc that intersects line x at point R. PR is the required line segment. The total length of the line segment is 5.6 + 5.6 = 11.2 cm.
Example 3: Construct a line segment XZ equal to the sum of the lengths of PQ and QR in the quadrilateral shown below without measuring it.
1 Draw a line l and mark a point X on it.
2 Position the compass with its pointer at P of the side PQ of the quadrilateral, and adjust the width to align the pencil tip with Q.
3 Without changing the compass' width, move the pointer to point X, and then draw an arc that intersects line l at point Y.
4 Position the compass with its pointer at Q of the side QR of the quadrilateral, and adjust the width to align the pencil tip with R.
5 Without changing the compass' width, move the pointer to point Y, and then draw an arc that intersects line l at point Z. XZ is the required line segment.
Construct a line segment equal to the sum of the lengths of all the sides of the triangle shown without measuring it.
Constructions of Shapes and Figures
We already know how to construct a line segment. Now, let us learn to construct shapes and figures.
Constructing Squares and Rectangles
Let us construct a rectangle PQRS of length 5 cm and width 3 cm.
Step 1: Draw a line segment of length 5 cm. Label it PQ.
Step 2: Use the protractor to draw a perpendicular at P.
Step 3: Use a compass to measure the length of the arc as 3 cm.
Step 4: From P, cut an arc with the compass, on the perpendicular. Label the point as S.
Step 5: Repeat steps 2 to step 4 for point Q.
Step 6: Join points S and R to get the rectangle PQRS.
Construction of Rectangles using a Diagonal
A rectangle can also be constructed given the length of one side and a diagonal. We know that the lengths of the diagonals in a rectangle are equal.
Let us construct a rectangle whose length is 5 cm and the length of the diagonal is 6 cm.
Repeat steps 1 and 2 from above.
Step 3: From Q, cut an arc with the compass width of 6 cm, on the perpendicular. Label the point as S. Join QS.
Step 4: Now draw a perpendicular on point Q. From point S cut an arc of length 5 cm on the perpendicular drawn at point Q. Join SR and RQ to get rectangle PQRS.
Construction by Splitting Rectangles and Squares
Let us now see how to construct a square of length 7 cm and split it into 2 rectangles.
Step 1: Construct a square ABCD of length 6 cm using the steps above.
Step 2: Measure 4 cm on the compass and draw an arc on line AB from A and an arc from D on line DC and mark points F and E.
Step 3: Join points E and F.
Hence, two rectangles of length = 4 cm and breadth = 6 cm and length = 2 cm and breadth = 6 cm are constructed.
Construction of Figures
Now, let us try to construct a house shape as shown.
Step 1: Construct the 3 sides of the rectangle using a ruler, protractor and a compass.
Step 2: Measure 4 cm on the compass and taking D and E as the centres draw two arcs cutting each other. Mark the point of intersection as C.
Step 3: Join DC and EC. Taking C as the centre, draw an arc cutting points D and E.
Example 4: Construct a square DEFG of side 8 cm. Split the square into 2 rectangles of length 3 cm and 5 cm.
1 Construct a square DEFG of side length as 8 cm using the same steps as shown above.
2 From points D and G, make arcs on DE and GF of arc radius 3 cm.
3 Join points A and B.
4 Two rectangles: length = 3 cm and breadth = 8 cm, and length = 5 cm and breadth = 8 cm are formed.
Example 5: Construct an eye using a ruler and a compass.
1 Draw a line segment XY using a ruler.
2 Mark points Z and W above and below the midpoint of the line segment.
3 Taking Z and W as the centre, draw the arcs above and below the line segment XY, thus getting an eye.
Given below is a rectangle. Split the rectangle into 2 equal-sized rectangles.
Use a ruler to draw line segments of the given length.
Construct line segments of the lengths given using a compass and a ruler.
Construct a copy of the line segments.
Given PQ = 2.3 cm and RS = 3.6 cm, construct line segments such that:
BC = 4RS − 3PQ
XY = RS + 2PQ
Construct a square of length 5 cm using a ruler and a compass.
Construct a figure made up of a square and a circle of any measure.
Construct a rectangle measuring 12 cm in length and 4 cm in breadth. Split the rectangle into 3 equal squares.
Challenge
Critical Thinking
1 Can we construct a square with a length of one diagonal given as 6 cm? Give a reason for your answer and show the construction if the construction is possible.
Construction of Angles
Emma is visiting her grandparents in the village. She sees a farmer ploughing his fields.
Emma: Grandfather, why is the plough shaped like that?
Grandfather: Emma, the farmer uses angles to plough his field effectively. He adjusts the angle of his plough to control the depth of the furrows, allowing him to plant his crops evenly.
Constructing Angles and Their Bisectors
Angles are geometric figures formed by two rays or lines that share a common end point, known as the vertex. Angles are typically measured in degrees. We can see angles all around us in trees, intersections of roads, tilted roofs, etc.
Constructing a Perpendicular Using a Compass
Now, let us see how we can construct a perpendicular to a line from a point on it.
Constructing a perpendicular to a line from a point on it
Let XY be the given line, and P be a point on it.
1 Place the compass needle at point P and draw a semicircle of any radius that intersects line XY at points A and B.
2 Using point A as the centre and selecting a radius greater than the distance from A to point P, draw an arc.
3 Using the same radius as used in step 2, draw an arc from B, cutting the arc drawn from A. Label the point where the 2 arcs cut each other as Q.
P A X Y
4 Connect points P and Q to form the line segment PQ. PQ is perpendicular to XY.
P Q A X Y
Constructing a line perpendicular to another line from a point outside it
Let XY be the line, and P be a point outside it.
1 Place the compass needle at point P and draw an arc of any radius that intersects line XY at points A and B.
2 Using point A as the centre and a radius larger than half the length of segment AB, draw an arc.
• Construction of Line Segments and Angles
Acute Angle (less than 90°)
Right Angle (90°)
Obtuse Angle (more than 90°)
B
B
3 Using point B as the centre and the same radius as in the previous step, draw another arc that intersects the previously drawn arc at point Q.
4 Connect points P and Q to form a line segment PQ, which intersects line XY at point L. PQ is the perpendicular to line XY.
Example 6: In ΔRST, a point Z is marked on RT. Construct a perpendicular to side RT passing through point Z, using a ruler and a compass.
1 Place the compass needle at point Z and draw a semicircle of any radius that intersects RT at points A and B.
2 Using point A as the centre and selecting a radius greater than the distance from A to point Z, draw an arc.
3 Using point B as the centre and the same radius as before, draw another arc, which intersects the previously drawn arc at point Y.
4 Connect points Y and Z to form the perpendicular YZ to RT.
Construct a line segment with UV = 7.1 cm. Mark a point P, 3 cm away from the end point U. Construct the perpendicular to the side UV passing through point P using a ruler and a compass.
Drawing Angles of Given Measures
Let us draw an angle using a protractor. Draw ∠XYZ of measure 56º.
1 Draw ray YZ. Place the protractor so that Y coincides with the midpoint of the baseline, and the baseline (0-180º. line) aligns with ray YZ.
2 Start with the starting point at Z and mark point X at 56º.
Example 7: Use a protractor to draw ∠DEF measuring 294º
3 Draw ray YX. Thus,
To begin drawing 294º, we will first subtract it from 360º. Hence, 360º − 294º = 66º. Now, we will draw 66º.
Draw an ∠RST of measure 216º. S Do It Together
Constructing an Angle Bisector
Make sure the extension of the compass isn’t affected while drawing the arc. It may lead to inaccuracies in x.
An angle bisector is a geometric line or ray that divides an angle into two congruent or equal angles. In other words, it is a line or ray that passes through the vertex of an angle and divides that angle into two smaller angles of equal measure.
OT is the angle bisecor to ∠POQ
Let us construct an angle bisector to ∠AOB.
1 Draw angle AOB of any measure. Using O as the centre and a chosen radius, draw an arc that intersects lines OA and OB, marking the points of intersection as P and Q, respectively.
2 Using point P as the centre and a radius greater than half the length of PQ, Draw another arc.
Q
3 Using point Q as the centre and the same radius as used previously, draw another arc intersecting the previously drawn arc at R.
4 Extend the line segment OR to a point X. OX is the bisector of angle ∠AOB. Verify that ∠BOX = ∠AOX.
Remember!
The arcs always need to have a radius greater than half the length of the line segment to be bisected; otherwise, they won’t intersect.
Example 8: Draw a 90° angle. Construct its angle bisector.
1 Draw ∠AOB of measure 90° .
2 Using O as the centre, draw an arc intersecting lines OA and OB at points X and Y.
3 Using points X and Y as centres, draw arcs intersecting each other at point P. Join OP and extend to form a ray.
4 OP is the angle bisector of ∠AOB. Construct an angle bisector to ∠QRZ. Do It Together
Constructing 60°, 120° and 30° Angles
Let us construct an angle of 30º, 60º and 120º using a compass.
Steps to construct a 60º angle.
1 Draw the ray OA.
A
3 Now, using point B as the centre and maintaining the same radius as previously, draw another arc to intersect the initial arc at point C.
Steps to construct a 120º angle.
2 Using point O as the centre and an appropriate radius, draw an arc that intersects OA at point B.
B A
4 Join O and C, extending the line to reach point D. ∠DOA is the required 60° angle.
D
B A
To construct an angle of 120° we can first construct an angle of 60° and then extend it to construct another 60° angle on one of the arms to get the 120° angle.
1 Draw the ray OA.
A
3 With point B as the new centre and maintaining the same radius, repeat the process to intersect the arc once more at C. Continue by using point C as the centre, again with the same radius, to intersect the arc once more at point D.
Steps to construct an angle of 30º
2 Using point O as the centre and a suitable radius, draw an arc that intersects ray OA at point B.
B A
4 Finally, join OD and extend this line segment to reach point E. ∠EOA is the required 120° angle. O B A
A 30° angle is half a 60° angle. So, a 30° angle can be constructed by first constructing the 60° angle and then constructing its angle bisector to get two 30° angles.
1 Construct an angle ∠XOY = 60°.
O P X Q Y
2 Draw the bisector OZ of ∠XOY. We get 2 angles of 30°. ∠XOZ = ∠YOZ = 30°.
O P X Q 30° Y Z
Example 9: Draw ∠XOY = 41° using a protractor. Construct an angle of 60° using OX as the base.
1 Draw ∠XOY = ∠41° using a protractor as explained previously.
2 Using point O as the centre and an appropriate radius, draw an arc that intersects ray OX at point P.
3 Now, using point P as the centre and maintaining the same radius as previously, draw another arc to intersect the initial arc at point Q.
4 Join O and Q, extending the line to reach point Z. Hence, ∠ZOX =∠60° and ∠XOY = 41°.
Draw ∠PQR = ∠47° using a protractor on the line segment QR. Construct an angle of ∠30° using PQ as the base.
Draw the given angles with the help of a protractor.
a 20° b 35° c 50° d 165°
The return of light from a medium is called reflection. The angle of reflection of a mirror is 60°. Draw the angle and its bisector.
Construct the following angles using only a ruler and compass.
a 60° b 90° c 120°
Draw the angles of given measures. Construct their angle bisectors. Write the measure of the two new angles formed.
a 80° b 136°
Draw a line segment PQ of length 6.5 cm and mark a point R on the line. Construct a perpendicular to PQ passing through R using a ruler and compass.
The large letter 'H' on a 'Helipad' provides a crucial visual cue during the landing. Draw the letter 'H' using a compass.
Copy the given angles in your notebooks. Construct their angle bisectors. Write the measure of the two angles formed.
Read the statement and choose the correct option. Give a reason.
Statement 1: There can be more than one angle bisector to any given angle.
Statement 2: The angle bisector of an angle is also its line of symmetry.
a Only Statement 1 is true.
b Only Statement 2 is true.
c Both Statement 1 and Statement 2 are false.
d Both Statement 1 and Statement 2 are true.
Points to Remember
• A ruler can be used to draw line segments, and a protractor can be used to draw angles.
• A rectangle can be constructed with either the lengths of its sides or of one of its sides and a diagonal.
• A compass can be used to construct a copy of a line segment or angle without actually measuring it.
• A rectangle can be split into identical squares. Similarly, a square can be split into rectangles.
• An angle bisector is a straight line or ray that divides an angle into two equal parts, creating two smaller angles with the same measure.
Math Lab
Setting: In groups of 3–4
Experiential Learning & Communication
Geometry Construction Challenge
Materials Required: Rulers, compasses, protractors, set squares, blank sheets of paper, pencils and erasers.
Method:
1 Provide a simple architectural floor plan or sketch on the board, highlighting a room or structure that requires perpendicular walls and angle bisectors. Discuss the measurements and angles needed for this design.
2 Have students work in groups to create their own architectural blueprint. Encourage them to design a room, a building, or any structure of their choice while incorporating the following:
• At least two perpendicular walls.
• An angle bisector for an angle within their design.
• At least two measured angles using a protractor.
3 Allow the students to present their blueprints to the class. Their peers can review the designs, identify the perpendicular lines, bisectors and measured angles.
4 Have a class discussion on the challenges faced and lessons learnt during the activity.
Chapter Checkup
Use a ruler and compass to construct line segments of the given measures.
a PQ = 3.6 cm b AB = 8.2 cm
Construct a line segment EF with a length of 10.2 cm. From EF, cut off EG, which measures 3.8 cm. Determine the length of FG.
Construct a line segment DE of length 4.6 cm. Construct a perpendicular to DE passing through a point on DE using a ruler and a compass.
Given ST = 4.7 cm and UV = 6.2 cm, construct a line segment as given:
a WX = 3UV – 2ST
b AB = 3ST + 2UV
The outline of the new parliament building in India is shown. Construct a single line segment that equals the total length of any two sides in the given shape using a ruler and compass.
Construct a rectangle of length 5 cm and a diagonal of length 9 cm using a ruler and compass.
Construct the given figures using a ruler and compass.
Draw angles of the given measures using a protractor.
Draw the given angles and construct their angle bisector.
The javelin throw is a track and field event where a javelin is thrown as far as possible. The angle of release of a javelin is 30°. Construct the angle using a ruler and a compass.
Draw a line segment JK = 5.1 cm. Mark two points X and Y on the line segment so that X is 1 cm away from J, and Y is 2 cm away from K. Draw perpendiculars to JK passing through X and Y.
Draw a quadrilateral ABCD with sides of any suitable length. Construct the perpendicular bisectors of all four sides using a ruler and a compass. Do the perpendicular bisectors intersect at a common point? Explain your findings.
Challenge
Read the assertion and the reason supporting the assertion. Choose the correct option.
Assertion(A): Two lines intersecting at a point at 90° can look like alphabet ‘L’.
Reason(R): Both lines are perpendicular to each other.
a Both A and R are true, and R is not the correct explanation of A.
b Both A and R are false.
c Both A and R are true.
d Both A and R are true, and R is the correct explanation of A.
Anna drew a linear pair of angles. She constructed angle bisectors to both angles. Are the bisecting rays perpendicular to each other? Give a reason for your answer.
Case Study
Cross Curricular & Art Integration
Angle Explorer
A sundial made up of a flat plate and a rod is a horological device that tells the time of the day based on the apparent position of the sun in the sky. The length of the rod in a sundial is 45 cm. At 4 o' clock an angle of 120° is formed by the shadow with the rod. Jantar Mantar, located in Jaipur, features the word's largest stone sundial.
1 If the scale is 5 cm = 1 mm, what would be the length of the rod in the sundial when drawn on paper?
2 Construct a line segment to show the rod of the sundial using the scale as 5 cm = 1 mm.
3 If the scale is 1 unit = 2°, how many units will be equal to the measure of the angle bisector of the angle mentioned above?
4 Construct the angle mentioned above. Also draw its angle bisector.
Answers
Chapter 1
Let’s Warm-up 1. False 2. True 3. True
4. False 5. True
Do It Yourself 1A
1. a. 7; 7,00,000 b. 7; 70,000 c. 9; 90,00,000 d. 1; 1,00,00,000 e. 8; 8,00,00,000 f. 2; 20,00,00,000
2. a. 20,00,00,000 + 4,00,00,000 + 80,00,000 + 5,00,000 + 60,000 + 7000 + 800 + 30 + 5 b. 50,00,00,000 + 6,00,00,000 + 40,00,000 + 2,00,000 + 80,000 + 6000 + 600 + 4
c. 90,00,00,000 + 3,00,00,000 + 20,00,000 + 7,00,000 + 50,000 + 9000 + 100 + 90 + 1
3. a. twenty-three crore thirty-four lakh twenty-six thousand five hundred seventy-eight b. seventy-four crore thirty-six lakh fifty-four thousand six hundred seventy-nine c. forty-six crore fifty-seven lakh sixty-eight thousand nine hundred seventy-nine
4. a. 16,26,92,143 b. 65,22,44,900 c. 80,99,40,201
5. a. No b. 50,00,000; 40,000 c. 49,50,000
6. 27,25,45,619 = 20,00,00,000 + 7,00,00,000 + 20,00,000 + 5,00,000 + 40,000 + 5000 + 600 + 10 + 9 Number name: twenty-seven crore twenty-five lakh forty-five thousand six hundred nineteen
7. sixty-four crore sixty-two lakh seventy thousand
Challenge 1. Smallest 9-digit number = 11,11,13,579
Eleven crore eleven lakh thirteen thosand five hundred seventy-nine
Do It Yourself 1B
1. a. Face value = 9, place value = 9,000,000 b. Face value = 2, place value = 200,000,000 c. Face value = 8, place value = 800,000 d. Face value = 8, place value = 80,000,000
e. Face value = 0, place value = 0 f. Face value = 7, place value = 70,000 2. a. 200,000,000 + 40,000,000 + 8,000,000 + 500,000 + 60,000 + 7000 + 400 + 60 + 5 b. 700,000,000 + 80,000,000 + 7,000,000 + 100,000 + 30,000 + 5000 + 400 + 30 + 6
c. 900,000,000 + 30,000,000 + 2,000,000 + 500,000 + 40,000 + 2000 + 100 + 90 + 1
3. a. three hundred forty-five million seven hundred sixty-eight thousand, nine hundred seventy-nine b. Five hundred seventyfive million one hundred twenty-three thousand three hundred eighty-nine c. nine hundred seventy-eight million seven hundred sixty-four thousand two hundred twenty-four
4. a. 826,120,066 b. 192,504,016 c. 718,710,156
5. Number: 4,824,238 Number name - four million eight hundred twenty-four thousand two hundred thirty eight
6. Two hundred ninety nine million seven hundred ninety two thousand four hundred and fifty eight Challenge 1. The number is 953,286,714
Do It Yourself 1C
1. a. XCIX b. CLXXXIV c. CCLXXXV d. CDXXXVIII
e. MDLXIV f. MCCLXXXVII
2. a. 466 b. 96 c. 115 d. 365 e. 1864 f. 999
3. a. 100,356,782 < 500,040,367 < 887,210,460 < 931,124,820
b. 360,841,910 < 692,180,350 < 826,020,031 < 927,516,890
c. 500,216,138 < 604,503,821 < 650,241,567 < 945,241,823
4. a. 871,926,345 > 826,374,510 > 670,814,256 > 450,070,921
b. 962,115,108 > 801,210,450 > 678,203,001 > 423,516,789
c. 967,208,891 > 788,216,134 > 578,206,010 > 543,343,867
5. a. Greatest 9-digit number = 98,75,43,210, smallest 9-digit number = 10,23,45,789 b. Greatest 9-digit number = 87,65,43,210, smallest 9-digit number = 10,23,45,678
c. Greatest 9-digit number = 98,76,54,321, smallest 9-digit number = 12,34,56,789
6. a. Greatest 9-digit number = 88,76,54,210, smallest 9-digit number = 10,24,56,788 b. Greatest 9-digit number = 99,87,65,431, smallest 9-digit number = 13,45,67,899
c. Greatest 9-digit number = 99,75,43,210, smallest 9-digit number = 10,23,45,799
7. a. 99,99,99,998 b. 99,99,98,765 c. 99,99,99,876
8. Brazil, US, China and India
9. Challenge 1. Yana is older by 18 years.
Do It Yourself 1D
1. a. i. 40,540 ii. 94,270 iii. 2,54,770 iv. 20,45,540 b. i. 8,09,800 ii. 23,98,800 iii. 3,80,08,000 iv.
89,74,53,500
c. i. 4,90,000 ii. 31,41,000 iii. 1,03,09,000 iv.
30,47,58,000
2. 4250 to 4349 3. 7499; 6500
4. a. 13,500 b. 1,12,800 c. 3700 d. 55,700 e. 2,00,000 f. 6,18,80,000 g. 7 h. 4.4
5. About ₹2000 6. 30 rows 7. ₹30,000
Challenge 1. 50,020 2. 5,67,785 and 5,67,794
Do It Yourself 1E
1. a. 5,55,181 b. 56,26,226 c. 12,05,568 d. 1,00,14,961
e. 6,70,172 f. 9,73,43,133
2. a. 21,90,63,480 b. 3,89,45,160 c. 21,92,866 d. 1,09,77,634 e. 2,39,09,667 f. 8,43,58,742
3. a. 932 b. 239 c. 1245 d. 133 e. 259 f. 12
4. 48,35,858
5. a. 2 2 2 7 5 0 2 + 7 2 8 1 2 2 8 0 = 7 5 0 3 9 7 8 2 b. 3 6 6 0 7 6 8 1
6. 19,58,996 7. 1,80,87,09,200
8. Ajay travels a distance of 1,86,850 m in 5 days.
9. 16,20,73,000 km 10. 17 glasses 11. ₹17,040
12. 7 gloves 13. 100 carnations
14. Answer may vary. Sample answer.
The number of men in a town is 2,13,765 and number of women is 3,98,761. ow many more women than men are there in the town?
Challenge 1. 4,02,304; 4,01,804
Each number is first decreasing by 1000; then increasing by 500.
2. a. (6 + 5) × 3 – 1 = 32 b. 3 + 4 × (8 – 4) = 19
c. 10 × (8 – 4) + 1 + 5 = 46
Chapter Checkup
1. a. Indian system: 35,04,27,681 = thirty-five crore four lakh twenty-seven thousand six hundred eighty-one, 30,00,00,000 + 5,00,00,000 + 4,00,000 + 20,000 + 7000 + 600 + 80 + 1
International system: 350,427,681 = three hundred fifty million, four hundred twenty-seven thousand, six hundred eighty-one, 300,000,000 + 50,000,000 + 400,000 + 20,000 + 7000 + 600 + 80 + 1
b. Indian system: 42,08,79,502 = forty-two crore, eight lakh, seventy-nine thousand, five hundred two, 40,00,00,000 + 2,00,00,000 + 8,00,000 + 70,000 + 9000 + 500 + 2
International system: 420,879,502 = four hundred twenty million, eight hundred seventy-nine thousand, five hundred two, 400,000,000 + 20,000,000 + 800,000 + 70,000 + 9000 + 500 + 2
c. Indian system: 63,56,58,421 = sixty-three crore fifty-six lakh fifty-eight thousand four hundred twenty-one, 60,00,00,000 + 3,00,00,000 + 50,00,000 + 6,00,000 + 50,000 + 8000 + 400 + 20 + 1
International system: 635,658,421 = six hundred thirty-five million, six hundred fifty-eight thousand four hundred twenty-one, 600,000,000 + 30,000,000 + 5,000,000 + 600,000 + 50,000 + 8000 + 400 + 20 + 1
d. Indian system: 90,15,00,084 = ninety crore, fifteen lakh, eightyfour, 90,00,00,000 + 10,00,000 + 5,00,000 + 80 + 4
International system: 901,500,084 = nine hundred one million five hundred thousand eighty-four, 900,000,000 + 1,000,000 + 500,000 + 80 + 4
2. a. 460,722,239 b. 63,12,58,143 c. 8,09,50,062
d. 100,100,039 3. a. > b. > c. < d. =
4. a. 19,08,04,365 < 67,23,56,475 < 68,91,63,896 < 76,90,87,687
b. 324,335,678 < 435,406,576 < 676,162,895 < 676,817,980
c. 65,45,12,845 < 65,78,15,325 < 87,12,63,256 < 97,12,36,125
5. a. DCXXXV b. MMXV c. MMLXXXVII d. MMDCCL
6. a. 317 b. 1409 c. 797 d. 218 7. a. Ten: 7,65,380; 65,410; 57,67,880; b. Hundred: 2,34,600; 43,300; 1,23,35,000; c. Thousand: 9,03,000; 83,000; 47,70,000; 8. a. 73 b. 5760
c. 4440 d. 53 9. a. 28,11,555 b. 55,03,319
c. 24,63,05,750 d. 2,27,093 e. 57,724 f. 77,587
10. 444,777,888; International number system: four hundred fortyfour million, seven hundred seventy-seven thousand, eight hundred eighty-eight; Indian number system: 44,47,77,888 = forty-four crore forty-seven lakh seventy-seven thousand eight hundred eighty-eight 11. Indian number system: 75,06,35,389 = seventy-five crore six lakh thirty-five thousand three hundred eighty-nine International number system: 750,635,389 = seven hundred fifty million, six hundred thirty-five thousand, three hundred eighty-nine 12. 9,24,000 books 13. 3,29,000 14. Answers may vary. Sample answers. (30 + 3) × 3 – 6 ÷ 3 + 3 = 100 (20 + 5) × 3 – 25 ÷ 5 + 5 × 6 = 100
15. Greatest 9-digit number: 98,65,43,210, smallest 9-digit number: 10,23,45,689; Greatest 9-digit number (except 0): 99,86,54,321, smallest 9-digit number (except 0): 11,23,45,689 Ascending order: 10,23,45,689 < 11,23,45,689 < 98,65,43,210 < 99,86,54,321
Descending order: 99,86,54,321 > 98,65,43,210 > 11,23,45,689 > 10,23,45,689
16. a. 5,84,850 tons b. 3,89,900 tons c. 9,74,750 tons
Challenge 1. Option c 2. 1,00,000 times
Case Study
1. 84,38,000
2. a. True b. False 3. 65,67,686 tonnes
4. 27,98,950 tonnes.
5. Four million, one hundred sixty-five thousand
Chapter 2
Let’s Warm-up
1. By 3
2. By 4
By 5
By 6
Do It Yourself 2A
1. 13,215; 13,216; 13,217; 13,218; 13,219
2. a. Predecessor = 66,777 Successor = 66,779
b. Predecessor = 54,556 Successor = 54,558
c. Predecessor = 86,456 Successor = 86,458
d. Predecessor = 89,664 Successor = 89,666
e. Predecessor = 97,563 Successor = 97,565
3. 1483, 1485
4. a. 0 b. 6567 c. 10 d. 50 e. 10, 14 f. 9875 g. 0
h. 224, 448 i. 999, 1000 j. 257 5. a. 30, 50, 70, 90;
b. 100, 150, 200, 225, 250 6. a. 20 + 10 +
b. 18 – 3 – 3 – 3 – 3 – 3 = 3
7. a. 22 b. 5 c. 12 d. 6 e. 10 f. 4 8. a. 1
b. y can be any non-zero whole number c. 1 d. 1
9. 153 million square km
10. a.
+
= 60
11. 8 cookies 12. ₹100 13. 0; 7
Challenge 1. Successor: 10,00,000; Predecessor: 9,99,998
2. Answers may vary. Sample answer. 6
Do It Yourself 2B
1.
2. a. 21 b. 1944 c. 64
3.
5. The rule is to add 3 more dots in the next shape in the sequence. The fourth shape in the pattern will have 15 dots.
6. 49 dots
7 . Answer may vary. Sample answer.
4. 1111 × 1111 = 1234321 11111 × 11111 = 123454321
5. 1 × 11 + 1 = 12 1 × 12 + 1 = 13 1 × 13 + 1 = 14 2 × 8 + 1 = 17 2 × 10 + 1 = 21 2 × 12 + 1 = 25 12 × 11 + 2 = 134 12 × 12 + 2 = 146 12 × 13 + 2 = 158 123 × 11 + 3 = 1356 123 × 12 + 3 = 1479 123 × 13 + 3 = 1602
6. ₹9,99,99,999
7. The first shape has 6 rectangles; the next shape has 11 rectangles and the next shape has 16 rectangles.
So, the rule followed by the pattern is to add 5 rectangles every time. The third shape has 16 rectangles.
The fourth shape will have, 16 + 5 = 21 rectangles.
The fifth shape will have, 21 + 5 = 26 rectangles.
So, the fifth shape will have 26 rectangles.
8. a. 405 b. 4455 c. 44955 d. 449955
Pattern: 405, 4455, 44955, 449955, 4499955, 44999955…
9. 13, 21 10. Answer may vary. Sample answer
8. 60 9. 2244 10. ₹50,360 11. 590; Multiplication by 9 12. ₹7875 13. ₹22,900 14. a. 16,500 b. 49,305 c. 5460
d. 6300 15. 4352 m, Distributive property Challenge 1. d
2.
Case Study
1. Triangles and circles
2. Option c
3. The rule is a man and a woman are repeated alternatively.
4. The next term will have 1 man and 1 woman.
5.
Challenge 1. 64 pencils
Chapter Checkup
1. a. 1,54,697; 1,54,699 b. 2,54,894; 2,54,896
c. 5,45,976; 5,45,978 d. 6,45,711; 6,45,713
2. a. 11111 × 1 – 2 = 11109; 111111 × 1 – 2 = 111109
b. 11111 × 2 – 2 = 22220; 111111 × 2 – 2 = 222220
c. 11111 × 3 – 2 = 33331; 111111 × 3 – 2 = 333331
d. 11111 × 4 – 2 = 44442; 111111 × 4 – 2 = 444442
3. a.
0
b.
c.
4. a.
0
Chapter 3
Let’s Warm-up Answers may vary. Sample answers: 1. 2, 5 2. 3, 5 3. 1,17 4. 4, 5 5. 4, 8
Do It Yourself 3A
1. a. 1, 2, 4, 5, 10, and 20 b. 1, 2, 13, and 26 c. 1, 2, 3, 4, 6, 9, 12, 18, and 36 d. 1, 2, 4, 8, 11, 22, 44, and 88 2. a. 37, 74, 111, 148, 185, and 222 b. 62, 124, 186, 248, 310, and 372 c. 84, 168, 252, 336, 420, and 504 d. 99, 198, 297, 396, 495, and 594
3. 2 and 12 4. 9 and 8
5. a. 1 × 15 3 × 5 5 × 3 1 × 15 b. 7 × 2 2 × 7 1 × 14 14 × 1
Challenge 1. 2 is a factor of 100 but 5 is not a factor of 67. So, Roy is wrong.
Do It Yourself 3B
1. a. E b. O c. E d. O 2. 28 is a perfect number. 3. a. False b. True c. True d. False
4. a. 41, 43, 47; Sample answer. 41 and 43 b. 61, 67; Sample answer. 61 and 67 c. 11, 13, 17, 19, 23, 29; Sample answer. 11 and 13 d. 2, 3, 5, 7; Sample answer. 2 and 3 5. 71, 73
6. 990, 2484, 41,870 and 9,12,048 are divisible by 2. 990, 2484 and 9,12,048 are divisible by 3. 2484 and 9,12,048 are divisible by 4. 990 and 41,870 are divisible by 5. 990, 2484, and 9,12,048 are divisible by 6. Only 9,12,048 is divisible by 8. 990 and 2484 are divisible by 9. 990 and 41,870 are divisible by 10.
7. Answers may vary. Sample answers: a. 14 b. 48 c. 99 d. 9
8. Factors of 14 = 1, 2, 7, 14. 14 is a composite number since it has more than 2 factors. 14 × 1 = 14 1
9. Prime numbers: 11 and Composite numbers: 24
10. Answers may vary. Sample answer. Is 1331 divisible by 11?
No, 20 and 8 are not co-primes.
Chapter Checkup
1. a. 1, 2, 3, 4, 6, 8, 12 and 24 b. 1, 2, 3, 5, 6, 10, 15 and 30
c. 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 d. 1, 2, 4, 19, 38 and 76
2. Students will draw factor trees. a. 267 = 3 × 89
b. 315 = 3 × 3 × 5 × 7 c. 539 = 7 × 7 × 11 d. 714 = 2 × 3 × 7 × 17
3. a. Yes; Yes b. Yes; Yes
4. 4 and 9; Yes. 9 8 7 6 5 4 3 2 1
5 a. (i) 12 (ii) 14 b. (i) 18 (ii) 11 6. a. (i) 4410 (ii) 360
b. (i) 84 (ii) 2800 7. 99,960 8. 864 9. 15 books
10. 7:04 a.m. 11. 6 friends 12. Answers may vary. Sample answer. 23154 13. 300 minutes 14. 2
Challenge 1. 771 2. Option c Case Study
1. Option d 2. 12 p.m. 3. Option c 4. Answer may vary.
Chapter 4
Let’s Warm-up 1. 3 2. 5 3. 8 4. 7 5. 4
Do It Yourself 4A
1. c 2. c 3. a. True b. False c. True d. False
4. a. 2.7 cm b. 3.9 cm c. 5.7 cm
5. Students will draw line segment of lengths, 5.2 cm, 7.7 cm, 61 mm, 35 mm.
6. b 7. a. 23,000 m b. 23,00,000 cm c. 2300 dam d. 230 hm 8. 9 stars 9. 4 cm
Challenge 1. 6 flight routes
Challenge 1. Answers may vary. Sample answers. 2 + 3 + 5 + 7 = 17; 2 + 3 + 5 + 13 = 23
Do It Yourself 3C
1. a. 126 = 2 × 3 × 3 × 7 b. 882 = 2 × 3 × 3 × 7 × 7 c. 6241 = 79 × 79
d. 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 192 2. a. No b. No
3. 3, 5, and 23 4. Students will draw the factor trees.
a. 210 = 2 × 3 × 5 × 7 b. 293 = 1 × 293
c. 816 = 2 × 2 × 2 × 2 × 3 × 17 d. 952 = 2 × 2 × 2 × 7 × 17
5. 560 = 2 × 2 × 2 × 2 × 5 × 7 Order: 2, 5, 7 Product: 10
6. 10,000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 7. 56 samples
Challenge 1. 210
Do It Yourself 3D
1. a. (i) 12 (ii) 15 (iii) 9 (iv) 36 b. (i) 10 (ii) 15 (iii) 15 (iv) 18 c. (i) 12 (ii) 21 (iii) 24 (iv) 1 2. a. (i) 60 (ii) 105 (iii) 24 (iv) 48 b. (i) 1680 (ii) 924 (iii) 6160 (iv) 300
c. (i) 396 (ii) 1920 (iii) 108 (iv) 2040 3. 42 4. No
5. 9 6. 18 7. 8
Challenge 1. 4 pairs
Do It Yourself 3E
1. 16 gift bags 2. 1400 minutes 3. 36 groups 4. 36 days
5. 18 cm 6. 8:48 am 7. Once 8. 250,755 days
Challenge 1. Option a
2. No, all intersecting lines are not necessarily perpendicular lines. They can meet at any angle.
Do It Yourself 4B
1. Option c 2. a. Acute angle b. Obtuse angle
c. Reflex angle d. Acute angle
3. Option b
4. Approximate measures ∠P = 34º; ∠Q= 66º; ∠R = 67º
5. ∠QON and ∠QOR
6. Approximate measures a. 25º b. 45º c. 19º d. 107º
7. a. 1 4 b. 1 3 c. 1 2 d. 3 4
8. a. X A D B N Z C M
9. Around 70º to 80º. Close to a right angle. 10. a. South b. West c. North
11. a. B A X
12. 18 right angles
13. West
Challenge 1. E,F, K 2. 45º 3. 30º
Do It Yourself 4C
1. Vertices: T, R, S Sides: TR or RT, TS or ST, RS or SR Angles: ∠T or ∠RTS or ∠STR, ∠R or ∠TRS or ∠SRT, ∠S or ∠TSR or ∠RST
2. a. Scalene b. Equilateral c. Scalene d. Isosceles e. Scalene f. Equilateral 3. a. acute-angled triangle
b. acute-angled triangle c. obtuse-angled triangle d. right-angled triangle e. obtuse-angled triangle f. acute-angled triangle
4.
5. First Garden – isosceles triangle Second Garden – scalene triangle
Challenge 1. Option i
Do It Yourself 4D
1. a. Sides (PQ, QR, RS, SP) vertices (P,Q, R, S); Adjacent sides: (PS, SR), (RQ, QP), (SR, RQ), (QP, PS) Opposite sides: (PS, QR), (SR, PQ) b. Sides(AB, BC, CD, DA) vertices (A, B, C, D); Adjacent sides: (AB, BC), (BC, CD), (CD, DA), (AD, AB) Opposite sides: (AB, CD), (AD, BC) c. Sides (JK, KL, LM, MJ) vertices (J, K, L, M); Adjacent sides: (JM, ML), (ML, LK), (LK, KJ), (KJ, JM) Opposite sides: (JM, KL), (ML, JK) 2. a. Diagonals: BD, AC b. Diagonals: SU, VT c. Diagonals: MK, JL 3. a. Square b. Parallelogram c. Kite
4. Figures may vary.
a.
b.
5. a. Convex quadrilateral b. No, since the angles do not measure 90° c. No, since the opposite angles of the quadrilateral are unequal.
Challenge 1. Option c
Do It Yourself 4E
1. a 2. Vertices: A, B, C, D, E, F, G, H Sides: AB or BA, BC or CB, CD or DC, DE or ED, EF or FE, FG or GF, GH or HG, HA or AH
3. Figures may vary. Sample figures. a. b. c. d.
4. Regular hexagon 5. Sarah, 4 sunflowers
Challenge 1. A Do It Yourself 4F
1. a & c 2. a. Closed curve b. Open curve c. Open curve d. Open curve e. Closed curve
3. a. b. c. d.
4. d 5. d 6. Boundary of the curve
Exterior of the curve P Q R
Interior of the curve
7. Figures may vary. Sample figure. a. b. c. d. 8. a. 3 cm b. 3.8 cm c. 4 cm d. 3.6 cm
9. Answer may vary, sample answer.
Simple curves Non-Simple curves
10. a. 44 cm b. 52.8 cm c. 132 cm d. 308 cm
11. a. 10.5 cm b. 14 cm c. 17.5 cm d. 24.5 cm
12. 0.5 m
Challenge 1. (b) Both A and R are true, but R is not the correct explanation of A.
Chapter Checkup
1. a. 9 b. 8 c. 5 d. 7
2. a. Quadrilateral b. Octagon c. Pentagon d. Heptagon
3. Students will draw line segments of length 4.6 cm, 77 mm, 6.9 cm and 2.2 cm.
Madhya Pradesh
Maharastra Odisha Telangana
Karnataka
4. Figures may vary. Sample Figure.
Do It Yourself 5A
1. a. ii b. ii c. i d. ii 2. a. Cuboid b. Cylinder c. Cone
d. Cube 3. a. cylinder b. cube c. 5
d. cube, square pyramid 4. 5 faces, 8 edges and 5 vertices.
5. Cuboid. 6 faces, 12 edges and 8 vertices.
6. a. Cuboid b. Hexagonal pyramid c. Cone d. Octagonal pyramid e. Pentagonal prism
7. Differences: a. Number of bases: An octagonal pyramid has one base, while an octagonal prism has two bases. b. Shape of the lateral faces: An octagonal pyramid has triangular lateral faces, while an octagonal prism has rectangular lateral faces.
b. Reflex angle
c. Acute angle
d. Obtuse angle
e. Obtuse angle
f. Reflex angle
5. Y P O Q X 6. a. Acute angle
g. reflex angle
h. obtuse
7. a. b. c. d 8. b, d
9. Closed curve: a, c Open curve: b, d
10. a. Isosceles b. Scalene c. Equilateral 11. a. acute-angled triangle b. acute-angled triangle c. obtuse-angled triangle 12. a. 90º b. 180º c. 0º d. 270º 13. a. East b. West 14. a. RT and SU b. ∠RST, ∠STU, ∠TUR, ∠URS c. Yes, UQ = TQ because diagonals of the square are equal and bisect each other.
d. (RU ∥ ST), (UT ∥ RS) 15. 3 cm diameter secant chord 4 cm
16. Scalene triangles: ∆AEF, ∆BFC, ∆ABF Equilateral triangle: ∆DEC Isosceles triangle: ∆FEC 17. a. Points 6, 7, 8 b. Point 2 c. Points 1, 4, 5 d. Points 9, 10, 11, 12, 13 18. a. Yes b. Yes c. Yes
Challenge 1. 15 2. Option B Case Study
1. Option a 2. Option b 3. morning 4. obtuse 5. True
Chapter 5
Let’s Warm-up 1. Square 2. Circle 3. Rectangle 4. Triangle
Similarities: a. Number of sides in the base: Both an octagonal pyramid and an octagonal prism have lateral faces.
b. Three-dimensional shape: Both an octagonal pyramid and an octagonal prism are three-dimensional shapes.
8. Differences: A nonagonal pyramid has a base with 9 edges, while a hexagonal pyramid has a base with 6 edges. Also, a nonagonal pyramid has 10 vertices and 9 lateral faces, while a hexagonal pyramid has 7 vertices and 6 lateral faces.
Similarity: Both nonagonal and hexagonal pyramids are 3D shapes with a polygonal base and triangular lateral faces.
Challenge 1. Heptagonal pyramid.
Do It Yourself 5B
1. a. Hexagonal pyramid b. Triangular pyramid c. Cube d. Cylinder 2. a. sphere b. 3 c. 5 d. 1, 4
3.
4. Answer may vary. Sample answer: a. b. c. d.
5. Answer may vary. Sample answer: a. b. c. d.
6. Triangular pyramid 7. Triangular prism
Challenge 1. 9 faces, 16 edges and 9 vertices.
Chapter Checkup
1. a. 6 faces, 10 edges, 6 vertices b. 6 faces, 12 edges, 8 vertices c. 2 faces, 1 edge, 1 vertex d. 10 faces, 24 edges, 16 vertices
2. a. 10, 6 b. 8 c. 21 3. a. Cone 4. a. Total number of faces = 6; 4 rectangular, 2 square shaped b. Total number of faces = 8; 7 triangular, 1 heptagon shaped c. Total number of faces = 8; 6 parallelograms, 2 hexagons
5. a. Red b. Grey
6. Answer may vary. Sample answer:
7. Answer may vary. Sample answer:
8. Triangular pyramid Cone Cylinder Square pyramid
9. a. Octagonal pyramid b. Cube c. Pentagonal pyramid d. Pentagonal prism 10. a. b.
11. Answer may vary. Sample answer: a. b. c. d.
Challenge 1. Option a 2.
Case Study
1. Option a 2. 3. Option d
4. All 6 rectangular faces in a rectangular prism, and only 1 rectangular face in a rectangular pyramid.
5. Answers may vary. Sample answer: The community centre can make the shed environmentally friendly by using recycled materials for building it.
Chapter 6
Let’s Warm-up 1. 7, 9 2. 14, 16 3. 28, 30
4. 37, 39 5. 105, 107
Do It Yourself 6A
1. a. +10 km, −10 km b. −₹1000, +₹1000 c. +5 apples, −5 apples d. −3 kg, +3 kg e. +3 runs, −3 runs f. −10 minutes, +10 minutes
2. 0 1 2 3 4 5 6 7 8 9 10 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
3. A = −6, B = −9, C = −2, D = 3, E = 6, F = 1 4. a. 16, 18 b. −13, −11 c. 24, 26 d. −89, −87 e. 124, 126 5. −3
6. a. −2, −1, 0, 1, 2, 3 b. −4, −3, −2, −1 c. 2, 3, 4, 5 d. −8, −7, −6, −5 e. −14, −13, −12, −11, −10, −9 7. a. 8 b. 10 c. 25 d. 35 e. 65
8. a. Three places on the left of the number line; –25 b. Six places to the left on the number line; 0 c. Two places on the left of the number line; –36 9. a. > b. <, c. > d. <, e. < 10. a. −10 < −8 < −2 < 5 < 9 b. −11 < −7 < −6 < 16 < 20 c. −15 < −9 < 13 < 14 < 23 11. −7°C < −3°C < 2°C < 5°C < 12°C < 18°C
12. Helium balloon +35 +10 0 −15
Parachute
Helium Balloon
Sea level
Submarine
Challenge 1. 6
Do It Yourself 6B
2. a. 10 b. −10
Challenge 1. a. 7 b. 1 + 2
Challenge 1. Anu reaches the top first.
Chapter Checkup
1. a. + + + + + + + + + + + 11
3. a. +31°, –5°, +18°, –8° b. Coldest: Sonamarg; Hottest: Chennai
4. E 5. a and d 6. a. −9, −7 b. −27, −25 c. −39, −37 d. 44, 46 7. a. +8 is larger than –16 8. None 9. b 10. a. 11 b. −190 c. −61 d. 300 11. 1186 12. Answer may vary. Sample answer: −25, +42 and −65 13. −859
14. 192°C 15. Burnt or lost 240 calories 16. 382°F
17. ₹375 18. Manoj obtained 234 marks in the test.
Challenge 1. Descending order = (C – B), (C – D), (D – E), (A – D), (A – E)
2. Option b
Case Study
1. Option c
2. Greenland, Russia, Austria, Brazil, Qatar
3. Russia is warmer than Greenland by 14.89°C
4. It gets warmer as the temperature rises because the energy and heat in the environment increases.
Chapter 7
Let’s Warm-up
Do It Yourself 7A
1. a. 3 4 b. 6 16 c. 10 20 d. 5 9
2. Answers may vary. Sample answers. a. b. c. d.
3. Proper fractions: 4 5 , 1 22 , 51 89 , 4 9 Improper fractions: 6 4 , 47 14 , 16 3 , 15 14 Mixed
Do It Yourself 7B
11. Jacky covers a longer distance than Pritam. Challenge 1. Answer may vary, sample answer:
Do It Yourself 7C
Each piece of the brownie will be 1 8 of the whole.
Do It Yourself 7D
Challenge 1. 4 6 × 1 3 = 4 18
Chapter Checkup
1. a. 21 2 b. 17 3 c. 100 23 d. 15 8 2. a. 33 4 b. 83 5 c. 206 7 d. 5 2 17
3. Answers may vary. Sample answers:
c. 33 363 , 22 242 , 44 484 d. 12 15 , 8 10 ,
5. a. Not equivalent b. Not equivalent c. Not equivalent d. Not equivalent 6. a. 2 5 b. 3 1 3 c. 5 9 d. 2 17 58
7. a. The difference of 8 3 8 and 3 1 4 is more than the sum of 2 1 4 and 1 2 15 b. Sum of 1 4 and 24 5 is less than the product of 1 2 3 and 4 5
c. 179 60 d. 12 5 , 120 8. 5 8 gram 9. 1 20 kg 10. 3517 20 m
11. Answer may vary. Sample answer:
1 5 of a chocolate is eaten by Rahul. He gives 1 2 of the remaining chocolate to his sister. How much chocolate is left with Rahul?
Challenge 1. ₹8320 2. Option d
Case Study
1. b 2. c 3. 41 6 4. Martial arts < Dance < Yoga
5. Answers may vary.
It Yourself 8A
2. a. 606.46 b. 39.087 c. 840.201 d. 1248.16 3. Part
4. a. 6 25 b. 59 100 c. 13 20 d. 1 8 e. 9 40 f. 64 125
5. A = 0.8, B = 1.4, C = 3.4, D = 4.6
6. a. 125; 0.375 b. 25; 1.75 c. 5; 6.5 d. 4; 1.48 e. 2; 2.4 f. 25; 2.225
7. a. 3 6 25 b. 5 3 25 c. 2 13 250 d. 4 3 40 e. 52 1 500 f. 25 3 50
8. a. 1.540 and 15.021 b. 54.000 and 48.021 c. 1.10 and 4.65 d. 18.054 and 97.000 9. a. 15.41 b. 178.2 c. 12.825 d. 15.51
10. a. 7.23 < 7.3 < 8.32 < 8.37 b. 58.37 < 58.45 < 58.54 < 58.73 c. 97.08 < 97.18 < 97.2 < 97.8 d. 18 < 18.1 < 18.101 < 18.11
11. a. 3.068 kg, 14.05 kg, 5.005 kg, 0.725 kg, 0.045 kg b. 4.015 km, 8.04 km, 3.004 km, 0.75 km, 0.015 km 12. First shop 13.
Tuesday Monday Thursday Friday Wednesday 34o C 34.5o C 35o C 35.0 34o C 33.2o C
33o C
33
14. In words: one point five six or one and fifty-six hundredths; twenty-one point five or twenty-one and one fifths; fourteen point five or fourteen and five tenths
Expanded form: 1 + 0.5 + 0.06 or 1 + 5 10 + 6 100 ; 20 + 1 + 0.5 or 20
+ 1 + 5 10 ; 10 + 4 + 0.5 or 10 + 4 + 5 10
15. 0.6; 3 5 16. 1.58 < 2.39 < 2.53 < 3.13, Day 4
17. 2.54 L > 2.45 L > 2.3 L > 1.75 L
Challenge 1. 8.7 inches
Do It Yourself 8B
1. a. 24.62 b. 62.34 c. 11.097
2. a. 2.5 b. 1.72 c. 0.229 3. a. > b. = c. <
4. a. 123.91 b. 172.11 c. 810.907 d. 99.871 e. 41.363 f. 76.163
5. 15.45 mm 6. 32.545 g 7. 2.02 km
8. ₹1386.7 9. ₹112.3 10. 18.0148 amu
Challenge 1. The extra weight was the weight of the container.
Do It Yourself 8C
1. a. 204.8 b. 113.4 c. 898.7 d. 1729
2. a. 6.36 b. 61.1 c. 631.113 d. 2.9264
3. a. 15.2 b. 23.9 c. 41.3 d. 16.5 4. a. 2.3
b. 8.9 c. 12.6 d. 3.68 5. a. 12.36 b. 89.7
6. a. 1244.808 b. 2068.304 7. 20.88 sq. feet
8. 14.4 quarts 9. 2.625 cups 10. ₹1273.50
11. Answers may vary. Sample answer.
Rani is preparing for a bake sale. She has 6.75 kg of flour, and each batch of cookies she makes requires 0.45 kg of flour. How many batches of cookies can Rani make with the amount of flour she has?
Challenge 1. c. Both statements together are enough to answer the question, but neither statement alone in enough.
Chapter Checkup
3. a. 43.1 b. 950.43 c. 649.891
4. a. 5 + 0.2 + 0.03 or 5 + 2 10 + 3 100
b. 10 + 2 + 0.05 + 0.006 or
5. a. 45.5 b. 18.125 c. 12.6 6. a. 2563 50 b. 3407
7. a. 15.1 > 15.05 > 15 > 14.95 b. 1.331 > 1.303 > 1.23 > 1.2 c. 153.32 > 153.233 > 153.23 > 152.1 8. a. 3.055 b. 0.286 c. 8.04 d. 82.06 e. 280.84 f. 2427.505 9. Nina 10. 1.8177 litres 11. ₹19776.75 12. ₹1455.75 13. 51.575 kg 14. ₹10,212.75 Challenge 1. ₹62.5 2. Type A = 53; Type B = 7 3. Option c
Case Study
1. b. 1.52 AU 2. b. 1.39 AU 3. 4.20 AU 4. Venus (0.72 AU), Mars (1.52 AU), Uranus (19.22 AU), Neptune (30.05 AU)
Chapter 9
Let’s Warm-up 1. 7 2. least 3. most 4. 31 5. 2
Do It Yourself 9A
1. a. False b. True c. False d. False
2. a. |||| |||| |||| ||||
b. |||| |||| |||| |||| |||| ||
c. |||| |||| |||| |||| |||| |||| |
d. |||| |||| |||| |||||||| |||| |||| |||| ||
Challenge 1. Week 1: 4.5 pictures; Week 2: 5.25 pictures; Week 3: 3.75 pictures; Week 4: 8 pictures
Do It Yourself 9C 1. a. 09:00-10:00 b. 08:00-09:00
3. Name of Mughal emperor Tally Marks Babur ||| Humayun ||| Akbar |||| Jahangir ||| Shah Jahan |||| Aurangzeb |||| | 4. Fruit Tally Marks Orange |||| ||
5. Age Number of Student Tally Marks
7 |||| ||
3 |||
|||| |||| || Banana |||| |||| | Guava |||| | Litchi |||| ||||
a. 7 students b. 6 students c. 17 students d. 14 students
Challenge 1. a. |||| b. |||
Do It Yourself 9B
1. Brand A Brand B Brand C Brand D Brand E Key: = 500 soaps
2. a. 1500 bulbs b. 300 bulbs c. 3900 bulbs d. Tuesday, 1500 bulbs e. 14,700 bulbs
3. Name Population of Tigers Ranthambore
Scale: = 20 tigers
3. a. ₹7000 b. Clothes; ₹6000 c. Rent; ₹9000 d. ₹37,250
4. a. 210 million doses b. 50 million doses c. May
5. Answers may vary. Sample answer.
a. How many families have more than 2 members in their family?
b. How many families have less than 5 members in their family?
Challenge 1. Option c
Chapter Checkup
1. Transportation Tally Marks Number of people
4. a. 4500 fans b. 5500 fans c. June, 8000 fans d. 500 fans
5. Year Number of earthquakes
Key: = 10 earthquake
6. Answer may vary. Sample answer. How many more earthquakes occured in 2015–2016 than that in 2013–2014?
7. Day Number of shirts
Key: = 10 shirts
3. a. Maths b. 14 students c. Science d. 17 54
4. a. 30 saplings b. 33 saplings c. Block B d. 173 saplings
5. a. Delhi b. Lakshadweep c. 1500 square km d. 2750 sq. km
6. Answers may vary. Sample answer. What is the area of Chandigarh?
7. a. 675 ice creams b. May; 1200 ice creams c. 750 ice creams d. 4275 ice creams
8. a. 30,000 books b. 40,000 books c. 11 58 d. 35:36
9. a. June; 40° C b. Yes, January and December have the same temperatures; 5° C c. 3 months d. August
10.
1 division = 100
11. Sohail
Ashraf
Mahmood
Raju
Vivek
Key: = 1000 mangoes
8. 72 9. ₹500 10. 468.75 sq. m
11. Area = 100 sq. m 12. Sailesh 13. 50 sq. m
14. Square, 30.25 sq. cm
15. Answers may vary. Sample answers: 4 9 12 3
Challenge 1. Option a 2. 24 cm
Challenge 1. c 2. a. 6 symbols b. The students participating in chess will be shown by 10 faces and the students participating in drama will be represented by 4 faces. c. 7 students Case Study
1. Option b 2. Option c 3. April; 40 4. May, 10 5. No. Answers may vary.
Chapter 10
Let’s Warm-up 1. 16 units; 12 square units 2. 18 units; 16 square units 3. 14 units; 8 square units 4. 26 units; 17 square units 5. 28 units; 13 square units
Do It Yourself 10A
1. a. 37 cm b. 34 cm c. 64 cm 2. 84 cm 3. 41 cm
4. 20 cm 5 a. 14 cm b. 6 cm 6. 9.34 m
7. Answers may vary. Sample answers: 2 cm 5 cm 4 cm 3 cm
8. 645 m 9. 25 cm 10. 4.8 cm 11. ₹2352
Challenge 1. The perimeter will remain the same. For example, if a square of side length 5 cm is cut from a bigger square of side length 10 cm, the perimeter which was originally 40 cm is now also 40 cm.
Do It Yourself 10B
5 cm
10 cm
1. 49 sq. cm 2. 15 cm 3. a. 30 sq. cm b. 11.5 sq. cm
c. 47 sq. cm 4. 90 cm 5. 25 6. ₹3375, ₹24,975
7. The square has the larger area. 8. a. P = 104 cm, A = 192 sq. cm b. P = 48 units, A = 63 sq. units
9. Rectangular plot since the area of the rectangular plot I is greater.
Challenge 1. Area increaes 4 times.
Do It Yourself 10C
1. 46 m 2. 441 sq. m 3. 25 m 4. 180 stickers
5. 63 cm, 220.5 sq. cm 6. 1050 cm 6. 8 m
7. 24 rounds 8. 230 m 9. Answer may vary. Sample answer: Maria has a rectangular garden. She knows the perimeter of the garden is 40 metres, and one of the sides measures 12 metres. To find out how much area she needs to plant flowers, calculate the area of the garden.
Challenge 1. 40 m
Chapter Checkup
1. a. 48 cm b. 22 cm c. 40 cm 2. a. 18.5 sq. units
b. 31 sq. units c. 27.5 sq. units 3. 399 sq. cm
4. 625 tiles 5. 75 sq. cm 6. 18 cm
7. Ramesh’s plot has the greater area and perimeter.
Case Study
1. Option d 2. 104 feet 3. 14 feet 4. ₹16,800
5. Answer may vary.
Chapter 11
Let’s Warm-up 1. 5 2. 12 3. 9 4. 10 5. 16
Do It Yourself 11A
1. a. 2n; 36 b. 3n; 54 c. 5n + 1; 91 d. 5n − 2; 88
2. a. 3n; 150
b. 7n; 350
3. a. 2n + 1 b. 3n + 2 4. Hours 1 2 3 4 5 6 7 8 Cupcakes 10 20 30 40 50 60 70 80 10n; 240 cupcakes
5. A = l × b 6. a. 6a b. 2(l + b) c. 9a d. 8a 7. D = 2r
8. Shape - Heptagon, Rule - 7a
Challenge 1. 7a
Do It Yourself 11B
1. a. Arithmetic expression b. Algebraic expression c. Algebraic expression d. Algebraic expression
2. a. x and y b. 2a, 3b, –c c. x, y, 2z d. x, –6z e. 2x, 5
f. 4 3 x, 2 g. 5abc, –2ab, 7ac h. 2ab, 4ac, –6c 3. a. 3 + 6m
b. n – 10 c. 15x d. 2xy e. 9y + 1 f. 3y + (x – 3)
4. a. 24 b. −11 c. −154 d. 1 4 5. 12u; 156 6. x + 7
7. 3x + 5 8. y = x + 6 9. x – 5 10. m – 5 2 11. `(21y + 3z)
12. ₹(67x – 152)
13. a. a a – 6 −3 3(a – 6) – 6 × 3 for a = 5 b. 15 15 + a 17 15 + a – 8 + a – 8 for a = 10
c. a a × a 51 (a × a) + 2 × a + 2 for a = 7
14. Answer may vary. Sample answer. On a birthday party, Rahul distributed 3 candies to all his friends and he was still left with 5 candies. How many total candies Rahul had initially if the total number of friends were x?
Challenge
1. a. k 10 k + 10
b. 9 k k k k k k k k k k
c. k − 94 94 k
d. k 4 k
Do It Yourself 11C
1. a. x + 3 = 12 b. x – 7 = 2 c. 2x – 7 = 3 d. 4x + 2 = 26
2. X + 3 x + 8 = 3 4 3. 6x = 72 4. 2 × (3x – 7) = 52
5. Answers may vary. Sample answers: a. When we add 5 to the product of 4 and a number, we get 9. b. When we add 1 to twice a number, we get 5. c. When 7 is taken away from the product of 12 and a number, we get 5. d. Three times the sum of a number and 1 gives 12.
6. a. 1 b. 1 c. 4 d. 3 e. 6 f. 9 7. a. 8 b. 7 c. 12
8. a. Yes b. No; 1 c. Yes d. Yes e. No, 6 5 f. Yes
9. a. 4 b. 3 c. 15 d. −11 e. 5 f. 4 10. 12 and 13
11. 25 mm and 75 mm 12. 16 years and 40 years
13. 100 and 50 14. ₹17
Challenge 1. The numbers are 56, 64 and 72.
Chapter Checkup
1. a.
b.
c.
Rule: 4n
Rule: 4n + 2
Rule: 4n + 1
2. 0 + x = x = x + 0 3. a. 3x + 4 b. 2y – 2 c. x 3 – 4 d. 6 + y 5
4. a. Linear b. Non-Linear c. Linear
5. a. 78 b. 66 c. 128 d. 416 6. a. (6x – 5) cm b. 6 cm
7. Statements may vary. Sample statements. a. Raj has some money and after receiving ₹30, he has a total of ₹50. How much money does he have originally?; 20 b. Six times a number reduced by 3 gives 9. What is the number?; 2 c. Ravi distributes 24 pieces of candy equally among some friends, and each friend gets 4 pieces. How many friends were there?; 6 d. Four times the sum of three times a number and 7 equals 40. What is the number?; 1 8. 25 marbles
9. Perimeter of the skating rink is 2(l + w) 10. a. z + 8 b. 5z + 9 11. 2x + 4; 3x + 4 12. 2x – 4, 2x 13. b 14. `32
15. 19 years
Challenge 1. Option a 2. 15 years Case Study
1. Option b 2. ₹420
3. D = 2C
4. The quantity has to be 10.
Chapter 12
Let’s Warm-up 1. 1 2 2. 2 3 3. 4 10 4. like 5. 3 4
Do It Yourself 12A
1. a. False b. True c. True d. False e. False
2. a. 2:3 b. 2:3 c. 2:1 d. 100:3 e. 1:1 f. 1:1
3. Answer may vary. Sample answer: a. 10:14, 15:21, 20:28 b. 8:18, 12:27, 16:36 c. 24:32, 36:48, 48:64 d. 12:30, 18:45, 24:60 e. 16:22, 24:33, 32:44 f. 18:34, 27:51, 36:68
4. a. > b. > c. < d. < e. < f. = 5. a. 15:1 b. 18:5 c. 46:37 d. 3:20 6. 16:24 7. 20:60 8. a. 15:10 b. 5:15 c. 10:20
9. 1:10 10. 1:2 11. 3:2 12. 2:7 13. 15 and 20 14. 9 metres by 6 metres 15. 14.4 metres 16. 6 litres 17. ₹4000 18. 4 cups
Challenge 1. 1.4 times
Do It Yourself 12B
1. a, c, d, 2. a. False b. True c. True d. True
3. Answer may vary. Sample answer: a. 3:13::9:39 b. 7:42::3:18 c. 7:21::4:12 d. 2:14::8:56
4. a. 4 b. 18 c. 8 d. 4 5. a. 3 b. 3 c. 2 d. 6
6. a. 1 b. 8 c. 81 d. 49 7. 15 8. 27 9. 9 metres
10. 65 kg 11. 25 gallons 12. 3 moles
Challenge 1. Riyaʹs opponent
Do It Yourself 12C
1. a. ₹180 2. 4 hours 3. 60 postcards 4. 9 months
5. 4℃ 6. 25 buses 7. ₹10,500 8. Kiran
9. 220 words
Challenge 1. 9.9 cups
Chapter Checkup
1. a. 45:32 b. 66:143 2. a. 8:7 b. 6:7 c. 20:3
3. Answers may vary, sample answer:
a. 6:16, 9:24, 12:32, and 15:40 b. 10:18, 15:27, 20:36, and 25:45
c. 22:30, 33:45, 44:60, and 55:75 4. a. < b. < c. < 5. b, c
6. 2:3 7. 72 8. 16 9. ₹90 10. 45 11. 11.25, 18.75
12. 3 hours 13. 450 bottles 14. 27 volunteers
15. 36 years 16. 240 minutes 17. 8:1
18. Answers may vary. Sample answer.
A factory produces 120 toys in 5 hours. How many toys can the factory produce in 8 hours at the same rate?
Challenge 1. ₹2400 2. Option a
Case Study
1. a. 30 2. a. 5 3. 3:1 4. True
5. We should observe wildlife from a distance, avoid disturbing their natural habitats, follow park rules and regulations, and learn about conservation efforts to protect these animals and their environments.
Chapter 13
Let’s Warm-up 1. Symmetrical 2. Non-symmetrical
3. Symmetrical 4. Symmetrical 5. Non-symmetrical
Do It Yourself 13A
1. One line of symmetry 2.
3. Pihu is right. The chessboard has two lines (diagonals) of symmetry. The horizontal and vertical lines will not divide the checks symmetrically.
4. a. Both l1 and l2 b. Both l1 and l2 c. Neither l1 or l2
5. a. b.
c. d.
6. H, I, O and X
7. Answers may vary. Sample answers:
8. a. 1 b. 0 c. 2 d. 4
Challenge 1. The lower right portion is missing a hole towards the centre. We can make it symmetrical by drawing another hole.
Do It Yourself 13B
1. b. 2. a.
b. c. d.
3. MOM 4. GOVERNMENT OF INDIA
5. a. MIRROR b. SYMMETRY c. REFLECTION d. MATHEMATICS
6. a. b.
c. d. e. f.
7. a. 2 b. 6 c. 3 d. 1
8. The logo has 3 angles of symmetry. The angles of symmetry are 120°, 240°, 360°. Challenge 1. Blue and Yellow Chapter Checkup
1. Yes, the figure has one line of symmetry.
2.
Z
S 3
D 8 U
3. no line of symmetry 1 line of symmetry more than 1 line of symmetry F G J L N P Q R S Z A B C D E K M T V Y W U H I O X
4. a. True b. True c. False d. True
5. No, because a rainbow has a semi-circular shape; hence, it has only one line of symmetry (vertical).
6. a. 4; b. 2; c. 4; d. 4;
7. a. Yes b. Yes c. No
8. a. b. c.
6. Figure may vary. Sample figure.
9.
10. a. b.
c.
11. The smallest angle of symmetry is 90°.
So the other angles of symmetry are: 180°, 270°, 360°
12. The outline of the sign board is in the shape of a rhombus.
Order of symmetry of a rhombus = 4
Angle of symmetry = 90°
Challenge 1. Option a 2. 2
Case Study
1. Option b 2. Option b
3.
4. Figures may vary. Sample figure.
Chapter
14
Let’s Warm-up 1. 5, 5 2. 5, 6
Do It Yourself 14A
1. a. A B 5.4 cm b. P Q 6.7 cm
c. X M 6.1 cm d. L Y 7.2 cm
2. a. A B 4.3 cm b. D E 5.6 cm
c. G H 3.9 cm d. P Q 2.7 cm
3. Students will construct line segments of the same length as AB, PQ and MN.
4. a. B C 7.5 cm b. X Y 8.2 cm
Challenge 1. Yes, we can construct a square by drawing a perpendicular to the diagonal. y w 6 cm z x
Do It Yourself 14B
1. a. 20º b. 35º c. 50º
4. a. 40° 40° A B C O P b. B D O 68° 68° Q
5. Figures may vary. Sample figure.
Challenge 1. Option b
An angle can have only one unique angle bisector. The angle bisector divides the angle into two equal parts, making it the line of symmetry for that angle.
Chapter Checkup 1. a. P 3.6 cm Q
No A B
Challenge 1. Option d
2. Yes, the bisecting rays are perpendicular to each other. A linear pair of angles is formed when two adjacent angles add up to 180°.
Let’s denote the two angles as ∠A \angle A∠A and ∠B \angle B∠B.
Let ∠A = x and ∠B= 180° − x
The bisector of ∠A = 2 x and the bisector of ∠B = 180 2 x °−
Sum of angles formed by the bisectors = 2 x + 180 2 x °−
Since the sum of two angles formed by the bisectors is 90°, the bisecting rays are perpendicular to each other.
Case Study
1. Option b
2. A 9 mm B
3. Angle formed by the shadow = 120°
Measure of angle bisector = 60°
2° = 1 unit
60° = 60° 2° = 30 units
30 units will be equal to 60°
4.
P B D Q C
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
• Case Study: 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 15,000+ schools across India, Southeast Asia and the Middle East.
ISBN 978-81-979482-8-2
